Fundamentals of Preparative Preparatiue and of Chromatography Nonlinear Chromatography Second Edition
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Fundamentals Preparatiue and of Preparative Chromatography Nonlinear Chromatography Second Edition
Georges Guiochon
Attila Felinger
Department of Chemistry University of Tennessee University Knoxville, TN 37996-1600 USA USA
Department of Analytical Chemistry University of Pécs Pecs University Pecs H-7624 Pécs Hungary
Dean G. Shirazi
Anita M. Katti
Director of Analytical Development Development Laboratories Laboratories AAIPharma Wilmington, NC 28405 USA
Department Chemistryand andPhysics Physics Department of Chemistry Purdue University Calumet Hammond, IN 46323-2094 USA
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Preface The first edition of this book was written to consolidate the fundamental concepts which form the basis for the development of preparative separations by chromatography on the laboratory, pilot and process scales. Our goal had been to present the many issues one may encounter while working in the field of preparative and process scale chromatography, and to bridge the gap between the basic mathematics associated with the modeling of chromatographic separations and the experimental phenomena one observes during experimental exploits. Emphasis was put on the illustration of equations and concepts to provide a physical image of the different chromatographic phenomena along with written narrative explanations. Considerable attention was also given to the application of these concepts to the optimization of the separation conditions. We hope that we succeeded in providing readers with an understanding of the fundamental concepts of chromatography and with the methods needed to apply them to the solution of practical problems. Since the first edition is now out of print and much progress was made in the last ten years that is relevant to our topic, it was decided to prepare an updated second edition. This new book includes most of the important results acquired in the fundamentals of chromatography since the first one was published. The study of reversed phase liquid chromatography (RPLC) showed that the surface of the adsorbents that are used in this method is heterogeneous, with serious consequences on the nature of the equilibrium isotherms. The development of practical methods of determination of adsorption energy distributions informs on this heterogeneity. The real adsorbed solution theory affords improved accuracy in the prediction of competitive adsorption isotherms from the single- component isotherms of the components of a mixture. The new, inverse method of isotherm determination is gaining acceptance. Surface diffusion, competitive pore diffusion, the concentration dependence of the coefficients of mass transfer kinetics have been investigated in depth and this new knowledge proven useful in the prediction of band profiles. The simulated moving bed process has become main stream in preparative chromatography. All these issues and more are covered in this revised version. The reader will find here a complete mathematical development of the models of chromatography and other physical laws which direct the chemical engineer in the design and scale-up of chromatographic processes1. For preparative chromatographic separations, our ultimate purpose is the optimization of the experimental conditions for maximum production rate, minimum solvent consumption, or minimum production cost, with or without constraints on the recovery yield. The considerable amount of work done on this critical topic is presented in the 1
We did not intend to write a book on the mathematics of chromatography; this has been done excellently by Rhee, Aris and Amundson.
v
vi
Preface
last chapter that was considerably expanded. In this sense, it is the culmination of this book. The availability of powerful computers enables the use of sophisticated models of chromatography which take into account most of its complexity. Users must keep in mind, however, that all the parameters used in a model must be determined with an accuracy compatible with the improvement in prediction accuracy of the new model. Thus, the use of the general rate model remains still rarely warranted. The direct determination of the numerous rate constants involved is highly time consuming and difficult at best, inaccurate at worst. Their determination by curve fitting procedures gives an illusory feeling of confidence, but the parameters obtained are empirical and do not deserve the physical sense that is often afforded to them. The use of these parameters should be reserved to those applications involving interpolation of previously obtained experimental results, such as in the optimization of the experimental conditions of a given separation that has been studied in detail (with accurate measurements of competitive isotherm data in a wide range of relative and absolute concentrations and precise determination of rate coefficients). These considerations explain the relative importance that we have given to the models of chromatography, in relation to their different degrees of sophistication. The ideal model which requires only the determination of the equilibrium isotherms, permits a rapid estimation of the individual band profiles for convex upwards isotherms. At high loading factors, with modern, high efficiency packing materials, its results are strikingly accurate. Nevertheless, actual columns have a finite efficiency, axial dispersion and mass transfer resistances disperse the profiles, and often these effects need to be taken into account. They reduce yields and production rates of purified products. In almost all cases of practical importance in preparative chromatography, the equilibrium-dispersive model is satisfactory. In addition to the determination of the equilibrium isotherms, it requires only the measurement of the column HETP under linear conditions as a function of the mobile phase velocity. When the mass transfer resistances are important, as in some protein separations, or when particles of unusually large size or complex pore structure are employed, a simple kinetic model based on the use of a lumped kinetic parameter or the POR model should give excellent results. Some issues of importance in preparative chromatography are not discussed in this volume, such as instrumentation, column technology, column packing procedures, safety considerations, hardware layout, or the selection of solvents and stationary phases. Among the fundamentals, the thermodynamics of phase equilibria, its concentration dependence and competitive nature were given the primary importance. Recognizing the existence and the fundamental importance of these same characteristics, concentration dependence and competitive nature, for the diffusion and the mass transfer coefficients, the authors have discussed them in considerably more detail in this second edition than in the first one. However, they warn that these phenomena have limited practical importance in the preparative chromatography of most chemicals and drugs and that there is still a paucity of reliable data on the transport properties of proteins in chromatographic systems. The extension to the separation and purification of proteins of the concepts and tools applied so effectively to the separation and purification of regular
Preface Preface
vii
chemicals will remain an important and fruitful area of research for years to come. Also, the chromatographic column has been considered, throughout all our work, as linear. The radial dependence of the stationary phase density, the porosity of the packed bed and its permeability have been neglected, as well as the thermal effects associated with band migration. Experimental evidence available so far suggests that the consequences of these simplifications are minor in almost all practical cases. Approaches useful to handle these issues when needed are presented in Chapter 2. They have not been pursued. This book stems from the work done in the group of Professor Georges Guiochon at the University of Tennessee and at Oak Ridge National Laboratory in the late 1980's, 1990's and early 2000's. It contains the many contributions of the students, post-doctoral fellows and visiting scientists who came from all over the world to East Tennessee to contribute to the advancement of this field. Their contributions add to those of the many scientists who have worked in this area over the last sixty years and have produced innumerable, valuable publications.
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Acknowledgments The advancement of science involves the input of a multitude of people. Similarly, the production of this book required the support of numerous individuals. We gratefully acknowledge those who have over the years, and to various degrees, contributed by their comments, questions, suggestions, results, and insights to the progress of our cooperative work. Some of these include Tarab Ahmad (Knoxville, TN), Klaus Albert (Tubingen, Germany), Leonid Asnin (Perm, Russia), Henning Boysen (Dormung, Germany), B. Scott Broyles (Richmond, VA), Alberto Cavazzini (Ferrara, Italy), Frederic Charton (Nancy, France), Sophie Charton (Nancy, France), Yibai Chen (Philadelphia, PA), Djamel E. Cherrak (Woburn, MA), Martin Czok (Paris, France), Moustapha Diack (Baton Rouge, LA), Francesco Dondi (Ferrara, Italy), Eric V. Dose (Chicago, IL), Eric C. Drumm (Knoxville, TN), Zoubair El Fallah (Newport Beach, CA), Tivadar Farkas (Torrance, CA), Torgny Fornstedt (Uppsala, Sweden), Joerg Fricke (Dortmund, Germany), Wilmer Galinada (Union, NJ), Samir Ghodbane (Nutley, NJ), Gustaf Gotmar (Uppsala, Sweden), Fabrice Gritti (Knoxville, TN), Jun-Xiong Huang (Beijing, China), Laurent Jacob (Paris, France), Stephen C. Jacobson (Bloomington, IN), Francois James (Orleans, France), Pavel Jandera (Pardubice, Czech Republic), Alain Jaulmes (Thiais, France), Krzysztof Kaczmarski (Rzeszow, Poland), Marianna Kele (Milford, MA), Saad Khattabi (Exton, PA), Hyunjung Kim (Knoxville, TN), Ervin sz. Kovats (Lausanne, Switzerland), BingChang Lin (Anshan, China), Xiaoda Liu (Beijing, China), Zidu Ma (Lafayette, IN), David V. McCalley (Bristol, UK), Nicola Marchetti (Ferrara, Italy), Michel Martin (Paris, France), Daniel E. Martire (Washington, D.C.), Kathleen Mihlbachler (Indianapolis, IN), Kanji Miyabe (Toyama, Japan), Uwe Neue (Milford, MA), Joan Newburger (Trenton, NJ), Wojciech Piatkowski (Rzeszow, Poland), Igor Quinones Garcia (Union, NJ), Roswitha Ramsey (Chapel Hill, NC), Jeffrey Roles (Orlando, FL), Pierre Rouchon (Paris, France), Hong and Peter Sajonz (Edison, NJ), Matilal Sarker (Bellefonte, PA), Andreas Seidel-Morgenstern (Magdeburg, Germany), Andrew Shalliker (Sydney, Australia), Brett J. Stanley (San Bernardino, CA), Pawel Szabelski (Lublin, Poland), Ulrich Tallarek (Magdeburg, Germany), Nobuo Tanaka (Kyoto, Japan), Patrick Valentin (Solaize, France), Jennifer Van Horn (Indianapolis, IN), Claire Vidal-Madjar (Thiais, France), Tong Yun (Knoxville, TN), Guoming Zhong (Tonawanda, NY), Dongmei Zhou (La Jolla, CA), and Jie Zhu (Kenilworth, NJ). It was difficult to name all those who have contributed; our deepest regrets go to those who are unmentioned. We are thankful for the assistance and support of our colleagues, particularly Professors C. E. Barnes, F. M. Schell, and M. J. Sepaniak, and the staff (especially Mr. Bill Gurley) of the Department of Chemistry at the University of Tennessee in the solution of the countless problems encountered in the conduct of our research activities. GG would like to thank Lois Ann Beaver, who urged him to embark upon this endeavor and provided constant and welcome support toward its completion. His secretary, Cathy Haggerty, was helpful in many opportunities. DS ix IX
x
Acknowledgments Acknowledgments
would like to recognize his sister, Shari Shirazi, for her encouragement during this project. AF would like to thank Gabriella Felinger for her continued patience and understanding. AMK acknowledges A. Rachel Prakash (Atlanta, GA) for her assistance with the figures, JBK and LZK for their love and patience. We pay tribute to a colleague, a friend, and a mentor. Csaba Horvath was a leader whose work contributed greatly to the advancement of separation methods and whose remarkable experimental and theoretical contributions have had profound effects on our understanding of analytical and preparative liquid chromatography. We miss him. We acknowledge the constant support of the National Science Foundation over the last twenty years, through grants CHE-8519789, CHE-8715211, CHE-8901382, CHE-9201663, CHE-9701680, CHE-00-70548, and CHE-02-44693. These grants enabled us to obtain the results which are our contribution to the Fundamentals of Preparative and Nonlinear Chromatography.
Contents Preface
v
Acknowledgements
ix
1 Introduction, Definitions, Goal 1.1 History of Chromatography 1.2 Definitions 1.3 Goal of the Book References
1 3 12 15 17
2 The Mass Balance Equation of Chromatography and Its General Properties 2.1 Mass and Heat Balance Equations in Chromatography 2.2 Solution of the System of Mass Balance Equations 2.3 Important Definitions References
19 21 42 57 63
3 Single-Component Equilibrium Isotherms 3.1 Fundamentals of Adsorption Equilibria 3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria . . . . 3.3 Adsorption and Affinity Energy Distribution 3.4 Influence of Experimental Conditions on Equilibrium Isotherms . . 3.5 Determination of Single-Component Isotherms 3.6 Data Processing and Assessment References
67 70 80 109 117 122 135 144
4 Competitive Equilibrium Isotherms 151 4.1 Models of Multicomponent Competitive Adsorption Isotherms . . 153 4.2 Determination of Competitive Isotherms 191 References 216 5 Transfer Phenomena in Chromatography 221 5.1 Diffusion 222 5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media . . 240 5.3 The Viscosity of Liquids 257 References 275 6 Linear Chromatography 6.1 The Plate Models 6.2 The Solution of the Mass Balance Equation 6.3 The General Rate Model of Chromatography 6.4 Moment Analysis and Plate Height Equations 6.5 The Statistic Approach 6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography 6.7 Extension of Linear to Nonlinear Chromatography Models References xi XI
281 283 290 301 310 328 335 341 342
xii
Contents Contents
7 Band Profiles of Single-Components with the Ideal Model 7.1 Retrospective of the Solution of the Ideal Model of Chromatography 7.2 Migration and Evolution of the Band Profile 7.3 Analytical Solutions of the Ideal Model 7.4 The Ideal Model in Gas Chromatography 7.5 Practical Relevance of Results of the Ideal Model
347 349 351 363 377 379
8 Band Profiles of Two Components with the Ideal Model 8.1 General Principle of the Solution 8.2 Elution of a Wide Band With Competitive Langmuir Isotherms . . 8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms . 8.4 Method of Calculation of the Ideal Model Solution in a Specific Case 8.5 Dimensionless Plot of a Two-component Band System 8.6 The Displacement Effect 8.7 The Tag-Along Effect 8.8 The Ideal Model in Gas Chromatography 8.9 Practical Relevance of the Ideal Model References
387 390 395 401 407 414 416 419 421 423 436
9 Band Profiles in Displacement Chromatography with the Ideal Model 9.1 Steady State in the Displacement Mode. The Isotachic Train . . . . 9.2 The Theory of Characteristics 9.3 Coherence Theory 9.4 Practical Relevance of the Results of the Ideal Model References
437 439 450 461 467 468
10 Single-Component Profiles with the Equilibrium Dispersive Model 10.1 Fundamental Basis of the Equilibrium Dispersive Model 10.2 Approximate Analytical Solutions 10.3 Numerical Solutions of the Equilibrium-Dispersive Model 10.4 Results Obtained with the Equilibrium Dispersive Model References
471 473 476 492 509 527
11 Two-Component Band Profiles with the Equilibrium-Dispersive Model 531 11.1 Numerical Analysis of the Equilibrium-Dispersive Model 532 11.2 Applications of the Equilibrium-Dispersive Model 542 References 567 12 Frontal Analysis, Displacement and the Equilibrium-Dispersive Model 569 12.1 Displacement Chromatography with a Nonideal Column 570 12.2 Applications of Displacement Chromatography 587 12.3 Comparison of Calculated and Experimental Results 599 References 603
Contents Contents
xiii
13 System Peaks with the Equilibrium-Dispersive Model 13.1 System Peaks in Linear Chromatography 13.2 High-Concentration System Peaks References
605 606 626 647
14 Kinetic Models and Single-Component Problems 651 14.1 Solution of the Breakthrough Curve under Constant Pattern Condition 653 14.2 Analytical and Numerical Solutions of the Kinetic Models 669 14.3 Comparison Between the Various Kinetic Models 680 14.4 Results of Computer Experiments 687 14.5 Numerical Solution of the Lumped Pore Diffusion Model 689 14.6 The Monte Carlo Model of Nonlinear Chromatography 693 References 695 15 Gradient Elution Chromatography under Nonlinear Conditions 699 15.1 Retention Times and Band Profiles in Linear Chromatography . . . 701 15.2 Retention of the Organic Modifier or Modulator 705 15.3 Numerical Solutions of Nonlinear Gradient Elution 711 15.4 Gradient Elution in Ion-Exchange Chromatography 726 References 731 16 Kinetic Models and Multicomponent Problems 16.1 Analytical Solution for Binary Mixture; Constant Pattern Behavior . 16.2 Linear Driving Force Model Approach 16.3 Numerical Solution of The General Rate Model of Chromatography References
735 736 747 754 775
17 Simulated Moving Bed Chromatography 17.1 Introduction 17.2 Modeling of Simulated Moving Bed (SMB) Separations 17.3 Analytical Solution of the Linear, Ideal Model of SMB 17.4 Analytical Solution of the Linear, Nonideal Model of SMB 17.5 McCabe-Thiele Analysis 17.6 Optimization of the SMB Process 17.7 Nonlinear, Ideal Model of SMB 17.8 Recent Improvements in SMB Performance with New Operating Modes 17.9 Numerical Solutions for Nonlinear, Nonideal SMB References
779 780 783 785 806 808 809 816 826 836 845
18 Optimization of the Experimental Conditions 18.1 Definitions 18.2 The Economics of Chromatographic Separations 18.3 Optimization Based on Theoretical Considerations 18.4 Optimization Using Numerical Solutions 18.5 Recycling Procedures
849 851 857 867 883 915
xiv
Contents Contents 18.6 Practical Rules 18.7 Optimization of the SMB Process References
920 924 935
Glossary of Symbols
939
Glossary of Terms
949
Index
969
Chapter 1 Introduction, Definitions, Goal Contents 1.1 History of Chromatography 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5
Discovery by Tswett and Early Works The Rebirth of Chromatography The Manhattan Project and the Purification of Rare Earth Elements The API Project and the Extraction of Purified Hydrocarbons from Crude Oils . Preparative Chromatography as a Separation Process
1.2 Definitions 1.2.1 1.2.2 1.2.3 1.2.4
Linear and Nonlinear Chromatography Ideal and Nonideal Chromatography Separation, Extraction, and Purification The Various Scales of Preparative Chromatography
1.3 Goal of the Book References
3 3 4 5 6 8
12 13 13 14 15
15 17
Introduction Chromatography was born as a preparative technique [1] at the turn of the last century. This was a time when there were no physical methods of analysis, when the acquisition of physico-chemical data was slow and limited to a few parameters of low specificity (e.g., melting points, densities, refraction indices). Analytical methods were essentially based on chemical reactions, they were slow, and had a poor sensitivity. Originally designed to extract purified plant pigments from complex mixtures of vegetal origin, chromatography had to be used with sequential fraction collection, followed by the off-line analysis of these collected fractions. In the early 1950s, the development of sensors, transforming the variations of physical or physicochemical properties into a current or a voltage and now known as detectors, and that of recorders of the electric signals provided by these sensors transformed science. With the invention of these devices, modern instrumentation changed profoundly the way in which chemists use chromatography. In the early days of chromatography, the lack of sensors and the need to perform chemical reactions on the isolated fractions to identify and quantify their components imposed off-line detection. Then, the relative lack of sensitivity of most of the chemical methods of detection available at the time dictated the use of large-diameter columns, the injection of large samples, and the handling of only concentrated sample solutions. All these reasons combined made the chromatographic technique nonlinear, resulting in strongly unsymmetrical individual
2
Introduction, Definitions, Goal
band profiles. Moreover, under such conditions, the retention times and the band shapes depend not only on the amount of each component in the sample, but also on the composition of that sample. It was only in the mid-1950s that the development of gas chromatography [2] and the rapid progress made in the field of instrumentation permitted a considerable reduction in the column loading and made possible the operation of columns under linear conditions. Progressively, (1) the advancement of on-line detection, of on-line recording of the eluent composition, and later of the automatic integration of the peak area; (2) the development of new, extremely sensitive detectors; and (3) the progressive extension of on-line detection with very sensitive detectors to all the modes of chromatography led to the close association in the minds of analytical chemists between analytical chromatography and operation under linear conditions. Finally, a stage was reached where column overloading was considered to be an error, if not a sin, and was avoided at all costs. From there, preparative chromatography had to be rediscovered: a mental process that requires the painful revisitation of our knowledge and the reassessment of many rules and principles. For the last forty years, there has always been interest in the use of chromatography for the purification of valuable compounds. Several attempts at commercializing and popularizing preparative gas chromatography were made in the 1960s and 1970s. They met with little success [3-5]. The main reasons for these failures were economic. Few compounds are both valuable enough to justify the extraction/purification costs of this process and volatile enough to be purified at an affordable price by preparative gas chromatography. The rapid development of the fine chemical, pharmaceutical, and biotechnology industries during the last twenty years has combined with the pressure of the regulatory agencies. Their effort has led to the production of many high-purity chemicals to be used as pharmaceuticals or pharmaceutical intermediates, to the identification of the metabolites of these compounds, and to the completion of systematic studies of the toxicological properties of potential drugs and of their metabolites prior to their approval. This endeavor has generated the need to separate, extract, and purify many chemicals in the laboratory or at the industrial scale. Moreover, traditional techniques such as distillation, counter-current or centrifugal extraction, and crystallization used by the petroleum and the commodity chemical industries do not meet the needs of the pharmaceutical industry. No industrial separation technique is more versatile than chromatography, nor better suited for the rapid production of milligram to ton quantities of highly pure products. None has a comparable separation power. In recent years, preparative chromatography has been adopted as an industrial process in the pharmaceutical industry. Units with a production capacity ranging from a few pounds to more than a thousand tons per year have been built. In the meantime, the chemical industry has developed numerous processes based on the use of adsorption. The complexity of these processes has been increasing constantly. Some of the recent ones are based on the use of chromatographic principles. This is the case for separation processes based on the simulated moving bed concept [6-8]. Initially developed for the extraction of a few
1.1 History of Chromatography
3
specific compounds from complex mixtures, such as para-xylene from reforming streams or fructose from corn syrup, these processes are competing with the simpler chromatographic processes evolved from direct scaling-up of the laboratory procedures. Currently, equipments for overloaded elution may achieve production rates of up to 500-1000 ton/year or above and can handle most complex mixtures. Simulated moving bed units have been built with production rates between a few pound/year to more than a million ton/year. These equipments are unsurpassed for the separation of binary mixtures {e.g., enantiomers). A variety of recycling processes involving elution can fill in the gap.
1.1 History of Chromatography Tswett (1872-1919) was ahead of his time. A long induction period followed the tragic interruption of his work during the Russian civil war and his untimely death. Not until the early 1930s did the importance of chromatography became recognized among chemists involved in the study of natural products [9] and biochemists who continued to play a critical role at several stages of development {e.g., in the discoveries of paper chromatography [10], gas chromatography [2], size exclusion chromatography [11], and affinity chromatography [12], among others). With the progress made in the development of sensitive detection methods, analytical and preparative chromatography parted in the late 1940s. The first major preparative chromatography projects were the purification of rare earth elements by the group of Spedding [13] for the Manhattan project, and the isolation of pure hydrocarbons from crude oil by Mair et al. for the API project [14]. Later followed the development of the simulated moving bed technology by Broughton for UOP [6]. Finally, in the 1980s, the pharmaceutical industry began to show interest in high-performance preparative chromatography and this interest has been increasing steadily over the last twenty years [15]. Chromatography is now acknowledged as an industrial unit operation for the extraction and the purification of fine chemicals, particularly those used as pharmaceutical intermediates.
1.1.1 Discovery by Tswett and Early Works The story of the discovery of chromatography is classical [16,17]. A most lucid analysis of Tswett's work from the point of view of the preparative applications of chromatography, has been written by Verzele and Dewaele [18]. The Russian botanist Tswett discovered around 1902 that plant pigments could be separated by eluting a sample of plant extract with a proper solvent on a column packed with a suitable adsorbent [1]. Did he name the technique chromatography because it separates pigment mixtures into a rainbow of colored bands, or because "tswett" means color in Russian, or both? Nobody knows. What is remarkable, however, is the extreme care with which Tswett selected the adsorbents he used [19-21]. For the famous separation of a- and /3-carotenes, he tried 110 different adsorbents and selected inulin (a water-soluble polyfructose plant reserve material) as the
4
Introduction, Definitions, Goal
stationary phase, and ligroin (the 70-120° C distillation cut of crude oil) as the eluent [18-22]. Thus, the first publications on chromatography are masterpieces in method development. From its inception, the method suffered from a relatively poor production rate (poor relative to the size of the column used and to the amount of solvent that is needed per unit amount of products extracted). This limited production capacity was the main criticism that Willstater made to the work of Tswett [17]. To this criticism, Tswett rejoined that production rate could always be increased by using larger bore columns. After a century, the dilemma remains unchanged. The strength of the objection, the scientific stature of Willstater, and the late recognition of the extreme separation power of chromatography delayed its adoption as a laboratory technique for a third of a century. Tswett understood very well the importance of the nature of the adsorbent and the key role of differential adsorption in the separation [1,19-21]. His most popular result, the separation of the carotene and xanthophyll isomers on a calcium carbonate column, turned out to be most difficult to reproduce by later workers. Tswett had understood the importance of a number of characteristics of the packing material that had escaped the attention of the early followers, in part because the Russian text of his Ph.D. thesis was unavailable to most of them. The characteristics discussed by Tswett were (i) the purity of the adsorbents, (ii) the average particle size (which should be fine), and the size distribution (which should be narrow), (iii) the packing homogeneity, and (iv) the integrity of the top of the packed bed (which should not be disturbed when the sample is injected). He had also recognized the importance of avoiding chemical changes catalyzed by the packing, a serious risk in the analysis of carotenoids, and the particular inertness of polyol-based adsorbents. There are very few serious claims to a glimpse at chromatography predating Tswett's work or even to an independent later discovery, in spite of the 30-yearlong induction period that followed Tswett's earlier publications. The one who probably came closest was Day [22], who attempted to fractionate petroleum by filtration through columns of powdered limestone or fuller's earth.
1.1.2
The Rebirth of Chromatography
Unfortunately, the rare scientists who have used chromatography in the beginning of the 20th century [23-26] did as Tswett had done. They purified only very small amounts of a few natural pigments in solution, for further spectroscopic studies. For nearly thirty years, chemists remained reluctant to use chromatography. Its production rate was very small, at a time when organic chemistry reactions were carried out on a large scale and when new chemicals were produced in amounts that seem, now, for us, to be amazingly large. Accordingly, chromatography had to be run at a scale that would now be that of laboratory preparative chromatography. It was an expensive method requiring large volumes of solvents. Nonlinear effects made the results of the method difficult to understand and control. The lack of detectors made its application to colorless compounds impractical. Furthermore, serious doubts had been cast by Willstatter and Stoll [27] on the integrity of the collected products. This was due to their use of inadequate adsorbents that
1.1 History of Chromatography
5
catalyzed the isomerization of the labile carotenoids and the other plant pigments that they were studying. Tswett had warned against the use of silicagel in these separations but the reasons for this advice had not been understood. So, Tswett's findings were not reproduced and attracted little interest until Kuhn and Lederer [28,29] demonstrated the power of chromatography as a preparative method for the separation of many carotenoids, performing the separation of a- and /3carotene from carrots and separating egg yolk pigments [30] on a 7-cm i.d. column packed with calcium carbonate, using carbon disulfide as the mobile phase. This last work yielded 30 mg of carotene. Gram amounts were prepared soon afterward [31]. The interesting story of the rediscovery of chromatography in Heidelberg in 1931 is told in quite impressive detail by Lederer [9]. Among many contributions dating from that period, we must note the clear and systematic definitions by Tiselius and Claeson in 1943 [32] of the three modes of chromatography, already known to Tswett [19-21,33] — elution, frontal analysis and displacement. Displacement became very popular during the late 1930s and the 1940s, later to fade away and reemerge in the 1980s under the creative impulsion of Csaba Horvath [34]. The properties and performance of this method are discussed in Chapters 9 and 13.
1.1.3 The Manhattan Project and the Purification of Rare Earth Elements The progressive realization that adsorption chromatography could provide a high selectivity that few other separation techniques could offer paved the way for several large-scale applications, such as the isolation of oxides of rare earth elements for nuclear applications and the purification of petroleum products [35]. In the mid-1940s, the development of large-scale purification schemes for the separation of rare earth metals [13,35-41] stemmed from the demands of physicists who became interested in the unusual nuclear properties of the actinide elements in the early 1940s [38]. These investigations were carried out under the auspices of the Manhattan Project. Purification of the rare earth elements by ion-exchange chromatography was primarily conducted by Spedding et ah [13,35-41], who were able to separate various salts by displacement chromatography. Figure 1.1 presents a chromatogram whereby they separated three of the rare-earth ions, Samarium, Neodymium, and Praseodymium on Amberlite IR-100, using three 22 mm i.d., 30, 60, and 120 cm long columns, and a 0.1% buffer solution of citric acid and ammonium citrate at pH 5.30 as the displaces Other experiments show that the separation and the retention factors increase with decreasing pH, resulting in an optimum pH around 5. Numerous investigations on the effects of the various operating parameters were carried out (see further explanations in Chapter 12). A pilot plant, involving sixty-eight 4- and 6-inch-diameter columns, in a fourstep cascade, was developed to achieve the production of high-purity metals in large quantities [35,38-40]. The concentration of the citrate buffer was minimized to reduce costs. The advantage of using fine resin particles was noted, as well as the rather long column length needed to achieve formation of the isotachic
Introduction, Definitions, Goal
r
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3 0 CM
100 -
50 ;
0 -.
/
.
.
i
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\
15
20
25
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Figure 1.1 Separation of three rare-earth elements, Sm, Nd, and Pr, by displacement chromatography. Columns: 22mm i.d., 30, 60, and 120 cm long, packed with Amberlite IR-100 resin. Displacer solution: 0.01% citrate solution, pH 5.3. Reproduced with permission from F.H. Spedding, E.I. Fulmer, T.A. Butler and /.£. Powell, } . Am. Chem. Soc, 72 (1950) 2354 (Fig. 8). ©1950 American Chemical Society.
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train, and the improvement of the separation with increasing column length at constant loading factor.1 One major source of difficulties was the growth of molds in the column, efficiently controlled by adding 0.1% phenol to the mobile phase, an addition which had no effect on the separation [35].
1.1.4
The API Project and the Extraction of Purified Hydrocarbons from Crude Oils
In the late 1940s and early 1950s, the American Petroleum Institute (API) used displacement chromatography to fractionate samples of virgin crude oil and petroleum distillates to determine their content of paraffins, naphthenes, olefins, and aromatics and to isolate many pure compounds that were identified for the first time in crude oil or crude oil extracts [42]. In addition to working on the laboratory scale, 1
Ratio of the sample size to the amount of solute needed to saturate the resin.
1.1 History of Chromatography
THIRD FLOOR-
Lib SECOND FLOOR.
PENTHOUSE
Figure 1.2 Assembly of a 52-foot adsorption column. Aj, A2, connections to a source of pressurized nitrogen. B, Vent to roof. Cj, C2, Copper tubes, 1-inch o.d. for filling reservoirs. Dj-Dio, Packless diaphragm valves. Ej, E2, brass reservoirs to contain hydrocarbon and alcohol. F, steel plate for supporting reservoir. G, copper tubing. H, 1-liter receiver for recovering alcohol regenerant. I, prismatic sight glass. Ji, ~\i, top and bottom flanges. Kj, K2, top and bottom gaskets. L, petcock. M1-M4, transite collars to permit vertical movement. N1-N3, angle iron supports. O, aluminum foil. P, magnesia insulation. Q, nichrome heating wire. R, asbestos coated with resin. S, 1-inch o.d. stainless steel column. T, 200-325 mesh silicagel adsorbent. U, thermocouple well. V, steel plate supporting entire weight of column. W, glass wool. X, brass plug supporting adsorbent. Y, standard tapper. Z, receiver. Reproduced with permission from B.D. Mnir, A.L. Gaboriault, and F.D. Rossini, Ind. Eng. Chem., 39 (1947) 1072 (Fig. 1). ©1947 American Chemical Society.
a team led by Mair and Rossini [14,43-45] assembled and tested six 3/4 inch (19 mm) i.d., 52.4 foot (16 m) long stainless steel columns, a schematic of which is shown in Figure 1.2 [14]. These columns were placed in an elevator shaft. They were temperature controlled. Each column (volume, 5.52 L) was packed with 3.7 kg of 200-235 mesh silica gel, the total porosity being 77%. Columns were operated in the displacement mode, at first using isopropanol as the displacer. The typical sample size was 0.5 L. Sample introduction in the column took 12 hours; the first component broke through after 34 hours. Sample collection (1550 mL) took 21.5 hours and the average liquid flow rate was 72 mL/hour, with an inlet pressure of 15 psi at the
Introduction, Definitions, Goal
1 -methylnaphthalene
160
:
155 triethylbenzene
150
r
:
145 -
\
1 \
n-dodecane
140 -
isopropanol
135
:
1
.
.
1—-,
400
,
,
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,
,
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,
,
,
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Figure 1.3 API Project, separation of a synthetic mixture of n-dodecane, triethylbenzene, and 1-methylnaphthalene with isopropanol as the displacer. Column as in Figure 1.2, packed with silica gel. Reproduced with permission from B.D. Mair, A.L. Gaboriault, and F.D. Rossini, Ind. Eng. Chem., 39 (1947) 1072 (Fig. 5). ©1947 American Chemical Society.
beginning of the experiment and 90 psi at the end. An example of the composition profile of the column eluent is reproduced in Figure 1.3. Because the mixed zone between the last component and the displacer was too wide, isopropanol was replaced as the displacer by 3-methyl-l-butanol, which provided a much sharper separation [14]. The importance of using a displacer more viscous than the feed is noted but not justified (this was the first observation of the effects of viscous fingering in chromatography; see end of Chapter 5 for explanations).
1.1.5
Preparative Chromatography as a Separation Process
In the early 1970s, Union Oil developed and patented a chromatographic system based on the principle of a simulated moving bed (SMB) [6-8]. A schematic of a SMB unit is shown in Figure 1.4. Streams of the mobile phase (the 'desorbent') and of the feed to separate are continuously injected into the column while streams of the less retained (the 'raffinate') and the more retained components (the 'extract') are continuously withdrawn, all at constant flow rates. The rotary valves switch periodically the positions in the columns where these streams enter or exit. The operation of SMB units is discussed in detail in Chapter 17. Manufacturing facilities have been built and are operated for the fractionation of various petroleum distillates, for example, the selective separation of p-xylene, o-xylene and ethylbenzene from the C7-C8 aromatic fraction of light petroleum reformates, the separation of olefins from paraffins in feed mixtures of hydrocarbons having 10 to 14
1.1 History of Chromatography
LEGEND AC = ADSORBENT CHAMBER RV = ROTARV VALVE EC = EXTRACT COLUMN RC = RAFFIHATE COLUMN
Figure 1.4 Chromatographic system based on a simulated moving bed. ReprintedfromD.B, Broughton, Sep. Sci. Tech., 19 (1984) 723 (Fig. 5) by courtesy of Marcel Dekker Inc.
carbon atoms, and the separation of fructose and dextrose from corn syrup. It has been shown that chromatography, which is normally a batch process in both the elution and the displacement mode, could be turned into a continuous process if the stationary phase were forced to move along the column. Physically moving the stationary-phase bed is impractical (Chapter 17). However, the moving bed operation can be simulated, as accomplished in the Union Oil process by the use of a number of columns placed on-line and connected to a rotary valve [6]. The position of the desorbent, extract, feed, and raffinate can be switched in a practical way, permitting continuous unit operation (Chapter 17). In the late 1970s, Elf Inc. (France) developed a process to extract a large proportion of the n-alkanes (mainly n-pentane and n-hexane) contained in light petroleum distillates to prepare high-octane gasoline [46]. This process is based on the injection of large pulses of feed on a Molecular Sieve column. The branched pentanes and hexanes are not retained and elute as a large unretained pulse. In this process, the n-alkanes are strongly retained and elute as a very wide band. In practice, when feed pulses are injected sequentially at a sufficiently high frequency, the eluent contains a constant, low concentration of n-alkanes, which is given by the product of their concentrations in the feed and the ratio of the pulse duration to the pulse period. The elution bands of isoparaffins are collected as a highoctane gasoline. With this gas chromatographic process, it is easy to decrease the n-alkane concentration in the gasoline by a factor that slightly exceeds two. The plans to build a 100,000 tons per year plant were canceled after the first petroleum price shock, due to financial difficulties independent of the economics of the process [44]. Several designs were suggested in the 1970s for the continuous operation of the
Introduction, Definitions, Goal
10
FIXED FEED INLET
ELUENT
ELUENT INLET
STATIONARY ELUENT WASTE COLLECTION
- ANNULAR BED
Figure 1.5 Chromatographic system based on a rotating annular column. Reproduced with permission from A.}. Howard, G. Carta and CM. Byers, Ind. Eng. Chem., 27 (1988) 1873 (Figs. 1 and 2). ©1988 American Chemical Society.
chromatographic process, by rotating the column around its axis while the feed injector and the fraction collector remain fixed (Figure 1.5 [47]). Baxter and Deeble used a rectangular cross-section column, with an open side sliding along a plate pierced by several holes for the feed injection and the collection of a "retained" {i.e., moving backward, in the same direction as the column) and an "unretained" {i.e., moving forward, in the same direction as the carrier gas) fraction [48]. Scott et al. used a rotating, annular column [49]. The feed is injected at a fixed point. The solutes follow helicoidal paths in the annular column and exit at fixed places where the corresponding fractions are collected. Although this process is continuous, similar to elution chromatography, only a fraction of the column volume works at any given time. The implementations of these designs are complex. Perhaps because this complexity was considered excessive by chemical engineers, or because its performance was no better than that of competitive processes, or because the design was early for the times, or for some other reason, this process has not been accepted yet. New designs, including that of simulated rotating annular columns, are proposed occasionally in the literature or in communications made at meetings and a few implementations are made commercially available for some time. It seems that this solution is still looking for the problems that it can solve usefully. Over the last twenty years, the use of semipreparative and preparative chromatography has expanded considerably. A vast number of applications has been reported, mostly in the pharmaceutical industry, where preparative chromatography is an important general-purpose separation process. The amounts of purified
1.1 History of Chromatography Figure 1.6 Effects of a progressive consolidation of a column bed. (a) Photograph of a process-chromatography flanged-end column (no compression) after removal of the flange, (b) Chromatogram of a test sample in this column voids and after its repacking, (c) Chromatogram of a test sample in a chromatographic columns with and without compression. The exact amounts injected are different for all chromatograms.
11
Repack With Compression
Without Comprettiion
^ Voids
Time
products that are required for these applications are compatible with the use of columns ranging from a few inches to a few feet in diameter, which are technically quite feasible to build, pack, and operate. The purity requirements are often easier to meet in chromatography than with other separation methods. Most fine chemicals, particularly those to be used for the production of pharmaceuticals, can bear the cost of chromatographic separations. The purifications of enantiomers, peptides, and proteins are the most widely published applications, but many others have been reported. Several reviews [50-52], numerous books [53-58] and many special volumes of the Journal of Chromatography have discussed these applications [59]. One of the critical problems of large scale applications of preparative chromatography has been the achievement of large diameter columns having a separation power comparable to that of analytical columns. Long-term bed stability, high column efficiency, and low hydrodynamic resistance are important qualities required of industrial columns. Manufacturing groups have long complained that after a certain period of satisfactory operation runs lasting from a few days to several months, the performance, particularly the efficiency, of wide columns decreases sharply. When the column is opened, large voids are seen at the top of the bed. Filling them with fresh packing material can restore the initial column efficiency, provided suitably intense vibrations are applied. Figure 1.6 illustrates a large process-scale column with voids, as indicated by the arrow on the RHS of the photo. Below are chromatograms showing channeling and tailing as a result of void formation. To prevent this phenomenon and its often costly consequences, various implementations of bed compression have been proposed. Compression can be static or dynamic, the latter providing the large advantage of automatically eliminating the voids as they form, before their size can make them harmful for the column efficiency. With static methods, bed compression has to be applied by the operator when the column efficiency has dropped below a threshold and this reduces the performance of the unit compared to dynamic compression. Compression can be annular [60], axial [61] or radial [62]. Currently, implementations are commercially available for dynamic axial and radial compression and for static annular compression for semipreparative units but only for dynamic axial compression for large-scale industrial units.
12
Introduction, Definitions, Goal
The hardware employed in preparative liquid chromatography typically has the capability for feedback control of the pumps, automatic column switching, column backflushing, recycling, gradient mixing, online pressure, flow rate, UVabsorbance, and temperature monitoring, and automatic pneumatic actuation of fraction collection valves based on time, volume, or UV-absorbance thresholds. These capabilities are afforded by computer control. In conclusion, it is interesting to note that the early applications of preparative liquid chromatography in the 1940s and 1950s involved operating the column in the displacement mode [13,14,32,33,35-45], in which by definition chromatography is performed under nonlinear conditions. In the 1980s the approach to preparative scale chromatography was somewhat different, in fact nearly opposite. Twenty years of successful applications of liquid chromatography in the achievement of all kinds of analytical separations where column "overloading" was always scrupulously avoided had made chemists extremely reluctant to perform separations under conditions that deviate strongly from analytical practice. The early trend involved the nearly exclusive use of large-diameter columns operated in the elution mode, with the injection of rather large volumes of dilute solutions, in order to achieve "volume overloading" with no or only moderate "concentration overloading." In most cases, however, concentration overload is a far more economical approach. Realization of this situation led to thorough studies of the properties of nonlinear chromatography, and their mastery in the operation of columns under conditions of heavy volume and concentration overload. Still, elution is practically the only mode of chromatography used. In the last twenty years, proponents of displacement have relentlessly presented claims that pose serious advantages for certain separation problems [63,64]. However, demonstrations of practical applications of this technique in the regulated environment of the pharmaceutical industry have never been supplied. The error of the claim that displacement provides a 100% recovery yield because of the boxcar elution of its bands has been demonstrated (see Chapter 12). The progressive realization that isocratic elution, whenever possible, leads to larger production rates, higher recovery yields, and easier operation, albeit to the production of more dilute fractions than displacement [65-67] has ended the controversy. For proteins and moderate or large peptides, the balance of advantages and drawbacks of the displacement and elution methods has long remained uncertain as, in most cases, these biochemicals cannot be extracted or purified by isocratic elution. It seems now that the advantage goes to gradient or step gradient elution. The various recycling procedures available, as well as the simulated moving bed process, continue to undergo thorough investigations now that the fundamentals of nonlinear chromatography are better understood.
1.2
Definitions
In order to clarify matters, we present here a number of definitions of terms frequently used in this book.
1.2 Definitions
13
1.2.1 Linear and Nonlinear Chromatography In linear chromatography, the equilibrium concentrations of a component in the stationary and the mobile phases are proportional. In other words, the equilibrium isotherms are straight lines beginning at the origin. The individual band shapes and the retention times are independent of the sample composition and amount. The peak height is proportional to the amount of each component in the injected sample. Linear chromatography accounts well for most of the phenomena observed in the analytical applications of chromatography, as long as the injected amounts of the sample components are kept sufficiently low. Linear chromatography is discussed in Chapter 6. Since any isotherm can be expanded into a second degree polynomial, q(C) = aC + bC2, we consider any chromatographic experiment as carried out under linear conditions as long as bC
1.2.2
Ideal and Nonideal Chromatography
In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria, hi linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer
14
Introduction, Definitions, Goal
and of axial dispersion. Ideal chromatography is discussed in Chapters 7 to 9. In nonideal chromatography, the column efficiency has a finite, measurable value. Several models are distinguished, depending on assumptions made regarding the nature and importance of the band spreading. We can assume that axial dispersion is finite, while the mass transfer kinetics proceeds at an infinite rate. Alternately, we can neglect axial dispersion and select one source of band broadening of kinetic origin, for example, the kinetics of adsorption-desorption or the kinetics of diffusion through the porous particles. Finally, the general rate model of chromatography takes into account all the sources of nonideal behavior, molecular diffusion, eddy diffusion, and the kinetics of the various steps involved in the mass transfer process. The models of nonideal chromatography are discussed in Chapters 10 to 16.
1.2.3
Separation, Extraction, and Purification
Because of the different nature of the separation problems encountered in chemistry, a variety of names are used, sometimes interchangeably, e.g., separation, concentration, enrichment, extraction, purification. Analytical separations deal with di-
lute solutions, and separation is usually the only term used, complemented by resolution when the degree of separation achieved is quantified. In preparative chromatography, extraction usually refers to the collection, from the feedstock, of a certain component contained in that feed, or of most of it. Usually, the extraction is positive; that is, the extracted component is desirable. However, in some cases, an undesirable component can be eliminated from a mixture; the procedure is quite similar, and chromatography is equally well suited to extraction and to elimination. Rony [68] has proposed that enrichment be used when the mole fraction of the desired component remains below 0.1. Concentration should be used if the mole fraction is higher than 0.1 but lower than 0.9. In purification, the mole fraction of the desired component is raised above 0.9. Obviously, these levels are arbitrary. The use of Rony's conventions would contribute to a clarification of the language, which appears desirable. However, the imprecision of the words used reflects the complexity of the technical situation, since separation by chromatography is often accompanied by dilution. This is always the case in isocratic elution chromatography and often also in gradient elution. With these techniques, a large increase in the product purity occurs with dilution in the mobile phase. Thus, in general scientific terms, enrichment often refers to a moderate increase of the product purity, accompanied or not by a certain dilution. Purification refers to a similar effect but assumes a larger increase of the product purity. Concentration refers to an increase of the absolute concentration of the component in the final product relative to its feed concentration. In practice, concentration is a required step in the downstream processing of the fractions collected from a preparative chromatograph.
1.3 Goal of the Book
1.2.4
15
The Various Scales of Preparative Chromatography
Depending on the primary aim, two main areas in preparative chromatography are distinguished. It is possible to use preparative chromatography either to prepare the small amounts of purified products required for e.g., their structural identification, or to produce large amounts of an intermediate or a final product for commercialization. In the first area, the purified chemicals are needed as intermediates in the process generating the desired information. This is the case when small amounts of material are needed for structural identification and characterization, for the determination of some physical or physico-chemical property (e.g., spectral data), or for the acquisition of toxicological or pharmacological data. Generally, time is of the essence in such cases. Very little time is available for optimization. The application of a few simple optimization rules must do. Knowing the basics of nonlinear chromatography will help, and that may be the only help available. Furthermore, the time spent by the person charged with the development of the separation and its achievement is the essential component of the cost, and that time is what should be minimized. The contributions of the cost of solvents, stationary phase, chemicals, and instrument amortization will be small in comparison, due to the small amounts required. In such cases, there is not much time available and no incentive for isotherm measurements and modeling. In the second area, a purified compound is needed to obtain a final product, and the cost of the production of the compound is an important cost factor that will have to be minimized. The production will last a significant period of time, whether it is continuous or by batches run periodically, and the operation is relatively routine. The cost components, equipment, solvent, packing material, crude feed, and downstream processing become prominent and must be taken into account together. Then significant investment in the design of the separation process is required for a careful optimization of the experimental conditions. Optimization procedures are discussed in Chapter 18.
1.3 Goal of the Book The fundamental problems of nonlinear chromatography and the theory of preparative chromatography have been the topic of intense activity by chromatographers and chemical engineers. Each community has largely overlooked the activities, as well as the preoccupations and viewpoints, of the other one. Furthermore, this activity did not start abruptly and is not entirely finished. Theoretical problems in nonlinear chromatography have been discussed in the literature for more than 50 years and some are still today. Some of these works have been quite influential in some circles, while they were completely ignored in others. Preparative chromatography will continue to play a role of major importance as a separation process in the pharmaceutical and biotechnological industries in the next quarter of a century. A number of different processes are already used. Some of them will undergo considerable improvement. A few others will be invented or developed. There is presently no simple way to have easy access to the
16
Introduction, Definitions, Goal
abundance of literature of uneven quality on this topic. We have attempted to present here, in a rather condensed form, a view of the present status of the fundamentals of preparative and nonlinear chromatography. The fundamental problems and the various models used to model chromatography are discussed first (Chapter 2). As the thermodynamics of phase equilibrium is central to the separation process, whatever model is used, we devote two chapters to the discussion of equilibrium isotherms, for single components (Chapter 3) and mixtures (Chapter 4). A chapter on the problems of dispersion, mass transfer and flow rate in chromatography (Chapter 5) completes the fundamental bases needed for the thorough discussion of preparative chromatography. In the second part of this volume, the separation process is described according to the different models of chromatography. Since in linear chromatography the band profiles are controlled only by axial dispersion and the mass transfer resistances, the study of this particular case (Chapter 6) permits a detailed investigation of the interaction of band migration, band profiles, and the various kinetic phenomena in a column. Conversely, the ideal model which assumes that these kinetic effects are negligible permits the investigation of the single remaining contribution, that of thermodynamics, to the formation and migration of band profiles and to band separation. Chapters 7 to 9 discuss the solution of the ideal model for a single component, for a binary mixture and for a mixture separated in displacement chromatography, respectively. As we shall see, at high concentrations, the influence of thermodynamics on the results of chromatographic experiments is, in practice, much more important than that of kinetics and the various sources of band broadening. Nevertheless, chromatographic columns have a finite efficiency, and the equilibrium-dispersive model (Chapters 10 to 13) provides the solution of the problem in almost all cases of interest in preparative chromatography. It permits the calculation of the individual band profiles given the necessary physicochemical data. Two chapters (Chapters 14 and 16) deal with the kinetic models which permit the study of chromatography in cases in which the kinetics of mass transfer or that of adsorption-desorption (or, more generally, the kinetics of the retention mechanism) is slow. Gradient elution under nonlinear conditions is discussed in Chapter 15. In the last part of this book, we apply the different models discussed earlier, particularly the ideal model and the equilibrium-dispersive model, to the investigation of the properties of simulated moving bed chromatography (Chapter 17) and we discuss the optimization of the batch processes used in preparative chromatography (Chapter 18). Of central importance is the optimization of the column operating and design parameters for maximum production rate, minimum solvent use, or minimum production cost. Also critical is the comparison between the performance of the different modes of chromatography. This book is written for those who want to understand the mechanisms through which band profiles are generated and separated in preparative and process chromatography. It should also be useful to those who develop and optimize chromatographic separations.
REFERENCES
17
References [1] M. S. Tswett, Tr. Protok. Varshav. Obshch. Estestvoistpyt., Otd. Biol. 14 (1903, publ. 1905) 20. On the New Category of Adsorption Phenomena and their Applications in Biochemical Analysis. Reprinted and Translated in G. Hesse and H. Weil, Michael Tswett's erste chromatographische Schrift, Woelm, Eschwegen, 1954. [2] A. J. P. Martin, A. James, Biochem. J. 50 (1952) 679. [3] R. E Baddour, U. s. patent no. 3,250,058 (1966). [4] B. Roz, R. Bonmati, G. Hagenbach, P. Valentin, G. Guiochon, J. Chromatogr. Sci. 14 (1976) 367. [5] R. Bonmati, G. Chapelet-Letourneux, G. Guiochon, Separat. Sci. Technol. 19 (1984) 113. [6] D. Broughton, Separat. Sci. Technol. 19 (1984) 723. [7] R. M. Nicoud, G. Fuchs, P. Adam, M. Bailley, E. Kusters, F. Antia, R. Reuille, E. Schmid, Chirality 5 (1993) 267. [8] C. B. King, K. H. Chu, K. Hidajat, M. Uddin, AIChE J. 38 (1992) 1744. [9] E. Lederer, J. Chromatogr. 73 (1972) 361. [10] A. J. P. Martin, R. L. M. Synge, Biochem. J. 35 (1941) 1359. [11] G. H. Lathe, C. R. J. Ruthven, Biochem. J. 62 (1956) 665. [12] P. Cuatrecasas, C. B. Anfinsen, in: S. P. Colowick, N. O. Kaplan (Eds.), Methods in Enzymology, Vol. XXII, Academic Press, New York, NY, 1971, p. 345. [13] F. H. Spedding, A. F. Voight, E. Gladrow, N. R. Sleight, J. E. Powel, J. M. Wright, T. A. Butler, P. Figard, J. Am. Chem. Soc. 69 (1947) 2786. [14] B. J. Mair, A. Gaboriault, F. Rossini, Ind. Eng. Chem. 39 (1947) 1072. [15] E. P. Kroeff, R. A. Owens, E. L. Campbell, R. D. Johnson, H. Marks, J. Chromatogr. 461 (1989) 45. [16] K. Sakodynskii, J. Chromatogr. 49 (1970) 2. [17] K. Sakodynskii, J. Chromatogr. 73 (1972) 303. [18] M. Verzele, C. Dewaele, Preparative High Performance Liquid Chromatography, Chapter I., TEC, Gent, Belgium, 1986. [19] M. S. Tswett, Ber. Deut. Botan. Ges. 24 (1906) 316. [20] M. S. Tswett, Ber. Deut. Botan. Ges. 24 (1906) 384. [21] M. S. Tswett, Khromofilly V Rastitel'nom Zhivotnom Mire [Chromophylls in the Plant and Animal World], Izd. Karbasnikov, Warzaw, Poland, 1910, partly reprinted in 1946 by the publishing house of the Soviet Academy of Science, A. A. Rikhter and T. A. Krasnosel'skaya, Eds. [22] D. T. Day, Congr. Int. Petrol. Paris (France) 1 (1900) 53. [23] C. Dhere, W. Rogowski, C. R. Acad. Sci., Paris, France 155 (1912) 653. [24] L. S. Palmer, C. Eckles, J. Biol. Chem. 17 (1914) 191. [25] C. Dhere, G. Vegezzi, C. R. Acad. Sci., Paris, France 163 (1916) 399. [26] H. F. Coward, Biochem. J. 18 (1924) 1114. [27] R. Willstatter, A. Stoll, Untersuchungen iiber Chlorophyll, Springer, Berlin, 1913. [28] R. Kuhn, E. Lederer, Naturwissenschaften 19 (1931) 306. [29] R. Kuhn, E. Lederer, Ber. Deut. Chem. Ges. 64 (1931) 1349. [30] R. Kuhn, A. Winterstein, E. Lederer, Hoppe-Seyler's Z. Physiol. Chem. 197 (1931) 141. [31] A. Winterstein, K. Schon, Z. Physiol. Chem. 130 (1934) 139. [32] A. Tiselius, S. Claeson, Arkiv Kemi Mineral Geol. 16A (1943) 18. [33] L. S. Ettre, in: Cs. Horv&th (Ed.), Fligh Performance Liquid Chromatography: Advances and Perspectives, Vol. 1, Academic Press, New York, NY, 1980, p. 25. [34] Cs. Horvath, A. Nahum, J. H. Frenz, J. Chromatogr. 218 (1981) 365.
18
REFERENCES
[35] E H. Spedding, E. I. Fulmer, T. A. Butler, E. M. Gladrow, P. E. Porter, J. E. Powel, J. M. Wright, J. Am. Chem. Soc. 69 (1947) 2812. [36] E H. Spedding, E. I. Fulmer, T. A. Butler, J. E. Powel, J. Am. Chem. Soc. 72 (1950) 2349. [37] E H. Spedding, E. I. Fulmer, J. E. Powel, T. A. Butler, J. Am. Chem. Soc. 72 (1950) 2354. [38] E H. Spedding, J. Powel, J. Am. Chem. Soc. 76 (1954) 2545. [39] F. H. Spedding, J. Powel, J. Am. Chem. Soc. 76 (1954) 2550. [40] E H. Spedding, J. E. Powel, H. J. Svec, J. Am. Chem. Soc. 77 (1955) 6125. [41] E H. Spedding, J. E. Powel, in: F. C. Nachod, J. Schubert (Eds.), Ion-Exchange Technology, Academic Press, New York, NY, 1956, p. 359. [42] D. L. Camin, A. J. Raymond, J. Chromatogr. Sci. 11 (1973) 625. [43] B. J. Mair, A. J. Sweetman, F. D. Rossini, Ind. Eng. Chem. 41 (1949) 2224. [44] B. J. Mair, J. W. Westhaver, E D. Rossini, Ind. Eng. Chem. 42 (1950) 1279. [45] B. J. Mair, M. J. Montjar, E D. Rossini, Anal. Chem. 28 (1956) 56. [46] D. M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, NY, 1984. [47] A. J. Howard, G. Carta, C. H. Byers, Ind. Eng. Chem. Research 27 (1988) 1873. [48] P. E. Baxter, R. E. Deeble, Chromatographia 8 (1975) 67. [49] C. D. Scott, R. D. Spence, W. G. Sisson, J. Chromatogr. 126 (1976) 381. [50] G. Guiochon, A. Katti, Chromatographia 24 (1987) 165. [51] J.-X. Huang, G. Guiochon, BioChromatography 3 (1988) 140. [52] E. Francotte, A. Junker-Buchheit, J. Chromatogr. 576 (1992) 1. [53] P. Belter, E. L. Cussler, W.-S. Hu (Eds.), Bioseparations: Downstream Processing for Biotechnology, Wiley-Interscience, New York, NY, 1988. [54] J. Asenjo (Ed.), Separation Processes in Biotechnology, M. Dekker, New York, NY, 1990. [55] C. T. Mant, R. Hodges (Eds.), High Performance Liquid Chromatography of Peptides and Proteins, CRC Press, New York, NY, 1991. [56] G. Sofer, D. Zabriskie, Biopharmaceutical Process Validation, M. Dekker, New York, NY, 2000. [57] A. S. Rathore, A. Velayudhan (Eds.), Scale-Up and Optimization in Preparative Chromatography: Principles and Biopharmaceutical Applications, M. Dekker, New York, NY, 2003. [58] R. G. Harrison, P. Todd, S. R. Rudge, D. P. Petrides (Eds.), Bioseparations Science and Engineering, Oxford University Press, Oxford, UK, 2003. [59] J. Chromatogr. 461,484, 590, 658, 702, 707, 734, 760, 796, 827,908, 944, 989,1036. [60] A. S. Lawing, L. Lindstrom, C. M. Grill, LC-GC 10 (1992) 778. [61] E. Godbille, P. Devaux, J. Chromatogr. 122 (1976) 317. [62] J. N. Little, R. L. Cotter, J. A. Prendergast, P. D. McDonald, J. Chromatogr. 126 (1976) 439. [63] J. Frenz, Cs. Horvath, in: Cs. Horv&th (Ed.), High-Performance Liquid Chromatography — Advances and Perspectives, Vol. 5, Academic Press, New York, NY, 1988, pp. 211-314. [64] G. Subramanian, S. Cramer, J. Chromatogr. 484 (1989) 225. [65] A. M. Katti, E. Dose, G. Guiochon, J. Chromatogr. 540 (1991) 1. [66] A. Felinger, G. Guiochon, J. Chromatogr. 591 (1992) 31. [67] A. Felinger, G. Guiochon, Biotechnol. Bioeng. 41 (1992) 134. [68] P. Rony, Chem. Eng. Progr., Symp. Ser. 68 (1972) 89.
Chapter 2 The Mass Balance Equation of Chromatography and Its General Properties Contents 2.1 Mass and Heat Balance Equations in Chromatography 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7
Derivation of the Differential Mass Balance of a Compound Discussion of the Fundamental Assumptions Relationship Between the Concentrations in the Stationary and Mobile Phases . Initial and Boundary Conditions of Equation 2.2 Near-isothermal and Nonisothermal Systems Mass Balance in a Radially Heterogeneous Column Separate Mass Balance Equations for the Two Fractions of the Mobile Phase . . .
2.2 Solution of the System of Mass Balance Equations 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6
The Ideal Model of Chromatography The Equilibrium-Dispersive Model The Lumped Kinetic Models The General Rate Model of Chromatography The Lumped Pore Diffusion Model Equivalence Between Equilibrium-Dispersive and Kinetic Models
2.3 Important Definitions 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6
21 21 23 27 28 35 36 39
42 46 47 49 51 54 54
57
The Definitions of the Different Porosities The Definitions of the Densities of Packing Materials The Definitions of the Mobile Phase Velocities The Definitions of the Equilibrium Constants Example of Possible Confusions Definitions, Conclusion
References
58 60 60 61 62 63
63
Introduction Chromatography is a complex phenomenon, which results from the superimposition of a number of different effects.1 A mobile phase percolates through a bed of porous particles. It carries the components of a mixture that interact to different degrees with the stationary phase. These components diffuse in and out of the particles, undergo molecular interactions with the stationary phase or form transient chemical bonds with it, and are eventually swept out of the column. Fluid dynamics, mass transfer phenomena, and equilibrium thermodynamics play an 1
Basic concepts and relationships can be found in the Glossary of Terms, at the end of this volume.
19
20
The Mass Balance Equation of Chromatography and Its General Properties
important role in the outcome of the separation. Depending on the experimental conditions, the relative importance of the thermodynamics of phase equilibria and of the kinetics of mass transfers changes. There is one parameter, however, that is conserved throughout any chromatographic separation — the mass of each of the components of the injected mixture. Admittedly, a few chromatographic systems have been designed in which reactions take place while the products of the reaction are separated from the reagents and from each other. However, this is an exceptional case, which has not yet been reduced to practice, and it will not be considered further in this book [1,2]. There are also cases in which some components (e.g., proteins) may, during the separation, undergo isomerization, dimerization, association or aggregation, dehydration, or other similar reactions. This is rare under analytical conditions and should certainly be avoided when a phase system is selected for a preparative application. Although we do not discuss this case in any great detail, we present in Chapter 16 the VERSE kinetic model, which can successfully address such complicated cases and is applicable to the study of reaction/liquid chromatography. This chapter provides the fundamental theoretical basis of chromatography and is organized as follows. Since it is assumed that no reactions take place in the chromatographic system, we may write the differential mass balance for each component (Section 2.1.1). In this derivation, a few well-specified assumptions are made. These assumptions are discussed in the following section (2.1.2). The mass balance contains the amounts of each component in each phase and must be completed by an assumption regarding the relation between the concentrations in the stationary and the mobile phases (Section 2.1.3). As for all partial differential equations, the mass balance equations must be completed by a set of initial and boundary conditions, of which there is ample choice to suit the wide variety of experiments carried out in chromatography (Section 2.1.4). If the chromatographic process is not isothermal, a set of heat or energy balance equations must be considered in addition to the mass balance equations (Section 2.1.5). Finally, more complex mass balance equations must be considered to account for the behavior of heterogeneous columns (Section 2.1.6) or of slow mass transfer kinetics (Section 2.1.7). Depending on the assumptions made in Section 2.1.3, several models of chromatography can be derived and studied (Section 2.2). These models include the ideal model (Section 2.2.1), the equilibrium-dispersive model (Section 2.2.2), lumped kinetic models (Section 2.2.3), the general rate model (Section 2.2.4) and the lumped pore diffusion model (Section 2.2.5). The solutions supplied by these different models are discussed in detail in Chapters 6 to 16.
2.1 Mass and Heat Balance Equations in Chromatography
21
2.1 Mass and Heat Balance Equations in Chromatography The mass balance equation was derived and studied first by Wicke [3] and later and independently by Wilson [4] and DeVault [5], under different forms. In the derivation of this equation several assumptions are made. For the sake of clarity, these assumptions will merely be stated in the text of Section 2.1.1. They are discussed separately in Section 2.1.2. As we show, their consequences are minor and do not limit significantly the range of validity of the system of mass balance equations finally derived.
2.1.1 Derivation of the Differential Mass Balance of a Compound First, the column is assumed to be unidimensional. This means that it is radially homogeneous. All the column properties are constant in a given cross section and so are the concentrations of the individual components. Thus, the mass balance equations have two independent variables, the time t and the column length z. We shall consider two functions of these variables, the concentrations Cm>i = Q in the mobile phase and CS/I- in the stationary phase. If V is the volume of mobile phase passing through the column, the integral mass balance states that the area of the band profile of a component in the coordinate system (Q, t) at the outlet of the column of length z (i.e., under these conditions, CS/l- = 0) is independent of z and is equal to the area of the injected profile of this component. This extends to the area of the elution profile (Q, t), at the end of the column of length L, provided that the mobile phase velocity remains constant during the elution of the band. Because the equilibrium isotherm is not linear in most cases, however, the area of the profile in the coordinate system (Q, z) along the column is not constant. Only the sum of the areas of the two profiles (Cm/,-, z) and (CS/,-, z) is constant. The integral mass balances do not permit any predictions regarding the band profiles. The determination of these profiles requires the study of the differential mass balance. The differential mass balance in the mobile phase (Figure 2.1) states that the n u
*
Ni,Z Z*" v
9b ^«^ "O5 ^ t
^NI,Z|Z+AZ
1 s ^
*—AZ-* Z+AZ
Figure 2.1 Differential mass balance in a column slice.
22
The Mass Balance Equation of Chromatography and Its General Properties
difference between the amount of the component i that enters a slice of column of thickness Az during the time At and the amount of the same component that exits this slice in the same time is equal to the amount accumulated in the slice: The flux, NjiZ, of component i that enters the slice (Figure 2.1) is (2.1a)
NiiZ = eS z,t
where e is the total porosity of the column packing, S = nR\ = rrd^/i the column geometric cross-sectional area, u the local average mobile phase velocity, Q the local solute concentration in the mobile phase, D^-, the axial dispersion coefficient of the compound in the mobile phase (due to molecular and eddy diffusion), and z the distance along the column. The first term between the parentheses in Eq. 2.1a is the convection term, the second is the axial dispersion term. The axial dispersion coefficient includes the contributions to the axial dispersion of the band due to molecular diffusion and to the nonhomogeneity of the flow. The latter is often called "eddy diffusion." The flux of solute that exits from the slice is (2.1b)
Ni :+Az,t
By definition, the rate of accumulation in the slice of volume SAz is
dt
dt
(2.1c) z,t
where z is the average value of z for the slice. Hence, the differential mass balance for the component i in the mobile phase is eS
uQ -
Du
3Q dz
- eS uQ -
dz
z,t
z+Az,t
p.id) z,t
This equation can be rewritten dC{ dt z
£
dt z+Az,t
Az
zj_
_
u
Qlz+Az,f
Az
U
Cj\Z/t
2.1 Mass and Heat Balance Equations in Chromatography
23
Assuming that u and D^, are constant along the column, and making Az tend toward 0, gives 3Q
^
+ f 3CM + M 3Q = DL 3 Q
^r
^
''^
(Z2)
where F is the phase ratio, Vs/Vm, equal to (1 — e)/e, and Vs and Vm are the volumes of stationary and mobile phases. Equation 2.2 is a partial differential equation of the second order. The first two terms on the left-hand side are the accumulation terms, in the mobile and stationary phases, respectively. The third term is the convective term. The term on the RHS of Eq. 2.2 is the diffusion term. One equation such as Eq. 2.2 must be written for each component of the system, including those of the mobile phase. Thus, even in the case of a pure component band eluted by a pure mobile phase, two mass balances are in principle needed to calculate the band profile. This is the situation in gas [6,7] and in supercritical fluid chromatography where, because of the mobile phase compressibility and the difference between the partial molar volumes of the components in the two phases, the mobile phase velocity and the carrier gas partial pressure vary significantly along the column, and along the band profile itself. We do not discuss here the situation encountered in gas chromatography. hi liquid chromatography, we can make the following two assumptions which greatly simplify the problem [3-7]: • As stated above, we can neglect the compressibility of the mobile phase. Thus, its density along the column and its velocity remain constant and are independent of the pressure. • The partial molar volumes of the sample components are the same in both phases. As a consequence of the first assumption, we have removed the velocity from under the differential operator when deriving Eq. 2.2 from Eq. 2.1e. Furthermore, as a consequence of these two assumptions, and provided the proper convention for the reference state of adsorption is chosen [8,9], the mass balance equation of the mobile phase can be dropped, at least if this phase is pure. If the mobile phase is a mixture, the mass balance of the weakest solvent can be dropped (see Chapter 13 for further elaboration). Since Eq. 2.2 contains two functions, Q and CS/j, another equation or relationship between them is necessary for its solution. Depending on the model of chromatography used, Eq. 2.2 will be accompanied by a mass balance in the stationary phase and a kinetic equation, by a lumped mass transfer kinetic equation, or by an adsorption isotherm (Section 2.1.3).
2.1.2 Discussion of the Fundamental Assumptions In the previous section, several assumptions have been made in the derivation of the differential mass balances of the components. These assumptions are important and deserve a detailed justification which is presented here.
24
The Mass Balance Equation of Chromatography and Its General Properties
Assumption 1: The column is assumed to be radially homogeneous. Experiments show that it is possible to pack wide columns (up to at least 80 cm in diameter) and achieve packed beds that are nearly as homogeneous as those in analytical columns 1/4 inch in diameter. This homogeneity is witnessed by the values of the reduced height equivalent to a theoretical plate2 achieved, sometimes less than 3 [10]. Such a result is possible only if the input profile is radially homogeneous. Accordingly, appropriate flow distributors should be used [11,12]. This also requires that the column be operated isothermally or adiabatically. For the influence of a radial gradient of the column temperature, see [13]. We assume (see Assumption 6) that the column is isothermal. There are few fundamental studies on the homogeneity and stability of column packings [1418]. This is certainly a topic of great practical importance. It is addressed in more detail later, in Section 2.2.6. Miyabe [19] has discussed the influence of the column heterogeneity on the elution band profiles and suggested methods to account for this effect. In Section 2.1.6, we discuss the mass balance equation in the case of a heterogeneous column having a bed with a circular symmetry. Assumption 2: The compressibility of the mobile phase is always neglected in preparative chromatography and it is almost always negligible. Note first that, for mechanical reasons, it is rare to use in preparative liquid chromatography inlet pressures that exceed 200 bar. It has been shown that the influence of the compressibility of the liquid and the solid phases on the column hold-up volume in liquid chromatography is proportional to the pressure and is very small [20,21]. So, we can neglect this effect which, in the pressure range of 0 to 200 bar, might affect the column hold-up volume by 0.5 to 2%, depending on the mobile phase compressibility [21]. Thus, the mobile phase velocity can be considered constant along the column. It is proportional to the pressure gradient (Darcy law, Chapter 5), which is itself constant [22]. Similarly, the coefficients of the isotherm are considered as independent of the pressure and constant along the column. This is in agreement with what we know regarding the constancy of the retention factors under linear conditions in liquid chromatography. Careful studies, however, have shown that this is approximate. Fundamental thermodynamics shows that [23]: dink'
^T
AV =
W
ldlnF +
F^T
,„ , (Z3)
where k = Fa is the retention factor of the compound considered, a is its equilibrium constant3, and AV is the change of partial molar volume associated with the passage of this compound from one phase to the other. The importance of the effect depends on AV, hence on the size of the molecules involved. It is small 2
The reduced height equivalent to a theoretical plate, h, is the ratio of the height equivalent to a theoretical plate, H, to the particle diameter, dp. Values of h below 3 are considered as characterizing good columns. 3 Throughout this book, a denotes the initial slope of the isotherm. When the isotherm is linear, many authors prefer to use K, the equilibrium constant instead. We do not do so for the sake of consistency.
2.1 Mass and Heat Balance Equations in Chromatography
25
or negligible, 10 to 20 mL/mole, for small molecules (e.g., those with a molecular weight less than 200 Dalton) [24] but it may be far more important for macromolecules [25]. The retention factor of insulin on a column packed with Qgbonded silica is doubled when the average column pressure is increased by 100 atm [26]. The column saturation capacity increases also with increasing column average pressure [26]. In contrast, the separation factors of the insulin variants are unaffected. Finally, the mobile phase viscosity is also considered to be constant, which is acceptable in preparative liquid chromatography. The viscosity may increase by 5 to 10% between the atmospheric pressure and 100 bar for most conventional solvents, except for water for which the effect is still far smaller [21,27]. However, the feed viscosity can be larger than that of the eluent and the mobile phase viscosity will actually change during the passage of the band and go through a maximum (Chapter 5, Section 5.3.4). The pressure gradient also varies at the same time. Since most pumps used in chromatography deliver a constant flow rate and the mobile phase density is practically constant, this has no effect on the mass balance. For the same reason, the pressure dependence of the mobile phase viscosity, which affects the profiles along the column of the pressure and the mobile phase velocity, has no effect on the mass balance equation and need not be discussed here. When a viscous feed is used, an instability of the band front may be observed and viscous fingering may take place, potentially ruining the separation (see Chapter 5, Section 5.3.5). This phenomenon cannot be taken into account in a chromatographic model. Assumption 3: We consider the axial dispersion coefficient as constant. It has been shown that the dependence of diffusion coefficients in liquids on the pressure is not very important [20]. It has been reported that the HETP of insulin increases slightly with increasing column average pressure, from 50 to 250 bar. We neglect this effect here. The coefficient DLj in Eqs. 2.1a to 2.1e accounts for the amount of component i that leaves the slice at the abscissa z by dispersion. We also consider that the diffusion coefficients do not depend on the solute concentration, which is true for chemicals of low or moderate molecular weight in the concentration range used in liquid chromatography, rarely exceeding 5% by weight. This may not be true when higher feed concentrations are used, when highly viscous feeds are processed, or when proteins are separated. Then, the concentration dependence of the diffusion coefficient may become important and must be taken into account. In multicomponent nonlinear chromatography, the diffusion coefficient of one component may depend on the presence and concentration of the other ones. Few experimental studies have addressed this coupling, which was investigated recently by Kaczmarski et al. [28]. We will ignore this effect here. The dependence of the mass transfer kinetics on the concentration is handled using the general rate model and will be discussed later (see Chapters 14 and 16). In Eq. 2.2, all the mechanisms that contribute to axial mixing are lumped together into a single axial dispersion coefficient. Thus, D^,- is not the molecular diffusion coefficient, but an axial dispersion coefficient that includes the influence of the packing tortuosity (which slows axial molecular diffusion by increasing the
26
The Mass Balance Equation of Chromatography and Its General Properties
path length), as well as its anastomosis, which causes turbulent mixing by forcing the splitting and recombination of local streams flowing around the packing particles [29]. This latter effect is often referred to as eddy diffusion. As a first approximation, these contributions are additive. Thus, the axial dispersion coefficient Diti accounts for the terms B/u and Au1/3 of the Knox equation, or the terms A and B/u of the Van Deemter equation. The relationship between Di and the experimental parameters is discussed in Chapter 6 (Section 6.4.2) and in Chapter 10 (Section 10.1). Several models use the mass balance in Eq. 2.2 (ideal and equilibrium-dispersive models, Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accounts also for the effect of the mass transfer resistances. This is legitimate under certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models, Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or POR model, see later Sections 2.1.7 and 2.2.4). Assumption 4: The partial molar volumes of the sample components are the same in both phases. This is legitimate in liquid chromatography, where the differences between these volumes do not exceed a few percent [8,9], as discussed above (Assumption 2) with numerical data in Chapter 3, Section 3.4.1. Thus, we can consider that the mass transfers taking place during the adsorption process are made at constant volume. This is a main difference from gas chromatography, where the partial molar volumes of the components in the mobile phase are at least two orders of magnitude larger than in the stationary phase. The sorption effect, which is very important in gas chromatography, is negligible in liquid chromatography. In supercritical fluid chromatography, the sorption effect is probably quite significant but has yet to be properly documented and studied. Assumption 5: In the definition of the isotherm, the convention is adopted that the solvent (if pure) or the weak solvent (in a mixed mobile phase) is not adsorbed [8]. Riedo and Kovats [9] have given a detailed discussion of this problem. They have shown that the retention in liquid-solid (i.e., adsorption) chromatography can best be described in terms of the Gibbs excess free energy of adsorption. But it is impossible to define the surface concentration of an adsorbate without defining the interface between the adsorbed layer and the bulk solvent. This in turn requires a convention regarding the adsorption equilibrium [8,9]. The most convenient convention for liquid chromatography is to decide that the mobile phase (if pure) or the weak solvent (if the mobile phase is a mixture) is not adsorbed [8]. Then, the mass balance of the weak solvent disappears. If the additive is not adsorbed itself or is weakly adsorbed, its mass balance may be omitted [30].
2.1 Mass and Heat Balance Equations in Chromatography
27
Assumption 6: In the general case, we shall assume that there are no thermal effects and neglect the influence of the heat of adsorption on the band profile. In principle, heat or enthalpy balances for the mobile and stationary phases should be included in the system of equations, as discussed in Section 2.1.4. In practice, however, the temperature excursion associated with the migration of a band seems to be small and no detectable effect has been demonstrated [31]. Accordingly, we ignore the thermal effect in the rest of this book, except in Section 2.1.5. We assume that the column is operated isothermally. A temperature gradient along the column would require appropriate adjustments in Eq. 2.2 to account for the variations of the isotherm and the kinetic parameters, of u, DLj, and, improbably, F with T, hence, in this case, z. Assumption 7: We assume that the column is operated under constant conditions, e.g., under constant temperature, pressure, mobile phase flow rate, so that all the physico-chemical parameters remain constant (e.g., diffusion coefficients). In writing Eqs. 2.1b to 2.Id, it was assumed that the porosity remains constant, which is not always true (see later, Section 2.1.6). If the phase ratio, the mobile phase velocity, and/or the axial dispersion coefficient are not constant but depend on the space variable or on the solute concentration, it is easy to modify Eq. 2.2 by leaving the corresponding term under the appropriate differential operator. In summary, the series of assumptions made does not significantly limit the range of validity of the conclusions drawn from the consideration of the system of Eq. 2.2. Far more serious problems will result from the selection of the proper relationship between the concentrations of component i in the mobile and the stationary phases (Section 2.1.3).
2.1.3 Relationship Between the Concentrations in the Stationary and Mobile Phases The two phases of a chromatographic system can never be truly in equilibrium. The continuous stream of mobile phase pushes the sample components along the column, and the mobile phase drift thwarts the drive toward equilibrium resulting from the difference in the chemical potentials of each component in the two phases. Both the kinetics of mass transfer between the bulk mobile phase stream and the inner surface of the pores of the packing particles and axial dispersion proceed at a finite rate. The relationship between the local concentrations of a compound in both phases should be given by a kinetic equation (or by a set of several kinetic equations) relating dCst/dt to the mobile and stationary phase compositions. These equations are still the topics of intense investigations if not controversies (see Chapter 14). They are not completely known and even their functional structure can only be approximated. However, chromatography is known to be carried out under conditions where equilibrium is reached quite rapidly. So, it can be assumed in most cases that the two phases are either always at equilibrium, close to, or near equilibrium. In the first case, the relationship between the local stationary phase concentration of a compound and the mobile phase composition
28
The Mass Balance Equation of Chromatography and Its General Properties
is given by the equilibrium isotherm. In the second case, the difference between the actual composition of one phase (stationary or mobile phase) and its composition at equilibrium with the other phase can often be considered as a perturbation, and the kinetics of mass transfer between phases assumed to be linear. If the mass transfer kinetics between and across the mobile and stationary phases in the column are very fast, these phases are very close to equilibrium. As a first approximation, it can be assumed that the mass transfer kinetics is infinitely fast and that the two phases are in equilibrium. Then, accordingly: CSri = qi=fi{C1,C2r--,Ci,---tCn)
(2.4)
where CS/! is the instantaneous concentration of the component i in the stationary phase and qi is its stationary phase concentration when in equilibrium with the concentrations Cj [j £ (1, n)] in the mobile phase. The different functional relationships that can be used to represent the adsorption isotherms, / , , are discussed in Chapters 3 and 4, in the single component and competitive cases, respectively. If it is assumed that the equilibrium between the two phases is instantaneous and, at the same time, that axial dispersion is negligible, the column efficiency is infinite. This set of assumptions defines the ideal model of chromatography, which was first described by Wicke [3] and Wilson [4], then abundantly studied [5-7,32-39] and solved in a number of cases [7,33,36,40,41]. In Section 2.2.2, which deals with the equilibrium-dispersive model, it is shown how small deviations from equilibrium can be handled while retaining the simplicity of Eq. 2.4 and of the ideal model. When the kinetics of mass transfer is slow {e.g., in some applications of liquidsolid chromatography, ion-exchange chromatography or affinity chromatography to the separation of large peptides, proteins, or polynucleotides), a relationship of the following form is written to take this into account. -..
' = gi\C\,C2,
• • •, Q , • • •, Cn, C S/ i,C S/ 2, • • •, C s i, • • •, Cs,n)
(2-5)
where the function g,- depends on both the rate constants of adsorption and desorption and the isotherm equations (through the differences between the actual concentrations and the equilibrium ones). The different functional relationships used for gi are discussed in the last section of this chapter on the kinetic models (Section 2.2.3). The properties, the advantages, and the drawbacks of the ideal, equilibriumdispersive, and kinetic models are discussed later, in Section 2.2.
2.1.4 Initial and Boundary Conditions of Equation 2.2 These conditions are necessary to permit the choice, among the infinite number of solutions of a system of partial differential equations, of the solution of the problem considered. The initial and boundary conditions define the exact solution needed. They are the mathematical translation of the experimental procedure followed.
2.1 Mass and Heat Balance Equations in Chromatography
29
a
C mg/mL
30 -r
30
15 15 --
15
0 0
b % % Organic Organic
80 T 80 40
0 0
3
6
0 0
10 10
20 20
c 90 T 60 '-30 :0 :0
10 10
20 20
d 90 j 60 '-30 :0 :0
time, min time, min
4
8
Figure 2.2 Classical boundary conditions of common chromatographic problems, (a) Elution of a rectangular pulse, (b) Multiple gradient elution of a rectangular pulse, (c) Staircase frontal analysis, (d) Displacement. Dotted line: component 1. Shaded line: component 2. Solid line: displacer.
2.1.4.1 The Initial Conditions The initial conditions describe the state of the column when the experiment begins, i.e., at t = 0. For example, in conventional elution (no mobile phase additive), gradient elution, frontal analysis or displacement, the column is filled with a weak mobile phase which does not participate in the equations [8]. Hence, in general, the initial condition is Q(z, t = 0) = 0
for
0
(2.6a)
where L is the column length. In the elution mode when the mobile phase contains a competitive additive, in gradient elution when the initial concentration of the strong solvent is different from zero, or in frontal analysis when the elution of successive concentration steps is carried out, the column contains a constant concentration of a component interacting with the stationary phase at the beginning of the experiment. Thus, the initial condition is Ca{z,
t = 0)= C °
for
0
(2.6b)
where C^ is the constant concentration of additive in the mobile phase contained in the column. 2.1.4.2 Classical Boundary Conditions of Common Chromatographic Problems The different classes of boundary conditions of the chromatography problems are discussed in detail in Chapter 6, Section 6.2.1, to which the reader is referred. We
30
The Mass Balance Equation of Chromatography and Its General Properties C mg/mL ing/mL C
8
a
00
Time, Time,
0
sec sec
1
3.0
2.52.5
AU
2.0' 2.0
Figure 2.3 Boundary conditions. (a) The Dirac boundary condition. (b) Comparison of a rectangular pulse injection (solid line) and an experimental injection profile (symbols). Valco valve, volume injected: 0.1 mL at 1 mL/min. Re-
mL/min. Reproduced with permission from A. Felinger et ah, } . Chromatogr. A, 1024 (2004) 22 (Fig. 1).
oo
o
1.01.0
o 0.50.5
o
o
000.0
0
produced with permission from A.M. Katti et ah, J. Chromatogr., 556 (1991) 205 (Fig. 1). (c) Comparison
0.2
0.4
0.6
0.8
1
1.2
Time 70
c
60
50
40 C (g/L)
of an experimental pulse injection of aniline (symbols) and a model arising from considering dispersion and tailing taking place in the extra-column volumes (solid line). Volume injected: 0.9 mL at 1
1.5 1.5 •
30
20
10
0 1
1.5
2.5
2
3
3.5
time (min)
present here a mere description of the conditions corresponding to the different chromatographic processes (Figure 2.2). The most common boundary condition for elution is Q(0,i) =Cf0 s (O
(2-7)
where Cf is the concentration of the component i in the feed and
2.1 Mass and Heat Balance Equations in Chromatography
31
high. For this reason, a rectangular boundary condition is often assumed, with C(x = 0, t) = Co C(x = 0, 0 =
0
0 < t < tp t
< 0
K 0>
t > tp
Thus, the maximum concentration, Co, can now be realistic. The injected amount is n = CotpFv, where Fv is the mobile phase flow rate. The loading factor along the column axis is Lj- = uotpCo/ (L(l — e)qs). The loading factor is the sample size referred to the monolayer capacity of the adsorbent in the column bed. It is often observed, however, that the actual injection profiles are far from the Dirac model, as illustrated in Figure 2.3b, which compares a rectangular pulse injection of 100 ]iL (solid line) and the injection profile recorded with a six-port Valco valve (Houston, TX) fitted with a 100-^L loop [42]. The Dirac injection is an acceptable model only if the width of the experimental injection is small compared to the standard deviation of the band profile under linear conditions. Usually, the experimental injection profile has a sharp front followed by a tailing decay (Figure 2.3b). This profile is also typical of those encountered in preparative chromatography, except that they include a concentration plateau lasting for a certain period of time (see Figure 2.3). Whenever numerical calculations are carried out to predict band profiles from equilibrium isotherms and kinetic data (see Chapter 10) or to derive the equilibrium isotherms from acquired band profiles (see Chapters 3 and 4), it is imperative accurately to model the actual boundary condition, i.e., to perform the calculations using the concentration profile of the feed as it enters into the column. The importance of the selection of the boundary condition, of its modeling in certain cases, has been demonstrated many times [42-45]. Figure 2.4 illustrates the importance of following this recommendation when comparing experimental and calculated band profiles. The extra-column band broadening—in the case of an infinitesimally narrow impulse injection—is usually accounted for by the exponentially modified Gaussian (EMG) function, which is the convolution of a Gaussian peak and an exponential decay function (see Chapter 6, Section 6.6.1). The former contribution describes the band broadening in the connecting tubes while the latter models the mixer-type extra volumes [46]. When the ideal inlet profile is a wide rectangular pulse, the true inlet concentration can be modeled by the convolution of the EMG function and a rectangular pulse of length tp. The resulting profile is _. . Cm
=
\ [ . m —t — < erfc—-= 2« I y/lff
tn + m — t ( o2 erfc——-= V exp —-= -\ V2CT * \2x2
m-t\
_ e r f,c / cr
V2T
m-t\ T +
x
)
m-
V2
where m is the residence time in the connecting tube, c is the Gaussian band width and T is the time constant of the mixer-type extra-column volume. The measured inlet concentration profile for a 0.9-minute injection of the content of a loop filled with a dilute solution of aniline is reported in Figure 2.3c (symbols). The fitted model described in Eq. 2.9 (solid lines) follows remarkably well the measured concentration profile.
32
The Mass Balance Equation of Chromatography and Its General Properties
2
3 4 Time (min)
6
7
3 4 Time (min)
6
7
Figure 2.4 Experimental and calculated band profiles of benzyl alcohol (a) and phenylethanol (b). Sample volume 0.5 mL, Co = 25 g/1. Column, Symmetry-Cis; mobile phase, methanol and water (1:1, v/v); flow rate, 1 ml/min. Experimental profiles (circles), profiles calculated with a rectangular injection (dashed line) and with the experimental injection profiles (solid line). Reproduced with permission from I. Quinones et al., Anal. Chem., 72 (2000) 1495 (Fig. 6). ©2000, American Chemical Society.
Different boundary conditions are needed for operation modes other than elution. Usually, there are no difficulties in translating the experimental procedure into a set of appropriate boundary conditions. If the injection is performed in a mobile phase containing an additive (initial condition as in Eq. 2.6), for example, the boundary condition for this additive after the injection is completed, is usually Ca{0,t)=C°a
for
t < 0 and tp
2.1 Mass and Heat Balance Equations in Chromatography
33
that this combination does not mean that Ca (z, t) will remain constant over the entire length of the column. Because of the coupling between the mass balance of the additive and those of the feed components, complex concentration profiles of the additive may appear under the proper set of experimental conditions [47,48] (see Chapter 13)4. The validity of these theoretical conclusions was demonstrated experimentally by Quinones et al. who, after demonstrating that benzyl alcohol (BA), 2-phenylethanol (PE), and 2-methylbenzyl alcohol (MBA) follow ternary competitive Langmuir behavior on a Cis-bonded silica column with water/methanol (1:1, v/v) as the solvent [49], analyzed the effluent of a column fed with a stream of constant concentration of MBA into which large amounts of a BA/PE mixture was injected [50] (see Chapter 11). In gradient elution or in displacement chromatography, the boundary condition given by Eq. 2.7 still applies to the sample components. However, the mobile phase contains an additive at a concentration which varies during the experiment. This additive is the strong solvent in gradient elution and the displacer in displacement chromatography. In gradient elution the boundary condition for the additive is (Figure 2.2b) Ca{O,t) = C°a(z,O) + <pa{t)
(2.11)
where C^(z, 0) is the concentration of the additive at the injection time, which is usually constant over the entire column, hence the initial condition (Eq. 2.6b), and <pa(t) is the gradient profile. It can be linear, a step function with a delay, an exponential function, or other. In frontal analysis, the boundary condition for the sample is for the first step: Q(O,f) = Cf
t>0
(2.12)
and then it models a staircase (Figure 2.2c). In displacement chromatography, the boundary condition for the feed components is given by Eq. 2.8 as in elution chromatography, while the boundary condition for the displacer is a step (Figure 2.2d) Cd(z,t)=0 Cd(O,t) = CJ
t < tp + 5t t > tp + 5t
where tp is the duration of the sample injection. There is usually a period of time, 6t, elapsed after the end of the injection and before the displacer is pumped into the column, in order to avoid mixing the sample and the displacer. For other experiments, more complicated boundary conditions may be necessary. This is particularly the case for simulated moving bed separations (see Chapter 17). 2.1.4.3 The Danckwerts Boundary Conditions There is a fundamental objection to the use of the boundary conditions described above. These conditions have vertical boundaries or shocks (Chapter 7) at both *Such phenomena take place only when the additive and some feed components are similarly adsorbed.
34
The Mass Balance Equation of Chromatography and Its General Properties
C
Figure 2.5 Illustration of the Danckwerts Boundary Condition. 1: Rectangular pulse injection. 2, 3, 4: Danckwerts injection conditions for the same sample amount; 2: D = 0.04 cm 2 /s; 3: D = 0.08 cm 2 /s; 4: D = 0.12 c m 2 / s . Reproduced with "permission from G. Guiochon, B. Lin, Modeling for Preparative Chromatography, Academic Press, San Diego, CA, USA, 2003 (Fig. 111-2).
ends. Axial diffusion proceeds at an infinite rate along vertical boundaries. Although chromatographic columns are usually highly efficient and shock layers are often observed on the front or rear side of bands (see Chapters 14 and 16), these boundary conditions are not realistic and their use raises serious fundamental difficulties. In practice, it is better to use the boundary conditions of Danckwerts [51], which are written: = uCn
(2.14)
=
(2.15)
x=0
dC
0
x=L
From a physical viewpoint, the Danckwerts condition states that the mass flux in the column at the column inlet where the injection is made, i.e., uC(0,t) — D^ , is equal to the mass flux that would be achieved in a pipe having the same diameter as the column, uCa(t). If the term D ^ is neglected in the equation above, we obtain the earlier coundary condition, C(0, t) = Co(t). This assumption constitutes the approximation of the ideal model. The Danckwerts boundary condition is illustrated in Figure 2.5 for three different values of the coefficient of apparent axial dispersion, D [2]. In actual columns, D is finite so the term D |^ is different from zero and dx
x=0
should not be neglected. In many numerical calculations, particularly those made for virtual chromatography, the boundary condition C(0, t) = Co(t) is often used instead of the Danckwerts condition. This is approximate but reasonable for efficient columns, e.g., those having more than a few thousand plates. By contrast, when the column efficiency is lower than a few hundred theoretical plates, significant errors may occur and the Danckwerts condition should be used. This situation is typically encountered in the numerical calculations carried out in Simulated Moving Bed separations (see Chapter 17). While the Danckwerts boundary
2.1 Mass and Heat Balance Equations in Chromatography
35
condition must always be considered for numerical calculations, it is rarely considered in fundamental studies. It is a mixed condition, involving values of both the function and its derivative in certain points (at the column inlet and exit). It is complex to handle in algebraic calculations and makes most integrations really difficult if not altogether impossible.
2.1.5 Near-isothermal and Nonisothermal Systems In linear chromatography, the solute concentrations are very low. The heat transfer resistances can be neglected and the chromatographic system considered as isothermal. This is no longer true in nonlinear chromatography, where the concentrations of the feed components are high. If the heat of adsorption is high enough, the system can deviate significantly from isothermal behavior. In such a case, it would be necessary to complete the system of partial differential equations of chromatography by two differential heat balance equations, for the mobile and the stationary phases. For a nonisothermal system, the differential heat balance for the mobile phase can be written as [52] d2Tm
dTm
dTm
dTs
(2.16) and the heat balance for the stationary phase as
where Cp/tn and CprS are the volumetric heat capacities of the mobile and stationary phases, respectively; Tm, Ts, and Tw are the absolute temperatures of the mobile phase, the stationary phase, and the column wall, respectively; and A^ is the axial thermal conductivity of the column, 4H, the heat of adsorption, hm the overall heat transfer coefficient at the column wall, h the overall heat transfer coefficient between stationary and mobile phases, dc the column inner diameter, and dp the average particle size. In an adiabatic system, the wall heat transfer coefficient, hw, is zero, so the last term on the RHS of Eq. 2.16 disappears. For nonisothermal systems, two main cases can be considered, when the system is near isothermal or adiabatic. In near-isothermal systems, it is assumed that the heat transfer between mobile and stationary phases is slow. This causes an additional band broadening contribution to appear [53]. Such a contribution can be especially important on the front of a sharp concentration profile. On the other hand, the heat transfer between the chromatographic column and the outside is fast enough to prevent the formation of a temperature front and of an associated secondary mass transfer zone. In adiabatic or near-adiabatic systems, since the coefficients of the adsorption isotherm depend on the temperature, there will be an additional mass transfer zone which propagates at the velocity of the thermal front. Thus, one may
36
The Mass Balance Equation of Chromatography and Its General Properties
consider that an n-component nonisothermal system is equivalent to an (n + 1)component isothermal system. However, Yun et al. have shown that this effect was negligible in almost all cases of practical importance in preparative HPLC [31].
2.1.6 Mass Balance in a Radially Heterogeneous Column Chromatographic columns are usually considered to be entirely homogeneous. Unfortunately, it is not possible to prepare such columns. By construction or because of the influence of the solute concentration, some physico-chemical properties of the column depend on the position along the column. Variations of some column parameters along the column length are easy to take into account by making small, straightforward changes in the mass balance equation (Eq. 2.2). For example, Piatkowski et al. accounted for a column porosity that depends on the sample concentration by keeping the phase ratio under the differential operator [54]. This work was done using the general rate model (see later) but the same approach could be used with the other models discussed in the next section. More difficult to handle is the well-known radial heterogeneity of chromatographic columns [55]. Because there is friction of the column bed along its wall during packing [56,57], the bed is not homogeneous [58]. The packing density, the external porosity, the phase ratio, the permeability, and the local velocity of the mobile phase are functions of the radial position in the column [55]. In other cases, the column is packed with two different lots of a given packing material [59]. The simplest model that deviates from an homogeneous column is an heterogeneous column having a cylindrical symmetry [59-61]. 2.1.6.1
Mass Balance Equation With Two Space Variables
In this case, the concentration distribution at a given time in a band migrating along the column depends on the position, z, and the distance, r, from the column axis. It does not depend on the azimuthal angle around the column axis. The differential element of the column is a ring of axial thickness dz, limited by the cylinders of radii r and r + dr [60,61]. In this element, the mass balance is written "^ _i_-pz5. i u ZJ! n " "" _i_ i l l v "* / (2 is) a 3t 9f 9z dz2 r dr where Dr is the radial dispersion coefficient. In the framework of the equilibriumdispersive model (see later, Section 2.2.2), it is natural to relate it to a radial plate height by Hr=2^L
(2.19)
The axial plate height is related to the apparent axial dispersion coefficient by Ha = 7^±
(2.20)
2.1 Mass and Heat Balance Equations in Chromatography
37
A relationship is expected between Hr and Ha. However, if the packing density depends on the radial position, the bed tortuosity and eddy diffusion may be different in the axial and radial directions. Furthermore, the mass transfer resistances do not affect Hr. Although, in the general case, Da and Dr could both be functions of the coordinates z and r and of the concentration, we assumed in writing Eq. 2.18 that they are constant. This would constitute the second order approximation of a model of physical columns, the other models discussed here being first order approximations since they all assume a homogeneous column. The presentation of numerical solutions of Eq. 2.18 and their discussion are greatly simplified if some reduced variables are introduced at this stage. These variables are the reduced axial position (x), the reduced radial position (p), the reduced time (T), the axial (Pea) and the radial (Per) column Peclet numbers (note that these two Peclet numbers are different from the conventional particle Peclet number or reduced velocity, v = udp / Dm), and the column aspect ratio (<J>), which are defined as follows
X = I
(2.21a)
p = Yc
(2 21b)
'
r = f Pea
=
(2.21c)
^ =
2
^
(2-21d)
Per = £=2±0
=
(2.21e)
A
(2.21f)
where L is the column length and Rc its radius. Combination of Eqs. 2.18 and 2.21a to 2.21f gives the mass balance equation in reduced coordinates:
ac
a, tc = 1_#c
dr
9T
dX
Pea dx2
^KP|) Per
dp
K
'
'
This equation has four parameters, F, Pea, Per, and <J>. The phase ratio, F, is a property of the packing material, related to its internal and external porosities. The internal porosity is a property of packing materials that is very difficult to adjust because it results from the preparation process of the material and any attempt at modifying it would be prone to result in some changes of the surface chemistry. The external porosity or fraction of the column volume available to the stream of mobile phase percolating through the column packing depends on the packing density but, in practice, this parameter is extremely difficult, if not impossible, to adjust. Recent results have shown that the packing density depends somewhat on the stress applied to the column packing during the preparation of the column [55-58]. Still, it does not seem possible to use an adjustable external stress to modify significantly, in a controlled fashion, the external porosity of the
38
The Mass Balance Equation of Chromatography and Its General Properties
column. By contrast, the other three parameters are easily adjustable. Note that both Dfl (Eq. 2.20) and Dr (Eq. 2.19) are functions of the flow velocity through the corresponding plate height equation. The ratio of the axial and radial Peclet numbers, Pea/Per, does not remain constant when this velocity is changed. 2.1.6.2 Initial and Boundary Conditions of Equation 2.22 The initial condition corresponds usually to a column empty of feed but containing mobile and stationary phases in equilibrium. A physical description of the column is the basis for writing the boundary condition. If the bed is radially heterogeneous, the general direction of the streamlines is no longer parallel to the column axis. These lines tend to avoid the low permeability regions and concentrate in the high permeability ones. At a time when sophisticated fluid dynamics programs were not yet available, Yun et ah [59-61] could not investigate the flow of mobile phase through a radially heterogeneous bed. So, they assumed that the column bed was mechanically homogeneous. With this simplification, there is no convective transport in the radial direction of the column and the radial profile of the mobile phase velocity is flat. This allowed the investigation of only columns in which there are no radial variations of the packing density, hence the column porosity and its permeability are constant throughout the bed. Two problems were investigated. In the first one, the injection profile is a cylindrical band with a diameter smaller than that of the column but coaxial to it [61]. The boundary condition is then: = O,t) = Co
= o,f) = o
o<
Rin] <
= 0,f)=0
r r
<
^C
< 0
0 < t < tp 0 < t < tp tp <tp
(2.23)
In the second case, the column is heterogeneous because it contains two lots of a packing material that slightly differ by their chemistry alone but have the same particle size distribution [60]. For the sake of simplicity, Yun et al. assumed that all the sections had the same surface area, which gives an arbitrary model of a column packed with a mixture of two different lots of a given packing material. These authors considered a series of coaxial annular columns, the column of rank k having 2k concentric rings. The radius of each successive circle is r^ = Rc\fkfn. The boundary condition in the axial direction was the injection of a rectangular pulse of sample of concentration Q and duration tp. This injection extends all across the column. The injected amount was n — CotpFv, where Fv is the mobile phase flow rate. The loading factor along the column axis is Lf = uotpCo/(L(l — e)qs). The loading factor is the sample size referred to the monolayer capacity of the adsorbent in the column bed. In both cases, there are also two boundary conditions in the radial direction. They state that (i) no concentration can penetrate inside the column wall and (ii) the concentration distribution is symmetrical around the column axis, i.e., that the radial gradient of concentration is 0 at the wall (i) and at the column axis (ii): ^- = dp
0
for r = Rc
(2.24a)
2.1 Mass and Heat Balance Equations in Chromatography
39
— = 0 for r = 0 (2.24b) dp This system of equations is an extension of the classical equilibrium-dispersive model to problems with two spacial dimensions, e.g., to the cases of a column having a cylindrical symmetry. It has no analytical solution but it is possible to write simple computational schemes for the calculation of its numerical solutions, using finite difference algorithms (see Chapter 10) [60].
2.1.7 Separate Mass Balance Equations for the Two Fractions of the Mobile Phase The detailed study of the mass transfer kinetics is necessary in certain problems of chromatography in which the column efficiency is low or moderate. Complex models are then useful. The most important ones are the General Rate Model [52,62] and the POR model (see next Section) [63]. To study the mass transfer kinetics, these models need to consider separately the mass balance of the feed components in the two different fractions of the mobile phase: the one that percolates through the bed of the solid phase (column packed with fine particles or monolithic column) and the one that is stagnant inside the pores of the packing material. These equations are conveniently expressed with the following three dimensionless kinetic parameters: • The axial Peclet number, Pe = uL/Di, where L is the column length and DL the axial diffusion coefficient, • The particle Peclet number (also called the reduced velocity of the mobile phase), Pep = udp/Dp, where dp is the particle size and D p the diffusion coefficient inside the particles, • The Biot number, Bi — kfdp/2£pDp, where kf is the film mass transfer coefficient and €p the internal porosity. In dimensionless form, the general rate model consists of the set of equations described in the rest of this section. Each of these equations must be written for each component of the system. These equations are then completed with an appropriate set of kinetics equations. The corresponding models are discussed in the next section. Before discussing these equations, however, we must address an issue that must be carefully clarified to avoid confusions and errors. 2.1.7.1 Important Definitions In general (there are many exceptions), chemists and chemical engineers tend to use two different definitions for the mobile phase velocity and for the internal or particle porosity, and different units for the concentrations in the liquid and solid phases (see Section 2.3), hence different expressions for the equilibrium constant. While most papers dealing with the simple models derived from Eq. 2.2 do not need to distinguish between internal and external porosities (there is no
40
The Mass Balance Equation of Chromatography and Its General Properties
ambiguity regarding the total porosity) and are mostly written by chemists or for chemists, papers dealing with the general rate model and the POR model must be clear about these distinctions and the conventions chosen. Because many of these works are written by chemical engineers, chemists must be most careful when reading these papers. The reader must be familiar with the different definitions. This issue is discussed in detail later (see Section 2.3), so the main definitions are briefly stated. Internal porosity. The total porosity, e or ej, is the volume fraction of the column that is available to the mobile phase. The external porosity, ee, is the volume fraction of the column that is available to the mobile phase percolating through the bed while the internal or particle porosity, e,, characterizes the volume available to the stagnant mobile phase. Chromatographers define the internal porosity of the column as the difference between the total and the external porosities. So, for chromatographers, e = ee + £{
(2.25a)
Chemical engineers define the internal porosity, e-p, as the volume fraction of the particles that is available to the mobile phase. Thus, we have e = ee + (l-
ee)ep
(2.25b)
The relationship between the two definitions of the internal porosity is: ei = (1 - ee)ep
(2.25c)
Mobile phase velocity. Chromatographers define the cross-section average velocity of the mobile phase as the ratio, u = L/tg, of the column length and the hold-up time or retention of a nonretained compound. This time should be corrected for the transit time through the extracolumn volumes between injection valve and column inlet and between column inlet and detector. Chemical engineers tend to prefer the superficial velocity, or ratio of the flow rate and the column cross-section area, UQ = Fv/S. Obviously, we have: toFv
Vm
Se
e
2.1.7.2 Mass Balance in the Bulk Mobile Phase This mass balance is written -ldzcbi
dchi
dcbi
+N
^ ^ + ^T + -W
, c
c
/'< « " ^
l}
,
=°
In this equation, Pe,- is the Peclet number for component i, and the other reduced parameters and concentrations are:
2.1 Mass and Heat Balance Equations in Chromatography c
p,i
=
41
^
(2.28b)
Ci(r) 3(1 - e,
ebR Pu
u x
=
|-
(2.28e)
r —
Lt u
(2.28f)
xp =
^
(2.28g)
where Q^, Cpj and C^-(T) are the concentrations of component i in the bulk mobile phase, in the pore volume, and in the feed, respectively, Cf is the maximum concentration of i in the feed, ee is the volume fraction occupied by the bulk mobile phase {i.e., external porosity), Rp is the particle radius, dispersion coefficient, and kfri is the external film mass transfer coefficient. Note that Q,- depends only on the time and on the distance along the column since we assume that the column is radially homogeneous, Cp/J- depends on the radial position inside the pore (which we assume to have spherical form to allow modeling), and Cy J(T) is a function of time only. The initial condition for Eq. 2.26 is T= 0
cbii = cbii{Q,x)
(2.29)
In most cases, ejy (0, x) = 0. The boundary conditions depend on the nature of the experiment performed. We always have the Danckwerts relationships as a first boundary condition
(2.30)
There is a second condition that depends on the mode of chromatography used: • For frontal analysis x= 0
cf4{T)=l
(2.31a)
• For elution X=
°
c
fM=
I
cfri(r) = 0
X
'% if
^
T > Tp
(2.31b) v
'
42
The Mass Balance Equation of Chromatography and Its General Properties
2.1.7.3
Mass Balance in the Pores
The mass balance in the stagnant solution is written 3
m
3 [ 1 | (V^)] =0
(2.32)
In this equation, we have:
(233a)
evDv ,-L
* = "w~ csVii
=
-
- ^
(2.33b)
where ep is the particle or internal porosity and Dp!- is the effective diffusion coefficient in the pores. Equation 2.32 is the same in linear and nonlinear chromatography, the only difference being in the relationship between the concentrations in the stationary phase, cs •, and in the liquid phase, cpi[. hi the former case, this relationship is linear, in the latter it is given by the competitive isotherm model. The initial condition for the integration of Eq. 2.32 is T= 0
cP/i = cPii(Q,x)
(2.34)
and the boundary conditions are r= 0
- ^ = dr
0
r=l
- ^ =
Bii(cb/i -
(2.35a) cVii/r=i)
with B
'i =
)
T^r-
(2-35b)
This set of equations (Eqs. 2.25 to 2.35) constitutes the general rate model of chromatography.
2.2 Solution of the System of Mass Balance Equations The band profiles will be obtained as the solution of the relevant system of mass balance equations (Eq. 2.2), completed with a relationship between each stationary phase concentration, Gy, and the mobile phase concentrations, Q (Eq. 2.4 or 2.5), and with the proper set of initial (Eq. 2.6) and boundary conditions (Eqs. 2.8 to 2.15). There is an equation of each type for each component of the feed and of the mobile phase, except for the weak solvent in the mobile phase. However, the mass balance equations of the additives or strong solvents, whose retention factors
2.2 Solution of the System of Mass Balance Equations
43
Table 2.1 System of Equations of the Equilibrium-Dispersive Model. Binary Mixture and Pure (or weakly adsorbed) Mobile Phase Mass Balance Equations dqx
9Ci
_
n
Equilibrium Isotherms
/i(Ci,C 2 ) /2(Ci,C 2 )
Initial Conditions Ci(z, £ = 0) = 0 C 2 (z, £ = 0) = 0
for for
0
Boundary Conditions for Elution C1(z = 0,t)=
Ci
for
Q
Ci(z = 0, f) = C2(z = 0, f) =
0 C^ s (f)
for for
ip < t 0
C 2 (z = 0 , 4 =
0
for
tp
<tp
< t
in the pure weak solvent are at least five times smaller than those of the sample components of interest can be omitted (Chapter 13) [30]. These equations form a system of partial differential equations of the second order. Examples of two complete systems are given in Table 2.1 (a binary mixture and a pure mobile phase or a mobile phase containing only weakly adsorbed additives, a two-component system) and Table 2.2 (a binary mixture and a binary mobile phase with a strongly adsorbed additive, a three-component system). For the sake of simplicity, the equilibrium-dispersive model (see Section 2.2.2) has been used in both cases. The problem of the choice of the isotherm model will be discussed in the next two chapters. It has been shown that there is only one band profile for each component of the system [64]. Thus, the problem is well posed when the boundary and initial conditions are given as described in the previous section. In the general case of multicomponent or even binary mixtures, it is not possible to derive closed-form solutions of this system. It is even nearly impossible to make general statements about it and only numerical solutions are available. This makes it difficult to understand the behavior of interfering bands during their migration. Some general understanding of the solution properties can be reached, however, by considering the simplified ideal model which assumes that the two phases are in constant equi-
44
The Mass Balance Equation of Chromatography and Its General Properties
Table 2.2 System of Equations of the Equilibrium-Dispersive Model. Binary Mixture and Binary Mobile Phase Mass Balance Equations dt
dt
3z
0C2
j-, dcj2
0C2
dCa
~dqa
dCa
3t
3f
3z
^"Id dzz
Equilibrium Isotherms q\ =
/i(d/ d, d ) /2(d/ d/ d ) /a(d/ d/ d )
Initial Conditions C 1 (Z / * = 0) := 0 d ( z , t = 0) == 0 d(Z, i = 0 ) :
for for for
= eg
0 < z < L 0 < z < L 0 < z < L
Boundary Conditions for Elution d ( z = 0, t) d ( z = 0,i)
= =
d(z = 0, 0 = d(z = o, 0 =
d(z = 0, t) = d ( z = 0,t) =
Cfys(f) 0 C^s(i) 0 Ca,s
C°a
for for for for for for
0
<
t
0
<
0
<
t tp t tp
< tp < t < tp < t < tp < t
librium, that there are no sources of band broadening of kinetic origin (the column has an infinite efficiency). There are analytical solutions of the ideal model. These solutions show what it is that thermodynamics is trying to accomplish in the column. Similarly, analytical solutions of linear chromatography permit a thorough understanding of the role of kinetics in the absence of thermodynamic influence on the band profiles. The combination of kinetics and thermodynamics, as it arises in nonlinear, nonideal chromatography, is then easier to understand and can be illustrated by numerical solutions of simple models (equilibrium-dispersive and lumped kinetic models) which have now become easy to calculate. Figure 2.6 summarizes the relationship between the isotherm function in the case of a pure component, q = f(C) in Eq. 2.4, and the band profile. The top row (Figures 2.6a,f,k) shows the three major types of isotherms encountered in chromatographic systems: linear (a), convex upward or Langmuirian (f), and convex
2.2 Solution of the System of Mass Balance Equations
a
Linear
q
Langmuirian
q
f
45
Anti-langmuirian
k k
q
C
b
dq/dC
dq/dC
C
h
C
m
C
dq/dC
d
C
t
C
l
C
dq/dC
C
dq/dC
g
C
c
C
C
C
e
t
dq/dC
i
C
t
t
C
j
t
n
C
O
t
Figure 2.6 Relationship between the equilibrium isotherms and the band profiles for single components. Top row: isotherm; second row, isotherm differential; third row, same as second row, but symmetry around the first bisector; fourth row, band profiles with the ideal model; bottom row, band profiles with axial dispersion.
downward or anti-Langmuirian (k) isotherms. The second row (Figures 2.6b,g,l) shows the differentials of these isotherms. The third row is obtained by reversing the second one, exchanging C and dq/dC. In the fourth row (Figures 2.6d,i,n), scale factors are introduced for the detector response on the vertical axis and for the mobile phase velocity on the horizontal axis. This reflects the fact that a velocity is associated with each concentration in the ideal model (next section);
46
The Mass Balance Equation of Chromatography and Its General Properties
the high concentrations are as strongly adsorbed as the low ones (first column), less strongly adsorbed, hence less retained than the low ones (second column), or more strongly adsorbed and more retained than the low ones (last column). Thus, the peaks solutions of the ideal model are obtained in Figures 2.6d,i,n. Axial dispersion is introduced in the last row, giving solutions of the equilibriumdispersive model corresponding to a column efficiency of 750 theoretical plates (Figures 2.6e,j/o). All these intuitive results must now be established on a rigorous basis.
2.2.1 The Ideal Model of Chromatography This model of nonlinear chromatography, the simplest model, was formulated and studied first by Wicke [3], Wilson [4], and DeVault [5]. It assumes that the column efficiency is infinite. There is no axial dispersion and the two phases are constantly at equilibrium. The stationary phase concentration is given by the isotherm equation (Eq. 2.4). The mobile phase concentration is denoted simply by Q. Since the axial dispersion coefficient is nil, the mass balance equation for component i (Eq. 2.2) simplifies to
§ + F^
+ M fi=0
(2.36)
dt dt dz This model neglects completely the influence of the mass transfer kinetics and of axial dispersion on the band profiles. Accordingly, it can propagate concentration discontinuities, or shocks (the vertical sides of the profiles in Figures 2.6i and n). The ideal model focuses attention on the influence of the nonlinear thermodynamics of phase equilibria on the individual band profiles. It is shown in Chapter 7 that Eq. 2.36 for a single component in overloaded elution chromatography can be solved in closed form for many equilibrium isotherm models [40,65]. For a binary mixture, it is shown in Chapter 8 that the solution of a system of two Eqs. 2.36, one written for each component, can be solved in closed form almost completely, but only in the case of competitive Langmuir isotherms [35,41]. In all other cases, numerical5 solutions have to be calculated, which is difficult but not impossible for the ideal model (the difficulty in this case is due to the fact that the numerical algorithms used for the calculation of the solution handle poorly the discontinuities of the solution, see Chapter 7). On the other hand, this calculation becomes easy for real columns because all numerical methods introduce some degree of numerical dispersion (see Chapter 10). In concordance with the basic assumption of the ideal model, the band profiles obtained as solutions of this model are in good agreement with the experimental chromatograms recorded for large samples, with highly efficient columns, in which case the deviation of the isotherm from linear behavior is important and the dispersive effect of the finite column efficiency is small. On the other hand, the agreement with experimental results is poor in the case of low-efficiency columns 5
Throughout this book, a "numerical" solution refers to a solution of the problem derived by calculation, as opposed to an analytical solution.
2.2 Solution of the System of Mass Balance Equations
47
and when the sample size is low or moderate, since in this case the contributions of the mass transfer kinetics and the axial dispersion to the band profiles become significant or important compared to the influence of a nonlinear isotherm [66]. Chapters 7 to 9 review the closed-form solutions of the systems of Eq. 2.36 for a single component (Chapter 7) and a multicomponent mixture in elution (Chapter 8) and in the displacement mode (Chapter 9). These solutions have great importance because they describe clearly what it is that thermodynamics tries to accomplish in a chromatographic column. The understanding of these solutions gives precious clues to the behavior of high-concentration chromatographic bands in actual columns.
2.2.2 The Equilibrium-Dispersive Model It has been shown by Giddings [67], Van Deemter et al. [29] and Haarhoff and Van der Linde [68] (see Section 2.2.6) that when the mass transfer kinetics are fast but not infinitely fast, the system of mass balance equations (Eq. 2.2) and kinetic equations (Eq. 2.5) can be replaced by the following equation: (237)
at dt az ' ozl where qi is the equilibrium concentration of component i in the stationary phase (Eq. 2.4) and the apparent dispersion coefficient, Daj is given by _HL_Hu U
a,i — -WT — -^~ Z.TQ
[1.3b)
L
where H is the column height equivalent to a theoretical plate (HETP) for the component considered, and to is the holdup time of the column (to = L/u) or retention time of a mobile phase tracer. Equation 2.38 stems from the relationship giving the variance (in length units), of, of a Gaussian peak obtained in chromatography, under linear conditions, for a Dirac pulse injection (Figure 2.3a): of = HL = 2Dat0
(2.39)
The equilibrium-dispersive model assumes that all contributions due to the nonequilibrium can be lumped into an apparent axial dispersion term. It further assumes that the apparent dispersion coefficients of the solutes remain constant, independent of the concentration of the sample components. As concentrations in nonlinear chromatography remain low, rarely exceeding 10% (w/w), this is a reasonable assumption in most cases, except for some sugars, polymers, or proteins. The solutions of these compounds in the mobile phase are much more viscous than the pure mobile phase [69], hence the diffusion coefficients of these solutes are strongly dependent on their concentration. Under such circumstances, the equilibrium-dispersive model does not remain valid. The equilibrium-dispersive model is correct if mass transfer in the chromatographic column is controlled only by molecular diffusion across the mobile phase flowing around the packing particles and if the exchange of feed components between the stationary and mobile
48
The Mass Balance Equation of Chromatography and Its General Properties
C [mg/mL]
length [cm]
Figure 2.7 Solution of Eq. 2.37 for a pulse concentration input (elution). The boundary condition is a 0.5-min rectangular pulse as shown in Figure 2.2a, with the injected sample concentration of 5 g/L. The procedure used for the numerical calculation is discussed in Chapter 10, Section 10.3.4. The numerical parameters of the Langmuir isotherm are a = 10 and b = 0.1. The two solid lines in the plane C = 5 mg/mL show the projections in this plane of the trajectories of the concentration C = 0 given by the ideal model.
phases is very fast (Chapter 14, Section 14.1.7). Otherwise, it is an approximation. It happens that, because of the high efficiency achieved in modern column chromatography using fine-particle packing materials, this is most often an excellent approximation. In nonlinear chromatography, there are no known solutions to the equilibriumdispersive model in closed form. Numerical solutions are easily obtained, using computation methods such as finite differences, finite elements, or collocation [70-73]. In the former case, extensive results are available on the band profile and how it changes with various experimental conditions [74,75]. Many comparisons have also been made with experimental data [30,76-79]. An example of the solution of Eq. 2.37 is given in Figure 2.7, in the case of a rectangular pulse (elution) input. The solution is presented as a plot of C(x,t), the mobile phase concentration versus the position in the column and time. The analogy of the solution with the propagation of a wave is obvious in this figure. The contour lines above the solution indicate the trajectories of the concentration C = 0 that are
2.2 Solution of the System of Mass Balance Equations
49
predicted by the ideal model. Concentration discontinuities cannot take place with a mass balance equation such as Eq. 2.37. The diffusive term causes erosion of the discontinuity. Because the concentration gradient in a shock would be infinite, the diffusive mass flux would also be infinite. There is a dynamic equilibrium between the trend of thermodynamic origin to build up a concentration discontinuity, because of the dependence of the migration rate on the concentration, and the smoothing trend of the axial dispersion.
What is observed instead is called a shock layer (the steep sides of the profiles in Figures 2.6j and o). This is a thin region of space where the concentration varies rapidly, yet continuously. The importance of the shock layer concept results from the fact that the shock layer velocity is the same as the shock velocity (Chapter 14, Section 14.1.4). In preparative chromatography, a shock layer has properties that are similar to those of the shock. However, it has a finite thickness, which depends on the kinetics of mass transfer and on the axial dispersion coefficient, but it has the same propagation properties as the shock. Chapters 10 to 13 review the solutions of the equilibrium-dispersive model for a single component (Chapter 10), and multicomponent mixtures in elution (Chapter 11) and in displacement (Chapter 12) chromatography and discuss the problems of system peaks (Chapter 13). These solutions are of great practical importance because they provide realistic models of band profiles in practically all the applications of preparative chromatography. Mass transfer across the packing materials currently available (which are made of very fine particles) is fast. The contribution of mass transfer resistance to band broadening and smoothing is small compared to the effect of thermodynamics and can be properly accounted for by the use of an apparent dispersion coefficient independent of concentration (Chapter 10).
2.2.3 The Lumped Kinetic Models In these models, the mass balance equation (Eq. 2.2) is combined with a kinetic equation (Eq. 2.5), relating the rate of variation of the concentration of each component in the stationary phase to its concentrations in both phases and to the equilibrium concentration in the stationary phase [80-93]. Although in principle kinetic models are more exact than the equilibrium-dispersive model, the difference between the individual band profiles calculated using the equilibrium-dispersive model or the linear driving force model, for example, is negligible when the rate constants are not very small {i.e., when the column efficiency exceeds a few hundred theoretical plates), as shown in Chapter 14 (Section 14.2). The different kinetic models used have been summarized by Ruthven [52]. The simplest models used in nonlinear chromatography are the following. 2.2.3.1
The Langmuir Kinetic Model
This classical model, which appears in the "kinetic derivation" of the Langmuir isotherm, is written ^
= ka{qs,i - CsA)Q - kdCsj or
(2.40)
50
The Mass Balance Equation of Chromatography and Its General Properties
where qsj is the column saturation capacity (in the same units as Cs), as explained in the next chapter, and ka and k^ are the rate constants of adsorption and desorption of the component considered. This model has been used by Thomas [83], Goldstein [84], and Wade et al. [80]. Thomas has derived an analytical solution for a step function boundary condition {i.e., a breakthrough curve or frontal analysis problem) [83]. Goldstein [84] and Wade et al. [80] have derived analytical solutions for pulse boundary conditions (i.e., the overloaded elution problem). 2.2.3.2
The Linear Kinetic Models
There are several such models. Lapidus and Amundson [85] assumed a first-order kinetic model: ^=*«Q-*dCs,t-
(2.41)
where ka and kj are the rate constants for adsorption and desorption, respectively. Two similar but simpler models have also been used. The more popular of them is the solid film linear driving force model: d
^ C
s
,
i
)
(2.42)
where qi is the equilibrium value of CS/; for a mobile phase concentration equal to Q and kf is the lumped mass transfer coefficient. This model has been used by many authors, including Glueckauf and Coates [86], Hiester and Vermeulen [87], Lin et al. [88], Golshan-Shirazi et al. [89], Phillips et al. [90]. The other model is the liquid film linear driving force model: ^M=C(Q-C*)
(2.43)
where C* is the solute concentration in the mobile phase that is in equilibrium with the solid phase concentration CS/!-. Thus, C* is the root of the isotherm equation solved for the mobile phase concentration in equilibrium with the stationary phase concentration q^. Under linear conditions, C* = qj/a where a is the slope of the linear isotherm (a = ko/F, F, phase ratio), and km is the apparent (lumped) mass transfer coefficient. In linear chromatography, these last two linear kinetic models are particular cases of the model used by Lapidus and Amundson [85] (Eq. 2.22). By contrast, the different lumped kinetic models give different solutions in nonlinear chromatography. Investigations of the properties of these models and especially of the relationship between the band profiles and the value of the kinetic constant have been carried out for many single-component problems. Numerous studies of the influence of the mass transfer kinetics on the separation of binary mixture have been published in the last ten years. These results are discussed in Chapters 14 and 16, respectively.
2.2 Solution of the System of Mass Balance Equations
51
2.2.4 The General Rate Model of Chromatography The preparative separations of certain polar (e.g., strongly basic) compounds and of many large molecular compounds (e.g., peptides and proteins) usually involve a complex mass transfer mechanism that is often slower than the mass transfer kinetics of small molecules. This slow kinetics influences strongly the band profiles and its mechanism must be accounted for quantitatively. The accurate prediction of band profiles for optimization purposes requires a correct mathematical model of the various mass transfer processes involved. The purpose of the general rate model (GRM) is to account for the contributions of all the sources of mass transfer resistances to the band profiles [52,62,94,95]. The mass transfer of molecules from the bulk of the mobile phase percolating through the bed to the surface of an adsorbent or the mass of a permeable resin particle involves several steps that must be identified. The general principle of the GRM consists in considering separately the two fractions of the mobile phase: the fraction that percolates through the column bed and carries the sample components along the column and the stagnant fraction contained in the porous particles. A mass balance is written in each of these two phases. Appropriate initial and boundary conditions are selected. Suitable kinetic equations for the equilibration between the two liquid phases and between the stagnant liquid and the stationary phases complete the model. These equations account for the kinetics of the different steps involved in the chromatographic process: • The transfer between the two mobile phases, across the boundary between the stream of percolating mobile phase and the solution stagnant inside the pores. • Molecular diffusion through the porous space, toward the adsorbent surface. • Surface diffusion of the molecules of the sample components. • The kinetics of adsorption/desorption. Depending on the specific problem studied, one or several steps can be considered as much faster than the other ones and the corresponding contribution to the overall kinetics of mass transfer neglected. The GRM is the most comprehensive model of chromatography. In principle, it is the most realistic model since it takes into account all the phenomena that may have any influence on the band profiles. However, it is the most complicated model and its use is warranted only when the mass transfer kinetics is slow. Its application requires the independent determination of many parameters that are often not accessible by independent methods. Deriving them by parameter identification may be acceptable in practical cases but is not easy since it requires the acquisition of accurate band profile data in a wide range of experimental conditions. This explains why the GRM is not as popular as the equilibrium-dispersive or the lumped kinetic models. 2.2.4.1
Formulation of the General Rate Model
The two mass balance equations of the GRM model were given earlier in nondimensional coordinates (Section 2.1.7). However, in many references found in
52
The Mass Balance Equation of Chromatography and Its General Properties
the literature, these equations are used in more conventional dimensional coordinates [52,96,97]. The mass balance equation in the mobile phase stream is: ee-^
+u
0
^ + (1 - ee)kextriap[Q
- CPii(r = Rp)] = eeDL4-^-
(2.44a)
If the intraparticle mass transfer is supposed to arise from a combination of the parallel contributions of molecular diffusion through the liquid filling the pores and of surface diffusion, the mass balance equation for the stagnant liquid can be expressed as [28,52,95] ^
+ (1 " eP)|
= ^ [ e
p
D
v J
^
+ (1 -
ep)Ds,M])
(2.44b)
In these equations flp is the external surface area of the particles of packing material, Q, CP/; are the bulk and stagnant mobile phase concentrations of compound i, qi is the stationary phase concentration of compound i, q* is the stationary phase concentration at equilibrium with the local mobile phase concentration at the adsorbent surface, Defy is the effective diffusion coefficient (which combines the influences of pore and surface diffusion), Dirj is the axial dispersion coefficient of compound i (D^j is assumed to be independent of i in the second model), Dpj is the pore diffusion coefficient of component i, Ds{ is the surface diffusion coefficient of component i, ee is the external porosity of the column bed, £p is the internal porosity of the packing particles, kexi i is the fluid to particle mass transfer coefficient, r, Rp are the radial position in and the radius of the particle (all particles are assumed to be spheres of uniform diameter, Rp), UQ is the superficial velocity of the mobile phase, p is the density of the particles, In Eq. 2.44b, surface diffusion is assumed to originate from the concentration gradient of the adsorbate within the particle, dq*/dr = dq* /dC x dC/dr. As long as the external mass transfer and the effective diffusion coefficients are independent of the local concentration, the equation is correct. However, Kaczmarski et al. showed that this assumption does not always hold [28,98]. Using the dusty gaslike model of surface diffusion [99,100], they were able to improve the accuracy of calculated band profiles by combining the generalized Maxwell-Stefan model of diffusive mass flux, the surface-restricted model of diffusion, and the general rate model [98].
2.2 Solution of the System of Mass Balance Equations
53
Alternately, pore and surface diffusion can be lumped into an effective diffusion, with Deff = evDp + (1 - ep)Ds|£
(2.44c)
Then, Eq. 2.44b can be rewritten:
A simpler version of the GRM model assumes that the mass transfer kinetics is controlled by mere pore diffusion and by the external film mass transfer resistance. The mass balances in the two fractions of the mobile phase are then written [52,62,101]:
JT + ^-^k^Q-C,^)
= Du^±
(2.45a)
where: «/, is the interstitial velocity of the mobile phase, Pp is the density of the particles of packing material, kffi is the fluid to particle mass transfer coefficient, «; is the adsorption equilibrium constant of compound i. See later, Section 2.3, for further explanation regarding the different definitions used. 2.2.4.2 Solutions of the General Rate Model The solution of the general rate model of chromatography has been discussed extensively in the literature in connection with the theory of linear chromatography (see Chapter 6). The solution of this model in the Laplace domain was derived by Kucera [102] and by Kubin [103] for a linear isotherm. This work cannot be directly extended to nonlinear cases. Although it is computer intensive, the calculation of numerical solutions of the general rate model of chromatography has become a conventional method. Many authors have described procedures, using a variety of initial and boundary conditions corresponding to practically all the modes of chromatography (with the notable exception of system peaks). Orthogonal collocation on finite elements seems to be the most popular method for these calculations [28,96-98,104]. This model has been used by Lee et al. [91] and Yu and Wang [92] in nonlinear chromatography. Further discussions of these solutions are found in Chapters 14 and 16.
54
The Mass Balance Equation of Chromatography and Its General Properties
2.2.5 The Lumped Pore Diffusion Model This model is a simplification of the GRM and was suggested by Morbidelli et al. [63]. It assumes that the kinetics of adsorption/desorption is negligible. By integrating both sides of Eq. 2.44b in the particle and using the linear driving force approximation, it is possible to approximate this term on the basis terms of the average composition. This approximate model (POR) is less accurate than the general rate model, but it is more accurate than the linear driving force models. The POR model will be further discussed in Chapter 16.
2.2.6 Equivalence Between Equilibrium-Dispersive and Kinetic Models The relationships between these two types of models have been reviewed recently [67,81]. Mass transfer is usually fast in chromatographic columns, especially in the modern HPLC columns packed with small silica or chemically modified silica particles which have been developed to permit the achievement of highresolution, fast separations. Since mass transfers are fast, the influence of their kinetics on the band profile becomes secondary at high concentrations and acts essentially in dampening the strong concentration gradients created by the thermodynamic effects. In a concentration discontinuity, the concentration gradient is infinite, hence the mass flux due to diffusion is also infinite. A dynamic equilibrium takes place between the influence of the thermodynamics, which tends to create and build up concentration discontinuities, and the influence of the finite rates of axial dispersion and of mass transfers in the column, which tends to erode strong concentration gradients. The result is band profiles exhibiting shock layers, as explained earlier [4-7,33-37,105]. Lapidus and Amundson [85] showed that, in the case of a linear isotherm, it is possible to derive a closed-form solution to the system of partial differential equations combining the mass balance equation and a first-order mass transfer kinetic equation. This solution is valid only for analytical applications of chromatography and cannot be extended to nonlinear isotherms. Van Deemter et al. [29] demonstrated that when the mass transfer kinetics in the column is not very slow, the analytical solution of Lapidus and Amundson reduces to a Gaussian profile. In this case, the exact analytical solution derived by Lapidus and Amundson is equivalent to the solution of the simple plate theory of Martin and Synge [106]. Van Deemter et al. [29] provided the equivalence by deriving an equation relating the standard deviation of the Gaussian profile to which the Lapidus and Amundson solution reduces and the standard deviation of the Gaussian profile given by the plate model, through the value of the height equivalent to a theoretical plate of the column:
This means that when the sum of the heights of the mixing stage (i.e., 2Di/u) and of the mass transfer stage (i.e., 2uk'0/(l + k'Q)2kf) is small compared to the column
2.2 Solution of the System of Mass Balance Equations
55
length, L, the equilibrium-dispersive model and the lumped kinetic model are equivalent in linear chromatography. The condition is satisfied if the column has more than about 100 theoretical plates (see Chapter 6, Figure 6.2), which is a very low value in almost all practical applications. When the plate number becomes too low, the band profile is still given by the equation derived by Lapidus and Amundson [85], although the Van Deemter equation is no longer valid. Kucera [102] and Kubin [103] have derived a solution to the general rate model of chromatography, still within the framework of linear chromatography. The mass balance of his model includes the axial dispersion term; the kinetic equation involves the contributions of the resistances to mass transfers that are due to diffusion across the boundary layer between the flowing mobile phase around the packing particles and the stagnant mobile phase inside these particles, the molecular diffusion inside the particles, and the kinetics of the retention mechanism, assumed to be first order. The system cannot be integrated but a closed-form solution can be derived in the Laplace domain. Only a numerical conversion can be performed into the time domain. When the elution profile is Gaussian, the values of the second moment (proportional to the band variance, Eqs. 2.39 and 6.82) predicted by the general rate model and by the linear driving force model can be compared. This shows that the lumped mass transfer coefficient, kf, is related to the pore diffusion and the boundary layer mass transfer resistance by the equation k'^okf
^ (kf
J60e p D p
(2.47)
where F is the packing phase ratio, e/ (1 — e), e is the total porosity of the packing, ep is the internal porosity of the particles of the packing material, dp is the average particle size, kf is the boundary layer or film mass transfer coefficient, and Dp is the intraparticle diffusion coefficient [107]. This result shows that the lumped kinetic model and the general rate model are equivalent in linear chromatography. Horvath and Lin [108] and Huber [109] have investigated in detail the influence of the various contributions to band broadening in linear chromatography. They have derived complex equations relating the parameters of the mass transfer and the axial dispersion to the column height equivalent to a theoretical plate height. Comparisons between the predictions of these equations and the experimental results are difficult, due to the number of parameters introduced in the model and the uncertainty that surrounds the determination of part of them. As shown by Glueckauf [107], Van Deemter et ah [29], and Giddings [67], as long as the mass transfer kinetics is not very slow, all the kinetic models are equivalent in linear chromatography. They are also equivalent to the equilibrium-dispersive model and to the plate theory model, provided that the apparent overall equivalent parameters are used. In the case of a nonlinear isotherm, there is no general solution available that would compare to the equation derived by Lapidus and Amundson. Thus, we do not have a tool comparable to the column height equivalent to a theoretical plate (HETP) for the investigation of the influence of the column performance on the band profiles.
56
The Mass Balance Equation of Chromatography and Its General Properties
In his nonequilibrium theory of chromatography, Giddings [67] attempted to derive a general relationship between the broadening of a chromatographic zone due to the mass transfer kinetics and the experimental parameters. Central to his approach, however, is the recognition that the two phases of the chromatographic column are always near equilibrium. For the sake of simplicity, we shall refer the concentrations in the next few equations to the total column volume, not to the volume of either the stationary or the mobile phase as is customary. These particular concentrations are denoted with an overline. C denotes an equilibrium concentration and C an actual concentration. We can write, assuming near equilibrium Cm = C*m(l + em)
(2.48a)
Cs = C*(l + es)
(2.48b)
and
where em and es account for the small departure of the system from equilibrium. Although concentration gradients may be large locally, they apply only over very short distances, so these quantities remain small. They are related through emC*m + esC*s=0
(2.49)
which expresses mass conservation. Ignoring the contribution of the longitudinal flux due to molecular diffusion in the stationary phase, we can write the longitudinal flux, /, of a compound per unit cross-sectional area of the column as follows: } = uCm = uC*m + uC*mem
(2.50)
The second term in the RHS of this equation is due to the departure from equilibrium. It is formally equivalent to a dispersion term, with a dispersion or pseudodiffusion coefficient which, in accordance with Fick's law (Chapter 5), is given by Da = - = ^ L
(2.51)
dC/dz where C is the total concentration. It is related to the equilibrium mobile phase concentration by ~C*m — C/(l + k'o). Therefore, Giddings [67] has demonstrated that nonequilibrium effects resulting from the finite rate of mass transfer kinetics can be treated as a contribution to the axial dispersion, itself the result of axial molecular diffusion, the tortuosity of the packing, the anastomosis of the network of interparticle channels where the stream of mobile phase flows, and the nonhomogeneity of the column packing. The axial diffusion and column tortuosity account for the B term of the classical Knox equation: h = - + Av1/3 + Cv
(2.52)
2.3 Important Definitions
57
where h is the reduced plate height (h = H/dp) and v is the reduced velocity, or particle Peclet number. The nonhomogeneity of the flow structure accounts for the A term and the nonequilibrium effects {i.e., mass transfer resistances and kinetics of the retention mechanism) for the C term. The apparent dispersion coefficient in Eq. 2.30 is related to the column HETP by [69] Da = ^
(2.53)
Although derived in the context of investigations of linear chromatography, the demonstration of this result is more general and it applies to nonlinear chromatography, although the relationship between Da and the column HETP loses its meaning. Haarhoff and Van der Linde [68] have given a more direct mathematical demonstration of this result in the case of a moderately overloaded column, with a parabolic isotherm. It leads to the fundamental equation of the equilibriumdispersive model, in which the diffusion coefficient in the diffusive term of the mass balance equation (Eq. 2.2) is replaced by the apparent dispersion coefficient (Eq. 2.38). Replacing the complex kinetic models with their kinetic equations, which are difficult to write and contain uncertain dependences on a variety of factors, some hardly understood, by the simpler equilibrium-dispersive model is a great simplification in the study of preparative chromatography. It is an excellent approximation as long as the nominal efficiency of the column (i.e., the efficiency under analytical conditions) exceeds approximately 100 theoretical plates [81], as is the case in most practical applications. It should be well understood, however, that this conclusion is not rigorously correct, and that some minor deviations may arise at efficiencies below about a thousand plates and also at high concentrations (Chapter 14, Section 14.1). For example, the equilibrium-dispersive model does not apply to ligand exchange chromatography when the rate of associationdissociation of the complex formed is very low, nor does it apply to a significant portion of the implementations of affinity chromatography.
2.3 Important Definitions A chromatographic column consists essentially in a bed made of a porous material, the solid phase, through which percolates a liquid, the mobile phase. The porous material is either a bed of more or less consolidated particles or a monolith. It has most often a bimodal pore size distribution (rarely, it may have a trimodal pore size distribution, see Chapter 5, Section 5.2.2). The larger mode corresponds to the through-pores or macropores through which percolates the stream of the mobile phase. The smaller mode corresponds to the mesopores in which the mobile phase is stagnant or through which it moves most slowly (an exceptional case). The large surface of the adsorbent that is needed for retention and capacity is essentially that of the solid-liquid interface in these mesopores. The solid itself is generally unaccessible to the analytes and to the mobile phase (although these
58
The Mass Balance Equation of Chromatography and Its General Properties
compounds may diffuse through the solid phase of certain particles, e.g., those made of weakly cross-linked resins). All the pores are accessible to the mobile phase and to the analytes (although some pores may not be accessible to certain solutes, e.g., because of the size exclusion or of the Donan effects). In conventional RPLC columns, the unaccessible solid phase is made of three different fractions: (1) the skeleton made of solid silica; (2) the bonded ligands, often but not always alkyl chains, chemically bonded to the silica surface; and (3) the closed pores, i.e., pores that have formed during the synthesis of the material and were closed at a later stage of the particle growth. The value of the closed pore porosity is small. It has negligible effects on column performance but it cannot be entirely neglected in estimates of the density of the packing particles. Although these closed pores have no practical effect on the column performance, their presence affects the apparent density of the particles and prevents from calculating the volume of the solid silica in the column from its weight and the known density of solid silica [110,111]. Fundamental discussions of nonideal, nonlinear chromatography may reach an almost theological degree of obfuscation due to the use of different definitions of many of the important parameters used by the different groups of scientists involved. Different definitions of the internal porosity of the particles, the density of the material used, the mobile phase velocity, and the equilibrium constant are commonly used in chromatography and in chemical engineering. Because chemical engineers use chromatographic definitions at times while chromatographers may use some chemical engineer definitions, it is recommended to check the definitions used before reading a new publication. We have reached the threshold where creative freedom gives way to confusing anarchy. For the sake of clarification, we summarize here some of the most important of these definitions.
2.3.1 The Definitions of the Different Porosities The void volume fraction or total porosity of the column (er) includes: (1) the external (or interparticle or interstitial) porosity (ee), which is the fractional volume of the cavities in the bed that are between and around the particles or of the through-pores in monolithic columns; and (2) the internal porosity (e, in chromatography, often £p in chemical engineering), which is the volume fraction occupied by the mesopores inside the particles (or in the monolithic rod). Note that it is not easy to define the boundary between internal and external porosities. In principle it is the surface inside which the mobile phase velocity is zero. However, nobody really knows how to define this boundary when a stream percolates through a bed of rugose particles, the surface of which is covered with hundreds of pits, the openings of the internal pores. Furthermore, for doughnut-shaped particles, the mobile phase would percolate at a much lower velocity through the large mesopores (which could, as well, be called the small through-pores) than in the main network of through-pores. The very definition of the external porosity is fuzzy and this parameter cannot be measured with any great accuracy. The internal porosity has two different definitions. In chromatography, it is the fraction of the column volume occupied by the mesopores contained inside the particles. In
2.3 Important Definitions
59
chemical engineering, it is the fraction of the particle volume that is occupied by the mesopores inside the particles. Accordingly, we have the following relationships: e
*
=
%
(Z54a)
yG
=
nRlL
(2.54b)
e;,chr
=
y-
(2.54c)
^ = ^ 7 ^ r = ^ L Vp VG(l-ee) l-ee
(2.54d)
£i,CE = eT
=
ee + ei/Chr = ee + (1 - ee)eiCE
=
'e + Vj = Vm VG VG
where Vm is the total volume of the mobile phase contained in the column, Ve is the volume of the interparticle voids or of the through-pores in a monolith, V{ is the volume of the intraparticle voids, VQ is the geometrical volume of the column, Vp is the total volume of the particles of packing material, Rc is the column inner radius, L its length, and to the hold-up time or retention time of an unretained tracer. Note that Vm is the actual volume of liquid phase inside the column. Usual methods of measurements give the actual volume of liquid between the injection valve and the detector cell. The results of these measurements must be corrected for the extra-column volumes of the instrument used. The chromatographic definition of the internal porosity has little physical sense. It should be avoided as much as possible. Unfortunately, it is used by many manufacturers of packing materials in their data sheets. At the time of this writing, monolithic columns are becoming popular in analytical chromatography but the preparation of large-diameter rods appears difficult. Yet, it is worth drawing attention to the possible ambiguity of the distinction between the external and the internal porosities in monolithic columns. The former porosity should be the integral of the pore size distribution of the macro- or through-pores, the latter that of the mesopores. This distinction is clear only if the distribution has two modes that are sufficiently distinct. Silica being quite rigid, the volume of the particles, hence the internal porosity and the unaccessible volume fractions should remain constant during the consolidation of the bed, independently of the pressure applied to the packing material [55,112,113], while the external porosity depends on the degree of the consolidation of the bed. In contrast, the consolidation of resins under excessive pressure may lead to a significant decrease in the external and the internal porosities. This effect may be compounded by the swelling/shrinking of certain resins when the composition of the mobile phase is changed. Note, finally, that the internal porosity of Cis-bonded silicas (and most probably of other chemically bonded porous silicas) often changes significantly with the concentration of the organic modifier in an aqueous mobile phase [114].
60
The Mass Balance Equation of Chromatography and Its General Properties
2.3.2 The Definitions of the Densities of Packing Materials Different definitions are also used in the scientific community for the densities of the packing material in a column. We summarize some of these definitions 1. The skeleton density, ps^, is the density of the solid silica skeleton, assumed to have no closed pores. This density depends on the solid structure of silica, crystalline or amorphous. This is not the most useful density in chromatography. 2. The true solid density of the particles, ps, is the ratio of the particle weight to the unaccessible particle volume, or sum of the volumes of its silica skeleton, its alkyl chain layer, and its closed pores. Note that the volume of the alkyl chain layer is ill-defined (see below). 3. The apparent particle density, pp, is the ratio of the particle weight to the particle volume, or sum of the unaccessible volumes and the internal pore volume. This is the most practical definition. However, it cannot be as accurate as densities usually are because of the difficulty associated with defining accurately the external column porosity (see previous subsection), 4. The bonded alkyl chains density, p\,, is the density of the bonded alkyl chain layer. It is often assumed to be the density of liquid octadecane in the case of Ci8 columns. Strictly speaking, however, the bonded layer is not a liquid since all the octadecyl chains are bonded, so their average distance is higher than in a liquid, and their entropy lower. This density (and the volume occupied by the layer) depends also on the mobile phase composition, depending on whether these chains are poorly soluble in the mobile phase and collapse on the surface or whether they are soluble and are swollen by the mobile phase or by the organic modifier in the case of the aqueous solutions most often used in RPLC.
2.3.3 The Definitions of the Mobile Phase Velocities Another level of complexity is added by the use of different definitions of the velocity of a stream of liquid percolating through a chromatographic column. The mobile phase flow rate is constant and it is the physical parameter that can be measured most easily. In kinetic studies, however, the velocity of the liquid stream is the critical parameter. Obviously, in a packed column as in a monolith, the local velocity of the stream varies rapidly from place to place, depending whether the velocity is considered in a mesopore or in a through-pore and, in this last case, at which distance from the surface of the nearest particle. It is impossible to determine the mobile phase velocity everywhere and this would not make any sense. Cross-section average velocities are more useful. In certain cases, when studying columns the packing of which is not radially homogeneous, a local average velocity (defined for a fraction of the cross-section area of the column) is used [115]. The three main definitions of the mobile phase velocity are:
2.3 Important Definitions
61
1. When studying flow through porous media, chemical engineers consider the superficial velocity «o = ^
(2.55)
The superficial velocity is straightforward to calculate from conventional chemical engineering data (flow rate and tube dimensions) but does not inform on the rate at which bands move along the column because the volume of the column available for the liquid stream varies widely, depending on the nature of the bed and its porosity. 2. Chromatographers have always used a velocity that is straightforward to calculate from chromatographic data, the linear mobile phase velocity or chromatographic velocity
«
§
<256>
Because chemical engineers and chromatographers are involved in investigations of preparative chromatography, confusions between these two definitions are frequent. It is all the more so because the two conventional symbols used for the velocity, u and UQ, are commonly reported in both fields, in spite of their two different definitions. Care should be taken to avoid confusion. 3. The irony is that both velocities are derived directly from data easily acquired using the conventional methods of the field in which they are used but that, unfortunately, neither informs well on the actual kinetics of the band convection, which is the primary concern in mass transfer investigations. Since, for all practical purposes, the stream of mobile phase flows only through the macropores, a more useful definition of the velocity is the interstitial velocity uh = ^
(2.57)
This definition is more complex to use because it requires the actual knowledge of the external porosity that is difficult to measure directly.
2.3.4 The Definitions of the Equilibrium Constants Chromatographers and chemical engineers use different values for the equilibrium constant, K. The definitions are the same in both fields. The equilibrium constant is the ratio of the equilibrium concentrations of the solute under study in the two phases of the system. However, chromatographers generally report concentrations as the mass of solute per unit volume of the phase considered, whether liquid or solid, with the result that the equilibrium constant is dimensionless. By contrast, chemical engineers report liquid phase concentrations as the mass of solute per unit volume of the liquid phase but solid (or rather adsorbed
62
The Mass Balance Equation of Chromatography and Its General Properties
compound) concentrations as the mass of the adsorbate per unit mass of the adsorbent, with the result that the equilibrium constant has the dimension of L3M~1 (usually cm 3 /g in chromatography). Therefore, the numerical values of the same constant are different and a conversion is needed. The following equation relates the two constants Kchr
PP(l-ee)
In this equation, the chemical engineering definition of pp is used. To make matters more complex, some authors report the concentration of the adsorbate at equilibrium as the amount adsorbed per unit of surface area of the adsorbent used rather than to its weight.
2.3.5 Example of Possible Confusions The analytical solution of the general rate model under linear conditions is discussed elsewhere (Chapter 6). It can be derived in the Laplace domain but the reverse transform of the analytical solution into the time domain is far too complex to be useful. However, the moments of this analytical solution can easily be derived in the time domain. For example, when using the Suzuki model (2.45a), the first moment is given by
)]
(2.59b)
where ep = e^cE a n d &CE a r e the internal porosity and the equilibrium constant as defined by chemical engineers (see above), respectively, and UQ is the superficial velocity. The hold-up retention time is the retention time of an inert compound for which K = 0. Hence it is given by L
r
•f (1 -eejep\
Un
L
= —€T Mo
(2.60)
By subtracting the last two equations and rearranging, we obtain the equilibrium constant as
t0
pp(l-ee)
Chromatographers define the equilibrium constant otherwise and write
tR
= n = k [l + ^T^Kchrom]
(2.62a)
t0
=
(2.62b)
^ = ^1 U
Mo
REFERENCES
63
Accordingly, the equilibrium constant is given by _}ll-t0 ^chrom — 7
€T *
, (Z.b3)
n
—
to l — er Which is consistent with Eqs. 2.58 and 2.61. Therefore, readers attempting to use values of equilibrium constants derived from measurements of first moments should pay careful attention to the exact model and definitions used by authors. Not only are their units different (KQE is in dimension of Lr'M^1 and Kcfa^ is dimensionless), but the numerical conversion requires the values of ej, ee, and pp which are not always reported in the literature.
2.3.6 Definitions, Conclusion The confusion that a combination of these different definitions can generate, together with the difficulties encountered in the determination of some of the column characteristics involved (particularly the internal and the external porosities) makes useful a careful consideration of these issues. Given the stage of sophistication that the modeling of chromatography has now reached, it is not possible to tolerate errors, confusions, or approximations in the definitions nor in the estimations of the critical parameters related to the porosities, the velocities, and the equilibrium constants, nor to accept that more errors be made in the estimation of these parameters than those that are always involved in any measurement process. In this book, we use alternately the chemical engineering and the chromatographic definitions of the internal porosity, the mobile phase and the equilibrium constant, depending on the choice made by the original authors whose work is discussed. We try to indicate clearly which definition has been used in each case.
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Chapter 3 Single-Component Equilibrium Isotherms Contents 3.1 Fundamentals of Adsorption Equilibria 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.1.8 3.1.9
Basic Thermodynamics of Adsorption and the Gibbs Isotherm The Linear Isotherm The Langmuir Isotherm in Gas-Solid Equilibria The Volmer Isotherm The Van der Waals Isotherm The Virial Isotherm Statistical Thermodynamics of Adsorption Liquid-Solid Equilibria Surface Excess and Excess Isotherms
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
70 71 73 74 75 75 75 76 78 78
80
3.2.1 Isotherm Models for Ideal Adsorption on Homogeneous Surfaces 3.2.2 Isotherm Models for Ideal Adsorption on Heterogeneous Surfaces 3.2.3 Isotherm Models for Nonideal Adsorption on a Homogeneous Surface 3.2.4 Isotherm Models for Nonideal Adsorption on a Heterogeneous Surface 3.3 Adsorption and Affinity Energy Distribution 3.3.1 Affinity Energy Distribution 3.3.2 Experimental Validation of the EM Method 3.3.3 Applications of AED in the Study of Retention Mechanisms in RPLC
81 89 98 107 109 110 113 114
3.4 Influence of Experimental Conditions on Equilibrium Isotherms
117
3.4.1 3.4.2 3.4.3
Influence of the Pressure Influence of the Temperature Influence of the Mobile Phase Composition
3.5 Determination of Single-Component Isotherms 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6 3.5.7 3.5.8
Frontal Analysis (FA) Frontal Analysis by Characteristic Points (FACP) Elution by Characteristic Points (ECP) Pulse Methods The Retention Time Method Computation of Elution Profiles (CEP) Method or Inverse Method (IM) Nonlinear Frequency Response Static Method
3.6 Data Processing and Assessment 3.6.1 3.6.2 3.6.3 3.6.4
117 119 121
122 123 125 126 127 130 131 133 134
135
Processing Experimental Data into an Isotherm Equation Determination of Critical Experimental Parameters Accuracy and Precision Comparison of the Main Chromatographic Methods
References
135 136 138 140
144
67
68
Single-Component Equilibrium Isotherms
Introduction In the previous chapter (Section 2.2), we showed that the integration of the differential mass balances of the eluites requires prior knowledge of their equilibrium isotherms.These functions are the relationships between the concentrations of each compound in the two phases of the chromatographic system, at equilibrium, at constant temperature and pressure. These thermodynamic properties depend on the molecular interactions of the solutes with the two phases involved. Knowing the equilibrium isotherms is required whatever the model of chromatography that is used. It is critical because the equilibrium isotherms determine the profiles of bands at high concentrations, as we show later (Chapters 6 to 17). Deviation from equilibrium condition is rarely very important in chromatography, because phase systems in which this deviation would be significant are carefully avoided, their separation performance being poor. Also, the kinetics of mass transfer is related to the deviation from equilibrium (Chapter 2, Section 2.2.3). In analytical applications, we generally consider that the concentration of the analytes in the mobile phase is very small. In this case, a linear isotherm can be assumed; i.e., at equilibrium, the concentration of an analyte in the stationary phase is proportional to its concentration in the mobile phase. This is the fundamental assumption of linear chromatography. Although useful and valid in most analytical applications of chromatography, this assumption falters in most preparative applications. The concentration range of interest is wide and the proportionality assumption is no longer valid. The functional relationship between the equilibrium concentrations of a compound in the two phases of a chromatographic system can become most complex, depending on the concentration range of interest, the eluites considered, the retention mechanism selected, and the phase system used. Since chromatography is a separation method, we consider mostly mixtures of eluites. In many cases, however, the mobile phase itself is not a pure solvent but is a mixture and it may contain additives which themselves interact more or less strongly with the stationary phase. As soon as the concentration of one or more of the eluites or that of an additive is not small, the equilibrium concentrations of all of these compounds depend not only on their own concentrations in the mobile phase but also on the concentrations of all the other components involved. At high concentrations, the molecules of the various components of the mobile phase and of the feed compete for their involvement in the retention mechanism, which has a limited capacity. For example, in liquid-solid chromatography, the competition comes from the finite capacity of the adsorbent surface. The phenomenon of competition is not limited to liquid-solid chromatography, however, it extends to all retention mechanisms, e.g., to ion-exchange chromatography. The problem of relating the stationary phase concentration of each component to the mobile phase concentrations of all the other components of the system, feed components as well as additives in the mobile phase, is complex. In most cases, however, it suffices to take into account the concentrations of only a few other species to cal-
Single-Component Equilibrium Isotherms
69
culate the concentration of one of the components in the stationary phase at equilibrium. The bands of most components of a mixture are rapidly separated from their neighbors after the injection is complete and only the local concentrations of those few components that are eluted close to a given band have a significant influence on its profile. This is because only its close neighbors interfere with the band of a component during its elution. In order to understand the fundamentals of the retention mechanisms at finite concentration, it is necessary to study first the equilibrium of a single component in a simple chromatographic system (i.e., using a pure mobile phase or a solution without strongly retained additives that would compete with the compound studied). This is the topic of this chapter. The important phenomenon of competition for interaction with the stationary phase at finite concentrations will be considered in detail in the next chapter. This phenomenon is critical because it has a major influence on the individual band profiles and, in the separation of mixtures, it eventually controls the throughput, the production rate, and the recovery yield of any high concentration chromatographic process. Unfortunately, the study of phase equilibria in solution, e.g., liquid-solid adsorption, is not a highly popular area of research. Gas-solid adsorption and vapor-solution equilibria have been studied in far more detail, although most of the information available concerns the fate of single components in a diphasic system. Liquid-solid adsorption has benefited mainly from the extension of the concepts developed for gas phase properties to the case of dilute solutions. Multicomponent systems and the competition for interaction with the stationary phase are research areas that have barely been scratched. The problems are difficult. The development of preparative chromatography and its applications are changing this situation. In this chapter, we review the fundamentals of adsorption (Section 3.1), the main models currently used to represent the equilibrium isotherms of single components (Section 3.2), the adsorption energy distribution on the surface of heterogeneous adsorbent (Section 3.3), the experimental parameters that affect the adsorption isotherms (Section 3.4) and the methods of acquisition of equilibrium data for the derivation of single-component isotherm data (Section 3.5). In a final section (Section 3.6), we discuss briefly the problems of accurate and precise measurements of adsorption isotherm data and compare the methods of measurements currently in use. Problems arising from the surface heterogeneity of adsorbents may have considerable importance in practical applications. They are still poorly understood from a fundamental point of view. Although chemists and engineers have worked systematically to design preparation procedures of adsorbents giving highly homogeneous surfaces, "active sites" and quasi-irreversible adsorption of strongly acidic or basic compounds continue to plague the separation scientists. Fundamental study of this issue may permit a better understanding of the tailing phenomena.
70
Single-Component Equilibrium Isotherms
3.1 Fundamentals of Adsorption Equilibria The study of phase equilibria is a part of thermodynamics that concerns the equilibrium composition of two phases and the influence of various parameters on this composition (e.g., temperature, pressure). We are concerned here with phase systems for liquid chromatography, in which one phase is a solvent or a solution, the other a solid. This solid can be an adsorbent, it can be covered with chemical groups capable of complexing the components of the feed or otherwise interacting with them (ion exchange, ligand exchange, or complex formation). It could also be an inert solid matrix holding a liquid in its pores (note that liquid-liquid chromatography presents practical difficulties that have made it unpopular, but that weakly cross-linked polymers or gels immobilized in macroporous silica may provide an attractive approach). The equilibrium isotherm is a plot of the concentration of a component in the stationary phase versus its concentration in the mobile phase at equilibrium. Throughout this text, emphasis is placed on adsorption and ion-exchange equilibria because they correspond to the most important retention mechanisms used in preparative chromatography. The concepts are easily extended to other retention mechanisms. Gas-solid equilibria have been studied for over 200 years, since Fontana showed that activated charcoal adsorbs gases and vapors at room temperature [1]. A considerable amount of theoretical and experimental literature is available. The Gibbs isotherm [2] and the multilayer adsorption theory of Brunauer, Emmett and Teller [3], provide serious theoretical guidelines and support in understanding the results of experimental studies. Although, gas-solid isotherms are difficult to predict quantitatively [4], this branch of adsorption thermodynamics is much easier than liquid-solid adsorption because of the relative simplicity of the gas-solid interface as compared to the liquid-solid interface. The Gibbs equation relates the amount of a compound adsorbed per unit surface area of a liquid-gas or a liquid-liquid interface and the surface or interfacial tensions [2]. This relationship provides a useful theoretical framework. The situation is more complex for liquid-solid equilibria. The surface tension of solids is not readily measured. In liquid-solid equilibria the adsorbate always has to compete, at least with the solvent which is in large excess, for access to the solid surface, a phenomenon that does not take place in gas-solid equilibria under the conditions prevailing in gas chromatography [5]. As a consequence, most of our understanding of liquid-solid equilibria remains empirical. Gas-solid equilibrium isotherms have been divided into five classes, according to their shape, as shown in Figure 3.1 [6]. The type I (Langmuir or semiLangmuir) isotherms are the most frequent. Isotherms of class II and III have a vertical asymptote corresponding to the vapor pressure of the adsorbate. This asymptote is explained by capillary condensation and by the fact that when the partial pressure is increased and becomes close to the vapor pressure, condensation takes place preferentially in the most strongly curved region, i.e., in the smallest pores. There is no equivalent to these phenomena in liquid-solid equilibria. When the solubility of a compound in the solvent is limited and adsorption data are measured at concentrations becoming close to the saturation limit, there
3.1 Fundamentals of Adsorption Equilibria
71
Type! TypeE
Type JH
TypcX
Figure 3.1 The five types of van der Waals adsorption isotherms. Reproduced with permission from S. Brunauer, L. S. Doming, W. E. Deming and E. Teller,}. Amer. Chem. Soc, 62 (1940) 1723 (Fig. 1), ©1940 American Chemical Society.
is no significant change in the slope of the isotherm, which ends at the saturation concentration without exhibiting any abnormal behavior. Gas-solid equilibrium isotherms of class IV and V are due to the existence of small pores which are filled after only a few adsorption layers are formed. It is unusual in liquid-solid equilibria for adsorption to extend beyond the formation of a monolayer [7]. Mainly because of adsorbate-adsorbate interactions, however, isotherms of this class are sometimes encountered [8] and several examples of this behavior have been reported [9,10]. This case might be more frequent than most chromatographers believe. The Langmuir adsorption behavior does not seem to be the prevalent case that it was believed to be twenty years ago (see e.g., [11]). The fundamental references in gas-solid adsorption are the works by Fowler and Guggenheim [12], Everett [13], and Hill [14,15], and the books by Young and Crowell [16], de Boer [17], Kiselev [4], and more recently by Ruthven [18] and Toth [19], who gives a clear, logical, and simple presentation of this topic. We present first a few theoretical results obtained in the study of gas-solid adsorption, results that have been extended semiempirically to liquid-solid adsorption [18]. Then, we describe the various isotherm models that have been used in the study of retention mechanisms in liquid chromatography.
3.1.1 Basic Thermodynamics of Adsorption and the Gibbs Isotherm In the case of adsorption, the fundamental equation summarizing the first and the second law of thermodynamics as applied to the adsorbed component may be
72
Single-Component Equilibrium Isotherms
written [2,16-18] = -SadsdT
+ VadsdP - $dna + Fads^ads
(3-1)
where Gads, Sads, Vads, <J>, and p.ads are properties of the adsorbate in the adsorbed phase, the na moles of adsorbent being considered as thermodynamically inert [2]. Gads i s the molar Gibbs free energy of the compound in the adsorbed phase, Sads its molar entropy in the adsorbed phase, Vads its molar volume, }iads its chemical potential, T is the absolute temperature of the system, P its the pressure, n« is the number of moles of adsorbent, nad$ is the number of moles of adsorbate, and
represents the change in internal energy per unit mole of adsorbent due to the spreading of the adsorbate over the surface of the adsorbent or in its micropore volume. <£> is related to the spreading pressure, K, by
(3.2)
where A is the surface area. The spreading pressure itself is defined by the following equation: n=-[
-^j-
(3.3)
where Uads is the molar internal energy of the compound in the adsorbed phase The spreading pressure corresponds to the difference of the surface tensions of the clean surface and the surface covered with a dense monolayer of adsorbate. Constant temperature and pressure are the conditions prevailing in a chromatographic column, at least as a first approximation (see Section 3.4.1). We neglect the influence of the adsorption energy from the solution (this would be more questionable in gas chromatography than in liquid chromatography) and of the variation of pressure along the column, i.e., the compressibility of the liquids. Under such conditions, Eq. 3.1 can be integrated with the intensive variables held constant to give Gads = ~®na + Pads«ads
(3-4)
Differentiation of Eq. 3.4 and subtracting Eq. 3.1 from this differential gives: nad<£> = Adn = nadsd}iads
(3.5)
The procedure used to derive Eq. 3.5 is the same as for the derivation of the GibbsDuhem equation. Substituting the expression for the chemical potential of an ideal gas in Eq. 3.5 provides the classical Gibbs isotherm: RT (3 6)
'
with p pressure. From the Gibbs isotherm and assuming an equation of state for the adsorbed phase, a number of classical isotherm equations can be derived [18].
3.1 Fundamentals of Adsorption Equilibria Figure 3.2 Linear, Langmuir, and biLangmuir isotherms. The Linear, Langmuir, and bi-Langmuir isotherms are used as models to fit the experimental adsorption data for N-Benzoyl-D-phenylalanine. Parameters: Linear isotherm: a = 13.1; Langmuir isotherm: a = 13.1, b = 241; biLangmuir isotherm: a.\ = 9.6, b\ = 1920, «2 = 7.1, &2 = 33.8. Data from S. Jacobson, S. Golshan-Shirazi and G. Guiochon, AIChE ]., 37 (1991) 836. Reproduced by permission of the American Institute of Chemical Engineers. ©1991 AIChE. All rights reserved.
73 Q moles/L packing
0.04
0.02
/
If / 0
/ ^
- - Linear "•*"* Langmuir Bilangmuir •
°-003
N-Benzoyl-D phenylalanine
Cmoies/L 0 - 006
3.1.2 The Linear Isotherm If we assume that the equation of state for the adsorbed phase is similar to the equation of state of an ideal gas, we have (3.7)
which combines with the Gibbs isotherm (Eq. 3.6) to give: (3.8)
Thus, the isotherm, which relates the stationary phase concentration, q, with the fluid concentration, C, becomes
?-¥
(3.9)
where a is the slope of the isotherm (Henry's constant of adsorption), k' is the retention factor (k' = {tR — to) /1$, with £#, retention time, to, dead time), F is the phase ratio, F = (1 — e)/e, and e is the total porosity of the column (or volume fraction available to the mobile phase). Equation 3.9 is a linear isotherm. The concentration of the solute in the stationary phase is proportional to that in the mobile phase. This isotherm is widely used in analytical chromatography, where it gives results that are most often satisfactory. When the concentration becomes high, however, deviations from linear behavior take place, competitive interactions between the different components of the feed appear, and a more complex model becomes necessary to account for these experimental results. As an example, Figure 3.2 [20] shows a comparison between a linear isotherm and two nonlinear models (see below). The difference is small at concentrations below 0.05 mM, but significant deviations take place at 0.2 mM. They are sufficient to cause an important decrease in the band retention and a marked asymmetry of its profile [20].
74
Single-Component Equilibrium Isotherms
3.1.3 The Langmuir Isotherm in Gas-Solid Equilibria We may postulate for an adsorbed gas an equation of state that is slightly more complicated than Eq. 3.7 and which takes into account the finite surface occupied by a molecule adsorbed on the surface [18]. The following equation is similar to an equation of state for gases which would take into account the volume occupied by the gas molecules [i.e., P(V — b) = RT, with P, pressure]. /t^vAi
LJ j
— "ads
\O.LKJ)
Differentiation of Eq. 3.10 gives n\
_
nadsRT
Combination of Eq. 3.10 and the Gibbs isotherm (Eq. 3.6) gives
Integration of this equation is possible if we assume that jS is small compared to A and ignore /32 in the denominator of Eq. 3.12, which is correct at moderate concentrations. Then:
where 9 = 2fi/A = q/qs is the fractional surface coverage and qs the column saturation capacity. Equation 3.13 is the Langmuir isotherm [21], which can be rewritten in its classical form:
This thermodynamic derivation emphasizes that the range of validity of the Langmuir isotherm does not extend to high concentrations. Otherwise, Eq. 3.12 could not be integrated into Eq. 3.13. The major advantage of the Langmuir isotherm (Eq. 3.13) is that it can be inverted and solved for q in closed-form. This cannot be done with the Volmer isotherm (see next section), or with many other important isotherms, such as the Fowler or the virial isotherms (see below, Sections 3.1.6 and 3.2.3.1). Such isotherms are often called implicit isotherms. The other conditions of validity of the Langmuir model are the ideal behavior of the gas phase, the absence of adsorbate-adsorbate interactions, and the localized character of adsorption. The Langmuir isotherm can also be derived by statistical thermodynamics [15], assuming (i) an ideal lattice gas in equilibrium with the surface; (ii) only one molecule binding per site; (iii) a set of equivalent, distinguishable, and independent sites; and (iv) no interactions between bound molecules. This is a firstorder approximation. A second-order approximation is described below (Section 3.2.3.2). Finally, a kinetic demonstration of the Langmuir isotherm can be given [21]. We assume that the surface is covered by a finite number of independent binding
3.1 Fundamentals of Adsorption Equilibria
75
sites, each site being able to accommodate only one molecule. The gas phase is ideal. There are no adsorbate-adsorbate interactions. The rate of adsorption of the molecules from the gas phase is proportional to the vapor pressure and to the fraction of the surface that is not covered and thus is available for binding: = kaP(l-9)
(3.15)
where 9 is the surface coverage. The rate of desorption is proportional to the surface coverage
At equilibrium the two rates are equal, hence the isotherm given by Eq. 3.13, with b = ka/kd.
3.1.4 The Volmer Isotherm If we make no assumptions regarding the magnitude of /S compared to A, the direct integration of Eq. 3.12 gives the Volmer isotherm [18] bP = ^ e ^ B
(3.17)
where 9 — pi A. In liquid-solid chromatography, 9 — q/qs and the concentration, C, is used in place of the pressure, P. This equation cannot be solved for 9, making this model impractical for the modeling of nonlinear chromatography.
3.1.5 The Van der Waals Isotherm If we assume that the gas phase follows van der Waals equation behavior instead of the ideal gas phase equation, with: (7T+-^)(A-p)
= nadsRT
(3.18)
and then we follow the same procedure as above, we obtain the van der Waals isotherm that is written: 9
9
2aqs8
bP = -—-e^=ee- -ir 1—9
(3.19)
3.1.6 The Virial Isotherm Much success has been achieved in semiempirical studies of the properties of fluids by using a virial equation of state. Similarly, a virial equation of state can be assumed for the adsorbed layer [4,18]:
1 +A
+A
2 s
+ --- + A p n l d s + ---
(3.20)
76
Single-Component Equilibrium Isotherms
Combined with the Gibbs isotherm (Eq. 3.6) it gives the virial isotherm equation bP
2
"ads
This equation is widely used in gas-solid adsorption studies [4], especially to derive an accurate value of the Henry constant from experimental data, which can rarely be acquired at low enough partial pressures [18]. Equation 3.21 shows that a plot of log bP/nacjs versus na^s is linear at low values of na&s. The Henry constant is derived from its intersection with the ordinate axis.
3.1.7 Statistical Thermodynamics of Adsorption The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase; hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermodynamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic functions are then derived from these partition functions, using the usual relationships of statistical thermodynamics. The average number of molecules in the system is
where: A = evr
(3.23)
]i is the chemical potential of the component considered, and E is the grand partition function, which is the sum of all canonical partition functions of the system weighted according to e~£r ; (3.24)
where T is the canonical partition function and N is the number of molecules in the stationary phase. If we consider that N molecules are adsorbed on a set of M equivalent sites (N < M) and that there are no interactions between adsorbed molecules, the grand partition function is [18]
f
(3.25)
where / is the molecular partition function of an individual adsorbed molecule and the term w ,J)^ N w is the degeneracy factor or number of ways in which the N identical adsorbed molecules can be arranged on the M identical, localized, and physically distinguishable adsorption sites.
3.1 Fundamentals of Adsorption Equilibria
77
From Eq. 3.22 —
AM/
N =
\+f\
( >
and a_
N
JA 1+/A
_
M
(327)
The chemical potential ji is given by p = p° + kT In a
(3.28)
where ji0 is the reference potential and a is the activity of the compound in the mobile phase which we assume here to be ideal, so H = F ° + kT In C
(3.29)
According to Eq. 3.23, we have
A = Cefr
(3.30)
9
(3 31)
hence
= iTbc
-
where b = few is the equilibrium constant. Equation 3.31 is the Langmuir isotherm. This isotherm has been derived under the assumption that there are no interactions between adsorbed molecules. Let us now consider the adsorption of N molecules on a set of M independent pairs of identical sites, such that there is an interaction energy equal to 2w between the two molecules that are adsorbed on a pair of sites, when such a pair is fully occupied. The grand partition function becomes [18] 2 M w\ vr A)2 )
Z = (^1 + 2/A + fe-
(3.32)
According to Eq. 3.22, we have
M
l
2/V»A 2fA p X
2
and from Eq. 3.30 (we still assume an ideal behavior of the mobile phase), we obtain
M
b
b
*g2
(3,4, 2
where b = / e » is the equilibrium constant.
78
Single-Component Equilibrium Isotherms
Equation 3.34 can be derived from a completely different model. Let us assume that we have two different types of sites on the surface and that these sites are independent, so a molecule that occupies a site of one type does not interact with a neighbor. Adsorption on these two types of sites is noncooperative. Let us assume that adsorption on each site follows a Langmuir adsorption model: ff=q_=
qs
hC b2C l + hC + l + b2C
(b1 + b2)C + 2b1b2C l + ih + b^C + hhC2
{
'
Equations 3.34 and 3.35 are identical with b\ + b2 = 2b and b\b2 = b2e~vr. However, the physical meaning of the two models is quite different. The isotherm coefficients have a completely different interpretation.
3.1.8 Liquid-Solid Equilibria The theoretical approach to the investigation of liquid-solid equilibria is more complex and much less advanced than the study of gas-solid equilibria, which has just been reviewed to the extent that it supplies a convenient background to understand the fundamentals of liquid-solid equilibria. The Langmuir isotherm has been readily extended to liquid-solid equilibria, first on an empirical basis, then on a more fundamental one. This problem is discussed in the next section (Section 3.2.1.1). The Freundlich isotherm (Section 3.2.2.4), first used for gas-solid isotherms, has also been extended to liquid-solid equilibria. These isotherms have permitted a correct description of experimental results in a variety of experimental studies involving dilute solutions of a strongly adsorbed component in a pure solvent. The pressure is replaced by the concentration in the equation of the isotherm. As expected from the derivation already discussed, the Langmuir isotherm appears to account fairly well for adsorption data acquired at low or moderate concentrations. At high concentrations, on the other hand, the activity coefficients of the species in solution are concentration dependent and systematic deviations from Langmuir adsorption behavior are observed. A number of more sophisticated theories have been developed to describe the adsorption of solutes from solutions in the whole concentration range. Most significant is the work published by Everett [22], Kipling [23], Schay and Nagy [24], Larionov el al. [25], Kiselev and Chopina [26], Siskova et al. [27], Minka and Myers [28], and Rusanov [29]. Unfortunately, this problem is far from being solved and most equations used in this area are empirical and approximate.
3.1.9 Surface Excess and Excess Isotherms Actually, the extent of adsorption of a compound i on the surface of an adsorbent in equilibrium with a multicomponent solution is related to the difference between the concentration at the interface, X;, and the bulk concentration, Xf. The excess adsorbed is n\ = n°(Xf - X{) = n°AX{
(3.36)
79
3.1 Fundamentals of Adsorption Equilibria Type E
Type I
Type II 3
\
2
\
170
05
I
/ 3
_ *. X|
0.1
\ i fI
_
0
\x,
0-9
i
W,
Type E
TypeS
2 i
1
\
/
O.I
0,5
v
=^-
O.I
A
°'5\
/
*l
Figure 3.3 The different types of excess isotherms. Plots of the surface excess concentration, fj" (mmol/g), with n total number of mole of components 1 and 2, versus the mole fraction Xx (except Figure 3.3-II, plot of If (mg/g) versus weight fraction, w\). (I) 1,2Dichloroethane (1) and benzene (2) on alumina gel at 25°C. (II) Benzene (1) and M-heptane on (a) alumina gel, (b) silica gel at 25°C. (Ill) Ethanol (1) and water (2) on charcoal at 25°C. (IV) Benzene (1) and ethanol (2) on charcoal at 25°C. (V) 1,2-Dichloroethane (1) and benzene (2) on charcoal at 25°C. ReproducedfromG. Schay, Surf. Coll. Set, 2 (1969) 155 (Figs. 1 to 5), with kind permission of Springer Science and Business Media.
where n° is the number of mole of liquid mixture in contact with the unit surface area of the solid adsorbent. Since the sum of all mole fractions is 1, we must have
Y_n\ = 0
( 3 - 37 )
For a pure liquid, n\ = 0, and the adsorption of a pure liquid on an adsorbent is a meaningless problem, since its surface concentration is constant, unlike the adsorption of a pure vapor. We can rewrite Eq. 3.36 as n\
= m - XiY^n?
(3.38)
The surface excess tends toward 0 when n, tends toward 0 or 1. Classically, the plot of the surface excess of one component of a binary mixture versus its concentration (mole fraction) in the mixture can belong to one of five different types, as illustrated in Figure 3.3. In type I, the plot has a maximum around X = 0.5. In type II, there is still a maximum, but it takes place at a lower mole fraction. In type III, the curve has an inflection point but does not intersect the concentration axis, and the maximum occurs at a low value of the mole fraction.
80
Single-Component Equilibrium Isotherms
The type IV plot is similar to type III, but the surface excess concentration of component 1, Ff, becomes negative at high concentrations, before returning to zero. The plot is quite unsymmetrical. In type V, the curve intersects the concentration axis toward the center of the graph and is closer to symmetrical. The curves in types II to IV have a rather important part which is nearly linear. A simple model of excess isotherm [30] is
which reduces to the Langmuir isotherm at concentrations below X( = 0.1, a condition that is almost always valid in liquid chromatography. However, Toth has shown that all isotherm equations that include the expression (1 — 6) contradict the Gibbs thermodynamics [31]. The inconsistency of these equations and the Gibbs isotherm stems from the fact that they use the absolute amount injected while Gibbs considers the excess adsorbed amount, as shown above. These two amounts are equivalent only if the concentration of the compound studied is very small, which is obviously not the case when a nonlinear isotherm equation is applied. However, the following equation (3.40)
would be acceptable since the limit of d\ when Xj tends toward 1 is unity [31]. Riedo and Kovfits [30] have derived general relationships for the derivation of surface excesses from chromatographic measurements. They do not seem to have been applied yet in nonlinear chromatography.
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria A considerable number of adsorption isotherm models have been suggested in order to account for an extremely varied adsorption behavior, depending on the nature of the adsorbent and of the adsorbate. Even with stationary phases for HPLC which are now carefully manufactured to be as homogeneous as possible (with low levels of impurities such as aluminum, boron, and iron), experimental measurements suggest a wide variety of adsorption isotherm behaviors, depending on the adsorbate. For example, it was found that the isotherm data acquired by frontal analysis on a Kromasil Cis column eluted with an aqueous solution of methanol were best accounted for by a Jovanovid isotherm for aniline, a T6th isotherm for theophylline, a bi-Langmuir isotherm for phenol, caffeine and propranolol (with a buffered mobile phase at pH = 4.7), and an extended liquid-solid BET isotherm for ethylbenzene [32]. We consider here only those isotherms that have been used to account for nonlinear liquid-solid equilibrium data. These isotherm models can be divided into four main categories: (1) the isotherm models for homogeneous surfaces on which there are no significant adsorbate-adsorbate interactions; (2) the isotherm models for heterogeneous
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
81
surfaces on which there are no significant adsorbate-adsorbate interactions; (3) the isotherm models for homogeneous surfaces on which there are significant adsorbate-adsorbate interactions; and (4) the isotherm models for heterogeneous surfaces on which adsorbate-adsorbate interactions take place to a significant extent. Many more sophisticated models can be found in the literature [18,33]. Not all of them can be cited. The adsorption energy distribution may help in selecting the isotherm model that best accounts for a set of experimental data among isotherm models belonging to the first two categories. There is currently no method available to derive from these data the adsorption energy distribution (AED) of solutes on surfaces when their isotherms belong to the last two categories. Finally, the Langmuir isotherm is a two-parameter isotherm model. Accordingly, it is always possible to determine a Langmuir isotherm that is tangent to any complex isotherm model at the origin and has the same curvature around the origin (provided the model is convex upward). If the adsorption data are measured in a narrow concentration range, it will not be possible to distinguish between these different isotherms. To be meaningful, the more complex the isotherm model used, the larger the number of data points required, the more accurate they should be, and the wider the concentration range that should sample for the isotherm coefficients derived by a fitting of the experimental data to the isotherm model.
3.2.1 Isotherm Models for Ideal Adsorption on Homogeneous Surfaces There are only two important models in the first category, the Langmuir and the Jovanovic isotherms. There are no adsorbate-adsorbate interactions and the adsorbate are localized. 3.2.1.1 The Langmuir Isotherm in Liquid-Solid Equilibria The kinetic derivation of the Langmuir isotherm presented in the case of gas-solid equilibria (Section 3.1.3) can be readily extended to the liquid-solid equilibria. This isotherm can also be derived using a thermodynamic approach, in the case of a dilute solution [34]. Let us consider a two-component system, with a solvent (S) and a solute (A). The liquid-solid equilibrium can be written: ^sol + Sads ^ ^ads + Ssol
(3.41)
This equilibrium corresponds to the exchange of one molecule of solute and one molecule of solvent adsorbed on the surface. At equilibrium, we have ,,ads flsol
K=
a
A^s
af*
If we assume that both phases are ideal, we may replace the activities a? of component i in phase p by the corresponding concentration in Eq. 3.42, which can be
82
Single-Component Equilibrium Isotherms
rewritten [34] as X
A
K
01
=
( 3 43)
1 + {K-I)xs°l
x^
Consider now a solution containing HQ moles, with mole fractions xA and Xg of solute and solvent, respectively. After this solution has been equilibrated with a mass m of adsorbent, there are « ads and nso1 molecules (of either solute or solvent) in the adsorbed layer and in the solution, respectively. The amount of a component adsorbed by a solid adsorbent in a closed system corresponds to the change in the solution concentration, Ax A — x°A — x^. Since the solution contains at least two components, the solute and the solvent, the isotherm is a composite isotherm and, for a two-component system, the following relationship [23] exists between Ax A and the number of moles of component A and solvent adsorbed per unit mass of adsorbent, «^ds and n| d s , respectively:
= nfs (l-xA)-
"o ^
nf*xA
(3.44)
To include the experimental result (i.e., UQAXSA1 fm), Everett [34] has rearranged Eq. 3.43 into
In the case of a dilute solution, xfl w 1 and from Eq. 3.44, ^ ^ w nf3. Thus, Eq. 3.45 becomes n ads
nad
If the solute is strongly adsorbed (K 3> 1), Eq. 3.46 is equivalent to the Langmuir isotherm. The adsorption isotherm obtained is more conventionally written: «C
bqsC
(3 47)
* = TTbc = TTbc Tbc
"
where a and b are numerical coefficients. With the units generally chosen, the limit retention factor at infinite dilution is k'o = Fa (F = Vs/Vm = (1 — e)/e, phase ratio). The limit, qs = a/b, of the stationary phase concentration at high mobile phase concentration is called the saturation capacity of the stationary phase and can be related either to the unit volume of this phase (specific saturation capacity) or to the amount contained in the column (column saturation capacity). The ratio 9 = q/qs is called the fractional surface coverage. We show in Figure 3.4a the Langmuir isotherm in its reduced form: b C
8 = ±= q
l
^ = —
+ b C l
+r
(3.48) v
'
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
e
83 b
a 0 1 T
1-
„»»«-»
/b=5 , . |
—i.
"
**
*——-^^
0.5 J
0.5
""" b=0.05
( .
I
n
U ^
0
5
be
10
0
10
r-
/
i
2 0
Cmg/mL Q mole/L packing
0.03
c b=100 j .
a=13.1
Q moles/L packing
d
0.1 b=241 a=54
0.02 b=600 -~
0.01
0
a=32j4,.
0.05
fd
a=13.1 ^
r 0
u
•
0.0015
Cmote0^03
0
"""
0.0015
ai.4 0.003
C moles/L
Figure 3.4 Illustrations of the Langmuir isotherm, (a) Nondimensionalized form of the isotherm, (b) Effect of the curvature b at constant qs. (c) Effect of b and qs at constant initial slope, (d) Effect of a and qs at constant curvature b.
All Langmuir isotherms give the same curve if plotted in dimensionless units, i.e., if 9 is plotted versus F = bC, as shown in Figure 3.4a. We note that surface coverages of 0.50, 0.80, and 0.90 are achieved for values of bC equal to 1, 4, and 9, respectively. As we have shown, however, Eq. 3.13 and Eqs. 3.47 and 3.48 are not valid at high surface coverages. Thus, there is a fundamental contradiction. The Langmuir isotherm equation is used to fit sets of experimental data that are acquired under such conditions that the assumption of X^ 1 < 1 is incorrect, thus invalidating the physical meaning of the Langmuir model. Therefore, no physical interpretation of the values obtained for the saturation capacity should be attempted. To keep the model valid, Eq. 3.48 should be used preferably with values of 9 below 0.1. Although the fundamental assumptions made in deriving the Langmuir isotherm are usually not verified, it is very often observed in practice that experimental adsorption data are in good agreement with the Langmuir isotherm in a rather wide range of concentrations. As can be seen in Figures 3.4b, c, and d, adjustment of the parameters a and b allows the adjustment of the initial slope,
Single-Component Equilibrium Isotherms
84
Figure 3.5 Coverage of a uniform surface by hard spheres. The circles show the footprint of each adsorbed molecule; the area outside these circles is the excluded part of the surface. Reproduced with permission from X. Jin, N.-H.L. Wang, G. Tarjus and}. Talbot,}. Phys. Chem., 97 (1993) 4258 (Fig. 2b), ©1993 American Chemical Society.
initial curvature, and saturation capacity of the isotherm. The Langmuir model appears as the first-choice empirical equation to fit experimental results regarding the adsorption of single components. Finally, the Langmuir isotherm can be linearized in several ways. For example: C q
=
1 C ^- + bqs qs
(3.49)
1
=
-hq
(3.50)
bqs
Thus, plots oiC/q versus C or of q/C versus q (the latter known as the Scatchard plot [35]) should be linear. Although these plots are convenient for checking the validity of the model in a particular case, nonlinear regressions should be preferred for the determination of the best values of the parameters b and qs. Figures 3.4b to 3.4d illustrate the influence of the two parameters of the isotherm, b at constant qs (Figure 3.4b), b at constant a (Figure 3.4c), and a at constant b (Figure 3.4d), when the conventional expression of the isotherm is considered (Eq. 3.48). These figures illustrate how difficult it may be to achieve surface coverages permitting a reasonable accuracy in the values of b and qs. In many practical cases, the isotherm is used as a fitting function, and the values of qs derived in the fitting process have no real physical meaning. As a matter of fact there is a contradiction between the requirements that 8 should be lower than 0.1 to keep the model physically valid and that experimental measurements should be carried out at concentrations large enough to achieve values of 8 of the order of at least 0.5 to determine the parameter b with sufficient accuracy. Since the Langmuir isotherm model is widely applicable, it is normal to take the column saturation capacity as a unit to measure the degree of column overload achieved, by reporting the actual amount injected as a fraction of the column saturation capacity. The loading factor, Ljr, is defined for each component as the ratio of its amount in the sample to the column saturation capacity for that component: f =
n (1 - e)SLqs
=
nb eSLk'o
(3.51)
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
85
where n is the amount of the component injected, S and L are the column geometric cross-sectional area and length, respectively, and k'Q is the retention factor at infinite dilution (i.e., under linear conditions). As we have explained in the previous sections, the Langmuir model has been established on firm theoretical ground for gas-solid adsorption, a case where there is no competition between the adsorbate and the mobile gas phase. On the contrary, in liquid-solid adsorption, there is competition for adsorption between the molecules of any component and those of the solvent. Although we can choose a convention canceling the apparent effect of this competition on the isotherm [30,36], the conditions of validity of Eq. 3.47 are not met. These conditions are: (i) the solution is ideal; (ii) the solute gives monolayer coverage; (iii) the adsorption layer is ideal; (iv) there are no solute-solute interactions in the monolayer; (v) there are no solvent-solute interactions. These conditions cannot be valid in liquid-solid adsorption, especially at high concentrations. Finally, we should note that molecules cannot cover a surface completely. As a first approximation, the random site model [37] assumes that they can be represented by impenetrable spheres, adsorbing sequentially on a uniform surface, in locations selected randomly (see Figure 3.5). If the molecule adsorbing on a new trial position overlaps with a previously adsorbed molecule, it does not stick to the surface; otherwise, it is adsorbed. An initially fresh surface covers quickly, 1.5,
b 20
q mM
10
f/ V
TV 0
20
40
60
80
100
120
C mM t
20
\
1
40
Figure 3.6 Experimental isotherm data on ODS, fitted to a Langmuir model, (a) Hydrophilic organics. Mobile phase, 0.20 M Na + H2PO4" buffer, pH 6.3, for benzoic acids, water for other compounds; Fv = 70 /d/min, T = 25°C. 1, p-toluidine; 2, p-cresol; 3, o-toluidine; 4, m-cresol; 5, 2-amino-4-nitrophenol; 6, phenol; 7, wz-nitrobenzoic acid; 8, resorcinol; 9, hydroquinone; 10, benzoic acid, (b) Nucleosides. 1, Adenosine; 2, guanosine; 3, cytidine; 4, uridine. Spherisorb ODS-2, 25 mM Na + H^PO^, pH 7,25°C. Reproduced with permission from J.Jacobson,J.Frenz and Cs. Horvdth,}. Chromatogr, 316 (1984) 53 (Fig. 4a), and J. -X.Huang and Cs. Horvdth, ]. Chromatogr., 406 (1987) 275 (Fig. 5).
86
Single-Component Equilibrium Isotherms
cr.M A 0
» 0.001 » 001 * 01
/
2 CA,mM
Figure 3.7 Adsorption isotherms of Phenylalanine and Tyrosine on Amberlite 252. M. Sounders, }. Vierow, G. Carta, AICHE ]., 35 (1989) 53. (Figs. 5 and 6). Reproduced by permission of the American Institute of Chemical Engineers. ©1989 AIChE. All rights reserved.
but the rate of adsorption drops rapidly with increasing surface coverage and becomes negligible above a geometrical surface coverage of 55%, a value that is taken as the surface saturation capacity in classical adsorption measurements. The phenomenon becomes important when the adsorption sites are groups located at random on the surface (e.g., in affinity chromatography) [37]. Exclusion effects take place and only a fraction of the sites can be saturated. Figure 3.5 shows an example of a saturated configuration. This model provides some interesting clues to the behavior of phases used in affinity chromatography and to competitive adsorption (Chapter 4). Nevertheless, experience shows that the Langmuir isotherm equation (Eq. 3.47) is an excellent approximation for single-component adsorption equilibrium in LSC. Examples taken from the literature are given in Figures 3.6 to 3.9. Figure 3.6 [8,38] illustrates the adsorption isotherms measured for various small molecules on Ci8 bonded silica. Figure 3.6a shows the isotherms of various aromatic derivatives, Figure 3.6b those of nucleosides. Figure 3.7 shows the adsorption isotherms of two amino adds as a function of the chloride concentration in ion-exchange chromatography [39]. Since [Cl~] remains constant, the pH varies from one point to the next. The uptake of the amino acid is reduced by competition with H+ for the resin. When [Cl~] is small, the solution pH nears the isoelectric point of the amino acid, the resin capacity increases due to low competition with [H + ]. Normalization of the isotherm data at various chloride concentrations lead to a single line since the adsorption of the amino acid depends only on the ionic fraction of amino acid cations in solution. Thus, ion exchange is the primary mechanism and there is an insignificant amount of non-ionic retention mechanism. Figure 3.8 shows the isotherms of the enantiomers of 1-phenyl 1-propanol on a chiral phase. In this case, the isotherm data fitted well to the Langmuir model and the Scatchard plot is linear [40]. Figures 3.9a and 3.9b [41] illustrate the isotherms of lysozyme and a-chymotrypsinogen, respectively on a weak cation exchanger at different salt concentrations in the mobile phase. Higher salt concentrations result in decreased
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
87
Figure 3.8 Experimental adsorption data of the enantiomers of 1-phenyl-l-propanol (PP) on Chiracel OB from n-hexane/Ethyl-acetate, 95:5 (v/v). Left. Experimental data (symbols), Langmuir isotherms calculated with the best coefficients given by the regression of the experimental data to the Langmuir model (dashed lines), and Langmuir isotherms calculated with the coefficients derived from the regression of the combined competitive and single component data (solid lines). Right. Scatchard plots of these data. Reproduced with permission from S. Khattabi, D. E. Cherrak, f. Fischer, P. Jandera, G. Guiochon,}. Chromatogr, 877 (2000) 95 (Figs. 2 and 3).
adsorption. These results show excellent to reasonable fit of the experimental data to the Langmuir equation. The isotherms in bottom right subfigure of Figure 3.9 were obtained by batch measurements. The concentration of the supernatant was determined and the amount absorbed was calculated from the mass balance [42]. The equilibrium isotherm of a-Chymotrypsinogen was measured by a batch method on SP-Sepharose-FF, using a sodium phosphate buffer at pH = 6.5, with an ionic strength of 20mM (see Figure 3.10). The data fit well to a Langmuir isotherm. However, with the scatter observed in the data, a rectangular isotherm model would also be appropriate [43]. In most cases, significant differences are observed between experimental data and the best fit of these data to a Langmuir isotherm. Most deviations from Langmuir isotherm behavior arise from a failure of the system to meet one of the basic requirements of the Langmuir isotherm model, the surface is not homogeneous, there are significant adsorbate-adsorbate or adsorbate—solvent interactions. Then, the consideration of alternative approaches (see next few sections) is required. In a limited number of instances, serious disagreement between experimental data has been observed. For example, desorption hysteresis has been reported for /3lactoglobulin A on a weakly hydrophobic stationary phase. [44]. This leads to a murky situation, not easy to handle theoretically or empirically since there is no longer a reversible equilibrium between the two phases of the system and modeling of the separation becomes exceedingly difficult.
Single-Component Equilibrium Isotherms
200
150
too
Figure 3.9 Experimental isotherms of (Top left) Lysozyme in (1) 0.0 M (2) 0.05 M (3) 0.1 M NH+ SO^~ and 25 mM Na+ H 2 PO^. Best fit ( ) Jovanovic and (—) Langmuir model. WCX-300H DuPont, 25°C and (Top right) a-Chymotrypsinogen; 1, 0.0 M; 2, 0.05 M; 3,0.1 M in the same buffer, on SCX300. Reproduced with permission from J.-X. Huang and Cs. Horv&th, } . Chromatogr., 406 (1987) 285, (a) Fig. 4, (b) Fig. 3. Bottom right
Adsorption Isotherm of Fc-Fusion protein on a Protein A Type Column fit to the Langmuir Isotherm Model. Reproduced with permission from S. Ghose, D. Nagrath, B. Hubbard, C. Brooks, S. M. Cramer, Biotechnol. Progr, 20 (2004) 830 (Fig. 1). ©2004, American Chemical Society.
3.2.1.2
q, = 36 mg/ml b = 37.61 ml/mg O.f
0.5
0.3
0.1
0.5
0.6
07
3.8
CS
The Jovanovid Isotherm
This model was derived to account for the adsorption of a gas onto an homogeneous surface, with no adsorbate-adsorbate interactions [45]. It is similar to the 300
250 250
q (mg/mL)
200 200
/o
'
O
3
Figure 3.10 Single Component Isotherm of a-Chymotrypsinogen on SP-Sepharose-FF. G. Carta, A. Ubiera, AICHE ]., 49 (2003) 3066 (Fig. 5). Reproduced by permission of the American Institute of Chemical Engineers. ©2003 AIChE. All rights reserved.
1
150( 150
100 100
>
50 0A 0
0.5 0.5
11
(mg/mL) C (mg/mL)
1.5 1.5
2
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
89
Figure 3.11 Experimental isotherm (stars) and best Jovanovic isotherm (solid line) of aniline on a Kromasil-Cis column with methanol/water 45/55 (v/v) as the mobile phase. T = 295 K. Reproduced with permission from F. Gritti, G. Guiochon, J. Chromatogr. A, 1003 (2003) 43 (Fig. 5).
q* [g/L]
120
100
80
60
40
20
0 0
10
20
30
40
50
60
70
C [g/L] [g/L]
Langmuir model but it corrects for the error made in the derivation of this latter model when assuming that the adsorption and the desorption of the molecules of the solute considered are instantaneous. Since they are obviously not, the equilibrium compositions predicted by the two models are slightly different. The isotherm equation becomes o-bC
(3.52)
This equation has rarely been reported as accounting well for a set of experimental data. The adsorption of aniline on Kromasil Cig is a case in point [11,32], see Figure 3.11. Chromatographers seem to prefer the Langmuir model and rarely try to use this alternative. Accurate data acquired in a wide range of concentrations are needed to show that one of the two models is significantly better than the other one.
3.2.2 Isotherm Models for Ideal Adsorption on Heterogeneous Surfaces Actual surfaces are heterogeneous and the adsorption data on real adsorbents are usually not well accounted for by one of the two models discussed in the section above. The distribution of the adsorption energy (AED) on a homogeneous surface is a Dirac distribution. On an actual surface, the adsorption energy distribution has a finite width. It can be unimodal or multi-modal. The isotherm models discussed in this section correspond to such surfaces. Each of them can be related to a different AED. In all cases, however, there are no adsorbate-adsorbate interactions. In this group, the most useful models are the bi-Langmuir and multiLangmuir isotherms, the Toth isotherm, and the Freundlich isotherm. 3.2.2.1
The Bi-Langmuir Isotherm
In many cases the surface of the adsorbent used for chromatographic separations is not homogeneous. The simplest model for a nonhomogeneous surface is a mixed (patchwork) surface covered with patches made of two different homogeneous surfaces, i.e., covered with two different kinds of chemical groups.
90
Single-Component Equilibrium Isotherms
These two surfaces may give similar interactions with a given solute while exhibiting different interaction energies or they may give entirely different types of interactions. Chemically bonded Cig silica may provide examples of the former case [10,11], while many examples of the latter have been found with the chiral stationary phases (CSP) that have a moderate density of chiral ligands (e.g., immobilized proteins, Pirkle phases) [46-48]. Part of the surface of these CSPs is covered with chiral ligands, resulting in enantioselective interactions with the retained components and the selective retention of one enantiomer with respect to the other one [47]. In many cases, however, the density of nonselective sites is sufficient for their interactions with either enantiomers to contribute significantly to their retention [46]. This model has been successfully applied to many pairs of enantiomers on CSPs with a low or moderate density of enantioselective sites. It has been less successful for CSPs which have a high density of chiral sites ( e.g., for cellulose-based phases) on which the interactions with enantiomers are more complex (and do not necessarily involve the conventional 3-point chiral selective interaction mechanism but merely multiple, simultaneous chiral molecular interactions) [49]. With such a surface, covered with two different kinds of sites that behave independently (and on each of which the two basic principles of the Langmuir model, local adsorption and lack of adsorbate-adsorbate interactions, apply), the equilibrium isotherm results from the addition of the two independent contributions of the two types of sites [35]. Since in most cases the Langmuir isotherm is appropriate to account for single-component adsorption on a homogeneous surface (in practice if not in principle, see earlier), we have the following isotherm (see example in Figure 3.12.) l + b2C
(3-53)
where the subscript refers to the type of adsorption sites. The bi-Langmuir isotherm was first suggested by Graham [50] to account for adsorption behavior on certain inhomogeneous surfaces. It has been used successfully by Laub [51] in gas-solid chromatography. Cases in which the use of a bi-Langmuir isotherm has been found necessary to account for adsorption data include the enantiomers of mandelic acid, of the N-benzoyl derivatives of alanine (Figure 3.12) and phenylalanine, and the JV-benzoyl derivative of glycine adsorbed on bovine serum albumin (BSA) immobilized on silica [20,52], and many other pairs of enantiomers [46,47]. In Figure 3.12b [20], we see that the experimental data are too strongly curved to fit the Langmuir model (the ratio q/C should decrease linearly with increasing q, as depicted in the inset; see Eq. 3.50). In Figure 3.12a [20], the experimental data and the fitted model are plotted in the conventional format, q versus C [52]. By contrast, the bi-Langmuir isotherm model accounts very well for the experimental data. Furthermore, the retention mechanism justifies the use of this model. The physical interpretation of these data is as follows. With immobilized BSA, there are two types of sites, enantioselective (carried by the BSA molecules) and nonselective (the nonselective interactions between,
3.2 Models of Adsorption Isotherms in Liquid—Solid Equilibria
CM"
Q a1
91
0
* 2
• \
0
V 1
^ ^ «J
n0.0
0.2
0.4
0.6
0.00
0.02
0.04
b 0.08
Figure 3.12 Isotherms of two pairs of amino acid enantiomers on immobilized BSA. Left: 1 N-benzoyl-L-alanine; 2 N-benzoyl-D-alanine. Experimental data fitted to the bi-Langmuir model. Mobile phase, 10 mM phosphate buffer in water with 3% propanol-1. Right: Scatchard plot showing the best Langmuir (inset) and bi-Langmuir (main figure) fit of the experimental data for 1, N-benzoyl-L-phenylalanine; 2 N-benzoyl-D-phenylalanine. Mobile phase, 100 mM phosphate buffer in water with 7% propanol-1. Reproduced with permission from (Left) S. Jacobson, S. Golshan-Shirazi and G. Guiochon,}. Amer. Chem. Soc., 112 (1990) 6492 (Fig. 2), ©1990 American Chemical Society and (Right) S. Jacobson, S. Golshan-Shirazi and G. Guiochon, AIChE J. 37 (1991) 836 (Fig. 1) ©1991 AIChE. All rights reserved.
e.g., the benzoyl group and the BSA molecule or between the amino acid derivatives and the silica surface), respectively. The chiral selective sites (one per molecule as shown by reporting the saturation capacity in number of moles of enantiomer per mole of immobilized BSA [53]) are hydrophobic pouches in the protein molecule. The saturation capacity for the chiral-selective type of sites is nearly 40 times smaller than the saturation capacity for the nonselective sites. The adsorption enthalpy on the chiral-selective sites is nearly 4 kcal/mol larger than on the nonselective sites [54]. Similar adsorption data have also been reported for the adsorption isotherms of many compounds in various systems. For example, the adsorption data of several j3-blockers, particularly those of propranolol acquired in a 1 to 7000 relative concentration range, on an immobilized cellulase, Cel 7A, fit very well to the biLangmuir model, as illustrated in Figure 3.13 [47,55]). The enantioselective site was identified as a pair of amino acid residues in the tunnel formed by the main chain of the protein. The parameters of the isotherm depend on the pH as illustrated in Figure 3.14. A bi-Langmuir model was also found to account well for the separation of pairs of enantiomers on polymers molecularly imprinted with one of the enantiomers [56]. Note, however, that there are also many systems in which the adsorption isotherms of enantiomers are not accounted for by a biLangmuir model showing that enantioselectivity is often achieved by a complex
Single-Component Equilibrium Isotherms
92
C(M)
C(M)
Figure 3.13 Single-component equilibrium isotherms for (R)- and (S)- Propranolol on immobilized Cel-7A at increasing pH. Symbols: experimental data, o R- and * S-enantiomers. Lines: best bi-Langmuir isotherms (dashed for R-, solid for S-enantiomer). Left Low concentration data (in pM). pH-values: (1) 4.72, (2) 4.98, (3) 5.21, (4) 5.49, (5) 5.70, and (6) 5.93. Right High concentration data (in mM). Reproduced with permission from T. Fornstedt, G. Gotmar, M. Andersson, G. Guiochon, J. Am. Chem. Soc, 121 (1999) 1164 (Figs. 6a, 6c). ©1999, American Chemical Society.
mechanism [57,58]. The adsorption of two peptides, bradykinin and kallidin, on a Qg bonded silica [48] and that of chicken albumin on a weak anion exchange resin [59,60] follow also bi-Langmuir isotherm behavior. This is illustrated in Figure 3.15 [59]. In Figures 3.15a and 3.15b, the effects on the adsorption isotherm of the mobile phase flow rate and ionic strength, respectively, are illustrated. For small molecules, the flow rate at which adsorption isotherms are measured by frontal analysis has no influence on the amount adsorbed [38]. The result obtained in Figure 3.15a suggests that true thermodynamic equilibrium has not been reached at flow rates larger than 25 fiL/mia and that kinetic effects are present at the highest flow rate used. The adsorption of caffeine and phenol on many Cis-bonded silica stationary phases also follows bi-Langmuir adsorption isotherm behavior [61]. Quifiones et al. measured the adsorption isotherms of three basic drugs, the anxiolytic buspirone hydrochloride, the antidepressant doxepin hydrochloride, and the Ca 2+ blocker diltiazem hydrochloride. The adsorption data fitted well to the bi-Langmuir model (see Figure 3.16). This result suggests that adsorption takes place on two different types of sites and that there are two different retention mechanisms. Hydrophobic interactions take place on the low-energy sites (alkyl chains) and ion-exchange interactions on the high-energy sites (probably acidic silanols buried under the bonded alkyl layer). These results confirm other findings regarding RPLC separations of basic solutes. Further work is needed to ascertain a conclusion that is still tentative. Finally, it should be emphasized that the successful use of a bi-Langmuir isotherm model (as of any other combination of models that multiplies the number of model parameters) to account for a set of experimental adsorption data requires that these data are acquired in a wide concentration range [55]. This is even more
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria 40
•
70-
93
b SJI exp 1 b s I, fit
0 60-
b HJ1 exp
y
2 b Rjll Jrl +
30
b, exp 3 b, fit
50-
£ 40-
201 a
- 0.6 E •="30-
20-
.. _
,-.+•—- -
-*"
•y'~s)
10
"'"' +
*
"""• "+' Q
o
'
O
10-
/
0
04.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
PH
Figure 3.14 Equilibrium isotherms for (R)- and (S)- Propranolol on Cel-7A at increasing pH, see data in Figure 3.13. Plots of the saturation capacity (Left) of the three retention mechanisms and of their binding constants (Right) versus the pH of the mobile phase. Enantioselective interactions of S-propranolol (1) and of R-propranolol (2), and nonselective interactions of either enantiomers (3). NB. In both figures, the left y-axis corresponds to lines 1 and 2; the right y-axis to line 3. Reproduced with permission from T. Fornstedt, G. Gotmar, M. Andersson, G. Guiochon,}. Am. Chem. Soc, 121 (1999) 1164 (Figs. 7 and 8). ©1999, American Chemical Society.
important if a multi-Langmuir isotherm model is to be used. The nonlinear regression of adsorption data to a multi-Langmuir model would be an indeterminate or ill-posed problem if the concentration range is too narrow. It seems that the ratio of the larger to the smaller concentration for which the data had been acquired should be of the order of at least 1000 for the successful fit of the data to a bi-Langmuir model and at least 10000 in the case of the tri-Langmuir isotherm [46,62,63]. 3.2.2.2
The Toth Isotherm
Originally derived for the study of gas-solid equilibria, the Toth isotherm [64] accounts for adsorption on a heterogeneous surface, with no adsorbate-adsorbate interactions. It has three parameters. The heterogeneous surface has a unimodal adsorption energy distribution with a width related to the value of the parameter t. Like the Langmuir isotherm, it can be extended to the case of liquid-solid equilibrium. Its equation is
1 Is
(3.54)
Single-Component Equilibrium Isotherms
94 Q mg/mL packing
40 T
20
0* 4
C mg/mL
8
Figure 3.15 Experimental isotherms of chicken albumin. Stationary phase: TSK-DEAE5PW anion exchanger, (a) Effect of the mobile phase flow rate (50 mM Tris-acetate buffer at pH 8.6). Fv (fd/min): 1, 25; 2, 50; 3, 75. (b) Effect of ionic strength of the mobile phase. Concentration of added sodium acetate buffer: 1, 0 mM; 2, 25 mM; 3, 50 mM. Reproduced with permission from J.-X. Huang, ]. Schudel and G. Guiochon, } . Chromatogr., 504 (1990) 335 (Figs. 3 and 5).
20
Figure 3.16 Left Adsorption data of buspirone (o), doxepin (A) and diltiazem (•). Mobile phase is ACN:Buffer = 35:65; Buffer is 0.1 M phosphate, pH = 3.0, T = 25 CC. Right Scatchard plot of the adsorption data of buspirone (o), doxepin (A) and diltiazem (•). Reproduced with permission from I. Quinones, A. Cavazzini, G. Guiochon,}. Chromatogr. A, 877 (2000) 1 (Figs. 4 and 5).
This isotherm is similar to the Langmuir model, to which it becomes identical for t = 1. The parameters b and t permit independent adjustment of the initial slope and curvature of the isotherm. This model has been used successfully to account for experimental isotherm data regarding gas-solid adsorption [33]. It was used to account for the adsorption behavior of theophylline on a Kromasil Cis column eluted with an aqueous solution of methanol [11]. It is frequently used to account
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria Figure 3.17 Experimental adsorption data (symbols) and best Toth isothem (solid lines) for o porcine insulin, • , human insulin, A Lispro. Insert: isotherms at low concentrations. Reproduced with permission from X. Liu, K. Kaczmarski, A. Cavazzini, P. Szabelski, D. Zhou, ]. Van Horn, G. Guiochon, Biotechnol. Progr., 18 (2002) 796 (Fig 4). ©2002, American Chemical Society.
95
12108
Icr 6 4 2 0 0
36 Exp_R-indanol Exp_S-indnaol Best Toth Best Langmuir Best Bilangmuir
30
8
18 6
12
Q ( g /L )
tration data. Reproduced with permission from D. Zhou, D. E. Cherrak, A. Cavazzini, K. Kaczmarski and G. Guiochon. Chem. Eng. Sci, 58 (2003) 3257. (Fig. 2).
24
Q (g/L)
Figure 3.18 Single Component experimental isotherms for R- and S-1-indanol on cellulose tribenzoate fitted to three different models. The inset shows the low concen-
4
6
2
0
0 0
1
2
3
Concentration (g/L)
-6 0
4
8
12
16
20
24
Concentration (g/L)
for adsorption on heterogeneous surfaces. It has also been used for this purpose in connection with the determination of the adsorption energy distribution. Liu et al. showed that, when fitted to their experimental adsorption data, the T6th isotherm provided the highest value of the Fisher coefficient and thus modeled best these data (see Figure 3.17) [65]. In this case, the best value of the exponent of the Toth isotherm was markedly lower than unity, with values of 0.702, 0.514, and 0.534 for Lispro, human, and porcine insulin, respectively. These values indicate that the surface is more heterogeneous toward human and porcine insulin than toward Lispro. They also suggest that there could be more than one type of interaction sites where insulin can bind to the stationary phase. For Lispro, the B28 Lys and B29 Pro amino acid residues are in a different position compared to human insulin. This causes a significant, albeit small, decrease of the interactions of this protein with the adsorption sites and of its binding capacity, hence a lower saturation capacity and a higher homogeneity index than those for human and porcine insulin. Single component isotherms of the enantiomers of 1-indanol were measured on cellulose tribenzoate, using a solution of 2-propanol in n-hexane [66]. The FA data are shown in Figure 3.18 (symbols), with the results of their fit to the Toth (solid line), the Langmuir (dashed line), and the bi-Langmuir (dotted line) isotherm models. The Toth and the bi-Langmuir models gave curves that are essentially indistinguishable. The Langmuir model isotherm is different. Evaluation of the fit of the data made by application of Fisher coefficients indicates that both the
Single-Component Equilibrium Isotherms
96
T6th and the bi-Langmuir models give excellent fits; however, the bi-Langmuir isotherm is quantitatively better. The main figure illustrates all the data and the insert shows only the data at low concentrations. 3.2.2.3 The Unilan Isotherm The Unilan adsorption isotherm was also originally a gas-solid isotherm [67] and has been widely used to account for experimental results in this area [33]. It is based on a model of heterogeneous surfaces assuming a uniform distribution of adsorption energy. It can be extended to liquid-solid equilibria and used as an empirical model: „
1,
1 + bCe5
(3.55)
The Unilan isotherm tends toward a Langmuir isotherm when s tends toward 0. 3.2.2.4 The Freundlich Isotherm Boedeker [68] proposed the following empirical isotherm equation for the adsorption of polar compounds on polar adsorbents: q = aC l/n
(3.56)
where the exponent l / n is smaller than unity (it was 0.5 in Boedeker's results). This isotherm is illustrated in Figure 3.19. It is known as the Freundlich isotherm because of the great importance given to it by Freundlich [69] who popularized it. It accounts, for example, for the adsorption of acetic acid on activated charcoal, a surface that is certainly not highly homogeneous. More generally, the Freundlich isotherm accounts for the adsorption of strongly polar compounds on polar or strongly polar adsorbents (especially those that have an inhomogeneous surface) in low- or medium-polarity solvents. It seems to account properly for the adsorption of at least certain proteins on ion exchangers. Two serious difficulties are encountered with the Freundlich isotherm. First, from a fundamental point of view, this isotherm violates the Gibbs isotherm equation and it is not thermodynamically consistent. From a more practical point of O mg solute/mL packing
n=1
Figure 3.19 Illustration of the Freundlich isotherm, a = 13.1. Values of n: 1; 1.1; 1.25; 1.5; 2. Inset, n = 1 and 5.
20
30 Cmg/mL
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria Figure 3.20 Experimental data for dyes fitted to a Freundlich isotherm. Parameters: Basic Yellow 21, a = 67, 1/ n = 0.233; Basic Blue 3, a = 79.5,1/ n = 0.231; Basic Red 22, a = 46.7,1/ n
97
Q mg dye/g peat 400 -
= 0.246. Data from B. Al-Duri, Y. Khader and G. McKay, ]. Chem. Biotechnol., 53 (1992) 345. ©Society of Chemical Industry. Reproduced with -permission. Permission is granted by John Wiley & Sons Ltd on behalf of the SCI.
600 800 C mg dye/dmS
Figure 3.21 Adsorption isotherm of benzoic acid in water on Dowex Optipore L-493. Reproduced with permission from M. Breitbach, D. Bathen, H. Schmidt-Traub, Ind. Eng. Chem. Res. 42 (2003) 5635 (Fig. 2). ©2003, American Chemical Society.
view, the limit at low concentrations of the ratio q/C is infinite because the isotherm is tangent to the vertical axis. This means that the retention time under analytical conditions is infinite. Under nonlinear conditions, the band profile may have a sharp front, eluted at a finite time, but its tail is asymptotic to the time axis and it never ends. This property of the Freundlich isotherm has an important consequence. Phase systems in which some compounds follow Freundlich adsorption behavior are nearly impossible to use for the purification or for the extraction of later-eluting fractions. In addition, a column regeneration step is required before beginning a new run. The use of adsorbents under experimental conditions where the isotherm follows the Freundlich model should be avoided in chromatography, whether analytical or preparative. The Freundlich isotherm is shown in Figure 3.19 for several values of n between 1 and 5. For n = 1, a linear isotherm is obtained. The inset illustrates the initial tangent of the isotherm. In Figure 3.20, experimental data are fitted to the Freundlich model for three different dyes [70]. The isotherms are well accounted for by the Freundlich model, with values of n of the order of 4. The extremely strong adsorption at low concentrations is clearly visible.
98
Single-Component Equilibrium Isotherms
3.2.2.5 The Langmuir-Freundlich Isotherm This empirical isotherm is a combination of two classical models, the Langmuir and the Freundlich models discussed earlier (3.57)
At low concentrations, this model reduces to the Freundlich isotherm. Because p < 1, this isotherm is tangent to the vertical axis, its initial slope is infinite and it is impossible to elute all the amount of sample injected out of the column in a finite time. This is not an attractive behavior for a chromatographic packing material. This model has been used in simple studies of the adsorption behavior on heterogeneous surfaces [71]. In this application, it has the major drawback of imposing a unimodal adsorption energy distribution that does not necessarily reflect the actual properties of the heterogeneous adsorbent surface studied [72]. The isotherm of Radke-Prausnitz is somewhat similar, with q = KC/ [1 + (KC)?]. It has been used essentially in gas-solid adsorption. The adsorption isotherms of benzoic acid from an aqueous solution were measured on a polymeric Dowex resin (a styrene-divinyl benzene copolymer, surface area 1100 m 2 /g, average particle size 630-800 ]im), using FA between 25 and 43°C [73]. The data were fitted to the Freundlich and the Redlich-Petersen isotherm models. The temperature dependant Freundlich isotherm model fits correctly the data only at high concentrations (see Eq. 3.57 with a = koexp(A/T), A numerical parameter). An improved fit of the data in the whole temperature range was obtained with the Rydlich-Peterson isotherm model (see Figure 3.21) (3.58)
3.2.3 Isotherm Models for Nonideal Adsorption on a Homogeneous Surface Besides the heterogeneity of the adsorbent surface, the second major reason for the adsorption of a compound to deviate from Langmuir isotherm behavior is that the adsorbed molecules interact. In this category, we find the Fowler isotherm, the anti-Langmuirian isotherm, and several S-shaped isotherm models, including the quadratic isotherm, the extended BET isotherm models, and the Moreau model. 3.2.3.1 The Fowler Isotherm This isotherm model was designed by Fowler and Guggenheim [12] to correct for the first-order deviations from the Langmuir isotherm. It assumes ideal adsorption on a set of localized sites on a homogeneous surface, with weak interactions between molecules adsorbed on neighboring sites. It assumes also that the interaction energy between two sorbate molecules is small enough that the random
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria Q mg/mL packing
99
dQ/dC
0.6
C mg/mL
C mg/mL
Figure 3.22 Illustration of the Fowler isotherm, x = 0,1, and 2; fe = 1.61 mL/mg; (js = 300 mg/mL. (a) Fowler adsorption isotherm, (b) First derivative of the Fowler isotherm.
character of the sorbate molecule distribution on the adsorbent surface is not significantly altered. Under these assumptions, the following gas-solid isotherm is obtained: bP =
1-
2w6 -CRT
(3.59)
where 9 is the degree of surface coverage and 2w is the interaction energy between the molecules in a pair of neighbors. For liquid-solid equilibria, the Fowler isotherm is extended empirically and written: (3.60) 1-9 where x is an empirical interaction energy parameter. The Fowler isotherm has the big disadvantage of being an implicit isotherm. Like the equation of the Volmer isotherm (Eq. 3.17), Eq. 3.60 cannot be inverted in closed form, which makes it difficult to use in the calculation of band profiles in combination with the mass balance equation. Figure 3.22 illustrates the Fowler isotherm. For x = 0, we obtain the Langmuir isotherm (Eq. 3.48). For % - 1, there is an inflection located at the origin, hence the isotherm appears to be linear in an unusually wide concentration range (Figure 3.22a). For x between 1 and 4, the isotherm has an inflection point at a finite concentration. The inflection tangent becomes vertical for x - 4, and for higher values the Fowler isotherm equation loses physical meaning. Figure 3.22b illustrates the location of the inflection points in a plot of the slope of the isotherm versus the concentration. It has been reported [74] that the adsorption data for 2phenylethanol and 3-phenylpropanol on ODS silica, from methanol-water (50:50) solutions are better fitted to the Fowler than to the Langmuir isotherm. In Figure 3.23 [74], the experimental data (symbols) are compared to the best Fowler isotherm.
100
Single-Component Equilibrium Isotherms a mg/mL packing
50 Phonylpropano
Figure 3.23 Use of the Fowler isotherm to account for experimental data. The adsorption isotherms of 2-phenylethanol and 3-phenylpropanol between spherical ODS silica (Vydac, Hesperia, CA), 10 Jim, and a methanol/water 50:50 solution. Column dimensions, 2.1 x 250 mm. Data from }. Zhu,
40-
Fhenylethancl
A.M. Katti and G. Guiochon, ]. Chromatogr., 552 (1991) 71.
12
16
C mg/mL
3.2.3.2 S-Shaped Isotherms and the Quadratic Isotherm Model These isotherms are sometimes referred to as anti-Langmuir isotherms because their initial curvature is convex down. The true anti-Langmuir isotherm, however, would have a vertical asymptote for some finite value of the mobile phase concentration, which has no physical meaning in liquid-solid equilibria. The S-shaped isotherms belong to class V (Figure 3.1) and are often called S-shaped isotherms. They are frequently observed in gas-solid chromatography [4]. The curvature of these isotherms at the origin and at low concentrations is concave upward, which indicates that the amount adsorbed at equilibrium increases more rapidly than the concentration in the mobile phase. Such an effect results usually from strong adsorbate-adsorbate interactions, e.g., lateral interactions between hydrocarbon chains [75], stacking of nucleotides or of large, planar, polycyclic aromatic compounds. A few systems of this type have been reported. Isotherms of class IV have been observed by Huang and Horvath [8] for several nucleotides (Figure 3.24a) and for the protein a-MSH (Figure 3.24b) [41]. Changing the mobile phase composition from aqueous methanol to a more complex aqueous organic solution turns the S-shaped isotherm of a-MSH to a convex downwards isotherm. The same change in mobile phase composition reduces the saturation capacity of the displacer (BDDAB) by almost 50%. This illustrates the importance of the mobile phase composition on the isotherm shape and the saturation capacity (this is not particular to the S-shaped isotherm, see Section 3.4). S-shaped isotherms have been reported recently for several alkylbenzenes on particle and monolithic CJSbonded silica columns [9,10]. A Langmuir-type isotherm equation with a negative value of b could account qualitatively for the shape of convex downwards isotherms at low concentrations. This equation, however, accounts for an isotherm of class III and has a vertical asymptote of equation C = 1/b, whereas in liquid-solid equilibria the isotherm should rather exhibit a horizontal asymptote and a finite saturation capacity. The quadratic isotherm, with a proper set of numerical coefficients, can account for isotherms having up to three inflection points (Figure 3.1, isotherm of class IV). Simple statistical thermodynamic models of adsorption [14] suggest that the equilibrium isotherm should be written as the ratio of two polynomials of the
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
101
400
100
300 -
200
50
100
150
20
30
Figure 3.24 Experimental adsorption isotherms of nucleotides on ODS silica, (a) Nucleotides on ODS. 1, Guanosine 2'3'-cyclic monophosphate; 2, adenosine 2'3'-cyclic monophosphate; 3, adenosine monophosphate. Spherisorb ODS-2 100 mM phosphate buffer pH 7, 25° C. (b) 1, a-MSH; 2, benzyl-dimethyl-dodecyl ammonium bromide (BDDAB) on Spherisorb ODS-2,25°C, eluted with 15% v/v aqueous methanol (—); or with 21% aqueous acetonitrile, 0.2% formic acid and 0.4% triethylamine, pH 7 (—). Reproduced with permission from from J.-X. Huang and Cs. Horvath, J. Chromatogr., 406 (1987) 275, (a) Fig. 6, (b) Fig. 7.
same degree1 2b2C2
nbnCn
(3.61) qs The Langmuir isotherm is the first-order isotherm predicted by statistical thermodynamics. The second-order isotherm, obtained with n = 2 in Eq. 3.61, is called the quadratic isotherm [76]. We see in Eq. 3.61 that the limit of 6 for C infinite is nqs. This results from the model, which considers that each site on the surface can accommodate n molecules and that there are qs such sites on the surface. The bi-Langmuir isotherm [50-53] is a particular case of the second-order Langmuir or quadratic isotherm. It is characterized by the fact that, being the sum of two convex-upward isotherms, it must also be convex upward. Furthermore, the polynomial in the denominator of the bi-Langmuir isotherm, (1 + b\C) (1 + b%C), has two real, negative roots. In the general case of the quadratic isotherm, however, the denominator roots can be positive or imaginary as well. In the former case (a positive root), the isotherm has a vertical asymptote corresponding to that root. This isotherm is of class II or III and is not suitable for liquid-solid adsorption, a case in which vertical asymptotes have never yet been encountered. In the latter case (two imaginary roots), the quadratic isotherm is not equivalent to : Note that Eq. 3.61 constitutes a Pade approximation, known for being able to mimic almost any mathematical function. Thus, the physical meaning of the coefficients derived from a fit of experimental data to this equation is doubtful, unless few coefficients are used and many data points recorded.
Single-Component Equilibrium Isotherms
102
Figure 3.25 Experimental isotherms of Troger's base enantiomers on microcrystalline cellulose triacetate. Experimental data by frontal analysis (symbols) and best quadratic isotherm (solid line). Experimental conditions: column length, 25 cm; column efficiency, N 106 plates; phase ratio, F = 0.515; flow velocity 0.076 cm/s, pure ethanol. Column (250x4.6 mm) packed with cellulose microcrystalline triacetate (CTA, 15-25f<m), previously boiled in ethanol for 30 min. (a) Isotherm data. Top line, (+)-TB, bottom line, (-)-TB. (b) Plot of q/C versus C. Reproduced with permission from A. Seidel-Morgenstern and G. Guiochon, Chem. Eng. Sci.r 48 (1993) 2787 (Figs. 4 and 5).
a bi-Langmuir isotherm; it does not even have to be convex upward in the whole concentration range. The first differential of the quadratic isotherm is 1 dq _ b\ q~sdC =
(3.62)
(1 + hC + b2C2)2
and its second differential is 1 d2q _ 2(2b2-b2)-
- \2b\C2 -
(3.63)
The numerator of the second differential is a third-degree polynomial which may have 1 or 3 real roots and, thus may have 0,1, 2, or 3 positive roots and as many inflection points. The quadratic isotherm is the first equation to which the experimental data should be fitted when an inflection point is observed. A quadratic isotherm has been used by Guiochon et al. {77] to calculate the band profiles obtained in the case of an S-shaped equilibrium isotherm. The same isotherm has been used by Svoboda [78]. An example of an isotherm with one inflection point, accounted for by the quadratic model is shown in Figure 3.25 [79]. It corresponds to the adsorption of the (+) isomer of Troger's base on microcrystalline cellulose triacetate, while the (-) isomer follows a Langmuir behavior in
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
103
?%
-
•
8to.
produced with permission from M. Diack and G. Guiochon, Anal. Chem., 63 (1991) 2608 (Fig. 2), ©1991 American Chemical Society.
q/c
1
9-
'
5
SQ-
•f
S6.60
Figure 3.26 Isotherm of phenyldodecane on porous graphitized carbon. Column 150 x 4.6 mm; mobile phase: acetonitrile, 1 mL/min. Inset: plot of q/C versus C. Symbols, experimental data; —> model n't. Re-
7.10
20
/
/ 0
c.(mM) 2
4
6
8
10
12
M
c(mM)
the same concentration range. The inflection point is barely marked on the conventional plot (Figure 3.25a), but is obvious on the plot oiq/C versus C (Figure 3.25b). In Figure 3.26 [75], we show an isotherm with two inflection points. It corresponds to the adsorption of «-dodecylbenzene on graphitized carbon black [75,80]. The surface of this adsorbent has micropores. The adsorption of the solute in these pores is well accounted for by a Langmuir term which has a steep slope and a very low saturation capacity. When adsorbed on the flat surface of the graphite, the long alkyl chains interact, and the adsorption energy of a second molecule is higher than that of the first one. This phenomenon is accounted for by a quadratic isotherm [75]. The combination of these mechanisms explains the features of the isotherm (Figure 3.26) and of its derivative (inset). 3.2.3.3
The Moreau Isotherm
The so-called Moreau isotherm is the simplest model for a homogeneous adsorbent surface with lateral, i.e., adsorbate-adsorbate, interactions [81]. The equation is ib2c2 'l + 2bC + Ib2C2
(3.64)
where qs, b, and I are the monolayer saturation capacity, the equilibrium constant at infinite dilution, and the adsorbate-adsorbate interaction parameters, respectively. This equation had already been derived by Langmuir and reported in a rarely cited paper [82]. The equilibrium constant at infinite dilution b is associ-
104
Single-Component Equilibrium Isotherms
ated with the adsorption energy ea through the following equation b = bQe«r
(3.65)
where ea is the energy of adsorption, R is the universal gas constant, T is the absolute temperature and bo is a pre-exponential factor that could be derived from the molecular partition functions in the bulk and the adsorbed phases, bo is often considered to be independent of the adsorption energy ea [83]. The adsorbate-adsorbate parameter / can be written as [81]
1 exp
(3 66)
= (W)
-
where BAA is the interaction energy (by convention €AA > 0) between two neighbor molecules of adsorbate A. This model was found to account well for the adsorption data of propranolol from a buffer or salt solution onto a Cig-bonded silica surface [84]. The results obtained, however, suggested that the surface was not homogeneous and would be better modeled by assuming that it consists in patches of two different types of sites (see later, Section 3.2.4.2). 3.2.3.4 The Extended Liquid-Solid BET Isotherm The extended liquid-solid BET isotherm describes well the adsorption behavior corresponding to types II or III isotherms of the van der Waals classification of isotherms (see Figure 3.1). Its expression parallels that of the BET isotherm model which is often applied in gas-solid equilibria [3]. It assumes the same molecular description: the solute molecules can adsorb from the solution onto either the bare surface of the adsorbent or a layer of solute already adsorbed. The equation of the model is derived from kinetic adsorption-desorption relationships, assuming first order kinetics [10,85]. The expression obtained after a rather lengthy derivation is
c + ic)
{3 67)
-
where bs and b\ are the equilibrium constants of adsorption of the compound on the bare surface and on a layer of adsorbate previously adsorbed, respectively. If ii/C « 1, the system can be described by a simple adsorbed monolayer, that is, by the classical convex upward Langmuir isotherm model (see Section 3.2.1.1 and Eq. 3.47). When the strength of the interactions between two adsorbate molecules increases (i.e., when b\ becomes large), a reversal in the direction of the isotherm curvature takes place. This means that significant adsorbate-adsorbate interactions cause the isotherm to be of type II (unless these interactions are strong even at low concentrations, in which case it is of type III). When the amount adsorbed increases further, the effect of the finite surface area of the adsorbent may be felt, the isotherm tends toward saturation and it is convex upward at high concentrations. However, if the adsorbate-adsorbate interactions are strong, the isotherm may be strictly convex downward, like the isotherms of type III. This takes place
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
105
16
300
A
14
180
q/C
q [g/L]
B 240
120
12 60
10
0 0
6
C[g/L] C [g/L]
12
18
0
6
[g/L] C [g/L]
12
18
Figure 3.27 Adsorption data of butylbenzene measured by FA on Cis-Chromolith, with methanokwater, 80/20 (v/v) at 296 K. A) Plot of the isotherm data and best extended liquid-solid BET isotherm. B) Plot of the experimental isotherm chord. Note the strictly anti-langmuirian behavior (fc; > bs/2). when the second order term of the isotherm at zero concentration is positive, i.e., if: d2q (3.68) >o Isotherms of types II and III have a vertical asymptote. The reason is clear in gassolid equilibria, capillary condensation takes place in the smallest pores. This phenomenon, however, has no equivalent in liquid-solid equilibria. Anti-Langmuirian isotherms are characterized by an increase of the adsorbate concentration that is faster than that of the liquid phase concentration. This phenomenon is not unusual in liquid chromatography. However, its extent is limited because, by contrast with what is observed in gas-solid equilibria, experimental isotherms in liquid-solid equilibria do not exist beyond the solubility of the solute in the mobile phase. This solubility is often low or moderate for mildly polar (e.g., light alkyl benzoates) or apolar (e.g., alkylbenzenes) compounds under RPLC (reversed-phase liquid chromatography) conditions. For instance, the adsorption equilibrium data of butylbenzene on Chromolith-Cis fit best to a type III isotherm [9]. This is confirmed by (1) the plot of the slope of the isotherm chord versus the mobile phase concentration (see Scatchard plots, Figure 3.27 [86], and Eq. 3.50). This plot increases constantly with increasing concentration in the mobile phase (in contrast with what it does under Langmuir adsorption behavior), and (2) the best values of the BET isotherm parameters (qs = 182 g/L, bs = 0.0584 L/g and bx = 0.03084 L/g), with 7b\ > bs (see Figure 3.27). The adsorption isotherms of slightly more polar compounds (e.g., butyl- or propyl-benzoate on the Chromolith-Cis, Figure 3.28) have an inflection point at some intermediate concentration [10,32]. These equilibrium isotherms are S-shaped isotherms of type II. At low concentrations, the slope of the isotherm chord decreases with increasing concentration, it then goes through a minimum and increases at higher concentrations. The best parameters are consistent with bx
Single-Component Equilibrium Isotherms
106
25.0
240
B A
24.524.5
J
I •
*
O1
24.0 24.0-
80
* *
q*/C
q [g/L]
160 160-
* *
* *
*
*****
*
23.523.5
0 0
3
[g/L] C [g/L]
6
9
0
3
6
9
C [g/L] [g/L]
Figure 3.28 Adsorption data of butylbenzoate measured by FA on Cig-Chromolith, with methanol:water, 65/35 (v/v) at 296 K. A) Plot of the isotherm data and best extended liquid-solid BET isotherm.. B) Plot of the experimental isotherm chord. Note the S-shaped isotherm behavior (fc; < bs/2).
3.2.3.5 Adsorption of Ions There is considerable discussion in the literature regarding the adsorption mechanism of ions from aqueous solutions onto RPLC stationary phases [87-90]. It has been shown that, under certain conditions, organic ions are adsorbed as ion pairs [87,89,91], and that, under other conditions, they may be adsorbed as separate ions. In this case, the model derived by Stahlberg [92] may be useful. In his theory of the retention mechanism in ion-pair chromatography, Stahlberg focused on the derivation of the isotherm of the amphiphilic compound, that is, the counter-ion used in this technique to adjust the retention factors of the sample components and their separation factors (e.g., the cation tetrabutulammonium). The counter-ion (Br ~, Cl , H2PO^) may not be strongly associated with the cation in a mobile phase that is a mere aqueous buffer. Other cations, under other experimental conditions may adsorb as true ion pairs, in which case the isotherm behavior is quite different. The Stahlberg model is based on the Langmuir isotherm model, which describes the competition of the ions for available surface area but it assumes that ions are adsorbed as separate, individual ions that undergo electrostatic interactions. The intensity of these interactions is calculated using the Stern-GouyChapman theory. The isotherm equation obtained [92] is zF
nKCe-
^/RT
(3.69a)
where qo is the saturation capacity, K the equilibrium constant, z the charge of the ion, F the Faraday, and fg is given by I0(Kr) Keoer
(3.69b)
3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria
107
where £o is the permittivity of vacuum, er the dielectric constant of the mobile phase, Jo and I\ the modified Bessel functions of the first kind and order 0 and 1, respectively, r is the average pore radius, and 1/K is the Debye length, that is a measure of the thickness of the electrical double layer, with (3.69c) where J is the ionic strength of the solution. This is a quasi Langmuir isotherm, but the equation is no longer explicit (since ¥o is a function of q), which makes the isotherm model rather complex to use in the calculation of band profiles. Later, Stahlberg [93] published a different derivation of the isotherm equation, obtaining another implicit equation C = \eB* qQK
(3.70a)
with (3.70b)
Used as an empirical model to fit experimental data, this equation gave satisfactory results [93,94]. This is no proof of the validity of the model used to derive it.
3.2.4 Isotherm Models for Nonideal Adsorption on a Heterogeneous Surface 3.2.4.1
The Martire Isotherm
A unified theory of adsorption chromatography has been developed by Martire and Boehm [95-98], based on statistical thermodynamics and on the mean field lattice model. This adsorption model can be applied as well to gas, liquid, and supercritical chromatography, using a homogeneous or a heterogeneous surface, with or without lateral interactions of the adsorbate molecules. For example, this theory has been used [99] to analyze the retention behavior in gas chromatography of n-butane on graphitized carbon black (Carbopack C), modified by the adsorption of propane from the carrier gas stream [100]. The model was fitted to the experimental results, and good agreement was reported between these data and the values calculated with the model for the adsorption isotherm of propane and for the dependence of the retention volume of n-butane on the amount of propane adsorbed on the Carbopack C column. The single component isotherm is given by the following equation:
(i-<Wv<
(l-^)"'
(
}
where /, is the fraction of the total surface area of the solute molecule i that comes in contact with the adsorbent surface; 0,- m and 9( s are the volume fractions of the
108
Single-Component Equilibrium Isotherms
solute in the mobile and stationary phases, respectively; 1/, = rf = ^, with r; and rc, numbers of adsorbent sites occupied by the solute and solvent, respectively, and v* and v* van der Waals molar volumes of solute and solvent, respectively; K? is the ratio of the equilibrium concentrations of the solute in the stationary and mobile phases at infinite dilution of the solute: K ? = l i m ^ = ! ^
(3.72)
and DiiTn is an interaction term given by Di,m = -2riXi,c
= ^
[2£i,c ~ (ey + eC/C)]
(3.73)
where e,y, £CfC, and e,-/C are the attractive interaction energy between nearest neighbor segments for the solute-solute, solvent-solvent, and solute-solvent molecules, respectively; Xi,c i s the interaction parameter; ze is the number of nearest neighbors, or external contacts of a molecular segment; T is the absolute temperature; and k$ is the Boltzmann constant. One major feature of this isotherm model is its generality. If we assume that the solutions are energetically ideal, Xi,c = A' = 0. Assuming further that Vj = 1 and that the solution is dilute (i.e., 6irm >C 1) yields - \ -
= K°9i,m = biCiM
(3.74)
or (3 75)
9iA = Y^Q-
-
which is the Langmuir isotherm. If only the adsorbed phase is not ideal (D; s ^ 0, Djitn = 0), but still assuming that v,- = 1 and that the solution is dilute (i.e., 9i/7n
1
-
"i,s
or
which is the Fowler isotherm. If we assume that there are molecular interactions in both phases, but still that v; = 1 and that the solution is dilute (i.e., (?!;m < 1 ) , Eq. 3.71 reduces to 1 + b-C-
e~K'u'-st
The Martire isotherm model can also give rise to a competitive isotherm model which is discussed in Chapter 4.
3.3 Adsorption and Affinity Energy Distribution 3.2.4.2
109
The Bi-Moreau Isotherm
This model assumes that the surface is covered by two types of sites and that a different Moreau model applies to each type of sites, considered as homogeneous and acting independently. The equation of this isotherm is q
^ 1 + 2blC + hb\C2
+ qs 2
' 1 + 2b2C + %
where the subscripts 1 and 2 correspond to the low and the high-energy types of sites, respectively. This model is nearly identical to the Ruthven model developed for adsorption on zeolites [18] and for which the relationships between the coefficients in the numerator and denominator are slightly different. This model was used by Gritti and Guiochon to account for the behavior of propranolol on several Cis-bonded silica adsorbents, from methanol/water solutions. Similar results were obtained for Kromasil-Cis [101/102], XTerra-Cjs [84], Symmetry-Cig [103]. When the mobile phase contains a high concentration of a monovalent salt, the adsorption follows bi-Langmuir isotherm behavior [101]. In the absence of salt, at low concentrations of a monovalent salt or with di- (e.g., phthalate, succinate, naphthalene sulfonate) or tri-valent salts {e.g., citrate), the isotherm data are best modeled by a bi-Moreau isotherm [91,104].
3.3 Adsorption and Affinity Energy Distribution All adsorbent surfaces are heterogeneous, some are just more so than others. Chromatographers tend to use adsorbents that have as homogeneous surfaces as possible because surface heterogeneity is at the origin of several serious drawbacks affecting both analytical and preparative applications of the method. Considerable resources have been invested by manufacturers in the production of highly homogeneous batches of porous silica and of other adsorbents. Notable progress has been made in the last twenty years. Whatever care is applied to the preparation and/or selection of the proper packing material, however, their surfaces are heterogeneous for two main reasons. First, all the impurities existing in the bulk solid used to prepare an adsorbent {e.g., iron or boron in silica) tend to segregate at its surface because they are poorly soluble in the bulk solid network. Second, the surfaces of adsorbents are rough, down to the molecular level. This means that the valences of the atoms on the surface are strained and are not fully saturated. Accordingly, there is on each surface a distribution of the electrical field above the surface of any adsorbent. The replacement of silica gel by a porous silica obtained by the controlled hydrolysis of an emulsion of highly pure tetraethylsilicate has been one of the reasons for the great progress made in the preparation of packing materials for chromatography. The surface of the highly pure adsorbents contain only traces of foreign atoms. The heterogeneity of the surface of the adsorbents used in chromatography has two important consequences. First, adsorption isotherm data do not fit well to the simple isotherm models designed for homogeneous surfaces, i.e., the Langmuir
110
Single-Component Equilibrium Isotherms
and the Jovanovic isotherm models (see Sections 3.2.1.1 and 3.2.1.2). More sophisticated models are necessary that account for the surface heterogeneity. Depending on the assumptions made to model the surface heterogeneity, a variety of isotherm models, e.g., the Fowler, the Freundlich, the Toth, the Unilan isotherms or of combined isotherm models such as the Langmuir-Freundlich and the JovanovicFreundlich isotherms have been spawned. A more fundamental approach must be used, however, to investigate the heterogeneity of the surface of actual adsorbents. Second, the adsorption energy, the adsorption entropy, and the adsorption equilibrium constant are not constant all over the surface but there are, for any adsorbate on any surface, distributions of the adsorption energy (AED), the adsorption entropy, and the adsorption equilibrium constant, the last one being called the affinity energy distribution [83]. The finite width and the characteristics of the AED have important thermodynamic and kinetic consequences. While the latter are becoming accessible to investigations, but not without great difficulties (see later), the former are not yet understood. The systematic investigation of the properties of adsorbents for chromatography, using the AEDs of series of probes is just beginning.
3.3.1 Affinity Energy Distribution The study of heterogeneous surfaces has now become an important part of adsorption studies. It has been shown that, for gas-solid equilibria, the experimental or apparent or global isotherm is related to the AED by the following equation
q(P) = t f(e)0(e,P)de
(3.80)
Ja
where q(P) is the amount of solute adsorbed at equilibrium with a solute partial pressure P, /(e) is the adsorption energy distribution, 9(e,P) is the local model of adsorption (i.e., the adsorption isotherm on an homogeneous patch of the surface on which the adsorption energy is e), and e is the adsorption energy [83,105]. The integration limits a and b correspond to the minimum and maximum values possible for the adsorption energy. Initially, of course, this equation was derived for gas-solid adsorption. It was recently extended to liquid-solid adsorption as described in Ref. [106]. Equation 3.80 is a Fredholm integral equation and, like many integral equations, its solution is an ill-posed problem [107]. Several approaches have been suggested to solve this problem. Earlier methods involved the fitting of the adsorption equilibrium data to an isotherm model (e.g., the Langmuir-Freundlich isotherm [71]), selecting the one model that gives the smallest residuals or the best fit (see Section 3.6), and deriving the AED from this adsorption isotherm. A slightly different approach consists in assuming that the AED follows a certain functional relationship (e.g., a Gaussian, an exponentially convoluted Gaussian,...) and deriving the best parameters of this function using an optimization routine. However, both methods are doomed to failure because there is an AED function associated with each isotherm and conversely [83]. So, any assumption made regarding the functional dependence of the isotherm or of
3.3 Adsorption and Affinity Energy Distribution
111
the AED is self-fulfilling but misleading. If this kind of approach is considered, a careful statistical investigation of the residuals obtained for the isotherm modeling and for the AED must be undertaken. The numerical estimation of the AED of a compound from its adsorption isotherm data, without making any assumptions about the functional form of either the energy distribution or the measured isotherm, is a highly desirable approach to the calculation of the energy distribution [106,108]. In this manner, all the information encoded in the experimental data and this information only is transferred into the distribution function, as evaluated by the local model of adsorption. This is a far better approach than the alternative ones that consist in introducing into the calculation of the AED some arbitrary information which is encoded in any presumed AED or isotherm function(s). An algorithm to achieve just that was proposed by Adamson and Ling [109]. This algorithm was later improved and modified by House and Jaycock who demonstrated the utility of their program with various simulated distributions, as well as some applications to real experimental data [110]. Because all sets of experimental data have errors of measurements, smoothing routines were used to avoid these errors amplifying to the solution. In the core algorithm, the solution is represented as a cubic spline and must be subjected to certain conditions to prevent oscillatory behavior. This is a general problem in the estimation of ill-posed problems such as the inversion of Eq. 3.80. Since the adsorption energy, e, is related to the adsorption equilibrium constant, K, by the relationship K = Koee/RT
(3.81)
it is convenient to extend Eq. 3.80 to liquid-solid equilibria through the following equation that defines the adsorption equilibrium constant distribution [71]
q(C) = J'cpiln^^^dilnK)
= J* f(K)J^d(lnK)
(3.82)
In this equation, the local adsorption isotherm follows Langmuir behavior, (p (In K) is equivalent to the distribution of the Gibbs free energy of adsorption, and f(K) is the distribution of adsorption equilibrium constants while a and b are related to the highest and lowest values accessible through a = 1/Cmax and b = 1/Cmin, Cmax and Cmin being the highest and lowest concentrations for which adsorption data can be measured. Equation 3.82 has the advantage that there is no need to determine the value of KQ in Eq. 3.81 to obtain the distribution of adsorption equilibrium constants (also called affinity energy distribution) while it is necessary to derive the true AED. A common method of extracting f(K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth function describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range
112
Single-Component Equilibrium Isotherms
of K values is adequately sampled in the isotherm data. If adsorption data cannot be measured up to sufficiently high concentrations (usually because of the solubility limit), the low side of the distribution is inadequately sampled and an exponentially decreasing function, beginning at K^^ is obtained (Kj^ corresponds to the maximum concentration at which the measurements of isotherm data were carried out, C max/ see Eq. 3.82). The derivation of analytical solutions of Eq. 3.82 by differentiating smooth isotherm models has been accepted due to its simplicity. However, the utility of a numerical procedure that does not imprint any shape or form on the distributions nor inject any arbitrary information in its derivation is preferable. A numerical method gives an AED the shape of which is dictated solely by the data. Obtaining numerical estimates of affinity distributions requires an algorithm that inverts Eq. 3.82 to obtain f(K) given only the experimental set of values of q(C). Several methods have been proposed. The iterative maximum-likelihood method called expectation-maximization (EM) is particularly stable and relatively free from artifacts as long as the data contain a minimum of errors, particularly systematic errors [106,111]. This problem is general for any method used to obtain/(K). The EM algorithm is attractive in its simplicity. Equation 3.82 is discretized as: ^max
<7(Q)=
E
f(Kj)e(Ci,Kj)
(3.83)
Kj) = ^ ^ ^ ( l n i C )
(3.84)
where
is the model kernel matrix, defined for each concentration at which isotherm data were measured, Q, and for the association constant in the distribution range, Kj. A(h\K) is the constant spacing between logK values in the discretized space. It is divided up logarithmically with input parameters for Kmax, Kmjn and the number of points to be calculated in the distribution. An initial estimate for f(Kj) is required. Since the influence of the data on the shape of the distribution should be maximized, a constant initial distribution is best, with f(V\-
^high-glow m A inA
.
.
for all Kj, with qu^x and qiow being the highest and the lowest isotherm data, respectively. Usually, (0,0) is included as the initial (q, C) isotherm data point, so ?low = 0- Equation 3.85 models the data range (numerator) evenly across the entire distribution range (denominator). The amount q(Cj) of solute adsorbed at concentration Cj is iteratively estimated by
= E
Fk £
( i)
H Ae l +K r q
i e [I-*!]; ' e [i,N]
(3.86)
3.3 Adsorption and Affinity Energy Distribution
113
Figure 3.29 Adsorption distribution of propranolol at different eluent pH: (1) and (4) pH = 5.01, (2) and (5) pH = 5.51, and (3) and (6) pH = 6.02. Lines: dashed for (R)- (# 1 to 3), solid for (S)-enantiomers (# 4 to 6). Plot of the concentration of adsorption sites (mM) versus the natural logarithm of the adsorption equilibrium constant. Reproduced with permission from G. Gotmar, B. J. Stanley, T. Fornstedt, G. Guiochon, Langmuir, 19 (2003) 6950 (Figure 4b). ©2003, American Chemical Society.
with (3.87) (» - l)Ae N-l where the index k stands for the rank of the iteration of the calculation of the AED function [106]. The EM procedure protects better than most other methods against the consequences of possible experimental artifacts which can be incorporated in the calculation of the AED or against the effect of modeling the experimental data.
3.3.2 Experimental Validation of the EM Method Accurate measurements of adsorption isotherm data of several pairs of enantiomers on chiral selective stationary phases (CSP) have been published [47,112]. Using these adsorption data, the AEDs of these compounds were calculated [56, 113,114]. In the case of L-phenyl-alanine anilide, the method using the LangmuirFreundlich isotherm to model the experimental isotherm data (see [71]) gave a single adsorption energy mode for both enantiomers [113], while the EM method gave a bimodal AED [56], in agreement with the isotherm model fitting best the experimental data being the bi-Langmuir, not the LF model [112]. The AEDs of the two enantiomers are identical in the low energy range but have different intensities and energies in the high energy range. The imprinted polymer exhibits a larger, separate mode, corresponding to high energy adsorption sites that adsorb the template more strongly than its enantiomer [56]. A similar result was obtained in the case of propranolol on a CSP made of a protein, Cel7A, immobilized on porous silica [47]. There is a large tunnel in the structure of this protein that can accommodate the molecules of /3-blockers and selectively retains the S- more strongly than the R-enantiomer. The AED calculated with the EM method is shown in Figure 3.29 [114]. The difference between the energies of the higher energy modes obtained for the two enantiomers (see Figure) agrees well with the difference between their association energies with Cel7A which was independently measured by microcalorimetry [115]. Figure 3.29 also shows the influence of the pH on the high energy sites of these two enantiomers. It confirms that at low pH (5.0), the selectivity of the retention of the S- over the
114
Single-Component Equilibrium Isotherms
R-enantiomer arises from the larger saturation capacity of the selective sites (compare curves 1 and 4 in Figure 3.29) while at high pH (6.0), the saturation capacities for the two enantiomers are close and the selectivity arises from the higher adsorption energy for the S- than for the R-enantiomer (compare curves 3 and 6 in Figure 3.29), an observation that was made earlier [47]. Note that only a small part of the low-energy mode of the AED (the high energy region of that mode) was determined. For solubility reasons, it was not possible to acquire adsorption data at sufficiently low adsorption energies, i.e., at high enough concentrations. It should be emphasized that the EM method gives the distribution of adsorption equilibrium constants on a surface. This distribution is a property of the surface studied. Recent investigations have shown it to be reproducible for the columns coming from one lot or even from different lots of fabrication for some of the best brands of commercial columns. However, it is also a property of the solute used as a probe in these investigations and of the solvents making the mobile phase. The AED is an excellent tool for the investigation of the properties of the adsorbents used as stationary phases in HPLC. A much larger amount of data and of systematic results than what is currently available is needed to support definitely the valuable conclusions that can already be drawn regarding the general properties of heterogeneous surfaces.
3.3.3 Applications of AED in the Study of Retention Mechanisms in RPLC The AEDs of phenol and caffeine on Cis-bonded silica columns were derived from their convex upward isotherms, isotherms that are accurately modeled with a multi-Langmuir equation, i.e., with the sum of two or several Langmuir terms [11, 61,114,116,117]. Because these compounds are quite soluble in the water/methanol solutions used as the mobile phase, their isotherms can be measured in a wide concentration range, sampling well the high- and the low-energy sites. Those are ideal systems to which to apply the EM method of AED determination. One of the most useful results derived from the AED is that the best isotherm model selected to account for a set of adsorption data must be consistent with the AED derived from these data of Ref. [114]. The adsorption isotherms of phenol and caffeine on a Kromasil Cig from an aqueous solution of methanol could be modeled accurately using either the Toth or the bi-Langmuir model. The statistical figures characterizing the quality of the fit would suggest that the Toth model gives slightly better results. However, the AED derived directly from the isotherm data exhibits two well resolved, narrow energy modes, a result that is not compatible with the T6th isotherm model, since this model of isotherm corresponds to a unimodal AED. A later investigation showed that the conclusions of this study for phenol extend to a wide range of composition of the mobile phase, from 0 to 70% of methanol [116]. AEDs derived for six different compounds, under different experimental conditions, on the same Kromasil Qg allowed the selection of the isotherm model that, in each case, best accounts for the adsorption data of these compounds. Knowing the isotherm model made it possible to use the rapid numerical method of derivation of the isotherm coefficients that is based on the solution of the in-
115
35
35
28
28
28
Caffeine Caffeine
21
14
qS,i [g/L]
35
qS,i [g/L]
qS,i [g/L]
3.3 Adsorption and Affinity Energy Distribution
21
Hypersil
14
21
Waters
14
Vydac 7
7
0
0 -5
-4
-3
-2
-1
35
0 -6
0
Ln bi
-5
-4
-3
35
qS,i [g/L]
28
21
-2
-1
0
-6
Ln bi
28
21
7
7
7
0
0 -2
Ln bi
-1
0
-1
0
-2
-1
0
Merck 14
-3
-2
Ln bi
21
14
-4
-3
28
14
-5
-4
Phenomenex
Kromasil
-6
-5
35
qS,i [g/L]
-6
qS,i [g/L]
7
0
-6
-5
-4
-3
-2
Ln bi
-1
0
-6
-5
-4
-3
Ln bi
Figure 3.30 Affinity energy distributions calculated with one hundred millions iterations from the experimental isotherm adsorption data measured by FA on six commercial Cjgbonded stationary phases, with methanol/water (30/70, v/v) as the mobile phase. T = 296 K. Reproduced with permission from F. Gritti, G. Guiochon, Anal. Chem., 75 (2003) 5726, Figure 4b). ©2003, American Chemical Society.
verse problem of chromatography (IM, see Section 3.5.6) and to determine the parameters of this isotherm for a lot of ten different columns packed with the same grade of Kromasil Qg, five of them from the same production batch, the other five from different batches [11]. The statistics of reproducibility of these coefficients afford an excellent estimate of the reproducibility of the material itself, important information for engineers using preparative chromatography for the purification of intermediates in the pharmaceutical industry. Figure 3.30 compares the adsorption energy distributions of caffeine on six different commercial brands of Cis-bonded silica [61]. Similar results (not shown, see [61]) were obtained for phenol. For both compounds, the AEDs are bimodal, the two modes being well resolved, with a difference in the average energy of their two modes that is of the order of 5 kj/mole. This difference is too small to be explained by the presence of residual silanol groups on the silica surface. The only difference between the AEDs of phenol and caffeine is in the relative intensity of the two modes, the second energy mode being smaller for caffeine. A more detailed investigation of the thermodynamic properties of the stationary phases and of the chromatograms obtained at low and high column loadings suggests an interpretation of the two types of adsorption sites [61,117]. The low-energy sites would correspond to adsorption on the alkyl chains bonded to the silica, either on the tip of the chains or on the side of collapsed chains. The high-energy sites would correspond to a partition into the network of bonded chains in the adsorption field of the underlying silica. Residual silanols have no effect on the reten-
116
Single-Component Equilibrium Isotherms
Figure 3.31 Distribution of the equilibrium constants of adsorption calculated by means of the biToth model for 1-indanol on cellulose tribenzoate. The arrows indicate the equilibrium constants derived from the competitive bi-Langmuir isotherm. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, }. Chromatogr. A, 1005 (2003) 35.
nonsBladive / \ \
enanUoselective
\ f\ f\
tion of phenol or caffeine. This explains why phenol is more retained than caffeine, although the latter has a higher molecular weight and is more strongly hydrophobic. These results are tentative and must be confirmed by more extensive investigations of the retention mechanisms. Similar results were also obtained on monolithic silica, although the AED was trimodal for phenol and tetramodal for caffeine, quite an interesting result since the fourth mode has an intensity that is 400 times smaller than that of the first mode [117]. Nevertheless, it was demonstrated that this result is real, that it is not an artefact nor is it due to the modeling of noise. It is highly reproducible for a set of six different columns. It seems that the recent investigations made on the AED have resulted in the development of an important new tool for the investigation of mildly heterogeneous surfaces such as modern packing materials used in HPLC. This method has already allowed the derivation of important new results regarding retention mechanisms in reversed phase liquid chromatography [61,117] and on molecularly imprinted polymers [56,118]. Its application in preparative liquid chromatography should be fruitful. Another application of this approach is in the study of certain enantiomeric separations. In applying the method of the solution of the inverse problem of chromatography to the determination of the competitive equilibrium isotherms of the enantiomers of 1-indanol on cellulose tribenzoate, Felinger et al. found that the data were much better accounted for when a competitive bi-T6th isotherm model was used rather than a bi-Langmuir model [119]. The AED derived from the FA adsorption data is shown in Figure 3.31. It is bimodal but the spread of the adsorption energies is significant, larger than it is expected to be for the modes of a bi-Langmuir isotherm model. Note also that the AED mode corresponding to the nonselective sites is larger than the one corresponding to the enantioselective sites. This means that, on the enantioselective sites, the adsorption can be characterized with a narrower energy range. These results agree well with studies of the adsorption energy distribution with the expectation maximization method [120].
3.4 Influence of Experimental Conditions on Equilibrium Isotherms
117
3.4 Influence of Experimental Conditions on Equilibrium Isotherms The equilibrium isotherms depend on the parameters that characterize the physical state of the system and the chemical potential of the compounds studied in this system selected, i.e., its temperature and pressure, the chemical compositions of the mobile and the solid phases. The last of these parameters is far more difficult to investigate than that of the other three. Suffice it to say that important variations of the isotherm parameters have been observed for different brands of Cig-bonded silica. However, the best isotherm model accounting for the adsorption data measured on these different brands remained the same [61] while the column to column reproducibility of the data was excellent for at least one brand and probably for several others [11,121,122], with relative standard deviations for the parameters being of the order of a few percent [123].
3.4.1 Influence of the Pressure The influence of this parameter has rarely been investigated systematically while many authors have unwittingly drawn conclusions from measurements of isotherm data made under different pressures. It is not often realized that changing the mobile phase flow rate through the column results in a proportional variation of the pressure all along the column (see Chapter 5, Section 5.3). Similarly, changing the column temperature at constant flow rate results in what is believed to be an investigation of the dependence of the equilibrium parameters on the temperature to be actually a complex study of a convolution of the effects of temperature and pressure, since the mobile phase viscosity depends significantly on the pressure. Fortunately, the effect of pressure is usually small. Furthermore, it seems to be linear, at least in the range of pressures that are accessible in preparative liquid chromatography (usually less than 200 bar). Therefore, the influence of the pressure gradient along the column is the same as if the isotherm data were measured under isobaric conditions at the average column pressure [124]. Nevertheless, any investigation of the temperature dependence of the isotherm should be made at constant inlet pressure, in order to eliminate the effects of the variation of the pressure along the column. The influence of the pressure on the adsorption equilibrium constant derives from the classical fundamental relationship of thermodynamics for a system at constant temperature and pressure AG = -RT]nK = AE + PAVm-TAS
(3.88)
where AG, AE, AVm, and AS are the changes in Gibbs free energy, internal energy, molar volume, and entropy of the solute associated with its passage from the solution to the adsorbed phase [124,125]. Accordingly, there is an exponential relationship between the difference of the partial molar volumes of the solute in the two phases and the equilibrium constant AVn = - R T ^
(3.89)
118
Single-Component Equilibrium Isotherms
Figure 3.32 Influence of pressure on the isotherms of porcine insulin on a Ci§bonded silica. Experimental adsorption data (symbols) and best isotherms of porcine insulin. The inset shows the low concentration data, o 56.5 bar, o 118.5 bar, A 178.5 bar, • 237.5 bar. Reproduced with permission from X. Liu, D. Zhou, P. Szabelski, G. Guiochon,}. Chromatogr. A, 988 (2003) 205 (Fig. 2).
It is important to observe that large molecules tend to have a larger difference in their molar volumes than smaller ones. Accordingly, the effect of pressure on the isotherm is usually rather small, often almost negligible, with compounds having a small molecular weight such as phenol or small-molecule aromatic compounds [126]. As examples, AVm is of the order of —1 mL/mole on low density monomeric Cis bonded phases and of —15 mL/mole on high-density polymeric bonded phases for linear fatty acids in the Cio to C20 range while that of polynuclear aromatics is between + 10 and —30 mL/mole on high density polymeric phases [126]. In contrast, the effect of pressure is rather large with peptides [48] and still more important with macromolecules such as insulin (AVm ~ —100 ml/mole) [127,128]. The isocratic retention factor of insulin in RPLC increases two-fold when the average column pressure increases by 200 arm (see Figure 3.32) [128]. Experimental measurements confirmed in this last case that measuring the retention factor at constant flow rate rather than constant inlet pressure lead to large errors in the estimates of the Gibbs free energy, enthalpy and entropy of adsorption. In spite of the large variation of the retention factor with the average column pressure, the separation factor of the insulin variants (e.g., porcine insulin and Lispro) do not vary by 1%. The equilibrium isotherms of insulin variants between a YMC ODS-A column and an aqueous solution of acetonitrile (30% ACN, 0.1% trifluoroacetic acid(v/v) was accounted for by a bi-Langmuir model [127]. The total saturation capacity of the column increases by 40% for a 200 atm increase of the average column pressure, allowing a corresponding increase in the potential production rate of a preparative unit. In RPLC, the influence of pressure on the chromatographic behavior is related to the hydrophobic interactions involved in the retention mechanism and to the change upon adsorption in the numbers of acetonitrile and water molecules in the solvent shells of the protein molecule and of the bonded layer. The importance of the changes in the retention factor and the saturation capacity with a change in the average column pressure will thus depend on the retention mode used and will vary with the hydrophobicity of the molecule [128]. In RPLC, it is larger with polymeric than with monomeric bonded phases [126].
3.4 Influence of Experimental Conditions on Equilibrium Isotherms
119
3.4.2 Influence of the Temperature The relationship between the equilibrium constant and temperature is given by van't Hoff equation
There have been many studies on the temperature dependence of the retention factor under linear conditions but few studies on the influence of the temperature on the saturation capacity or on the other numerical coefficients of an isotherm. It is probable that the best isotherm model remains the same in a wide temperature range while its coefficients vary continuously with temperature. Temperature is not a very popular parameter to adjust in order to optimize the results of a separation in liquid chromatography. Given the obviously great difficulties to be expected when attempting to run a column that is not completely isothermal (particularly if there is a radial temperature gradient, hence radial gradients of viscosity and mobile phase velocity), it is far more practical to run preparative columns at room temperature than to control their temperature at a set value. So, it is understandable that separation engineers are reluctant to change the column temperature in preparative applications. In these cases, wide diameter columns are used, and it is difficult and time consuming to change the temperature of such a column and to do it in such a way that the column remains radially homogeneous. This explains why there is a drastic paucity of thermodynamic data in this area. A recent study comparing the dependence on the temperature and on the mobile phase composition (water/methanol) of the parameters of the bi-Langmuir isotherm of phenol on a Cjs-bonded silica gave informative results but they are insufficient to base general conclusions. Examples of the temperature dependence of equilibrium isotherms are shown in Figure 3.33. Note that, as explained in the previous section, experiments conducted in order to determine the adsorption enthalpy or, more generally, the adsorption behavior of a system, should be run at constant inlet pressure (hence with a flow rate than will increase with decreasing viscosity, hence with increasing temperature). This has the great advantage of affording sets of data for which the effect of temperature is independent of that of pressure [124]. By contrast, during measurements carried out at constant flow rate, the pressure profile varies along the column, decreasing with increasing temperature. The data obtained exhibit the combined influence of temperature and pressure. The parameters derived from these measurements are not, as erroneously stated, the adsorption enthalpy and entropy. Depending on the value of AVm, the error made may be either negligible or significant. This error tends to increase with increasing molecular weight of the compound studied and it becomes quite important with proteins. Quiftones et ol. have measured equilibrium isotherm data for benzyl alcohol, 2-methyl benzyl alcohol and 2-phenylethanol on a Cig-bonded silica at different temperatures between 20 and 60 °C [129]. The results for benzyl alcohol are shown in Figure 3.34 left. From these isotherms, they derived the Clausius-Clapeyron
120
Single-Component Equilibrium Isotherms b o
4.0
in -
q
q CO '
y\S/ / / / /
q
o
W// $//
W// 100
120
Temperature o 273 K o 278 K » 283 K • 293 K
C
. 303 K • 313 K '
O-l
00
5.0
10.0
16 0
Figure 3.33 Effect of temperature on adsorption isotherms, (a) Adsorption of p-cresol on ODS silica. Column dimensions: 1.2 x 50 mm. Temperature: 1, 25°C; 2, 40°C; 3, 60°C. Reproduced with permission from ]. Jacobson, J. Frenz and Cs. Horvdth, J. Chromatogr., 316 (1984)
53 (Fig. 9). (b) Adsorption of N-Benzoyl-D-alanine on immobilized bovine serum albumin. Reproduced with permission from S. Jacobson and G. Guiochon, J. Chromatogr., 522 (1992) 23 (Fig. 2).
20
30
40
c, (g/i)
Figure 3.34 Left Adsorption data for benzyl alcohol (BA, symbols) at different temperatures, 20 o, 30 D, 40 o, 50 A, and 60 °C *. The solid lines are the correlations given by the Flory-Huggins model. Right Clausius-Clapeyron plots for BA, with data at different loadings, 10 o, 20 • , 30 o, and 40 g/1 A. Reproduced with permission from I. Quinones, J. C. Ford and G. Guiochon, Chem. Eng. Sci., 55 (2000) 909 (Figs. 7 and 8).
equation [54]: d\nA\ _ R
(3.91)
3.4 Influence of Experimental Conditions on Equilibrium Isotherms
121
where A^ is the activity of the z-th solute considered in the bulk liquid phase (see Figure 3.34 right). The isosteric heats of adsorption are derived at different loadings for all the solutes, at different loadings. The authors found that the values of the isosteric heats are basically constant for each adsorbate at different loadings, a result in agreement with previous findings for other RPLC systems [130]. These provide evidence that RPLC surfaces are nearly homogeneous under the conditions of their study. Although adsorbate-adsorbate interactions may contribute to the value of the isosteric heats, but activities being used in Eq. 3.91, the main contribution should come from the adsorbate-adsorbent interactions. Finally, the obtained (all approximately 10 kj/mol) are close to those determined from the Van't Hoff plots at low concentrations for the standard heat of adsorption. The very good agreement observed between data obtained at low and high concentrations further confirms the conclusion of an homogeneous adsorbent surface.
3.4.3 Influence of the Mobile Phase Composition There is a considerable amount of data regarding the influence of the mobile phase composition on the retention factor in the different modes of HPLC and many reports discussing these data. The influences of the concentration of the stronger organic solvent in normal phase HPLC, that of the organic modifier in RPLC, the pH and the ionic strength in ion-exchange chromatography are well known and are discussed in detail in basic books on analytical chromatography. The influence of these factors on the other isotherm coefficients and particularly on the saturation capacity are much less well known. Some recent work has discussed the effect of the methanol concentration in its aqueous solution on the isotherm of phenol on Kromasil Qg [116]. The model that best accounts for these adsorption data is the bi-Langmuir model. The saturation capacities of the two Langmuir modes decrease with increasing methanol content, the decrease being most important for the saturation capacity of the high-energy sites. The two adsorption constants also decrease significantly with increasing methanol concentration, that of the high-energy sites decreasing most strongly. The same model applies to the adsorption of phenol from an aqueous solution of methanol on five other stationary phases: 218TP Cis, a polymeric bonded phase from VYDAC (Hesperia, CA), HyPURITY Elite Ci8 from Hypersil (Thermo Hypersil, San Jose, CA), Symmetry Qg from Waters (Milford, MA), Luna from Phenomenex (Torrance, CA), and Chromolith Performance, a monolithic Cis silica from Merck (Darmstadt, Germany). For a constant methanol concentration (30:70), the numerical coefficients of the model are relatively close, except for the polymeric phase. The strong dependence of the isotherm parameters on the composition of the mobile phase confirms that adsorption of small molecules on chemically bonded Ci8 silica is more complex than is usually believed. Most prior work has been based on data acquired under linear conditions, that is, at infinite dilution. Chromatographers have long ignored the way in which consideration of the whole equilibrium isotherm and its modeling may inform on the detail of the retention mechanisms involved [131].
Single-Component Equilibrium Isotherms
122
Table 3.1 Best bi-Langmuir parameters for Phenol in RPLC
Vydac Hypersil Symmetry Kromasil Luna Chromolith
h
qs,2
(mol.1- 1 )
(Lmol" 1 )
(mol.1"1)
0.583 1.439 1.457 1.355 1.371 1.958
1.11 0.71 0.98 1.37 0.945 1.28
0.292 0.713 0.563 0.627 0.870 0.748
b2 (Lmol" 1 ) 7.31 7.72 10.94 11.8 9.11 11.82
3.5 Determination of Single-Component Isotherms The primary use of isotherm data measurements carried out on single-component elution profiles or breakthrough curves is the determination of the single-component adsorption isotherms. This could also be done directly, by conventional static methods. However, these methods are slow and less accurate than chromatographic methods, which, for these reasons, have become very popular. Five direct chromatographic methods are available for this purpose: frontal analysis (FA) [132,133], frontal analysis by characteristic point (FACP) [134], elution by characteristic point (ECP) [134,135], pulse methods {e.g., elution on a plateau or step and pulse method) [136], and the retention time method (RTM) [137]. These chromatographic methods are essentially attempts at solving the general inverse problem of chromatography, given the mass balance equation (see Eqs. 2.2 or 2.44), a set of experimental data characterizing a solution of the system of equations of chromatography, and the corresponding set of initial and boundary conditions. The direct and the inverse problems can be formulated as follows: the solution of the direct problem consists in calculating the column response to an input concentration signal knowing the equilibrium isotherm and the rate constants (or the apparent dispersion coefficient). The column response is either the transient concentration profiles at the column outlet or these concentration profiles along the column at a given time. The inverse problem consists in determining the isotherm and the rate constants, knowing one or several solutions of the system of equations {i.e., band profiles acquired under known experimental conditions, i.e. with known initial and boundary conditions). There is a paucity of mathematical results to guide this last quest. However, the selection of proper experimental conditions {i.e., of simple boundary conditions) can profoundly simplify the solution of the inverse problem.
3.5 Determination of Single-Component Isotherms
123 •«— Load
Pump
Valve A
•> Waste
Waste Column
Figure 3.35 Diagram of a chromatograph designed for the determination of equilibrium isotherms by FA, FACP, or ECP, using narrow-bore columns.
3.5.1 Frontal Analysis (FA) Frontal analysis was first developed and used independently by James and Phillips [133] and by Schay and Szekely [132] for the determination of adsorption isotherms. A solution of the studied compound, at a known, constant concentration, is percolated through the column. Successive, abrupt step changes of increasing concentration are performed at the column inlet and the breakthrough curves are determined [138,139]. Figure 3.35 [38] shows the diagram of a typical experimental setup for the determination of adsorption isotherms by frontal analysis [8,38]. It is designed especially for operation with narrow-bore columns. These columns offer considerable savings by reducing the amounts of valuable chemicals needed for the measurements and the volumes of solvent wasted. The preparation of the series of solutions of known concentrations needed is labor intensive. As an alternative, it is possible now to use a liquid chromatograph with a computer-controlled gradient delivery system that can deliver step gradients of known concentrations which are not only precise but also accurate. There are two methods available for the determination of the mass of solute adsorbed at equilibrium with the solution percolated through the column. Too often, this mass is estimated from the 'retention time of the breakthrough curve', itself derived from the retention time of the inflection point of this curve or from the elution time of the concentration at half-height of the plateau, duly corrected for the extra-column hold-up time. This procedure is incorrect. The accuracy of FA derives from it being a titration method: the amount adsorbed at equilibrium must be derived from the integral of the breakthrough curve. Figure 3.36 shows the principle of the method and illustrates the type of systematic errors that other methods can introduce: the breakthrough curve is rarely symmetrical, particularly at high concentrations.
Single-Component Equilibrium Isotherms
124 Figure 3.36 Determination of frontal analysis data. The breakthrough curve is the thick solid line. The two-hatched surfaces on its right and left sides have the same area and fix the volume of equivalent area used for the calculation. A large error may be made if the inflection point is considered.
Reproduced with permission from F. Gritti, W. Piatkowski and G. Guiochon. } . Chromatogr. A, 978 (2002) 81, Figure 1.
AU 1200 i
b
a
[
n
1200 i
j-J 1 FACP FA
800-
^J
\
800
400
\
400
\
/
o0
ECP
050
100
0
10
t, mi n 20
Figure 3.37 Typical experimental chromatograms obtained in the determination of equilibrium isotherms by chromatographic methods, (a) Frontal analysis staircase. FACP on the diffuse rear boundary after the last frontal step, (b) ECP. Data recorded with an HP 1090 (Hewlett-Packard, Palo Alto, CA) liquid chromatograph. Reproduced with permissionfromS. Golshan-SMrazi and G. Guiochon, Anal. Chem., 60 (1988) 2630 (Figs. 1 and 2), ©1988 American Chemical Society.
Figure 3.37 shows a typical experimental recording obtained for the determination by frontal analysis of the isotherm of phenol between dichloromethane and silica [140]. To estimate the amount adsorbed at equilibrium, the best approach is to calculate, by integration of the detector signal, the area from the hold-up time to the time when the signal has reached the value corresponding to the concentration of the solution injected into the column [18]. This is the only rigorous approach [141]. It is illustrated in Figure 3.36. When the column efficiency is high, an acceptable approximation is given by: (3.92) Va where Q,- and Q,+i are the amounts of compound adsorbed by the column packQi+i =
3.5 Determination of Single-Component Isotherms
125
ing after the fth and the (i + l)th step, when in equilibrium with the concentrations Q and Q_|_i, respectively, VpI + i is the retention volume of the inflection point of the (i + l)th breakthrough curve, VQ is the column void volume, and Va is the volume of adsorbent in the column. Frontal analysis is a very popular method of isotherm determination, and rightly so. It has been applied to the determination of a great number of equilibrium isotherms, in many modes of chromatography. Among others, it has been used for the measurement of the isotherms of peptides [38] and proteins [8,41,65]. It is the method most often used in our group. The advantages and drawbacks of FA relative to those of the other methods of isotherm determination are discussed later, in Section 3.6.4.
3.5.2 Frontal Analysis by Characteristic Points (FACP) Measurement of single-component isotherms can be derived from the profile of the rear boundary recorded in FA, or by making a negative step change from a finite concentration to the pure mobile phase (e.g., when the column is regenerated after a classical FA series of measurements, Figure 3.37). This method is called frontal analysis by characteristic points or FACP [134]. One large concentration step is pumped into the column. After equilibration with the column has taken place and a plateau has been eluting from the column for a sufficiently long time, pure mobile phase is again pumped into the column. The diffuse profile is recorded and the isotherm is derived from this profile, using Eq. 3.93. When accurate determinations in a wide concentration range are needed, several FACP experiments can be carried out with increasing step height. In the case of Figure 3.37a, the FACP experiment was performed at the end of a series of FA runs. Contrary to conventional practice, it is not necessary to wait until the steady response of the detector indicates that the eluate concentration is the same as the concentration in the stream of mobile phase pumped into the column to return to a pure mobile phase stream. In fact, a wide rectangular injection is usually sufficient, provided that a concentration plateau is eluted at the column exit. Admittedly, in such an implementation of the method, it becomes difficult to decide whether it is FACP or ECP (see next section) but this is irrelevant. If the width of the rectangular pulse is kept reasonably narrow, the amount of sample needed is comparable to that required in ECP for the determination of the isotherm in the same concentration range. A mass balance can be written for each point on the rear part of the elution profile of this rectangular injection pulse. The amount adsorbed is calculated by integration of the area under the peak, starting from the tail end (C = 0). Using partial sums, we may write:
= 7F Va JO
fC(V-V0)dC
(3.93)
where Q(C) is the amount of compound adsorbed by the column packing when in equilibrium with the concentration C, Va is the volume of adsorbent in the column,
126
Single-Component Equilibrium Isotherms
and V is the retention volume of the point of the diffuse profile at concentration C, or characteristic point.
Thus, each point of the rear profile gives one point of the isotherm. With modern systems of data acquisition, several hundred points of an elution profile can be conveniently recorded and stored, providing as many data points for the isotherm. This procedure tends to supply precise data. The primary disadvantage of FACP is due to the effect of the column efficiency on the rear boundary. The finite column efficiency introduces a systematic error on the derived isotherm [142,143], and prevents getting meaningful data at very low concentrations. The method should be applied only with highly efficient columns, having at least several thousand theoretical plates [142]. The careful experimentalist will not forget, however, that every time a wide rectangular band is injected to use the FACP method and to derive an arc of the isotherm from the rear of the band profile, a point of this isotherm can be determined from the retention volume of the steep band front (FA method). This provides a free opportunity to check the consistency of the results. The influence of the column radial heterogeneity on the determination of equilibrium isotherm data by the ECP method was studied by Miyabe et al. [144]. The authors calculated overloaded elution peaks using the equilibrium-dispersive model, a Langmuir isotherm, and taking into account typical radial distributions of the mobile phase flow velocity and the column efficiency across a column. The results of the ECP method depend on four parameters: the number of theoretical plates at the center of the column, the two Langmuir isotherm coefficients, and the loading factor. The influence of the mass transfer resistances and the radial heterogeneity of the column on the ECP data was analyzed by comparing the true isotherm and the one estimated from the diffuse profile of these overloaded peaks. The error made increases with increasing degree of radial heterogeneity. The analysis of these results allows a correction that improves the precision of the ECP method.
3.5.3 Elution by Characteristic Points (ECP) In the method of elution by characteristic points [134,135] the isotherm is derived from the rear part of the overloaded elution profile obtained upon injection of a large sample (Figure 3.37a [140]). The isotherm is calculated using Eq. 3.93. The method of ECP is very similar to FACP. In ECP, the isotherm is determined from the rear part of an overloaded elution profile, and in FACP it is determined from the rear of a wide rectangular band or from the profile obtained when washing a concentration plateau off the column (these are equivalent). Both methods are thus based on the demonstration that a velocity can be associated with a concentration and that this velocity depends only on the concentration, through the isotherm (see Chapter 7, Section 7.2.1). These methods are based on the determination of uz and the integration of Eq. 7.4 (Chapter 7). Again, the primary disadvantage of this method is that the amount adsorbed, q(C), is calculated from an equation derived from the ideal model, assuming that the column efficiency is infinite. Therefore, the ECP method should be used only with highly efficient columns, where the contribution to band broadening is neg-
3.5 Determination of Single-Component Isotherms
127
ligible. Moreover, the data points close to the top of the profile and those that are far along the tail, close to the baseline, are more affected than those in the middle by the sources of band broadening. They should not be used in the determination of the isotherm. Unfortunately, it is not possible to reject the data points close to the baseline, since the integration of the profile has to be made from C to 0. Thus, the systematic error due to the use of the inaccurate low-concentration data for the determination of the high-concentration points of the isotherm is the major drawback of ECP and FACE The errors made in the determination of single-component isotherms by ECP have been studied in detail by Guan et al. using a combination of calculations [142] and experiments [145]. They concluded that ECP (hence FACP) introduces a model error. The measured isotherm does not satisfy the Langmuir isotherm model if the experimental isotherm is truly a Langmuir isotherm. The systematic error on the determination of the first coefficient (a) of the isotherm is still of the order of 1% when the column efficiency is 5000 theoretical plates. It is lower for the second coefficient if the loading factor is of the order of 20% or larger. The ratio of the maximum peak height to the standard deviation of the baseline noise should exceed 500. Later, Ravald and Fornstedt [146] confirmed this result but showed also that, if the calculations are made with a more complex isotherm than the Langmuir model (e.g., if the compound studied follows bi-Langmuir isotherm behavior), the error is more important. A column efficiency of 5000 plates would still not guarantee errors smaller than 5% on the values of the isotherm coefficients. The ECP method should not be used for accurate isotherm determinations [146].
3.5.4 Pulse Methods There are two types of pulse methods, the elution of a pulse on a plateau and the elution of an isotopic (or possibly enantiomeric) pulse on a plateau. These two methods differ because in the first case no detector can recognize the molecules injected when they are eluted, while in the latter case this is possible, using a radiochemical detector, a mass spectrometer, or in the case of enantiomers, a polarimeter. The method of elution on a plateau was first suggested by Reilley et al. [147]. A steady stream of a solution of the studied component in the mobile phase is pumped through the column until equilibrium is reached, i.e., until the breakthrough of the constant concentration plateau has been reached. Then a small pulse of the component is injected. The velocity and the retention time of that pulse are related to the isotherm through the equations uz =
U 1 + F dq/dc
(3-94)
and ~F%)
(3.95)
Single-Component Equilibrium Isotherms
128 C mg/mL
UV or Non-selective Detector
Selective Defector for Tracer
Tracer Injection
16 Individual Profiles
Figure 3.38 Typical chromatograms obtained in the determination of equilibrium isotherms by pulse chromatographic methods, (a) Injection on a concentration plateau, and response of a selective detector for a tracer, (b) Response of a nonselective detector for the tracer injection made in (a), (c) Individual profiles of the labeled tracer (gray line) and the unlabeled component (black line).
C mg/mL c
21
20 Tracer Injection
19 time, mini 6
These equations are derived in Chapter 7 (Eqs. 7.3 and 7.4, respectively). They give the pulse velocity (equal to the velocity associated with the plateau concentration) and the pulse retention time, and permit the derivation of the slope of the isotherm at the plateau concentration. The isotherm is determined by repeating the procedure a number of times while progressively increasing the concentration of the solution and by numerical integration of the plot of dq/dC vs. C obtained. The method of elution of an isotopic pulse on a plateau was developed by Helfferich and Peterson [136] (see Figure 3.38). If labeled and unlabeled molecules are injected simultaneously on a concentration plateau, the unlabeled molecules travel at the velocity associated with the plateau concentration (Eq. 3.94), while the labeled molecules travel at the velocity associated with the concentration shock: (3.96) 1+F-: or (3.97) AC These equations are also derived in Chapter 7 (Eq. 7.5). The injection of a mixture of labeled and unlabeled molecules provides the simultaneous determination of a point of the isotherm and its tangent. Thus, the method is mathematically straightforward. Experimentally, it is simple provided that there is no isotopic effect, i.e., that the labeled and the unlabeled components have the same isotherm,
3.5 Determination of Single-Component Isotherms
129
which is generally the case. The injection and the elution profiles of the labeled compound or tracer (e.g., 13C-labeled compound detected in mass spectrometry) are given by the black line in Figure 3.38a, while the peak associated with the injection of the unlabeled compound (perturbation peak; see Chapter 13) is given by the shaded line (Figure 3.38b). This chromatogram is obtained with a detector that is not isotopically sensitive (e.g., a UV detector). Finally, the selective detector would record the shaded line in Figure 3.38b for the 12C (or unlabeled component) and the black line for the labeled 13C component. The sum of the two traces in Figure 3.38c is the signal of the nonselective detector in Figure 3.38b. The use of labeled molecules is extremely difficult and costly, however, except for the simplest molecules. The synthesis of properly labeled compounds requires special skills. U.S. Federal regulations make the use of radioactive isotopes most impractical, time-consuming, and expensive. The use of stable isotopes as labels is possible only in connection with mass spectrometric detection, which is neither an easy nor an inexpensive proposition in liquid chromatography. For these reasons, the pulse methods have been limited to the elution of a pulse on a plateau (or step and pulse method) and are not yet quite popular in liquid chromatography. The primary difficulties encountered are experimental. At low solution concentrations, the pulse size must be small to ensure that it corresponds to a small perturbation of the local composition and propagates at the velocity given by the tangent to the isotherm or by its chord. A concentration vacancy may also be injected, and the retention time of either a positive or a negative small concentration pulse must be the same. If the difference between these retention times is small, their average may be taken. At high solution concentrations, the signal noise must be low enough that a small pulse can be detected. Since the injected pulse must be small, its elution profile should be Gaussian. The minimum amount injected can be derived from the noise level, the detector response, and the relationship between sample size, m, and the maximum concentration of the peak, CM-
CM=
^m
( ]
where N is the column efficiency and VR the retention volume of the pulse. Two successive pulses, a positive and a negative one, can also be injected at a time interval of the order of several times the elution band width of each of these pulses. Measuring the retention time at the zero crossing between the two signals permits the reduction of the dependence of this retention time on the pulse size to the second order. The experimental problems encountered at high concentrations were discussed recently [32]. Figure 3.39 shows a plot of the retention times of small pulses injected on plateaus of increasing concentrations, using a monolithic Cissilica column and 4-tert-butyl phenol as the sample [148]. The plot is the derivative of a Langmuir isotherm. The method is advantageously combined with FA because the elution of a pulse on a plateau requires prior equilibration of the column with a constantcomposition stream, i.e., the elution of a breakthrough curve, and neither method needs detector calibration (unless one uses the integration of the breakthrough curve to calculate the amount of compound adsorbed at equilibrium). The method
Single-Component Equilibrium Isotherms
130 Figure 3.39 Plot of the isotherm derivative versus the mobile phase concentration. Symbols, experimental data. Line, derivative of the best Langmuir isotherm. Reproduced with permission from A. Cavazzini, A. Felinger, G. Guiochon, J. Chromatogr. A, 1012 (2003) 139 (Fig. 2).
has the advantage that it calculates directly the isotherm slope as a function of the concentration plateau. The isotherm is straightforwardly derived by integration of the plot of this slope versus the concentration and the phase ratio of the column is the only parameter needed. The amount of packing material in the column is not required. Like FACP or FA, its major disadvantage is the large amount of material one must use to generate the concentration steps required.
3.5.5 The Retention Time Method If the isotherm is Langmuirian, the analytical solution of the ideal model (see Chapter 7, Section 7.3) can be inverted and used to derive the best isotherm parameters from one large and one very small size sample injection [137]. The very small injection gives the retention time under linear conditions, proportional to the initial slope of the isotherm. For a Langmuir isotherm, the elution band profile has a very steep, sharp front whose retention time is a function of the loading factor only, i.e., the ratio of the sample size to the column saturation capacity. The retention time of a large sample permits the determination of the corresponding value of the loading factor, hence the column saturation capacity. The values of the coefficients of the Langmuir model are given by the following equations: a Lf
=
bqs =
=
1-
b =
Lf
j
(3.99)
y Ft0 -tp-
- to)
tP
(3.100) (3.101)
The parameters a, b, and qs are the parameters of the Langmuir isotherm (see Eq. 3.47). The times tp, t^p, and tf are the hold-up time of the column, the retention time of the compound under analytical conditions and the retention time of the front of a high concentration band (i.e., of its inflection point). The experimental parameters are Fv, the mobile phase flow rate and nm, the amount of sample injected. Since the Langmuir model often gives a reasonable or acceptable represen-
3.5 Determination of Single-Component Isotherms
131
35 14
3
experiment frontal analysis analysis inverse method
: & \ 2.5
Cmax
30
12
C (g/L)
2
25 1.5 1
8
Cmax
S-1-indanol
20 q (g/L)
C (g/L)
10
0.5
15
6
0 8
8.5
9
9.5
10
10.5
11
time (min) (min)
11.5
R-1-indanol 10
4
5
2
0
0 7
8
9
10 time (min)
11
12
0
5
10
15
20
C (g/L)
Figure 3.40 Illustration of the method of isotherm measurements by computation of elution profiles. R-1-indanol on cellulose tribenzoate chiral stationary phase. Mobile phase, nhexane and 2-propanol (92.5:7.5, v/v). (Left) Calculated (using the bi-Langmuir isotherm) and experimental chromatograms recorded for 46.25 (main figure) and 9.251 mg (insert) of R-1-indanol. The isotherm was determined from the band profile obtained for 46.25 mg. (Right) Bi-Langmuir isotherms obtained by the inverse method (lines) and by frontal analysis (symbols) for the R- and S-1-indanol enantiomers. C max indicates the maximum elution concentration. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, ]. Chromatogr. A, 35 (2003) 1005 (Figs. 2 and 3).
ration of the experimental equilibrium data in the single-component case2, the method might appear to be rather general. The method fails, however, if the isotherm is not Langmuirian. This is especially the case in liquid-solid chromatography when the surface is not homogeneous but contains two or several kinds of sites and the isotherm is well represented by a bi- or multi-Langmuir isotherm. For this reason it is always advisable to record several band profiles corresponding to different sample sizes. Any systematic deviation between the parameters of the Langmuir isotherm determined by the RTM method when using band profiles corresponding to different sample sizes should be tentatively ascribed to a model error. This method is very convenient for rapid isotherm determination. Like ECP, it can be performed with small amounts of material and could be convenient for studies involving biochemicals or other costly products. Unlike ECP, the RTM method does not require detector calibration because only retention times are measured. However, the method lacks redundancy and is neither as precise nor as accurate as FA, FACP, ECP, or the pulse methods.
3.5.6 Computation of Elution Profiles (CEP) Method or Inverse Method (IM) In this novel approach to isotherm determination a simplex algorithm permits the minimization of the error between the measured profile of a high-concentration band and the profile calculated from the approximate isotherm and the column 2
Unfortunately, this is rarely valid in RPLC. In this mode, multi-Langmuir isotherms are prevalent.
132
Single-Component Equilibrium Isotherms
Figure 3.41 Langmuir isotherm of 4-tertbutyl phenol measured by FA and by IM on two Chromolith columns. The inset illustrates the column-to-column reproducibility of the measurements. Reproduced with permission from A. Cavazzini, A. Felinger, G. Guiochon, J. Chromatogr. A, 1012 (2003) 139 (Fig. 1).
efficiency, using a numerical method of calculation of chromatographic profiles (see Chapter 10, Section 10.3). The method requires an initial estimate of the parameters, which could be obtained from the RTM method, for example. Using a satisfactory isotherm equation and a robust minimization algorithm permits the determination of the best set of isotherm parameters [149-151]. However, with this method, an adsorption isotherm model is needed a priori, while in the other methods (e.g., FA, ECP, or the tracer pulse method) the amount of component adsorbed per unit volume of stationary phase is obtained directly from the experimental data and model fitting is done afterward [150]. This is the main source of difficulty with this heavy computational approach. We show in Figure 3.40 an example of results obtained with this method [119]. Figure 3.40a shows that a good fit to the chromatographic data is achieved, i.e., that the convergence criteria are well met. The isotherms derived from the peak shape data shown in Figure 3.40b (solid line) agree well with the frontal analysis data (symbols). Because of the rapid dilution of the bands during their migration along the column, the isotherm derived by this method can be accurate only within the range of concentration extending from zero to the band maximum. As with all isotherm determination methods, the isotherm obtained should not be extrapolated much beyond this value. This method is becoming extremely useful with the recent work of Felinger et al. [152]. The investigations of the reproducibility of adsorption data measured on commercial columns of the same lot or the same brand, the comparison of the equilibrium isotherms of a compound on different Qg-bonded silica columns, the study of the influence of systematic changes in the experimental conditions under which equilibrium is studied on the parameters of the isotherm are all considerably facilitated by the systematic use of this method. The isotherm data are first acquired by FA on a typical system, under a given set of experimental conditions. These data are modeled. Then, the IM method is used to acquire rapidly the data needed. The basic assumption made in this method is that moderate changes of the experimental conditions (e.g., fluctuations of the production conditions of the packing material, changes in the column temperature or in the methanol concentration of the mobile phase) do not affect the isotherm model, but merely change the numerical values of the best parameters. Recent examples of applications of this approach are easily found [61,103,116,153,154]. The results obtained with two different methods, FA and CEP, on two Chro-
3.5 Determination of Single-Component Isotherms
133
molith columns of the same batch, having hold-up times of 1.38 ± 0.01 mL and 1.39 ± 0.01 mL are compared in Figure 3.41. The isotherm follows Langmuir behavior and the values of the b coefficients differ by 1%, corresponding to differences in the column saturation capacities of 2.5%, which indicates that the reproducibility of the monolithic columns is quite satisfactory and that the results of the FA and CEP methods are consistent.
3.5.7 Nonlinear Frequency Response Frequency response is a commonly used method for the investigation of process dynamics and for model identification [155]. It has been used for the study of mass transfer kinetics in adsorption [156]. It was recently proposed for the measurement of equilibrium adsorption data [157]. The method consists in equilibrating the column with a stream of constant concentration of the studied compound (as in FA or FACP), then in varying the stream concentration periodically around this initial value, registering the detector response, and analyzing this periodic response, using the generalized Fourier transform, similar to the analysis of nonlinear electrical and mechanical systems submitted to periodic fluctuations of their parameters [158]. While the linear frequency response consists only of a constant term and the first harmonic, the nonlinear frequency response contains, besides a constant component, an indefinite number of harmonics of higher order. If the input concentration is Ca = CH/o + Cfl/i cos(a;£), the concentration in the column eluent is Q> =
Chfi + Cb/1 cos(cot +
(3.102)
For mass balance reasons, Q 0 — Q,o- The problem is to find a relationship between the isotherm parameters and the parameters Qy and (pi of the z-th harmonic. In the calculations made to relate the equilibrium isotherm and the response of the system (Eq. 3.102), the equilibrium-dispersive model is used (Chapter 2, Section 2.2.2) and the mass balance equation is integrated with the Danckwerts boundary conditions (Chapter 2, Section 2.1.4.3) and with the initial conditions C = Cfl,o, q = q(Cajo). The isotherm can be replaced by its Taylor expansion around the average concentration Cgfl
= f{C) =
^
, C3 (3 103)
"
-
2
2
3
3
The partial differentials of the isotherm, 3/(C) /dC, d f(C) /dC , d f(C) /dC , • • • for C = Cafi are numerical coefficients that can be derived from the parameters Qy and ty of the column response to the cyclic perturbation [156]. However, this derivation is a complex mathematical problem which goes well beyond the scope of this chapter. The reader is referred to the literature [155,156]. The values of
134
Single-Component Equilibrium Isotherms
Figure 3.42 Static sorption of bovine serum albumin onto alkylsilicas. Solvent: 2-propanol (40%)/0.05 M phosphate, pH 2.1. (A) 500-A pore, data using C$ and CJS alkyl groups superimposable; (A) 100-A pore, data using Cg and Cjg alkyl groups superimposable; (a) 100Apore, Cig; (•) 100-A pore, C$; (D) concentration in solvent determined by UV measurement, other data by fluorescence. T = 23 ± 1°C. Reproduced with permission from R.A. Barford, in "HPLC in Biotechnology," W.S. Hancock, Ed., 1992, p. 63 (Fig. 1), Academic Press, New York.
these coefficients are independent of the column efficiency which affects only the precision of the measurements of the experimental parameters Qy and (pi. The interest of this method is that it allows the direct derivation of the first few differentials of the isotherms from the experimental results, hence it affords valuable information on the shape of the isotherm and, particularly, an accurate determination of the concentration(s) at which the second derivative becomes equal to zero and the isotherm has one (or a few) inflection point(s). The analysis of the frequency response can also give valuable information on the mass transfer kinetics. On the other hand, the experimental measurements made with this method require more time and chemicals than FA, FACP, or even the pulse method. It requires first that equilibrium be reached for a number of successive plateau concentrations and that numerous, relatively long records of the response to a sinusoidal input be recorded, which makes the method slow and rather expensive. It is probably useful mostly for the difficult cases of isotherms having several inflection points.
3.5.8 Static Method This method consists of immersing a known weight of adsorbent in a solution of known concentration in a closed vessel and waiting until equilibrium is reached. After a time, which depends on the size of the vessel, the amount of materials involved, and the particle size of the adsorbent, but is usually between a few hours and a few days, a state close enough to equilibrium is reached. The solution is then analyzed. An accurately known weight of the component (or of the adsorbent) is added to the system, and the process is repeated until enough data points have been acquired and the adsorbent saturation approached closely enough. In many respects, the static method is comparable to FA, except that it is much slower and usually requires larger amounts of material. Figure 3.42 illustrates isotherm data for bovine serum albumin determined on alkyl-bonded silica by the static method [159]. The static method may also be carried out in the column or a precolumn, as described by Gerritse and Huber [160], for the study of gas-solid equilibria. A solution of known concentration is left to equilibrate with the stationary phase
3.6 Data Processing and Assessment
135
by switching off the pump or bypassing the column for a certain time. Then the stream is resumed through the column, and the plateau concentration of the equate is measured. An integral mass balance gives the amount adsorbed by the stationary phase at equilibrium with the concentration of the solution that leaves the column. The same method provides the possibility of measuring the rates of adsorption and desorption when this kinetics is slow.
3.6 Data Processing and Assessment Once acquired, isotherm data should be processed and fitted to an isotherm model. Depending on the complexity of the problem at hand and on the amount of computing power available, different procedures can be used. This may have a bearing on the accuracy of the model coefficients, a most important factor for some applications, such as the prediction of individual band profiles and computerassisted optimization studies.
3.6.1 Processing Experimental Data into an Isotherm Equation The traditional methods use linearization techniques that were described earlier (see Eqs. 3.50). When the isotherm data follow Langmuir isotherm behavior, plots of C/q versus C or of q/C versus q give straight lines. The latter is known as the Scatchard plot (see Eq. 3.50 and examples in Figures 3.8 right, 3.12right, 3.16 right, and 3.43). This plot transforms the a bi-Langmuir isotherm into a hyperbolic plot, whose asymptotes are the straight lines corresponding to the two Langmuir terms [35]. A similar result is achieved with another, similar linearization method, which is given in Eq. 3.49 and is illustrated in Figures 3.25b, 3.26 insert, 3.27b, and 3.28b). Assuming a random error of constant standard deviation over the measurement range, Harrison and Katti [161] have shown that the errors made on the parameters determined by a linearization procedure can be very important. Although their assumption of a constant absolute error over the entire concentration range where measurements are performed is somewhat unrealistic, their work points to some important questions. The use of nonlinear minimization algorithms allowed by the availability of modern computers permits improved accuracy in the determination of the isotherm parameters. Another important factor to consider is the objective function used in the fit. The best results seem to be obtained by minimizing the following objective function, using Marquardt's method [162]:
cr =
ND-P
(3.104)
where ND and P are the number of data points and of model parameters, respectively, while qfx and qf1 are the experimental data points and the corresponding values calculated from the model, respectively. The objective function in Eq. 3.104
136
Single-Component Equilibrium Isotherms
gives data points a weight inversely proportional to qf-, which forces the calculation to account as well for the low-concentration data points as for the highconcentration ones. This is important, because otherwise, with a conventional least-squares fit procedure, the isotherm is inaccurate in the low-concentration range. Then the rear boundary of the calculated band profiles is in error [20].
3.6.2 Determination of Critical Experimental Parameters In order to determine accurate isotherm data using most of the methods described earlier, it is necessary to measure two important parameters that characterize the column used for these determinations: the hold-up time and the column volume, or rather the amount of packing material that it contains. 3.6.2.1 Determination of the Hold-Up Time The definition of the hold-up time is simple in gas chromatography because the interactions between mobile and stationary phases are practically negligible. This is not so in liquid chromatography [163-165]. The density of the mobile phase is not the same in the bulk and in the monolayer in contact with the surface of the adsorbent. The situation is more complex in RPLC because the bonded layer swells when the proportion of the organic modifier in the mobile phase increases [32,166,167]. The organic modifier dissolves in the bonded layer and, when its concentration in this layer is sufficient, some molecules of water may also penetrate it. The hold-up time of a column is a function of the nature and concentration of the organic modifier in an aqueous solution [168,169]. In order to predict accurately the elution band profiles, it might be necessary to account for the dependence of the hold-up volume on the mobile phase composition, which requires the use of a different mass balance equation in which the phase ratio has been left in the differential elements [170]. The value obtained for the hold-up time depends on the choice of the inert or non-retained tracer selected to measure it. This choice is influenced by the need to detect this tracer at the column exit. The detection of isotopically labelled compounds being impractical, except for D2O, which raises other problems [167, 171], best results are obtained with organic compounds. In RPLC, thiourea or uracil are often selected. The product of the hold-up time and the flow rate of the mobile phase gives the hold-up volume from which, knowing the column dimensions, the total porosity of the column and the phase ratio can be calculated. The hold-up time can also be measured from the perturbation signals recorded upon injection of small amounts of water or methanol, although there might be detection problems. Although thiourea and uracil are generally not retained, there are reports that, under certain experimental conditions, they can be retained. This possibility must be checked. For example, Gritti [172] has measured the retention factor of thiourea on a Cis-bonded silica column and found values of the order of 0.15 in pure water, 0.05 in a 50:50(v/v) mixture of water and methanol and 0.12 in pure methanol. The hold-up volume must be consistent with the known properties of the column.
3.6 Data Processing and Assessment
137
The many different methods suggested to determine the hold-up time have been reviewed in detail [165]. The ions of salts like NaNO3/ KI, KCr2O7, NaCl are prevented by the Donnan salt-exclusion effect [173,174] from penetrating the intraparticulate pores of the packing material, at least if the mobile phase does not contain any salt in solution. The retention volume of a small pulse of a dilute solution of one of these salts (e.g., NaNC>3 in RPLC) can be used to determine the external porosity (ee) of the column. The internal porosity (£;•) is then derived from the total and the external porosities [175]. However, it has been reported that the method can present difficulties because the exclusion of the ions decreases with increasing salt concentration in the sample. Berendsen et al. [176] recommend the use of small pulses of solutions at concentrations below 1 x 10~4 M of NaNC>3 or KBr. Pulses of much larger amounts of salt have a retention time that tends toward the hold-up time with increasing concentrations. It would be inadvisable to use concentrated salt solutions for the measurement of the hold-up time. Unreliable results could be obtained if the mobile phase contains a high concentration of ions. Such high salt concentration may cause demixing of aqueous solutions of an organic modifier, except for methanol. Demixing would lead to the separation of the mobile phase into a water-rich and an organic modifier-rich fraction between which the solutes would equilibrate, preventing successful chromatographic separation. Finally, the hold-up time derived from the retention of a tracer compound must be corrected for the extra-column volumes, between the injection system and the column inlet and between the column outlet and the detector. Depending on the instrument, this correction may be small or significant. The correction is best assessed by measuring the hold-up time of the tracer with a zero-volume connector replacing the column. This correction is particularly important when isotherm data are measured in the laboratory, on a narrow-bore column, with a certain instrument, and are to be used for the calculation of the chromatograms that are expected to be obtained on a wider bore column, with a different instrument, in which case another correction is needed for that other instrument. 3.6.2.2 Column Dimensions The equilibrium isotherm relates the concentrations of a compound in the stationary and the mobile phase at equilibrium. The isotherm is independent of the amount of stationary phase contained in the column. However, this amount is needed in the FA, FACP, and ECP methods, to calculate the stationary phase concentration from the data measured, that is the amount of solute in the column at equilibrium (see Eq. 3.92). Because, in practical applications, most isotherm information is used for the purpose of scaling up a process, the best approach would be to refer this amount to the mass of packing material contained in the column. This is impractical because columns are usually purchased, the weight of the tubing is rarely supplied by the manufacturer, and only the dry weight of packing material would be useful. Since the dry weight of packing in the preparative column is also unknown, it is more practical to refer the data to the volume of the column, with the caveat that the actual mass of packing material in a column of known volume
138
Single-Component Equilibrium Isotherms
depends on the compression stress applied during the packing process [177]. The density difference may be significant between an analytical column packed by the filtration of a slurry of the packing material through the column end frit and a preparative column consolidated under axial compression. The importance of the systematic errors on the isotherm data that can be introduced by the use of an erroneous column diameter in the numerical calculations is illustrated by data from Zhou et al. [178] who compared the isotherm data acquired on three different columns (i.d. 1.07, 4.57, and 10.1 mm) packed with the same stationary phase. The data were eventually shown to be in close agreement, with the band profiles on the large column matching closely the profiles calculated with the isotherm data measured with the microbore column. However, the initial comparison of the isotherm data suggested a marked disagreement. This was explained by a difference of 0.07 mm between the nominal column i.d. of the microbore column supplied by the manufacturer and used in the initial calculations, and the true i.d., measured later with an electronic caliper. A quantitative discussion of the importance of the errors caused by a small error on the column diameter is available [178]. Jandera et al. [179] showed excellent agreement between the adsorption isotherm data obtained by the pulse method for benzophenone on a Qg-bonded silica using two columns of different diameters, 3.2 and 0.32 mm, made with exactly the same packing material. They showed that the pulse method might be far easier to use for accurate measurements than the FA method.
3.6.3 Accuracy and Precision The major reason why we give so much importance to equilibrium isotherms is that their accurate knowledge is necessary for any quantitative prediction of individual band profiles in all modes of chromatography and for optimization studies. As we show later, however, band profiles are sensitive to the shape of the isotherm. Thus, they must be determined with accuracy and precision. In connection with this issue, several points are important to discuss. The retention factor under linear conditions is related to the initial slope of the isotherm (kQ = Fdq/dC). It is easy to check that this relation applies when the isotherm coefficients are calculated. Deviations from this relation are not unusual, and can often be ascribed to the heterogeneity of the adsorbent surface that can be explained, for example, by a small concentration of underivatized silanol groups on the surface of chemically bonded silica. It may be that an additional Langmuir isotherm term would be necessary to account for the contribution of these groups, term that could become saturated at very low concentrations [172]. Isotherms depend significantly on the column temperature. We have shown earlier (see Figures 3.33a [38] and 3.33b [54]) isotherms determined at different temperatures for the same systems. From these results, it seems that the amount of low-molecular-weight components adsorbed at equilibrium with a constant mobile phase concentration increases by about 1% when the temperature decreases by about 1°C. For more complex isotherms, Van't Hoff law may not apply. For example, with the bi- or tri-Langmuir isotherms, the equilibrium constant of the
3.6 Data Processing and Assessment Figure 3.43 Scatchard Plot of Isotherm data acquired with different methods. Reproduced with permission from Y. Xie, B. Hritzko, C. Y. Chin and N.-H. L. Wang, Ind. Eng. Chem. Res, 42 (2003) 4055 (Fig. 14). ©2003, American Chemical Society.
139 cis-(-)
0
2 "
1.5
^
g
s
cis-(+)
© Pulse
* Pulse
A Frontal
A. Frontal
• SMB
• SMB
s B
1 15
high-energy sites decreases fast with increasing temperature while that of the lowenergy sites decreases more slowly. This explains a rapid decrease of the retention factor in a certain temperature range, followed by a slower variation at higher temperatures. The saturation capacity tends to decrease slowly with increasing temperature. Thus, it is important that, during the measurement of isotherm data, the column temperature should be kept constant within a few tenths of a degree and measured with an accuracy of about 1°C. This can now be achieved easily with the use of recirculating thermostatic baths or column ovens. The adsorption isotherms of cis (±)-FTC-ester), a precursor of a potential drug, were measured using three different methods: Pulse, FA, and simulated moving bed separation (SMB) [180]. The results are compared in Figure 3.43. The pulse method is an inverse chromatography method that uses two chromatograms of each pure enantiomer, one in the linear range of the isotherm, the other overloaded. In the particular case, the Langmuir isotherm gives satisfactory results, consistent with the FA data. The SMB method is another inverse method using the general rate model. The isotherm (here Langmuir) parameters and the intraparticle diffusivities are adjusted to fit several chromatograms. Other important experimental parameters are the flow rate (which does not affect the isotherms that are measured by FA as long as the data are derived from the integration of the breakthrough curve but may affect them significantly, when the flow rate is high, if the data are derived from the simple but erroneous method of measuring the retention time of the inflection point of the breakthrough curve), the holdup time (which must be accurately measured and corrected for the extracolumn hold-up time, between the pump and the column inlet and between the column outlet and the detector cell), and, for certain methods, the detector response. Accurate results are obtained by FA only if the detector response factor is accurately measured in the concentration range investigated, because this is needed to determine the area of the integral of the breakthrough curve. Pumps currently available for liquid chromatography give a flow rate stability that seems adequate. With some instruments (e.g., HP 1090, Hewlett-Packard, Palo Alto, CA), the quality of the gradient delivery system permits the achievement of accurate plateau concentrations by mixing two streams of known compositions and known flow rates, hence it allows the acquisition of highly accurate data [181]. Finally, the dead time should be measured with a component that is really unretained. When the mobile phase is a mixture, care should be taken in the determination of the dead time [182]. There are two contributions to the column hold-up volume: one that arises from the volume of the mobile phase contained inside the column and one that is due to the instrument hold-up volume (volumes of the
140
Single-Component Equilibrium Isotherms 1.5
Figure 3.44 Effect of flow rate on the reproducibility of adsorption isotherms, pCresol on a 50 x 1.2 mm column packed with ODS 5-jim Spherisob, eluted by 0.2 M sodium phosphate buffer at pH 6.3, and 25°C. Flow rates: 11, 34, 42, 74, and 165 ^L/min. Reproduced with permission from } . Jacobson, J. Frenz and Cs. Horvath, J. Chromatogr., 316 (1984) 53 (Fig. 8).
60
80
100
120
tubings and connections). These two parameters should be determined separately to obtain accurate values of the column hold-up volume. Finally, the validity of the chromatographic methods for the determination of isotherms is based on the assumption that phase equilibrium is reached rapidly during the experiment. The rate constant for phase equilibration must be large enough for the experimental results to be independent of the mobile phase velocity. When carrying out FA, FACP, or ECP measurements on proteins that tend to equilibrate slowly, it is advisable to check the influence of the flow velocity on the isotherms (Figures 3.15 and 3.44 [38]).
3.6.4 Comparison of the Main Chromatographic Methods It has been shown in many investigations that ECP, FA, and FACP give the same experimental isotherm as the static method. An example of such results is given in Figure 3.45a [8], which shows that data obtained by a static method and data derived by a dynamic (i.e., frontal analysis) method compare very well. Among studies comparing different chromatographic methods, those by Jacobson et ah [38] and Golshan-Shirazi et al. [137,140,183] are important. Such a comparison is illustrated in Figure 3.45a and 3.45b. The former shows that data points determined by a static method and by FA give overlaid isotherms. The second shows that FA, ECP, and FACP lead to nearly identical isotherms [140]. The results of the RTM and the computation methods have also been shown to agree with those of FA, FACP, and ECP [140,183], as illustrated in Figure 3.45b [183]. The choice of a method of isotherm determination depends on the nature of the problem, the chromatographic system chosen, the equipment available, and the cost of the components studied. In contrast to the other chromatographic methods used for the determination of isotherms, detector calibration is unnecessary in FA (when the column efficiency is high and the integration of the breakthrough curve is not necessary for an accurate estimate of the breakthrough volume of a concentration), which may be a great advantage. Moderate deviations of the detector response from linear behavior would not result in a significant error. However, calibration can be made
3.6 Data Processing and Assessment
141 too
Figure 3.45 A-(above) Comparison of isotherm measurement methods. Static methods and FA. RNase A on DuPont WCX-300, Langmuir isotherm, o, FA, a = 47.59 mg/mL, b = 0.5290 mL/mg; •, static method, a = 47.37 mg/mL, b = 0.5234 mg/mL. Reproduced with permission from J.-X. Huang and C. HorvUh, ]. Chromatogr., 406 (1987) 285 (Fig. 6).
B-(below) RTM, ECP, FA, and FACP of (1) acetophenone on silica with 2.5:97.5 ethyl acetate/ n-heptane; 2, benzyl alcohol on silica, with 15:85 THF/ w-heptane; benzyl alcohol on ODS silica with 15:85 MeOH/H 2 O; and (4) phenol on ODS silica with 20:80 MeOH/H2O. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal. Chem., 60 (1988) 2634 (Fig. 6), ©1988 American Chemical Society. C (mol/L)
concurrently from the plateaus of the step changes in concentration for other uses, as needed. In both FACP and ECP, the detector response signal must be converted to concentration by calibration of the detector. This can be done by direct injection of the sample into the detector cell at different concentrations, i.e., by injecting wide rectangular bands into the detector cell, possibly bypassing the column. In the latter case, time may be saved when both FA and FACP results are needed. The main advantage of FA is that the self-sharpening nature of the FA fronts permit accurate determination of the isotherm points even in cases where the mass transfer kinetics is relatively slow. The position of the equivalent point of the breakthrough curve is independent of the kinetics of mass transfers and of axial dispersion (Chapter 14). If the rate of this kinetics is very low, a slow mobile phase flow rate may have to be used, but the measurement will remain accurate provided that the breakthrough volume is determined by integration. In cases where the fronts are diffuse but the rear parts of the profiles are self-sharpening (concave upward isotherm), negative concentration steps should be used instead. The main inconveniences of FA are the large amount of material required, although this can be minimized by using microbore technology [8], and the long time required for
142
Single-Component Equilibrium Isotherms
Figure 3.46 Single solute adsorption data at 30°C obtained via FA (points) and FACP (lines). Compounds: benzyl alcohol (circles and solid line), phenylethanol (squares and dashed line) and methyl benzyl alcohol (triangles and dashed-dotted line) Reproduced with permission from I. Quinones, } . C. Ford and G. Guiochon, Chem. Eng. Sci., 55 (2000) 909 (Fig. 4).
the acquisition of the whole set of data needed for a single isotherm, since one concentration step provides only one point of the isotherm. With the FACP and ECP methods, each point of the rear profile gives one point of the isotherm. With modern systems of data acquisition several hundred data points of an elution profile can be conveniently recorded and stored, so many points of an isotherm can be acquired rapidly. Therefore, these methods are more precise than the FA method (although they are less accurate; see below). A significant advantage of FACP and ECP over FA is that they are approximately 25 times faster and require amounts of pure compounds and of mobile phase that are at least one order of magnitude less. The amount of wasted compounds and solvents generated by these methods is comparatively much smaller than for FA. Quinones et al. measured the isotherm data of three aromatic alcohols on Symmetry Ci8 column with methanol/water (50:50, v/v) as the mobile phase [129]. Data were derived from both FA and FACP experiments, at 30°C (see Figure 3.46). There is a very good agreement between the data obtained by the two different methods. The coincidence between FA and FACP demonstrates the excellent thermodynamic consistency of the measured single solute data (in part a consequence of the use of highly efficient columns). The main drawbacks of the FACP and ECP methods are: (i) the need for an accurate calibration of the detector response; (ii) the influence of the axial dispersion on the rear profile, due to the finite rate of the mass transfer kinetics and the axial diffusion; and (iii) the cumulative character of some measurement errors that occur in the calculation. Drawback (iii) is shared with FA (but only in the staircase mode of FA). If an error is made in the amount adsorbed at low compound concentration, it results in a shift of the entire isotherm. In ECP and FACP, this is all the more important because of the influence of the finite column efficiency on the band profile and especially on the end of the profile tail. This tail may be complicated sometimes at low concentrations by some excessive tailing resulting from the influence of possible high-energy, active sites which may be present on the adsorbent surface. A multisite isotherm model may be needed. The accurate determination of the parameters of the term corresponding to the active sites requires the acquisition of a sufficient number of accurate adsorption data points in the low or very low concentration range [75]. Note that the use of multi-Langmuir
3.6 Data Processing and Assessment Figure 3.47 Dependence of the isotherm determined by ECP (or FACP) on the column efficiency. The ECP method is based on the ideal model profile (cf Eq. 7.4). A Langmuir isotherm (solid line, b) is used to calculate the band profiles obtained with columns of different efficiencies ( Lj = 10%). The profiles (a) are used to derive the isotherm following the ECP method. The isotherms differ from the initial Langmuir isotherm. The best fit of the data to a Langmuir model generates significant model errors, with deviations of the order of 1% between the initial and the measured coefficients of the isotherm for N = 5000 plates, larger at lower efficiencies and loading factors.
143 Cmg/mL 20i
a
N=5000 ...
N=1000 N=500 N=200
10
•' X , t, min 6
10
14
18
am 9 solute/mL packing
b
200
100
Cmg/mL 0
5
10
IS
20
isotherm models is justified only if the data can be acquired in a sufficiently wide range of concentrations. Equation 3.95 is valid only in ideal chromatography. Real columns have a finite efficiency. The ECP and FACP methods do not take into account the band broadening due to axial diffusion and mass transfer resistances, which can be significant on the diffuse part of the band profile. This problem was recognized and discussed for gas-solid equilibria by Huber and Gerritse [184]. These authors provided a demonstration of the influence of the column efficiency on the apparent isotherm by comparing isotherms measured by various chromatographic methods using columns of different lengths. Thus, ECP and FACP methods are less accurate because the exact profile of the rear part of the band depends on the kinetics of mass transfer and on the axial dispersion [142]. This problem is illustrated in Figure 3.47, as ECP and FACP are based on the ideal model solution. Thus, FACP and ECP should be used only with high-efficiency columns. The main advantage of ECP over FA or traditional FACP is that a much smaller amount of material is needed to determine an isotherm. However, FACP can be advantageously carried out by injecting a wide enough rectangular band of component solution in the column, rather than pumping a constant-concentration stream through the column until breakthrough and then purging the column. FACP is the complement to FA and has the major advantage that comparison of FA and FACP data permits the determination of the possible extent of adsorption hysteresis. Finally, we note that the accuracy of the FA method depends in part on the purity of the sample of the studied compound that is available. If there are early eluting impurities, the results may be biased for either of two reasons. First, there might be a retainment effect or delay of the elution of the front of the studied compound due to the presence of the impurity and the influence of its competitive iso-
REFERENCES
144
Figure 3.48 Comparison of the purchased and cleaned Troger base mixtures. Analytical injection at wavelength of 308 nm (a) and 315 nm (b). Solid lines, original sample; dashed lines, sample purified by crystallization; dotted lines, sample purified by preparative HPLC separation on a Cigbonded silica. Reproduced with permission from K. Mihlbachler, K. Kaczmarski, A. SeidelMorgenstern, G. Guiochon, J. Chromatogr. A, 955 (2002) 35 (Fig. 4).
time [min] 1.0-1
b
t
1
0,8-
0.3-
0.4-
0.2-
0.0-
N
JJ w\ time [min]
therm on the elution of this front. This effect cannot be important unless the concentration of this impurity is quite significant. More frequently and importantly, an early eluting impurity for which the detector used has a large response factor may affect the record of the breakthrough curve and the estimate of the amount adsorbed at equilibrium. Thus, Mihlbachler et al. had difficulties in using FA to measure the single-component isotherms of the enantiomers of Troger's base [57]. They had to complete the FA data by data derived from the pulse method. Figure 3.48 shows the analytical chromatogram obtained for a sample of the racemic mixture used to measure competitive isotherms (see next chapter). Although the sample used was better than 99% pure, the large response factor of the impurity makes it difficult to determine the elution volume of the breakthrough curve, whichever definition is applied. The accuracy of the data afforded by the pulse method is not affected by this difficulty. Cavazzini et al. showed excellent agreement between isotherm data derived by FA and by the pulse method [148]. The best estimates of the two coefficients of the Langmuir isotherm were obtained with standard deviations of about 0.1% (the initial slope) and 0.3% (the saturation capacity) by FA and of about 0.7 and 1.5%, respectively, by the pulse method. The differences between the sets of values given by the two methods showed excellent agreement.
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Chapter 4 Competitive Equilibrium Isotherms Contents 4.1 Models of Multicomponent Competitive Adsorption Isotherms 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.1.7 4.1.8 4.1.9 4.1.10 4.1.11 4.1.12
Competition for Adsorption The Competitive Langmuir Isotherm Model The Competitive Bi-Langmuir Isotherm Simple, Thermodynamically-consistent, Competitive Langmuir Isotherm . . . . The Ideal Adsorbed Solution (IAS) The Real Adsorbed Solution (RAS) The Statistical Isotherms The Competitive Fowler Isotherm The Competitive Freundlich-Langmuir Isotherm The competitive T6th Adsorption Isotherm The Competitive Martire Isotherm Competitive Isotherms Models for Other Modes of Chromatography
4.2 Determination of Competitive Isotherms 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6
153 153 154 160 165 166 177 179 180 180 184 184 186
191
Competitive Frontal Analysis 191 Determination of a Multi-component Langmuir Isotherm by Frontal Analysis and the Reverse /i-Transform 196 The Pulse Methods 202 The Simple Wave or Hodograph Method 210 The Inverse Method 212 Measurement of the Competitive Isotherms of Enantiomers 214
References
216
Introduction All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is understood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-
151
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Competitive Equilibrium Isotherms
rium is a function of the concentration of this component, as for single-component isotherms, and also of the concentrations of all the other components present in the solution that are retained by the stationary phase.
The amount of any given component adsorbed at equilibrium with a solution almost always decreases with increasing concentration of any other adsorbed compound. This fact illustrates the competitive character of adsorption (we know one exception only, [1]). For this reason, multicomponent isotherms are often called competitive isotherms. The surface area of the adsorbent available to accommodate the adsorbate in the adsorption layer is limited and the total number of molecules that can be accommodated is finite. Then, as the concentration increases, molecules compete for access to the adsorption sites, and those of the more strongly adsorbed compounds tend to exclude the others. In chiral separations, the density of enantioselective sites limits the saturation capacity of a chiral stationary phase. Similarly, the capacity of the stationary phase is also limited when retention mechanisms other than adsorption are used. In ion exchange chromatography, for example, the number of charge carriers bonded to the resin is finite, limiting the number of ions that can be exchanged. In this chapter, we discuss first a number of models that have been used to account for competitive isotherm data. Although the multi-component extension of many of these models is straightforward, most of them have been used almost exclusively with binary mixtures, hi the second part of this chapter, we describe the methods of determination of competitive isotherms. Finally, we discuss the methods of data acquisition for multi-component adsorption and we present a few examples. An important part of the theoretical effort in the area of competitive isotherms is the quest for models that depend only on parameters accessible from singlecomponent determinations, i.e., on the parameters of the corresponding singlecomponent isotherms. From a practical point of view, such models are extremely attractive, especially since the determination of single-component isotherms (see Chapter 3, Section 3.5) is generally much faster and simpler, hence more accurate, than the measurement of competitive isotherm data. Isotherms whose parameters can be determined directly from single-component data include the competitive Langmuir, bi-Langmuir, and their extensions such as the LeVan-Vermeulen isotherms, the competitive Fowler isotherm, and the competitive Freundlich-Langmuir isotherms. Unfortunately, deviations from the competitive Langmuir isotherm model behavior are frequent. Their importance depends on the interaction energy between the molecules of the solutes and of the solvent and on their size and shape differences. Thus, success in this endeavor requires that the interaction energy between a molecule of one component and a molecule of the other be close to the interaction energy between two molecules of either one of these two components. This is true only in rare cases. Often, however, reasonable agreement between the experimental data and the predictions of simple models has been observed.
4.1 Models of Multicomponent Competitive Adsorption Isotherms
153
4.1 Models of Multicomponent Competitive Adsorption Isotherms Competitive isotherms have been studied later and less aggressively than singlecomponent isotherms. The understanding of mechanisms of competition between several components for access to the surface, or more generally for involvement in the retention mechanism, and the modeling of this competition still leave much to be desired. This situation is largely due to experimental difficulties encountered in measuring competitive isotherms with conventional methods. It was also due, until recently, to the lack of challenging problems. Preparative HPLC brings both convenient experimental procedures permitting the rapid acquisition of accurate data and important and challenging problems. This combination has revitalized this area of research.
4.1.1 Competition for Adsorption As in the previous chapter, we simplify the problem by considering the adsorption isotherm of the solute-adsorbate separately from the solvent, adopting a convention that the solvent is not adsorbed [2]. We do not consider the Gibbs excess isotherms. Admittedly, this would be more rigorous, but it would lead to a complicated presentation, and this complexity does not seem warranted in the present state of development of competitive excess isotherms and because the concentration of the solutions considered in HPLC rarely exceeds 10% (v/v). In practice, competition has profound importance on the thermodynamics of adsorption and on the shape of band profiles. Because the equilibrium isotherms of each component of the feed are competitive, i.e., the amount of any component adsorbed at equilibrium depends on the concentration of all the other components present in the solution, the mass balance equations (Chapter 2, Eq. 2.2) of these components are coupled. In other words, the migration rate of each component, already a function of the local concentration of this component, also depends on the concentrations of the other components present at the same place, i.e., of the components in the interfering bands. Thus, the bands will interact during their migration and separation, and the problems of understanding the mechanism of chromatographic separation, of predicting the individual band profiles, their migration along the column and their evolution, their elution as concentration histories at the column exit and of optimizing the experimental conditions become very complex. It must be noted that competition is not limited to the feed components. Socalled strong solvents are often added in small or moderate concentrations to weak solvents in order to adjust the retention range of the feed components {e.g., acetonitrile or methanol in reversed phase chromatography, with water as the weak solvent, or dioxane, ethyl acetate, methanol in normal phase liquid chromatography, with n-heptane or dichloromethane as the weak solvent). Additives are mixed with the mobile phase for different purposes, as complexing agents [3], as ion-pairing agents, or as detecting reagents (in the case of analysis by system
154
Competitive Equilibrium Isotherms
peaks [4]; see Chapter 13). Some strong solvents can also be used in gradient elution, to elute in a reasonable time the components of a complex mixture with a wide range of adsorption strengths [5]. Strong solvents can act and modify the retention of solutes in two different ways. In some cases, they compete with the feed components for adsorption. This is the mechanism through which they reduce the adsorption of the feed components and accelerate their elution. This is what occurs in normal phase chromatography when polar solvents are used as components of the mobile phase (e.g., propanol added to dichloromethane). Usually, the concentration at which these additives are used is low, often of the order of a few percent. However, not all additives act by competition for retention. Another mechanism by which additives or strong solvents modify retention times is by changing the solubility of the components in the mobile phase. For example, this is why ionic strength gradients are used in hydrophobic interaction chromatography. The solubility of proteins in water depends on the ionic strength of the solution. At high salt concentrations, proteins are more strongly retained because they are less soluble in salt-concentrated aqueous solutions than in pure water ("salting out" of proteins). In this case, there is obviously no possibility of competition: if there were any competition between the inorganic salts used and the protein, it would act in the other direction. Under linear conditions, the retention factors of acetonitrile and methanol on octadecylsilica in pure water are very low [6] and it is not possible for either methanol or acetonitrile to displace any organic compound from a C18-bonded silica stationary phase in reversed phase liquid chromatography (RPLC). Nevertheless, the addition to water of a significant concentration of one of these two solvents accelerates considerably the elution of many components on such a column because it increases their solubility in the mobile phase. These pure solvents can also be used for rapid regeneration of the column, because they are excellent solvents of organic compounds, much better solvents than water. Because they act as mere solvent of the eluates, not as competitors for adsorption, they are used in rather large concentrations, much larger than the polar additives added to the mobile phase of a normal phase chromatographic system. Other organic modifiers may act through a combination of the two effects, as both solvents and competitors. In the case in which the additive or strong solvent is strongly adsorbed by the stationary phase, it competes with the feed components. As a result, the individual band profiles can exhibit unique shapes and be considerably different compared to the band profiles obtained for the same compounds in a pure mobile phase with which the components have the same apparent isotherms (see Chapter 13).
4.1.2 The Competitive Langmuir Isotherm Model The Langmuir equilibrium isotherm model can be extended to multicomponent systems [7,8]. However, when several components are simultaneously present in the solution, these compounds interfere. The amount of each of them that is adsorbed at equilibrium is smaller than if this compound were alone. Although
4.1 Models of Multicomponent Competitive Adsorption Isotherms
155
it is easy to extend the Langmuir isotherm to a multicomponent system, we shall give the kinetic derivation of this isotherm in the case of a binary mixture for the sake of simplicity. At equilibrium, in the presence of a binary solution of components A and B, a part of the surface is free, that is, it is covered only by the solvent, part is covered by molecules of the first component, and the rest by molecules of the second component. Let the fractions of the surface covered by the molecules of solvent and components A and B be So, S^, and Sg, respectively. Then we have So + SA + SB = 1
(4.1)
At equilibrium, the rate of adsorption of each component is equal to its rate of desorption. The rate of adsorption is proportional to the concentration and to the free surface area, So- The rate of desorption is proportional to the surface area occupied by the molecules of that component. For each of them, we have at equilibrium: kdA = kaiiS0Ci
(4.2)
where the subscript i stands for either A or B and the coefficients k^ and fcfl/; are the rate constants of desorption and adsorption for component i, respectively. Eliminating So between Eqs. 4.1 and 4.2 gives kdA = kaM1
~
S
A-
SB)
(4.3)
The i Eqs. 4.3 make a system of two linear equations with two unknowns. Eliminating successively SA and Sg between them gives the solution: (4.4)
Obviously, the amount, g,-, of the component i that is adsorbed at equilibrium is proportional to the surface area covered by its molecule so the competitive isotherm is written
*= i +
ffiVc
(45)
where n is the number of components in the system. The coefficients U{ and ty are the coefficients of the single-component Langmuir equilibrium isotherm for component i (see Eq. 3.47, Chapter 3). The coefficient fo, is the ratio of the rate constants of adsorption and desorption, so it is a thermodynamic constant and its temperature dependence is of the Arrhenius type. The ratio flj/b; is the column saturation capacity of component i. From the model used, it is expected to be substantially independent of the temperature. It is obvious from Eq. 4.5 that the coefficients a,- and ty for component i in the competitive Langmuir isotherm are the coefficients of the single-component Langmuir isotherm of that component in the same chromatographic system.
Competitive Equilibrium Isotherms
156
Figure 4.1 Representation of a competitive Langmuir isotherm with 3-D plots. Plots of cj\ and CJ2 versus C\ and C-iThe competitive Langmuir isotherm has some important properties. First, we observe that the separation factor is constant, independent of the relative composition of the mixture. The separation factor for a binary mixture is by definition: « = ^ = ^
(4.6)
As a consequence, the competitive Langmuir isotherm model offers no possibility to account for a reversal in the elution order of two components with increasing concentration. On the contrary, experimental results show that such an inversion is possible, and that it is not even unusual when the column saturation capacities of the two single-component isotherms are different. For example, experimental adsorption data and chromatograms of mixtures of trans- and cz's-androsterone show an inversion of the elution order when the sample size increases (see later, Figure 4.8 and the related discussion below) [9-11]. Competitive isotherms such as the Langmuir isotherm (Eq. 4.5) are represented by surfaces in a suitable space. The single-component isotherm is represented by a curve in the (q, C) plane (see Chapter 3). Binary competitive isotherms are represented by two surfaces, one in the three-dimensional space (c\\, C\, C2), the other one in the space [f\i, Q/ Q2). Figure 4.1 shows an example of these plots [11]. These surfaces intersect, as is easy to show. Consider a vertical cylinder centered on the vertical axis of coordinates (Q — 0, C2 = 0). This cylinder intersects the plane (Q, C2) along a circle of radius C. It intersects the surface q\ (Q, C2) along a curve that decreases from q\ (C, 0) to 0 when we move from the axis C\ to the axis C2 on the cylinder. In the same time, the intersection of the isotherm surface q2{C\, C2) is a curve increasing from 0 (on axis C\) to q2(C, 0). These two curves intersect in one point of coordinates (Q, C2). In this point we have
(4.7a)
4.1 Models of Multicomponent Competitive Adsorption Isotherms = «2C2
157 (4.7b)
because then, in any given point of coordinates {C\, C2), the denominator of Eqs. 4.7a is the same for both components. Consequently, the two surfaces intersect along a curve that is contained in the vertical plane of Eq. 4.7a or C\ = «C2, Another possible representation of the competitive Langmuir isotherm surfaces is through the use of contour maps, that show the intersections of the isotherm surfaces by a series of equidistant horizontal planes, qj = k. Considering Eq. 4.5 for a binary mixture, we can write the following equation for the curve intersection of the isotherm surface for the first component by the horizontal plane qx = k:
Clearly, k cannot be larger than qS/x = ax/bx, otherwise Eq. 4.8 has no sense as there are no negative concentrations. Equation 4.8 defines a family of straight lines issued from the point of coordinates (Q = 0, C2 = —l/fc2). Only the parts of these lines that are in the first quadrant of coordinates (Q > 0, C2 > 0) have a physical sense and actually belong to the surface. A similar result is obtained for the other isotherm surface. The intersections of this surface by a series of horizontal planes q2 = k is a family of straight lines issued from the point of coordinates (Cj = 1/bx, C2 = 0). The equation of these lines is fl2 — kb2
a2
-kb2
These two surfaces are conoids [11]. The contour maps of the competitive Langmuir isotherm surfaces are not shown here because, being made only of straight lines, they are not informative [11]. The properties of the competitive Langmuir isotherm are described here in detail because of the importance of this isotherm model. Because this model is the only simple one, it is convenient for general discussions aimed at understanding competitive isotherm behavior and the relationships between competitive isotherm behavior and chromatographic band profiles under nonlinear conditions. The Langmuir model is often considered to be an acceptable first-order approximation of the experimental competitive equilibrium isotherms in a wide variety of cases. Its simplicity and the ease with which its parameters can be derived from experimental data make it very attractive. In many cases the deviations from the predictions of the competitive Langmuir isotherm are less than about 20%. A number of systems whose adsorption behavior is well accounted for by this model have been reported in the literature. A word of caution is required, however. In most cases, these systems were mixtures of similar compounds, e.g., geometrical isomers, or close homologous compounds (see example in Sections 4.2.2 and 4.2.2.2, and Figure 4.25). The range of concentration in which data were acquired was insufficient to detect deviations between a more complex isotherm model and its two-term expansion around the origin. The second coefficient of a Langmuir isotherm, hence its saturation capacity, can be accurately determined only if the product bCM of the b coefficient of either compound
158
Competitive Equilibrium Isotherms
and the maximum concentration of this compound in the solutions used in the experimental measurements is significant compared to unity. Sufficient attention is not always paid to this requirement. On the other hand, the quantitative prediction of competitive isotherm behavior for the components of binary mixtures is not possible using the competitive Langmuir isotherm model when the difference between the column saturation capacities for the two components exceeds 5 to 10%. For example, the adsorption isotherms of pure cis- and trans-androsterone on silica are well accounted for by the Langmuir model [9]. However, the two column saturation capacities differ by ~ 30%, due to the nearly flat structure of the trans isomer compared to the folded structure of the cis isomer. As a consequence, the competitive Langmuir model accounts poorly for the competitive adsorption data [9,10]. Much improved results are obtained with the more complex LeVan-Vermeulen isotherm (Section 4.1.5). Another approach could use the random adsorption site model, with different exclusion surface areas for the competing molecules [12]. Finally, it can be shown that the multicomponent competitive Langmuir isotherm (Eq. 4.5) does not satisfy the Gibbs-Duhem equation if the column saturation capacities are different for the components involved [13]. This profound inconsistency may explain in part why this model does not account well for experimental results. There are two very different alternative approaches to the problem of competitive Langmuir isotherms when the saturation capacities for the two pure compounds are different. Before discussing this important problem and an interesting extension of the competitive Langmuir isotherm, we must first present the competitive bi-Langmuir isotherm model. 4.1.2.1
Examples of Binary Langmuir Isotherms
In spite of these major fundamental objections, the competitive Langmuir model has been found to be useful in several practical cases. Most of these cases were found in normal phase HPLC [14,15]. Similar results were also reported in RPLC for solutes of closely similar sizes and structures. Figure 4.2 illustrates the best competitive adsorption isotherm model for benzyl alcohol and 2-phenylethanol [16]. The whole set of competitive adsorption data obtained using Frontal Analysis was fitted to obtain the Langmuir parameters column saturation capacity (qs =146 g/1), equilibrium constant for benzyl alcohol (bsA = 0.0143) and the equilibrium constant for 2-phenylethanol (bp^ = 0.0254 1/g). The quality of the fit obtained with this simple model is in part explained by the small variation of the activity coefficients of the two solutes in the mobile phase when the solute concentrations increased from 0 to 50 g/1. The Langmuir competitive adsorption isotherm simplifies also in the case where activity coefficients are of constant value in both phases over the whole concentration range [17]. Blumel et al. [18] measured the isotherms of (+) and(-) hetrazipine (WEB170) on cellulose triacetate, using the perturbation method operated in closed-loop mode to reduce the amount of chemicals needed. The data were satisfactorily modeled with the Langmuir equation. The separation of the enantiomers of a-ionone
4.1 Models of Multicomponent Competitive Adsorption Isotherms
159
Total i»ncentratioi> in liquid (g/l)
Figure 4.2 Experimental (+) and calculated (—) competitive Langmuir isotherms of benzyl alcohol (left) and phenyl-2-ethanol (right) on Cig-silica with MeOH/H^O as the mobile phase, at different relative concentrations. The best estimates of the isotherm parameters were obtained by a least-square fit of the data measured by competitive FA. (a) Benzyl alcohol; (b) Phenyl-2-ethanol. Reproduced from F. James, M. Sipillvida, F. Charton, I. Quinones, G. Guiochon, Chem. Eng. Sci, 54 (1999) 1677 (Figs. 5a and 5b).
by SMB, on a CSP made of a derivatized cyclodextrin bonded to silica and using water/methanol solutions was optimized under linear [19] and nonlinear [20] conditions, using gradient elution. The binary isotherms followed competitive Langmuir behavior. Using the /t-root method, Wang and Ching [21] measured the competitive isotherms of four of the stereoisomers of the /3-blocker nadolol on an immobilized cyclodextrin. These isotherms follow competitive Langmuir behavior. These isotherms were used for the modeling of a SMB unit. Kaspereit [22] determined the optimum conditions for maximum production rate by SMB (see Chapter 17) of the purified enantiomers of mandelic acid from the racemic mixture, using two different stationary phases and various mobile phases of different compositions. These authors found that their adsorption data fitted reasonably well to a competitive Langmuir model. 4.1.2.2
Examples of Ternary Langmuir Isotherm Systems
Few multicomponent competitive isotherms have been measured so far although the progress in the development of methods and the pressure arising from the development of preparative chromatography and the need better to understand competitive isotherms combine to render such investigations attractive. The experimental data of two ternary isotherms were measured by frontal analysis [17, 23] while those of a quaternary isotherm were determined by the perturbation method [24]. Quinones et al. [17] measured by frontal analysis multisolute adsorption equilibrium data for the system benzyl alcohol, 2-phenylethanol and 2-methyl benzyl alcohol in a reversed-phase system. Data were acquired for the pure compounds, for nine binary mixtures (1:3,1:1, and 3:1) and four ternary mixtures (1:1:3,1:3:1, 3:1:1, and 1:1:1). These data exhibited very good thermodynamic consistency. The thermodynamic functions of adsorption were derived from the single-solute ad-
160
Competitive Equilibrium Isotherms
sorption data measured at different temperatures. The values of the isosteric heats of adsorption measured suggested a nearly homogeneous surface for the adsorbent (Symmetry-Cis from Waters) under consideration. A nonideal adsorbed solution theory model applied to the experimental data provided an excellent prediction of the multicomponent adsorption equilibria based on the parameters derived from the single solute adsorption data. Deviations from competitive ternary Langmuir isotherm behavior were small. When used, this latter model provided reasonably accurate band profile predictions. Liquid-liquid equilibrium models were also used to describe the experimental data. In this case, the best representation of the experimental data was obtained when using 1-decanol to represent the stationary phase. Although the partition model provided a satisfactory representation of the experimental data at low concentrations, it failed to represent the curvature of the equilibrium data at higher solute concentrations and was unable to describe the temperature dependence exhibited by the single-solute adsorption data. The system of ternary isotherms was used later to calculate the band profiles obtained with rectangular pulses of ternary mixtures of various compositions and of several binary mixtures injected on a plateau of the third compound [25]. Lisec et ah [23] measured by frontal analysis the ternary isotherm data for phenol, 2-phenylethanol and 3-phenyl-l-propanol on Kromasil-Cis, with a water/methanol (1/1) solution. The data were fitted to the model equations of the competitive Langmuir and competitive bi-Langmuir models and to the IAS and RAS models derived from the Langmuir model. No substantial improvements were observed with the more complex models. Satisfactory agreement was observed between experimental band profiles and the profiles calculated from the ternary Langmuir isotherm.
4.1.3 The Competitive Bi-Langmuir Isotherm By analogy to the competitive Langmuir isotherm, when a surface is covered with two different kinds of sites and the feed components considered follow biLangmuir isotherm behavior, we can account for the competitive behavior of two components using a competitive bi-Langmuir isotherm [26]: qi
- i + bA/1cA + bB/1cB
+
,. 1 m
1 + bA,2cA + bB,2cB
As with the competitive Langmuir isotherm, the coefficients of this competitive isotherm are those obtained for the single-component isotherms of the two components. The conditions of validity of this isotherm model are the same as those of the competitive Langmuir isotherm, ideal behavior of the mobile phase and the adsorbed layer, localized adsorption, and equal column saturation capacities of both types of sites for the two components. The excellent results obtained with a simple isotherm model in the case of enantiomers can be explained by the conjunction of several favorable circumstances [26]. The interaction energy between two enantiomeric molecules in solutions is probably very close to the interaction energy between two R or two S molecules and their interactions with achiral solvents are
4.1 Models of Multicomponent Competitive Adsorption Isotherms
161
Figure 4.3 Experimental competitive isotherm data (symbols) and best bi-Langmuir competitive isotherm (solid lines) for bradykinin and kallidin. Reproduced with permission from D. Zhou, K. Kaczmarski, G. Guiochon, J. Chromatogr. A, 1015 (2003) 73 (Fig. 1).
1.5
2.0
C[g/L]
the same. The coefficients of one of the terms of Eq. 4.10, corresponding to the nonselective sites, are identical for the two enantiomers. The saturation capacity of the enantioselective sites is nearly the same for the two enantiomers, while the coefficients are very different. Finally, the chiral selective sites have a very low saturation capacity and are filled at a low concentration, so the mobile phase remains ideal. The identification of the two types of sites is comforted by the results obtained in the investigation of the adsorption energy distributions [27], as explained in the previous chapter. 4.1.3.1 Application of the Bi-Langmuir Competitive Model to Enantiomers This isotherm model has been used successfully to account for the adsorption behavior of numerous compounds, particularly (but not only) pairs of enantiomers on different chiral stationary phases. For example, Zhou et al. [28] found that the competitive isotherms of two homologous peptides, kallidin and bradykinine are well described by the bi-Langmuir model (see Figure 4.3). However, most examples of applications of the bi-Langmuir isotherm are found with enantiomers. The N-benzoyl derivatives of several amino acids were separated on bovine serum albumin immobilized on silica [26]. Figure 4.25c compares the competitive isotherms measured by frontal analysis with the racemic (1:1) mixture of N-benzoyl-D and L-alanine, and with the single-component isotherms of these compounds determined by ECP [29]. Charton et al. found that the competitive adsorption isotherms of the enantiomers of ketoprofen on cellulose tris-(4-methyl benzoate) are well accounted for by a bi-Langmuir isotherm [30]. Fornstedt et al. obtained the same results for several /3-blockers (amino-alcohols) on immobilized Cel-7A, a protein [31,32]. Similar results were obtained with the enantiomers of methyl mandelate separated on 4-methylcellulose tribenzoate immobilized on silica [30]. Figure 4.4a shows the experimental adsorption data for the two pure enantiomers (symbols), the best bi-Langmuir isotherms (solid lines) and the best LeVan-Vermeulen isotherms [33]. The data (symbols) were obtained by ECP. Figures 4.4b-d compare the competitive isotherm data measured with three mixtures of different composition and the isotherms calculated from the single component isotherms (Figure 4.4a) using the competitive bi-Langmuir model (Eq. 4.10). Results obtained
162
Competitive Equilibrium Isotherms
Figure 4.4 Competitive isotherms of the 0 and © enantiomers of methyl mandelate on 4-methylcellulose tribenzoate, with hexane/2-propanol (90:10) as the mobile phase, (a) Single-component isotherms at 30°C; solid lines, competitive Langmuir model; dotted lines, LeVan-Vermeulen isotherm, (b) Experimental (symbols) and calculated (lines) competitive isotherms; ratio C(+)/C(-) = 1.05. (c) Same as (b), but ratio C(+)/C(-) = 2.43. (d) Same as (b), but ratio C(+)/C(-) = 0.32. Reproduced from F. Charton and G. Guiochon,}. Chromatogr., 630 (1993) 11 (Figs. 2 and 3).
with enantiomers are reviewed in more detail in Section 4.1.3.2. The agreement that was observed between the experimental results and the prediction of a competitive Langmuir model based on the use of single-component Langmuir isotherms in the case of the adsorption of enantiomeric derivatives of amino acids on immobilized serum albumin [26] is unusual. It demonstrates the validity of the competitive Langmuir model based on the use of the parameters of the single-component Langmuir model. However, as explained before, the experimental conditions are exceptionally favorable since the column saturation capacities for the two enantiomers are equal. Nevertheless, Zhou et al. have shown that it is possible, in certain favorable cases, to derive the equilibrium isotherms of the pure enantiomers and to calculate isotherm equilibrium data for any mixture of
4.1 Models of Multicomponent Competitive Adsorption Isotherms
163
them from the mere competitive equilibrium isotherm data measured only with the racemic mixture [34]. The validity of the approach must be carefully checked for each application. 4.1.3.2 Competitive Isotherms of Enantiomers Jandera et al. [35] measured by frontal analysis the competitive isotherms of the enantiomers of mandelic add, phenyl-glycine and tryptophan on the glyco-peptide Teicoplanin, in water/methanol or ethanol solutions. The less retained L enantiomers of the two amino acids follow Langmuir isotherm behavior while the D isomers follow bi-Langmuir behavior. The enantiomeric separation factors increase with increasing alcohol concentration while the solubilities of these compounds decrease. Similar results were reported by Loukili et al. [36] for the separation of the enantiomers of tryptophan on a teicoplanin- based CSR The authors insisted on the importance of the nature of the ions in a supporting salt. Optimization of the experimental conditions for maximum production rate must take this effect into account. In a series of systematic investigations [1,37—45], it was shown that the band profiles of all the pairs of optical isomers studied could be accurately predicted using isotherms data acquired by frontal analysis, in a concentration range exceeding three orders of magnitude, using only the racemic mixture. This concentration range is barely sufficient to ascertain the numerical values of the isotherm coefficients when a bi-Langmuir model is used, because the numerical calculation would be indeterminate otherwise. The experimental findings confirmed the calculation results for all bands of single-component and for all binary mixtures of different compositions. The compounds studied were phenyl-alanine anilide (PA) on a polymer imprinted with L-PA [37], propranolol, alprenolol, and metoprolol on the cellulase protein CBH I immobilized on porous silica [31,38]; 3chloro-1-phenyl-l-propanol on cellulose tribenzoate [40,41], 1-phenyl-l-propanol [39,42] and 1- indanol [43,44] on cellulose tribenzoate coated on porous silica, Troger's base on amylose tri-(3,5-dimethyl-phenyl carbamate) coated on porous silica [1,45]. Simple Langmuir or bi-Langmuir models account for most of the isotherms obtained. However, a more complex result was observed for Troger's base (see Figure 4.28). While the concentration of the less retained Troger's base (—)-enantiomer does not affect the adsorbed concentration of the more retained one (there is no competition), a most unusual cooperative adsorption is observed for the less retained enantiomer, whose equilibrium concentration in the adsorbed phase increases with increasing concentration of the more retained enantiomer [1]. This observation is consistent with the results of molecular dynamics calculations [45] and with the notable agreement between (i) the experimental concentration profiles recorded in overloaded elution, (ii) the profiles measured along the simulated moving bed (SMB) unit, and (iii) the corresponding profiles calculated using an isotherm model incorporating these experimental findings (see later, Chapter 17) [46]. Finally, except for the Troger's base enantiomers, excellent estimates of the coefficients of the competitive isotherm model, valid in the whole range of
Competitive Equilibrium Isotherms
164
2.5 -
0.05
0.15
0.1
0.2
0.25
c[mM]
Figure 4.5 Single-component adsorption data for L- and D-methyl mandelate determined by FA and best isotherm model. Symbols: experimental data: LM (A) and DM (•). Lines: best bi-Langmuir isotherm models, LM (solid) and DM (dashed). The main figure shows the medium-concentration range (up to 0.25 mM), the top left inset the low-concentration range (below 12.5 ^M), and the bottom right inset the high- concentration range (up to 5.0 mM). Reproduced with permission from } . Lindholm, P. Forssen, T. Fornstedt, Anal. Chem., 76 (2004) 4856 (Fig. 5). ©2004, American Chemical Society.
relative concentrations, were derived from experimental data obtained using only the racemic mixture [38-44], as was explained earlier. This result simplifies the determination of the competitive isotherms of enantiomers, facilitating the optimization of their separation by preparative chromatography [43]. Figure 4.5 illustrates clearly a bi-Langmuir isotherm with a marked change in curvature around 0.3 mM. 4.1.3.3 The Quaternary Isotherms of Two Different Sets of Enantiomers Recently, Lindholm et al. [24] measured by the perturbation method the quaternary isotherms of the two enantiomers of methyl- and ethyl- mandelate on a chiral stationary phase. The method developed earlier by Forssen et al. [47] consists in injecting perturbation pulses that contains concentrations of the different components that are alternately higher and lower than the plateau concentrations of the different components (see Eq. 4.104). This method ensures that the different perturbations (four in the present case) are easily detectable and that their retention times can be measured accurately. The isotherm data were modeled using a quaternary bi-Langmuir model and excellent agreement was reported between experimental and calculated band profiles.
4.1 Models of Multicomponent Competitive Adsorption Isotherms
165
4.1.4 Simple, Thermodynamically-consistent, Competitive Langmuir Isotherm Unfortunately, the available experimental results suggest that the column saturation capacity is often not the same for the components of a binary mixture, so Eq. 4.5 does not account accurately for the competitive adsorption behavior of these components [48]. A simple approach was proposed to turn the difficulty (next subsection). Although it is applicable in some cases, more sophisticated models seem necessary. Numerous isotherm models have been suggested to solve this problem. Those resulting from the ideal adsorbed solution (IAS) theory developed by Myers and Prausnitz [49] are among the most accurate and versatile of them. Later, this theory was refined to account for the dependence of the activity coefficients of solutes in solution on their concentrations, leading to the real adsorption solution (RAS) theory. In most cases, however, the equations resulting from IAS and the RAS theories must be solved iteratively, which makes it inconvenient to incorporate those equations into the numerical calculations of column dynamics and in the prediction of elution band profiles. There is a simple approach to deal with the issue of building up a competitive isotherm model for the mixtures of two compounds that both exhibit singlecomponent Langmuir isotherm behavior but have different saturation capacities. This approach consists in suppressing the problem. It assumes that there are two different types of sites on the surface, nonselective sites that are accessible to both compounds and selective sites that are not accessible to the compound that has the lower saturation capacity. For the type of sites on which both species are adsorbed, the saturation capacities of both components are the same, ensuring thermodynamic consistency for the competitive Langmuir isotherm. For the compound that has the larger saturation capacity, the competitive isotherm is a bi-Langmuir model that simplifies into a Langmuir model when this compound is pure. This model was initially suggested by Bai and Yang [50] to model the adsorption of gas mixtures. It makes some physical sense in the case of chiral separations, in which case the enantioselective sites may be heterogeneous, some of them, but not all, interacting with the less retained enantiomer with a lower energy than with the more retained enantiomer. In the case of two compounds exhibiting both single-component Langmuir behavior, the model can be expressed by the following equations. These equations assume that component 1 can be adsorbed on all types of adsorption sites (saturation capacity qsj,) but that some of these sites (saturation capacity qSra — qSib) are not accessible to component 2. • Adsorption on the sites accessible to both compounds. These sites are noted b. The
classical competitive Langmuir model applies.
I2,h
=
. , r , h r 1 + t>iL\ + O2C2
1
(4-12)
Adsorption on the sites accessible only to the first compound. These sites are noted
166
Competitive Equilibrium Isotherms
a. The classical single-component Langmuir model applies to component 1, the only compound adsorbed on these sites.. (4.13) qi,a
=
0
(4.14)
Total concentrations in the adsorbed phase. , (qs,a - qs,b)hiCi
,, 1 C .
Obviously, when Ci = 0, the isotherm of the first component reduces to a Langmuir isotherm. This model was used to account for the adsorption of the enantiomers of 1-indanol on cellulose tribenzoate [51].
4.1.5 The Ideal Adsorbed Solution (IAS) Myers and Prausnitz [49] developed the ideal adsorbed solution (IAS) model in order to predict thermodynamically consistent multicomponent isotherms of gas mixtures, using only experimental data acquired for single solute adsorption. The initial equation of the IAS theory for gases is Pi = z,-PP(n)
(4.17)
where P,, z,-, and J7 are the partial pressure of compound i in the gas phase, its mole fraction in the adsorbed phase, and the spreading pressure of the mixture. The standard state pressure Pf is defined as the gas pressure when only the component i is present in the gas phase. It is given by the Gibbs adsorption equation as:
where q^Pf) is the single-component isotherm of i. Since the area occupied on a surface by ideally adsorbed solutions does not change upon their mixing, we have the following relationship:
<419)
^
where t^ot is the total amount of mixture adsorbed at equilibrium. Finally, the sum of all the mole fractions in the adsorbed phase must be equal to one, hence: ^
1
(4-20)
4.1 Models of Multicomponent Competitive Adsorption Isotherms
167
Equations 4.17 to 4.20 employ the component partial pressure and can therefore be applied to calculate the gas-solid adsorption isotherm. These equations can be solved numerically for any type of single-component isotherm. However, it is generally impossible to derive analytical solutions of these equations since Eq. 4.18 can rarely be integrated analytically and, even when it can, it is generally impossible to calculate the inverse, Pf (17), which is needed to solve Eq. 4.20. Finally even if Ff(TI) were to be known, it is usually not possible to solve Eq. 4.20 to find n(P1,P2, • • •)• However, if the function TI(PlrP2, • • •) is known explicitly, the multicomponent isotherm Cji{P\, Pi, • • •) can be determined explicitly, either by using Eqs. 4.17 and 4.19 and the relationship c\i = x^ to t or by noting that the Gibbs adsorption equation for a mixture,
YJ%d^i
(4-21)
i •*»
implies that *=p
(4 22)
-
Figure 4.6 illustrates the use of the IAS model to account for the competitive isotherm data of a ternary mixture of benzyl alcohol (BA), 2-phenylethanol (PE) and 2-methyl benzyl alcohol (MBA) in reversed phase liquid chromatography. The RAS model accounts for the nonideal behaviors in the mobile and the stationary phases through the variation of the activity coefficients with the concentrations. Figures 4.6d and 4.6e illustrate the variations of the activity coefficients in the stationary and the mobile phases, respectively. The solutes exhibit positive deviations from ideal behavior in the adsorbed phase and negative deviations from ideal behavior in the mobile phase. The IAS theory was later extended to account for the adsorption of gas mixtures on heterogenous surfaces [52,53]. It was also extended to calculate the competitive adsorption isotherms of components from liquid solutions [54]. At large solute loadings, the simplifying assumptions of the IAS theory must be relaxed in order to account for solute-solute interactions in the adsorbed phase. The IAS model is then replaced by the real adsorbed solution (RAS) model, in which the deviations of the adsorption equilibrium from ideal behavior are lumped into an activity coefficient [54,55]. Note that this deviation from ideal behavior can also be due to the heterogeneity of the adsorbent surface rather than to adsorbateadsorbate interactions, in which case the heterogeneous IAS model [55] should be used. 4.1.5.1
Explicit calculation of multicomponent adsorption isotherms of gas mixtures using the IAS theory
Frey and Rodrigues [56] developed a method using the IAS theory and some approximations that yields explicit relationships for the adsorbed concentrations of a mixture with an arbitrary number of components.
Competitive Equilibrium Isotherms
168 1.5 60 40
1.0
20 0
10
10
20
30
40
20 30 40 C i + C 2 + C 3 (gfl)
50
50
60
60
o s
0
10
20
30 c
40
50
40
50
Figure 4.6 Competitive isotherms of a ternary mixture of benzyl alcohol (BA), phenyl-2ethanol (PE) and methyl-benzyl alcohol (MBA) on Cig-silica with MeOH/H/jO as the mobile phase, at different relative concentrations. Adsorbed amounts of (a) benzyl alcohol, (b) 2-phenylethanol, (c) 2-methyl benzyl alcohol, (d) activity coefficient in the mobile phase versus concentration. In Figures (a), (b), (c), the open circles are for equal parts of BA, PE and MBA, the diamonds for three parts of BA, 1 part PE and 1 part MBA), the triangles for one part of BA, one part of PE and three parts of MBA and the stars for the singlecomponent isotherms. The solid line is the Flory-Huggins model and the dashed lines are the IAS model isotherms. In Figures (d) and (e), the circles are for BA, the squares for PE, the triangles for MBA and the diamonds for the solvent. ReproducedfromI. Quinones, J. Ford, G. Guiochon, Chem. Eng. ScL, 55 (2000) 909 (Figs. 12,13 and 14).
Fitting Pf (17) to a Pade Approximation The function Pf (IT) can usually be represented by a Pade approximation [57]. If a (1/2) Pade approximation is used to represent PP(JT), an analytical solution of Eq. 4.20 for TI(Pi,P2, • • •) can be obtained, from which the multicomponent isotherms can be determined explicitly. In summary, the following procedure was suggested to determine the multicomponent isotherms [56]: • Step 1. Determine, analytically or numerically, the functions TJ(Pf) and Pf(TI) for each component of the system, using the known single-component isotherm. • Step 2. Fit the fuction Pf(n) to a three-parameter (1/2) Pade approximation having the following form :
n i + fc,-IT + 1
(4.23)
4.1 Models of Multicomponent Competitive Adsorption Isotherms
169
• Step 3. Substituting Eq. 4.23 written for each component into Eq. 4.20 yields the following equation which can be solved for 77, using the quadratic formula
n2 EqPi + nfcfaPi - 1)] + £>P(- = 0
(4.24)
Since Eq. 4.24 is a quadratic equation, it yields two solutions for 77. Only one of them is an acceptable solution and it is selected by using the criterion that, at low partial pressures of all the components, the multicomponent isotherms must yield the same Henry's law constant as the single-component isotherms. • Step 4. Determine the explicit multicomponent isotherms using the relationship Fl{P\,P2, • • •) derived from Eq. 4.24 together with Eq. 4.22 or using the relationship TI{Pi,P2, • • •) derived from Eq. 4.24 together with Eqs. 4.17 and 4.19 and the relationship qi = x^totAlthough this procedure involves the fitting of the function pP (17) to a Pade approximation, it is generally simpler to fit the experimentally determined single component isotherm cff(Pf) to the approximate form. If Eq. 4.23 applies, Eq. 4.22 yields the following single-component isotherm for each component: 1 — b-P1 '
-2b{P?
-1
(4.25)
Even though Eq. 4.25 may yield a complex number for qf when Pf is large, this equation can fit a wide variety of experimental data at moderate values of Pf, and do so often more accurately than a Langmuir isotherm. Figure 4.7 shows experimental data for the adsorption of ethane on activated carbon at 311 K, fitted both to the Langmuir isotherm and to the isotherm given by Eq. 4.25. As this Figure indicates, the experimental data fit better to the isotherm given by Eq. 4.25 than to the Langmuir isotherm. 4.1.5.2
The LeVan-Vermeulen Isotherm
Using the IAS theory, LeVan and Vermeulen derived a competitive binary isotherm equation that accounts for differences in the column saturation capacities for the two components, when the single-component adsorption isotherms follow a Langmuir or a Freundlich isotherm model [33]. We assume that the single-component isotherms are Langmuir isotherms:
where qi/S is the specific column saturation capacity for component i. LeVan and Vermeulen [33] have derived an equation for the competitive isotherms of the components of mixtures of gases or vapors. Their model is valid when the pure components follow the Langmuir isotherm. This model can be extended to multicomponent liquid-solid equilibria by assuming that the solution is ideal and that
Competitive Equilibrium Isotherms
170
Figure 4.7 Langmuir isotherm and isotherm from Pad6 approximation (Eq. 4.25) used to fit experimental data for the adsorption of ethane on attrition-resistant activated carbon as reported by Kaul [58]and Valenzuela and Myers [53]. D. D. Frey and A. E. Rodriguez, AIChE } . , 40 (2994) 182. (Fig. 1). Reproduced by permission of the American Institute of Chemical Engineers. ©1994 AIChE. All rights reserved.
the adsorption of the solvent on the adsorbent surface is negligible. Replacing pressures by concentrations in their general equation, we obtain: (4.27)
where qs is a weighted average monolayer capacity. Furthermore, this isotherm equation can be expanded in a Taylor series which converges very rapidly. Depending on the number of terms used, we have a group of three practical isotherms. First, the classical competitive Langmuir isotherm could be obtained by writing that qs = q's = (qi/S + q2iS)/2 is constant. Then we have (4.28)
This equation is a correct Langmuir competitive isotherm only if qSii = qSi2. The second-order isotherm is probably the most useful. The common value of qs is then given by b2C2
(4.29)
The isotherms are written b'2U 2C2
(4.30)
and q*sb2C2
. , r ,+ + OiC-i
(4.31)
= {qi,s - qi,s)
(4-32)
11
where
4.1 Models of Multicomponent Competitive Adsorption Isotherms
171
For the three-term expansion of the isotherm, we have *
_
quhCj
+ q2,sb2C2 , ,, (q1/S - q2/S)2 + b2C2 qi,s + qi,s (hCi + b2C2)2
x
[ { h c ^ c 2 + *>^+h^ + ^ ~x\(4"33)
The three-term expansion of the LeVan-Vermeulen isotherm is
'•-i+*ffW**
<434)
and hC
1 + f0\L.\l+ 02L2
(4-35)
In these equations, A\^ is given by Eq. 4.32 and: A
= 1/3
A
Ml
=
qi,s - qilS 1 [ (b2C2)2
fe - qi,s
+ W+
1
[ (^lQ) 2 + Ihd - 4b2C2 - jb2C2)2
*2C2) + 3 ( ^ + 4 ^ , ^ 2 ^ , 2 0 , 0 1 1 + DiLi + 0 2 L 2
J
These isotherm equations may appear complex but they are easy to calculate since they are simple algebraic equations. Most importantly, these isotherm equations depend on only four parameters, the two specific column saturation capacities, qiiS and the two coefficients ty, which are respectively equal to the ratios Ui/qi,s = ^;o/(^!,s)/ where F is the phase ratio and k'iQ is the retention factor under linear (i.e., analytical) conditions. These are the four parameters of the two single-component isotherms, as long as these isotherms are accounted for by a simple Langmuir isotherm. Furthermore, the two-term expansion is usually quite sufficient and gives nearly the same results as the three-term expansion (Figure 4.8a) [59]. Figures 4.8a, b, and c compare the LeVan-Vermeulen two-term and three-term expansion isotherms and the competitive Langmuir model. Figure 4.8a shows the single-component behavior. The solid line is the best fit of the experimental data for each pure component to the single-component Langmuir model. The other two lines are the single-component isotherms obtained by fitting all the experimental data (competitive and single component data) to the LeVan-Vermeulen isotherms. The other parts show the competitive isotherms for 1:1 (Figure 4.8b) and 9:1 (Figure 4.8c) mixtures. The LeVan-Vermeulen isotherm has been used by Golshan-Shirazi et al. [10] to account for experimental results obtained with cis- and trans-androsterone, two
172
Competitive Equilibrium Isotherms
Figure 4.8 Comparison of the two-term and three-term LeVan-Vermeulen isotherms and the Langmuir competitive isotherm. Solid line: two-term LeVan-Vermeulen isotherm; dotted line: three-term LeVan-Vermeulen isotherm; dashed line: Langmuir competitive isotherm. F = 0.25, qSf\ = 1 mmol/ml, qS/2 = 2 mmol/ml; fcQ1 = 3; a = A^/fcoj = 1.2; ty = H{lc\Sii = kj/^Fqsi). (a) Single-component isotherms, (b) Competitive isotherms for the 1:1 mixture, (c) Competitive isotherms for the 9:1 mixture. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, ]. Chromatogr. 545 (1991) 1 (Figs. 6, 7, and. 8).
a [mj/mlj 300
Figure 4.9 Experimental adsorption isotherms of pure cis- and irans-androsterone. Stationary phase: silica treated with a phosphate buffer; mobile phase: acetonitriledichloromethane solutions. Symbols, experimental data; solid lines, best fit of experimental data to the Langmuir model. 1 and •, cz's-androsterone; 2 and o, tens-androsterone. Reproduced with permission from f.-X. Huang and G. Guiochon, } . Colloid and Interf. Sci., 128 (1989) 577 (Fig. 2).
stereoisomers that are difficult to separate in chromatography because the initial slopes of their isotherms are close, but which have quite different column saturation capacities. Figure 4.9 illustrates the single-component data. As shown in Figure 4.10 [10], good agreement was obtained between the competitive isotherms calculated for mixtures, using the LeVan-Vermeulen model and the parameters of the best single-component Langmuir isotherms and the competitive isotherm measured experimentally, using the method of binary frontal analysis [10]. The
4.1 Models of Multicomponent Competitive Adsorption Isotherms
C (mg/ml)
173
C (mg/ml)
Figure 4.10 Experimental competitive adsorption isotherm data of cis- and frans-androsterone. Same phase system as in Figure 4.9. Comparison of the Langmuir competitive model (bottom) and the two-term expansion of the LeVan-Vermeulen isotherm (top). In both cases, the best-fit parameters are used to calculate the lines. Experimental data: A: cz's-androsterone; o: irans-androsterone. Theory: cz's-androsterone (dotted lines); transandrosterone (solid lines), a and d: 2:1 mixture; b and e: 1:1 mixture; and c and f: 1:2 mixture. Reproduced with permission from S. Golshan-Shirazi, J.-X. Huang and G. Guiochon, Anal. Chem., 63 (1991) 1147 (Figs. 1 and 2), ©1991 American Chemical Society.
IAS theory and the LeVan-Vermeulen model were able to explain the reversal of the elution order observed at high concentrations; the Langmuir competitive model was unable to do so. This problem is discussed in more detail in Chapter 11 (Section 11.2.5). Like single-component isotherms, competitive isotherms depend on the composition of the mobile phase and the temperature [10]. Figure 4.11 [10] illustrates for a 1:1 mixture of cis- and irans-androsterone the effects of the mobile phase composition and temperature, respectively, on the adsorption isotherm. The general trend is for adsorption to decrease with increasing concentration of the strong solvent and with increasing temperature. These experimental parameters must be
Competitive Equilibrium Isotherms
174
20
C Gng/mD
Figure 4.11 Effect of the experimental conditions on the competitive isotherm of cisand tons-androsterone (1:1 mixture). Experimental data: cis-androsterone, A, transandrosterone, D; two-term LeVan-Vermeulen isotherm: tis-androsterone, dotted line, frans-androsterone solid line. (Left) Effect of the mobile phase composition. (Right) Effect of the temperature. Reproduced with permission from S. Golshan-Shimzi, J.-X. Huang and G. Guiochon, Anal. Chem., 63 (1991) 1147 (Figs. 3 and 4), ©1991 American Chemical Society.
carefully controlled to achieve accurate and reproducible results. Unlike the LeVan-Vermeulen isotherm, the IAS theory is not limited to the case in which the single-component adsorption behavior follows the Langmuir isotherm model. The formulation is more complex, and tedious numerical calculations must be performed. Troger's base has two enantiomers. On microcrystalline cellulose triacetate, the less retained (-) isomer follows a Langmuir isotherm, while the more retained (+) isomer follows a quadratic isotherm, see Figures 3.25 and, later, 4.12 [60] and 4.28 [1]. The IAS theory permitted the prediction of competitive isotherms of the two enantiomers. These isotherms were only in fair agreement with the experimental data (Figure 4.12 [60]) and did not result in band profiles always matching exactly the experimental profiles recorded. Extension of the Series Solution of the LeVan-Vermeulen Isotherm to multicomponent systems Frey and Rodrigues [56] have extended the binary LeVanVermeulen isotherm to the case of multicomponent systems. They showed that, if we assume that the adsorption isotherm of each single component follows Langmuir isotherm behavior, the multicomponent isotherm is given by [56] LibiPdEibiPi
(4.38)
4.1 Models of Multicomponent Competitive Adsorption Isotherms
175
c
a /
'
'
•
'
'
'
'
Figure 4.12 Competitive isotherms of the (-)- and (+)- enantiomers of Troger base on microcrystalline cellulose triacetate, with ethanol as mobile phase, (a) Single-component adsorption isotherm of (+)-TB (squares) and (-)-TB (triangles) at 40° C. Experimental data and best fit to a Langmuir, (+)-TB, and a quadratic, (-)-TB, isotherm model, (b) Competitive isotherms of (-)-TB in enantiomeric mixtures for increasing concentrations (0, 0.5,1,1.5,2, 2.5, 3 g/L) of (+)-TB, calculated with IAS theory, (c) Competitive isotherms of (+)-TB in enantiomeric mixtures for increasing concentrations (0, 0.5, 1, 1.5, 2, 2.5, 3 g/L) of (-)-TB calculated with IAS theory. Reproduced from A. Seidel-Morgenstern and G. Guiochon, Chem. Eng. Set, 48 (1993) 2787 (Figs. 4, 6, and!).
When qi/S =
(4.39)
Or
i = C?(n)Zi
(4.40)
and
l/^tot = D n j
(4.41)
where X,- and z,- are the mole fractions of solute i in the solution and the adsorbed phase, respectively, Q is the liquid-phase concentration of solute i, C®{TI) is the liquid-phase concentration of the solute i, when it is adsorbed from a solution in
176
Competitive Equilibrium Isotherms
which it is alone/ at the same temperature, and under the same spreading pressure, TZ, as those of the multicomponent solution, Cj is the total concentration of all solutes in the liquid phase, q® is the adsorption isotherm of the solute i, when it is adsorbed from a solution in which it is alone, at the same temperature, and under the same spreading pressure, n, as those of the multicomponent solution, and gtot is the total adsorption concentration of all the solutes. Equation 4.40 is the analog for liquid-solid adsorption of Eq. 4.17 for the gas-solid adsorption of gas mixtures. The terms P/RT and Pf/RT are replaced by CT and Cf, respectively. In order to use Eqs. 4.40 and 4.41, the spreading pressure must be known for the various single-component solutes in the mixture. The spreading pressure for liquid-solid equilibrium is given by [54]: Ci;
1 = 1,2
(4.42)
where A is the surface of adsorbent. By defining TliiCf) = n(Cf)A/RT
(4.43)
Where I7,(C?) is the reduced spreading pressure, one obtains: 1 = 1,2
(4.44)
where the concentrations Cf are fictitious concentrations of the pure components such that these components pure would have the same reduced spreading pressure as they do in the mixture, TJm^K. Thus, we have nmix=ni;
z = l,2
(4.45)
Antia and Horvith [61] attempted to use these equations to obtain multi-component isotherms in cases in which each individual solute follows single-component Langmuir isotherm behavior, hence qSri
1 + bf-i
For known values of the reduced spreading pressure (17), the so-called IAS/L isotherm is given by [61] " bfj ^
exp(n/qj)
qs,j
where IT can be obtained from the numerical solution of the equation:
£ y
/
( y ) - l )
=l
(4.48)
The numerical solution of Eq. 4.48 is obtained by an iterative process that is greatly facilitated if the value of 17 given by the LeVan and Vermeulen approximation [33] is taken as the initial value to start the iteration.
4.1 Models of Multicomponent Competitive Adsorption Isotherms
177
4.1.6 The Real Adsorbed Solution (RAS) From a theoretical point of view, the IAS isotherm should be applied only to dilute solutions. However, although the simplified assumptions of the IAS theory are generally valid as far as the solution is concerned, they are not necessarily so for the adsorbed phase because often, the adsorbate-adsorbate interactions cannot be neglected. The RAS theory takes into account the deviations of the adsorbed phase from ideal behavior by introducing into Eq. 4.17 the adsorbed phase activity coefficient, 7 [54,55]: Q = 7iZiCJ)(ninix); i = l,2 (4.49) The activity coefficients, 7;, depend on the concentration. For nonideal systems, it is necessary to develop a model and derive an equation for the activity coefficients in the adsorbed phase. The definition of these coefficients must satisfy three thermodynamic conditions: • The activity coefficient of any mixture component must tend toward 1 when its mole fraction tends toward 1. • At low surface coverages, the spreading pressure tends toward 0, the behavior of the adsorbate tends toward ideal behavior, and the activity coefficients of all components tend toward 1. • The experimental values of the activity coefficients must satisfy the GibbsDuhem relationship [55]. If the data are collected at constant temperature and spreading pressure, then Y^
= 0
for 11^ = const, T = const
(4.50)
For a binary system, this equation can be written =0 (4.51) For very dilute solutions, hence at low spreading pressures, the concentrations of the adsorbates are also low, and the activity coefficients in the adsorbed phase approach unity. For real solutions, a suitable model of the activity coefficients must be used. Several such models have been suggested. The following equation, proposed by Gamba et al. [62] and based on the regular solution theory [63], was applied by Kaczmarski et al. [51] to account for the competitive isotherm of 1indanol on cellulose tribenzoate:
InH = E B 4 - 0.54)*;**
(4.52a)
i * with ^{C]
4
i = l,N
(4.52b)
Competitive Equilibrium Isotherms
178
16 -
Figure 4.13 Comparison between the experimental competitive isotherm data of two enantiomers (symbols) and the best isotherm calculated with (a) the IAS model, (b) the RAS model. Bullet S-1-indanol; solid square S-1-indanol. Reproduced with permission from from K. Kaczmarski, D. Zhou, M. Gubernak, G. Guiochon. Biotechnol. Progr., 19 (2003) 455. (Figs. 2 and 3). ©2003, American Chemical Society.
where A^ and C^ are adjustable parameters that have to be determined by matching the theoretical and the experimental binary equilibrium data. Numerical calculations showed that Eq. 4.51 requires that, for a binary system, C\i — C21 [23]. According to the RAS theory [64], the following equation holds for the total amount of adsorbed feed material 1
N
So7 + E :
h
(4.53)
The correlation of the experimental equilibrium competitive data within the framework of the RAS theory is done by using the equilibrium isotherm model for a single compound and Eqs. 4.42,4.45,4.49,4.52a-c, and 4.53. Migliorini et al. [65] used both the IAS and the RAS theory to account for the experimental binary competitive isotherm data of the Troger's base enantiomers on microcrystalline triacetate cellulose (CTA), using ethanol as the solvent. For the calculations of the RAS theory, they used the Wilson model for the solution, including the empirical spreading pressure dependence [66]. They concluded that the IAS model underestimates the extent of the competition between the two enantiomers in this system while the RAS model accounts accurately for the complex competitive adsorption behavior exhibited by these enantiomers. Kaczmarski et al. measured by frontal analysis the competitive isotherm data of the racemic mixture of the R and the S enantiomers of 1-indanol and the isotherms of the two pure components [51]. The data for each enantiomer fitted well to the single-component bi-Langmuir isotherm model2003. The competitive experimental data were fitted to the Ideal Adsorption Solution model (IAS), the Real Adsorption Solution model (RAS), and the bi-Langmuir Thermodynamically Consistent model (BTC, Section 4.1.4). Figure 4.13b shows a good agreement between the experimental data and the RAS model for which there is only a slight difference between symbols and solid lines. The agreement is poor for the IAS model (Figure
4.1 Models of Multicomponent Competitive Adsorption Isotherms
179
4.13a). These conclusions based on visual observations are confirmed by comparison of the sums of squares of the differences between corresponding experimental and best theoretical concentrations. It is worth noting that the agreement is still better with the thermodynamically consistent bi-Langmuir model (bi-Langmuir extension of the thermodynamically consistent Langmuir model in Section 4.1.4) than with the RAS model, in spite of the fact that the RAS model has two more parameters which account for the nonideal behavior of the adsorbed phase.
4.1.7 The Statistical Isotherms James et al. published a detailed, rigorous, systematic investigation of statistical thermodynamics models for multicomponent isotherms when both phases are nonideal [67]. We will consider a more empirical approach. As mentioned in Chapter 3 (Eq. 3.61), simple models of statistical thermodynamics suggest than the competitive equilibrium isotherms, as well as the single-component ones, should be written as the ratio of polynomials of the same degree [68]. Beyond the competitive Langmuir isotherm, which appears as the simplest such model, the next approximation is the ratio of two second-degree polynomials as in Eq. 3.31. In the case of a binary mixture, Ruthven and Goddard [69] have derived the following competitive isotherms:
& 3 c 1 c 2 + 2&4cf 1
qs
1 + &iCi + b2C2 + &3QC2
Cl + bC\
and
"2
_qi_ qs
b2C2 + b3C1C2+2b5C 1 + biQ + b2C2 + &QC + bC\ + bC\
K
' '
These equations contain seven parameters instead of four in the Langmuir isotherm. These parameters include the two column saturation capacities in the case of an empirical isotherm. However, like the competitive Langmuir isotherm, this model is thermodynamically consistent only if these saturation capacities are equal. If we apply one of these equations to single-component isotherm data, we see that Eqs. 4.54 and 4.55 can be applied to the competitive adsorption data for a binary mixture only if Eq. 3.31 applies to the single-component data for each component. Then the six parameters can be derived from the single-component isotherms and only the coefficient &3 has to be measured with the mixture. Using more complicated models, Lin et al. [70] and Moreau et al. [71] have derived similar isotherms. Attempts at reducing the number of independent parameters as well as at determining these parameters from sets of experimental data have had limited success so far. Considerable attention is required to clarify this issue. Depending on the relative values of the numerical coefficients, the isotherm may be convex, concave or exhibit an inflection point. An isotherm similar to the one given in Eqs. 3.23 and 3.24, but simplified by omission of the term b^C\C2 in the denominator, has been used by Svoboda for the calculation of the individual elution band profiles of mixtures with an S-shaped isotherm [72].
180
Competitive Equilibrium Isotherms
4.1.8 The Competitive Fowler Isotherm This model applies to two compounds that both exhibit single-component Fowler isotherm behavior. Depending on the assumptions one can make regarding the interaction between the two competing components, several equations can be written. The simplest equation for a competitive Fowler isotherm is written: 0f 1
K
(4.56) tV
This isotherm equation has six parameters, the two coefficients b\ and hi, the two coefficients Xi and X2, and the two column saturation capacities, since 0, = <\il<\s,i-
The competitive Fowler isotherm is sometimes used in a simplified form which assumes the two coefficients b( to be equal, reducing to 5 the number of degrees of freedom. There is only one coefficient b for the binary mixture, while the singlecomponent Fowler isotherms of the two components will normally lead to two different values, b\ and bx- Accordingly, if one wants to use this simplified version of the competitive Fowler isotherm using single-component adsorption data, one must calculate the single-component Fowler equation parameters simultaneously for the two sets of data, using a nonlinear regression approach and adding the restrictive condition that &i = f>2- This reduction in the number of parameters will decrease the quality of the fit. Experimental data for the mixture of 2-phenylethanol and 3-phenylpropanol were acquired by FA on a Qg silica, with a water-methanol mobile phase [73]. These data are shown in Figure 4.14a,b (symbols). They were fitted to the 5parameter competitive Fowler Isotherm (with b\ = b-i). The Fowler isotherm gave a better fit than the competitive Langmuir or bi-Langmuir isotherms. An improvement of the fit with using the Fowler model was obtained by using the same value of b for both components. In terms of the Fowler model, both components seem to exhibit similar absorbate-absorbate interactions. These data were also fitted to the Fowler, Ruthven, Moreau and Kiselev isotherms with no significant improvement. An extension to the classical Kiselev model that accounts for specific lateral interactions accounts better for the competitive data. However, the Kiselev model, similar to the Fowler model, does not invert analytically and does not take into account other non-idealities for example, the adsorbent heterogeneity or the interactions in the bulk phase.
4.1.9 The Competitive Freundlich-Langmuir Isotherm Because of unsatisfactory results obtained with the competitive Langmuir isotherm model, several empirical equations have been suggested, based on hybrids of common models. One of the most popular of these isotherm models addresses the problem of strong adsorption observed at low concentrations and its rapid subsidence at increasingly large concentrations. This equation combines conven-
4.1 Models of Multicomponent Competitive Adsorption Isotherms Q mg/mL adsorbent
a
18
b
30 3-P henylpropanol
2 -P he ny l e thano l
12
20
6
10
0
0 0
5
10
Q (mg/ml}
0
55 10 10 Total Total Solute Solute Concentration, Concentration,mg/mL mg/mL
Q (mg/ml)
G
3-P henylpropanol
2-Phenyleihanol • RSS 3:1=1.904 RSS 1:1=1.282 • RSS 1:3=0.07021
0
181
tf >r f
5
X
A
10
1
Figure 4.14 Comparison between the experimental adsorption data of 2-phenylethanol (a, c) and 3-phenylpropanol (b, d) and the best competitive Fowler isotherms derived from the best single-component parameters. For mixtures, the mobile phase concentration is expressed via the relative composition of PE and PP. All concentrations in mg/ml. Symbols: data for single component, o; data for 3:1 mixtures (3/1 PE and PP), x; data for 1:1 mixtures (1/1 PE and PP), +; and data for 1:3 mixtures (1/3 PE and PP), *. The RSS are the residual sum of squares calculated for each set of experiments performed with a constant relation of the mobile phase concentrations of the phenylalcohols. ODS silica from Vydac and watermethanol mixture (50:50) at room temperature. I. Quinones, G. Guiochon, Langmuir, 12 (1996) 5433 (Figs 1 and 2) and }. Zhu, A. Katti and G. Guiochon,}. Chromatogr. 552 (1991) (Figs 7 and 8)71.
tional features of the Langmuir and the Freundlich isotherms: (4.57)
This equation has the same problem as the one noted for the corresponding single-component isotherm. We should avoid the use of adsorbents for which
182
Competitive Equilibrium Isotherms
Figure 4.15 Competitive adsorption isotherms of Basic Yellow 21 and a few other dyes on peat, (a) In single-component solution; (b) in a 1:1 binary mixture with Basic Red 22; (c) in a 1:1 binary mixture with Basic Blue 3; (d) in a 1:1:1 ternary mixture with Basic Red 22 and Basic Blue 3. Reproduced with permission from B. Al-Duri, Y. Khader and G. McKay, J. Chem. Tech. Biotechnol. 53 (1992) 345 (Fig. 1). 50
1OO 150 200 250 300 350 400 450 500 550 Cs. mg dm"3
Figure 4.16 Experimental isotherm data (symbols) of R and S-1-indanol obtained by competitive frontal analysis on a narrow bore column packed with Chiracel OB. The data were fitted to the Toth (dash-dot), the bi-Langmuir (dash), the LangmuirFreundlich (dot), and the Langmuir (solid) models. The inset shows low concentration data. Reproduced with permission from D. Zhou, K. Kaczmarski, A. Cavazzini, X. Liu, G. Guiochon, } . Chromatogr. A, 1020 (2003) 199 (Fig 4).
q [g/L]
20
a R-1-indanol S-1-indanol Langmuir BiLangmuir L-Frendlich Toth
15
10 4
5 2
0 0 0.0
0
2
4
6
0.6
8
1.2
10
CC[g/L] [g/L]
adsorption data are well described by Eq. 4.57. On such adsorbents, the elution bands tail indefinitely, so separations are difficult to perform, and purifications are nearly impossible to achieve because the limit retention time of each band is infinite and the column must be regenerated after each run. Most stationary phases now available for chromatographic separations exhibit low-energy surfaces of a rather high degree of homogeneity. Thus, it is exceptional that preparative chromatography has to be conducted with adsorbents on which the feed components exhibit this type of isotherm. In the cases in which isotherms of this type are found and no reasonable alternative is available (e.g., proteins) gradient elution provides a practical, though expensive, solution to the separation problem. The competitive Langmuir-Freundlich isotherm has been used to fit successfully experimental data obtained with various dyes [74]. Some data, measured by batch uptake experiments, are shown in Figure 4.15. In the case of the mixture of the enantiomers R and S-1-indanol, the competitive bi-Langmuir and the Langmuir-Freundlich isotherms are in excellent agreement with the experimental data over the entire concentration range. The Langmuir and the Toth isotherm models do not account well for the experimental data, as illustrated in Figure 4.16 [44]. Extensions to the Langmuir-Freundlich models have been made in order to account for surface heterogeneity and lateral interactions in the mobile
4.1 Models of Multicomponent Competitive Adsorption Isotherms
183
RSS 3:1=9.56 RSS 1:1=8.372 RSS 1:3=1.679
Total mobila phase eoncsnlratian
Figure 4.17 Competitive experimental isotherm data for 2-phenylethanol and 3phenylpropanol fitted to the competitive Fowler-Guggenheim/Langmuir-Freundlich model. Reproduced from I. Quinones, G. Guiochon, J. Chromatogr. A, 796 (1998) 15 (Figs. 3 and 4).
phase, as shown in Figure 4.17. These two figures compares experimental adsorption isotherm data (symbols) for 2-phenylethanol and 3-phenylpropanol, pure or in binary mixtures, adsorbed on Vydac Q s silica from a (50:50) methanol-water solution, and the competitive isotherms calculated using the competitive FG/LF model [75]. The coefficients of this model were derived from those of the singlecomponent isotherms of the two compounds. Although these models improve the fit to the experimental data they have the disadvantage of being impossible to invert. Interpretation of the model parameter values shows higher adsorbateadsorbent interactions for phenylethanol compared to phenylpropanol and similar values of the adsorbate-adsorbate interactions in the adsorbed phase for both these compounds [75]. Figure 4.18 Experimental competitive adsorption data (symbols) and best competitive T6th isotherms (solid lines) for the enantiomers of 1-Phenyl-l-Propanol on cellulose tribenzoate. Symbols: • , R-PP; \/, SPP. Reproduced with permission from K. Kaczmarski, A. Cavazzini, P. Szabelski, D. Zhou, X. Liu, G. Guiochon, J. Chromatogr. A, 962 (2002) 57 (Fig 1).
184
Competitive Equilibrium Isotherms
4.1.10 The competitive Toth Adsorption Isotherm The competitive equivalent to the single-component Toth isotherm model (see Chapter 3, Section 3.2.2.2) is written
«=
f^
(4-58)
where qs is the saturation capacity of the monolayer, <j,- and Q are the concentrations of component i in the solid and the fluid phase at equilibrium, respectively, bj is the equilibrium constant of component i, and v is the heterogeneity parameter. In order to obtain a thermodynamically consistent model, the same saturation capacity must be assumed for all the components. In writing this model (Eq. 4.58), we also assumed that the heterogeneity parameter, v, is the same for both enantiomers, which means that the stationary phase is equally heterogeneous with respect to both compounds. This model was used recently [76] to account for the experimental isotherm data obtained for the enantiomers of 1-phenyl-l-propanol on cellulose tribenzoate (see Figure 4.18). An excellent agreement was observed between these experimental data and the Toth isotherm. The agreement was as good as that obtained when applying the IAS theory to the bi-Langmuir model. The Toth model, however, is significantly different from the conventional bi-Langmuir model that often accounts for the adsorption behavior of enantiomers on chiral phases [31]. Both models consider that the surface is heterogeneous. However, the Toth isotherm model assumes that the surface has a continuous, unimodal adsorption energy distribution. In contrast, the bi-Langmuir model considers that the surface of the adsorbent contains two different types of homogeneous sites, one type that are high-energy, low-saturation capacity sites {e.g., sites that exhibit chiral selectivity, the other type that are low-energy, high-saturation capacity sites {e.g., nonselective sites). The adsorption energy distribution of this surface is bimodal, each mode being narrow. The competitive adsorption isotherm data of the enantiomers of 1-Phenyl-l-Propanol on cellulose tribenzoate were also fitted to four different isotherm models: Langmuir, bi-Langmuir, Langmuir-Freundlich and Toth. The fittings of the experimental data to all four models were satisfactory but it was better to the Toth models (see Figure 4.18. The heterogeneity coefficient, t, is 0.77, corresponding to a less heterogeneous surface than in the previous case. Overloaded elution profiles calculated with this isotherm were in good agreement with the experimental profiles in all the different experimental conditions investigated.
4.1.11 The Competitive Martire Isotherm Based on the unified theory of chromatography [77], Martire has derived a competitive multicomponent isotherm model, assuming that the adsorbent surface is energetically homogeneous. This model extends the single component isotherm model previously derived (Chapter 3, Section 3.2.4.1). As previously noted in the section dealing with this single-component isotherm, an important feature of this
4.1 Models of Multicomponent Competitive Adsorption Isotherms
185
model is that it includes, as particular cases, most of the simpler, yet quite useful isotherm models. The same applies to the competitive Martire isotherm. The isotherm equation is
(i - eiim - ehmfi
~
(i - eiiS - ejrSy>
{
• '
where Q{>m and Q{tS are the volume fractions of the solute (component a or V) in the mobile and the stationary phases, respectively; v\ — rf- — pr, with r,- and rc the numbers of adsorbent sites occupied by the solute and solvent, respectively, and v* and v* the van der Waals molar volumes of solute and solvent, respectively; K^ is the ratio of the equilibrium concentrations of the solute in the stationary and mobile phases at infinite dilution of the solute (Eq. 3.72); and D^m is an interaction term given by Dt,m = -2riXi,c
= ^f-
[2eirC - ( e w + e c , c )]
(4.60)
where e^-, eCiC, and e,/C are the attractive interaction energies between nearestneighbor segments for the solute-solute, the solvent-solvent, and the solute-solvent molecules, respectively; ze is the number of nearest neighbors, or external contacts of a molecular segment; T is the absolute temperature; fcg is the Boltzmann constant; %i,c is the solute-solvent interaction parameter, and E,- = ri{x%,] ~ Xi,c - Xj,c), where i,j = a, b. If we assume that all the solutions are energetically ideal, Dj = Ej = 0, that Vj = 1, and that the solution is dilute (fym m
®irS
fl
= Ahm = hCi,m
(4.61)
or
^ = T^te"
(462)
which is the competitive Langmuir isotherm. If we take into account the molecular interactions in the adsorbed state, but not in the solution, and assume that still v,- = 1, 0,-/m <S 1, Q^m
-—lifi
„ = biCime-(*lD*B*'+*tE»eiJ
1
P
(4.63)
which is the competitive Fowler isotherm. Finally, if we take into account the molecular interactions in both the stationary and mobile phases, but assume that V{ = 1 and that 0!/m
186
Competitive Equilibrium Isotherms
4.1.12 Competitive Isotherms Models for Other Modes of Chromatography The isotherms that have been discussed can be used to account for the retention behavior of components in normal and reversed phase adsorption chromatography and hydrophobic interaction chromatography. With other retention mechanisms such as ion exchange chromatography or when the stationary phase can form weakly stable complexes with the feed components, a stoichiometric exchange reaction takes place. This is at variance with liquid-solid adsorption, where the exchange between solvent and component molecules in the adsorbed layer is statistical. In ion-exchange, for example, charge neutrality should be maintained. The basic principles of ion exchange have been discussed by Walton [78]. However, this discussion was mainly limited to the case of small inorganic ions. For the separation of biomolecules, the stoichiometric displacement model (SDM, next subsection) is of particular interest. This model is based on the assumption that ion exchange is the only mechanism of retention of the components studied and that the ion-exchange process can be modeled as a stoichiometric "reaction" described by the mass action principle. 4.1.12.1 The Stoichiometric Displacement Model (SDM) The use of this model is convenient for investigations of the nonlinear separations of biomolecules by ion-exchange chromatography. The SDM model is based on the assumptions, first, that ion-exchange is the only mechanism available for the retention of the molecule studied (e.g., that there are no hydrophobic interactions involved beside ion-exchange) and, second, that the ion-exchange process can be modeled as a stoichiometric "reaction", described by the mass-action principle. This model was first suggested by Glueckauf and Coates [79], who observed that the isotherm for the exchange of Na + and K+ ions on a resin was less steep than the exchange isotherms of either Na + or K + with H + . It was rediscovered by Rounds, Drager and Regnier [80,81], who used it first to correlate protein retention (i.e., log k') under linear conditions with the mobile phase composition (i.e., the logarithm of the salt or buffer concentration) and the apparent or effective protein charge (the slope of a plot of the former vs. the latter) in linear chromatography. Later, Regnier and Mazsaroff [82] applied the SDM model to the study of the competitive adsorption of proteins. They derived the consequences of the three-dimensional structure of the macromolecules, of their nonideal behavior in the liquid phase, and studied the displacement effects between competing adsorbates during nonlinear operation. Velayudhan and Horvath [83] have published a detailed theoretical analysis of the SDM competitive isotherm model and showed the effect of the salt concentration and the competitive binding on the retention in the nonlinear region. The stoichiometric exchange between a protein and the counterions of ionic groups bonded to the surface of an adsorbent can be written: (4.65)
4.1 Models of Multicomponent Competitive Adsorption Isotherms
187
Figure 4.19 Example of single-protein isotherm in the presence of a salt, calculated using the ion-exchange formalism. Two-dimensional surface representation of the stationary phase concentrations of the protein (A) and the salt (S) as a function of their mobile phase concentrations. Reproduced with permission from A. Velayudhan and C. Horvdth, J. Chromatogr., 443 (1988) 13 (Fig. 2).
Assume the counterion to be monovalent, let the concentrations of the protein in solution and in the adsorbed phase be [Pm] = C and [Pa] = Q, respectively, and the mobile and stationary phase concentrations of the salt be [Sm] = Cs and [Sa] = Qs, respectively, with v the charge of the protein under the experimental conditions selected. The equilibrium constant Ka for the ion exchange process is defined as: Ka =
QQ
(4.66)
The electroneutrality of the adsorbed monolayer on the stationary phase requires that A = Qs + vQ
(4.67)
where A is the surface concentration of the binding sites or stationary phase capacity. The combination of Eqs. 4.66 and 4.67 permits the calculation of the concentrations of protein and counterion at the stationary phase knowing their mobile phase concentration and A. An example of the results obtained, illustrating the progressive passage from a quasi-rectangular isotherm to a Langmuir isotherm, is shown in Figure 4.19 [84]. The approach can be extended easily to multicomponent mixtures [83]. The exchange reactions are written nAm nBm
aSs bSs
nAs nBs
aSm bSm
(4.68) (4.69)
Competitive Equilibrium Isotherms
188
-0
B
Figure 4.20 Multicomponent protein isotherms. Two-dimensional surface representations of the stationary phase concentrations of two proteins A and B as a function of their mobile phase concentrations at fixed concentration of mobile phase additive. A has an apparent charge of 10, B one of 6. Reproduced from A. Velayudhan and Cs. Horvdth, J. Chromatogr., 443 (1988) 13 (Figure 5)
where n, a and b are the charges of the counterion and of the two studied ions. Ion-exchange equilibrium provides that: \A
(4.70)
Kb =
[n M e ]b
(4.71)
The electroneutrality condition gives: a[As] + b[Bs}+n{Ss] =A
(4.72)
A typical set of competitive isotherms for two proteins is shown in Figure 4.20 for a given salt concentration [83]. Note that the competitive isotherms exhibit a behavior that is farther removed from a rectangular one than the single-component isotherms. This is because the separation factor, Ky /Ka, is much smaller than each of these equilibrium constants. Later, Velayudhan and Horvath [85] investigated the relationship between adsorption and ion-exchange isotherms. There is a certain similarity between the interactions of a protein with the binding sites in ion-exchange resins and the hydrophobic patches on the surface of RPLC adsorbents. Helfferich had already shown the equivalence between the multicomponent Langmuir isotherm and the monovalent ion-exchange isotherm. The latter can be converted into the former by the addition of a dummy component [86]. The general case of adsorbates having
4.1 Models ofMulticomponent Competitive Adsorption Isotherms
189
different saturation capacities was shown to parallel that of the heterovalent ionexchange equilibrium. The application of a general adsorptive formalism to RPLC yields a relationship between the retention factor and the modifier concentration that reduces to well known results in two different cases. For small molecules, it gives the classical linear or quadratic dependence of lnfc0 on the modifier concentration, C. For large molecules, it gives a linear relationship between lnfc0 and lnC that is conventional for proteins [85]. Details of the calculation of complex competitive isotherms and of chromatograms can be found in ref. [87]. This model was used by Cysewski et al. [88] to account for the adsorption behavior of bovine serum albumin on an ion-exchange resin deposited on silica and to calculate the band profiles of single proteins under nonlinear conditions, in isocratic and linear gradient elution, with ion-exchange systems and to study the influence of the protein charge and of the salt concentration on the column performance. Bellot and Condoret [89] have extended this to binary mixtures of proteins. Good agreement between the elution band profiles measured in gradient elution and those calculated with the isotherm was reported. As noted by the authors, however, this result should not be generalized. Protein adsorption on the resin, molecular interactions, possible molecular associations, or changes in the conformation of the protein in the solution or in the adsorbed state play different roles with different proteins. The combination of all these phenomena results in the experimental isotherm and may make it most difficult to account for equilibrium data with reasonable accuracy. 4.1.12.2 The Steric Mass Action Model (SMA) The major weakness of the SDM is that it assumes that the binding of a solute to the stationary phase involves a number of sites equal to the characteristic charge of this solute. Whitley et al. [90] observed that "adsorbed protein molecules block many exchanger sites, preventing other molecules from ion-exchanging at those sites." At about the same time, Velayudhan made the same observation [87]. Both groups assumed that a macromolecule covers a much larger area of the stationary phase surface than the area occupied by the few ions they can bind to, thus effectively shielding a number of ion sites and reducing the binding capacity of the resin below its stoichiometric value. To account for the fraction of the exchanger charges accessible to the protein molecule, Whitley et al. introduced an empirical shielding factor or accessibility coefficient and reformulated the mass action model. This new model permits a very good representation of the ion-exchange isotherms of lysozyme, myoglobin, and bovine serum albumin. For the same purpose, Velayudhan suggested the use of a steric factor [87]. Brooks and Cramer [91] extended this work on the single-component mass action model to competitive isotherms. The steric factor (&) that they introduced to account for this effect depends on the nature of the resin (e.g., density of ions), the ionic strength and composition of the solution (which control the molecular structure of the protein), and the nature of the protein and its concentration (most proteins will probably have a larger apparent footprint at high dilution than at high concentration). Thus, the steric factor appears as an additional but empirical
190
Competitive Equilibrium Isotherms
degree of freedom in the isotherm model, allowing a better fit of the experimental data. For a three-component system consisting in a salt (component 1, with K\\ = 1, V\ = 1, and a\ = 0), and two solutes, 2 and 3, the multicomponent isotherm is
(
gi
C 2 + (cr 3 +V3) K13 ( &
A|iC12(^j
A [Kl3 ( Ci-
|C 2
c2 +
C i - h(cr 2 +v 2 ) K]
1
(^3 + v3)
k i 3 ( § ) V 3 1 C3
(4-73b)
"'-] Ca
h (
C3
(P"3 +
^3) \K13 \^j '
c3
The calculation of the equilibrium concentrations in a multicomponent system requires first an implicit solution of the first of these equations for the stationary phase concentration of the salt, Qi. The stationary phase concentrations of the two compounds can then be derived directly from the other two equations. 4.1.12.3 Adsorbate-Adsorbate Interactions Li and Pinto [92] have approached the problem differently. They used a stoichiometric displacement model which incorporates lateral, or adsorbate-adsorbate interactions. That these interactions are important in protein-salts-ion-exchangeresin equilibria is beyond doubt [91]. Li and Pinto consider the adsorbed layer as a nonideal solution in equilibrium with an ideal bulk solution. This is justified by the high equilibrium constant resulting in a surface concentration which is an order of magnitude higher than the mobile phase concentration at equilibrium. The difficulty is in estimating the activity coefficients in the adsorbed phase, when some components (the proteins) are enormously larger than others (water and the inorganic ions). This is done using an equation from the literature relating the surface activity coefficients to the surface fraction of the adsorbates, their shape factors, and lateral interaction energies which can be related to the isosteric heat of adsorption of the pure components. The model includes four parameters for each component: its effective charge, its equilibrium constant at infinite dilution, its isosteric heat of adsorption, and a shape factor. Experimental results demonstrate the importance of the adsorbate-adsorbate interactions in the control of the adsorption equilibrium constant [92]. Good agreement is achieved when estimating some of the parameters from the results of independent measurements. However, these estimates are quite approximate and the physical meaning of the model has not been demonstrated. The suitability of the model for competitive isotherm studies has not been established either. It would require that the competitive isotherm could be derived simply from the
4.2 Determination of Competitive Isotherms
191
coefficients of the single-component isotherms. Nevertheless, the approach has potential.
4.2 Determination of Competitive Isotherms The competitive behavior of the feed components for access to the retention mechanism constitutes the fundamental basis of nonlinear chromatography. In adsorption chromatography, this behavior is expressed by competitive isotherms. When the concentration of one component increases in a solution, the amount of the other components adsorbed at equilibrium decreases in general, because the room available on the adsorbent surface, i.e., the number of adsorption sites or the number of ligands with which interaction is possible, is limited. In some cases, it is possible to imagine molecular interactions that would either increase or decrease the average surface area needed by a molecule adsorbed on a surface. Similarly, and if we consider bulk solutions, fewer or more molecules of a mixture can be packed in a certain volume than molecules of the pure components, depending on whether the excess mixing volume is positive (the mixed molecules need more room) or negative. There is a dearth of competitive adsorption data, in a large part because they are difficult to measure, but also because little interest has been devoted to them, as, until recently, there were few problems of importance whose solution depended on their understanding. Besides the static methods, which are extremely long and tedious and require a large amount of material, the main methods of measurement of competitive isotherms use column chromatography. Frontal analysis can be extended to competitive binary isotherms [14,73,93-99], as well as pulse techniques [100-104]. The hodograph transform is a powerful method that permits an approach similar to FACP for competitive binary isotherms [105,106].
4.2.1 Competitive Frontal Analysis Frontal analysis can easily be extended to binary mixtures. The shape of the breakthrough profiles and the effect of axial dispersion on these shapes have been studied theoretically [93,94] and experimentally [14,73,95-99]. These profiles are characterized by the successive elution of two steep fronts (shock layers) for a binary mixture. The use of these profiles for the determination of the competitive isotherms of two components has been developed by Jacobson et al. [14]. Figure 4.21 [14] shows the breakthrough curves obtained in two-component frontal analysis with competitive Langmuir isotherms, with four successive concentration steps, and with a column efficiency of 5000 theoretical plates. The thin and thick solid lines correspond to the first and the second components, respectively. The first step gives a different profile from all the following steps because the column is initially equilibrated with the pure mobile phase only [initial condition, Q(x, t = 0) = 0]. For this first step, two shock layers signal the successive exit of the lesser and the more retained components. The first component subplateau is more concentrated than the feed; there is no subplateau for the second
Competitive Equilibrium Isotherms
192 C mg/mL
60 50
3
5 Component Component 11
2
4 45
35
Sub
3
20
30 90
105
2
120
Component 2
15 1
Injection Plateau
0 20
60
100 100
140 140
180 180
t, min
Figure 4.21 Schematic of the determination of competitive binary isotherms by frontal analysis. Main figure: Typical experimental chromatogram in two-component frontal analysis. Thin line, concentration profile of the first component; thick line, concentration profile of the second component. Inset: Expansion of one step in the main figure. "Sub" indicates the intermediate subplateau during the breakthrough of the binary mixture echelon.
component since the column is initially empty. However, after this first step the column is no longer empty [Q(x, t — 0) — Cf] and, because of the competitive interactions between the two components, another type of profile is observed for each of the consecutive steps. The inset in Figure 4.21 shows an expansion of a typical step, described in the following. For each successive step, the profile of the second component exhibits an intermediate plateau at a concentration that is intermediate between the initial plateau concentration and the feed concentration in the new step, while, simultaneously, the first component profile exhibits an intermediate plateau with a concentration that is greater than the feed step. This is the result in frontal analysis of the displacement effect, itself the result of competition. Upon arrival at the column exit of the second component front (whose plateau concentration is equal to the feed concentration, the first component concentration undergoes a drop to the feed concentration. Note that the concentrations of both components converge simultaneously to the feed composition. The first step of a two-component frontal analysis has been studied experimentally and theoretically by Carta et al. [107], for the breakthrough of two amino acids, and by Zhu et al. [73] for the breakthrough of 2-phenylethanol and 3-phenylpropanol (Figure 4.22). As illustrated in Figure 4.21, in two-component FA, a series of binary solutions of constant relative composition of the two components to be studied is pre-
4.2 Determination of Competitive Isotherms
193
~T*,e
E E
1 E
3 E
E
2
r
5
4
P
f
f primary plateau
5
4 sub plateau
^-yJ
3 .
t
—
^
Step * P— >p *
1
C
1
-"Time
Figure 4.22 Experimental concentration profiles in the column effluent for adsorption isotherm determination by binary frontal analysis. (Left) Bottom trace: solid line, experimental UV profile; dotted line: reconstructed profile of 3-phenylpropanol (P); dashed line: reconstructed profile of 2-phenylethanol (E). Arrows 1-5 indicate the time when the eluent sample was taken for on-line analysis. Top trace: On-line analysis of the sampled eluent. Reproduced with permission from J. Zhu, A. Katti and G. Guiochon, J. Chromatogr. 552 (1991)
71 (Fig. 1). (Right) Examples of one-step binary frontal analyses for the determination of the competitive isotherms of N-benzoyl-D,L-alanine. Injection of large volumes (5 mL) of solutions of increasing concentrations of racemic mixture. Reproduced with permission from S.C. Jacobson, A. Felinger and G. Guiochon, Biotechnol. Progr., 8 (1992) 533 (Fig. 1), ©1992 American Chemical Society.
pared. They are pumped through the column in order of increasing concentration, and the breakthrough curves are recorded. The retention times of the two steep fronts of the breakthrough curve do not contain enough information to derive a point on each of the competitive isotherms of the two compounds. To do that, it is necessary to determine the concentrations at the intermediate plateaus [14]. This can be accomplished by sampling the concentrations at specific points, using a second on-line, fast chromatograph. Figure 4.22 illustrates an experimental chromatogram with on-line sampling of the intermediate plateau. The lower
Competitive Equilibrium Isotherms
194
Load Pump A Pump A Valve B B
Valve A Valve A
->• Waste
Frontal Chromatograph Adsorption Column
Detector
Detector
Sampling Valve Analytical Column
Pump B
Waste
In-Line HPLC Analyzer
Flow Monitor
Waste
Figure 4.23 Schematic of an equipment designed for the experimental determination of competitive isotherms by binary frontal analysis. plot illustrates the breakthrough curve, the upper plot the chromatograms for the sampled intermediate plateau. The use of short, high-efficiency columns for the on-line analysis allows the rapid determination of the intermediate plateau concentrations. In order to determine accurately the intermediate plateau concentrations, the extra-column (tubing) volumes in the chromatograph must be small to avoid significant band broadening. Figure 4.23 illustrates schematically an apparatus for competitive isotherm determination by frontal analysis [14]. From the breakthrough times and the plateau concentrations, the isotherms of the two components can be determined along a line of constant relative composition (i.e., the intersection of the isotherm surfaces, Cfi{C\, C2), with the vertical plane C4/C2 = const) [14]. Mass balance of the two components (see Figure 4.21, inset) of the binary mixture gives (V2-V0)(Cirb-Cia)-(V2-
(4.74)
where VQ, V\, V2, and Vads are the column dead volume, the elution volumes of the two breakthrough fronts, and the volume of adsorbent in the column, respectively. CiA and Cij, are the concentrations of component i in the column ahead of the first front (i.e., in the solution with which the column was equilibrated before the new experiment begins) and behind the second front (i.e., the concentration of the new solution with which the column is equilibrated, respectively) and Q m is the concentration of component i in the intermediate plateau. The experiment is repeated with a series of solutions corresponding to other ratios of the concentrations, in order to determine the entire isotherm surface [14]. Rather than making successive step changes in the feed concentration, it is also
4.2 Determination of Competitive Isotherms
195
Figure 4.24 Reproductibility of the measurement of the competitive isotherms of cis- and tans-androsterone using a 2:1 mixture before and after a long series of experiments. 50 x 4.6 mm column packed with Partisil 10 treated with a phosphate buffer; mobile phase ACN/CHC13 (15:85) at 1 mL/min. Reproduced with permission from J.-X. Huang and G. Guiochon, ]. Colloid and Inter/. Sci., 128 (1989) 577 (Fig. 8).
possible to make a series of wide plug injections of increasing concentration, allowing the sample to elute completely from the column. Although more tedious and time consuming than the staircase method, this approach avoids the need for analysis of the composition of the intermediate plateaus. Also, cumulative errors are prevented. Finally, it is important to make sure that the exact composition of the intermediate plateau is measured. This can be done only if such a plateau exists. When the column efficiency is insufficient, or when the concentrations at which measurements are needed are too high, the intermediate plateau disappears and a peak is formed instead. Then the results are poor, as explained in detail elsewhere [106]. The staircase method could not be used for the determination of the competitive isotherms of N-benzoyl-D- and L-alanine on bovine serum albumin immobilized on silica because the low efficiency of the column and the low saturation capacity of the enantioselective site prevented the achievement of well developed intermediate plateaus at concentrations above 10 mg/mL. Figure 4.22b illustrates the breakthrough curves of three sample plugs of different concentrations. The frontal analysis method has been applied to measurement of competitive adsorption isotherms where an interchange of the elution order occurs at increasing sample sizes, discussed previously in Section 4.1.2 [9]. The major drawbacks of the frontal analysis method are the important number of measurements to be made, the considerable amount of time that it takes to determine a set of competitive isotherms and the large amount of sample required. The competitive isotherms are sets of n surfaces in an n + 1 space where n is the number of components. For a binary mixture, we have two surfaces, f\(C\, C2) and fi{C\, C2). These surfaces depend minimally on four parameters, often on more, depending on the isotherm model selected. Accurate determination of the isotherm parameters requires a rather large number of data points, corresponding to a wide range of absolute and relative concentrations. The acquisition of pure component data as well as equilibrium data for 1:3, 1:1, and 3:1 mixtures seems to be a bare minimum [14]. Compared to other methods of isotherm determination, the frontal analysis method has the advantage of being nearly independent of the kinetics of mass transfer and axial dispersion, i.e., of the column efficiency. On the other hand, the results achieved can be very reproducible if proper experimental care is taken. Figure 4.24 [9] illustrates, for cis- and rrans-androsterone, the reproducibility of two runs before and
196
Competitive Equilibrium Isotherms
after a long series of experiments. The FA method gives isotherm data. To be useful in preparative chromatography, these data must be fitted to an isotherm model. There are presently no numerical procedures available to smooth the data from multidimensional plots, similar to the 2-D splines or French curves and obtain purely empirical isotherms. Therefore, the major difficulty is the selection of adequate models. The Langmuir isotherm is too simplistic in most cases, and the LeVan-Vermeulen isotherm is complicated and difficult to use as a fitting function. Several methods have been described to extract the "best" set of Langmuir parameters which could account for a set of competitive adsorption data [108]. These methods have been compared. The most suitable method seems to depend on the aim of the determination and on the deviation of the system from true Langmuir behavior [108].
4.2.2 Determination of a Multi-component Langmuir Isotherm by Frontal Analysis and the Reverse /i-Transform In all this section, we assume that the two feed components follow exactly multicomponent Langmuir competitive isotherm behavior. Jacobson et al. [14] described a method for the derivation from frontal analysis data of the best values of the coefficients of a set of competitive Langmuir adsorption isotherms that is based on the Helfferich theory of multicomponent chromatography [86]. This procedure involves the measurement of the velocities of composition changes through the retention times of concentration fronts propagating along the column, for various sets of concentrations, and the regression of these values to the parameters of a competitive Langmuir isotherm using the reversed /z-transform. 4.2.2.1
The Coherence Theory and the /i-Transf orm
One of the fundamental aspects of the theory of nonlinear chromatography is the recognition that any perturbation (e.g., the injection of any sample of composition different from that of the mobile phase stream) caused to a column in which the mobile and stationary phases are in equilibrium results in the formation of perturbation waves that, after a time, consist in a series of concentration waves that migrate along the column. The coherence theory is based on the observation that the velocities of the waves obtained for different compounds are equal. «c« = «
(4-75)
When the system follows Langmuir competitive equilibrium behavior, the coherence condition defines a grid of coherent composition paths to which the system is restricted once the coherence condition is satisfied. Knowing the feed history, i.e., the boundary condition, one can use this grid, find the composition routes for the column and predict the column effluent history. Using a nonlinear transform, Helfferich and Klein [86] defined an orthogonalized composition space called the h-composition space. The /i-transform is actually a means of transforming the set of dependent variables Q into a set of independent variables h{ and, thus, of simplifying the mathematical analysis of the
4.2 Determination of Competitive Isotherms
197
chromatographic process. When the system follows Langmuir competitive behavior, the transformation from the C-composition space to the h-space involves the calculation of the roots of the following equation
£. —bjCi
1= Q
(4 76)
In the rest of this section (Section 4.2.2), we change our convention to call the least retained component of a group the component 1. For the sake of consistency with the convention of these authors, we assume that component 1 is the more retained component and that component n is the least retained component. Accordingly, we have a.\ > a^ > • • • > az- > • • • > an. Equation 4.76 is an n-th order polynomial and its roots are, by definition, ord e r e d as h\ > hi > • • • > hi > • • • > hn.
The velocity of each front shock (see Eq. 7.7 in Chapter 7) is given by usi = — t ^ - r SA
(4.77)
1 + Ft
The calculations can be simplified by rewriting this equation in terms of normalized velocities U{
1
"S,l
(4.78)
An
When written in the h-space, in terms of the hj, the equation for the normalized front velocity defined above becomes vSii = ——-———-—^-^—
—
(4.79)
In Eq. 4.79, h^ and hij, are the values of the variable hi ahead and behind the shock, respectively. Regression of the experimental values of V{ for different sets of concentrations yields the best Langmuir parameters of all solutes. This method can be used for a mixture of multicomponent solutes. This method, which is based on the measurement of composition velocities, was called MCV [14] (see Figures 4.26A and 4.26B). For a binary system, Eq. 4.79 becomes (4.80)
v2 =
h
For binary systems, Eq. 4.76 becomes a quadratic expression and its roots are given by [14] hi =
%
198
Competitive Equilibrium Isotherms with: and:
Qi
=
Q2 =
a1/a2 + l + b1Ci + aib2C2 ^ ^
+
(4.83)
b2C2+ l)
«2
In a binary system, a nonlinear regression to these last five equations can be performed using the values of 1/,- determined by measuring the retention volumes of breakthrough curves in frontal analysis experiments. This method affords the determination of the parameters of the binary Langmuir isotherms a2- and b[ of the two components of the mixture. Jacobson and Frenz [108] have used this approach to develop two new methods. The Method of Mezzanine Concentration (MMC) According to the /i-transform theory, the composition of the mezzanine bands 1 is related to the h values through " the reversed ^-transform". <-i,m — 7-7—i
TT
(4.85)
b1(a1/a2-l)
In the MMC method, a non-linear regression of the values of the mezzanine concentrations measured during the FA experiments is made to Eqs. 4.80 to 4.86. This regression gives estimates of the parameters of the binary Langmuir isotherm, «; and bi, for each component of the mixture. This method is cumbersome compared to MCV because it requires the measurement of series of values Q/OT of the mezzanine concentrations, which is time consuming. In addition, if values of Qi1H are measured, it is much better to use the MMB method which gives the actual isotherm graph, a graph that is independent of the model while the MMC is valid only for the Langmuir isotherm model. The Hybrid Method of Mass Balance (HMMB) This method is a modification of the MMB method in which, instead of measuring a series of values of C;m, which is cumbersome, these concentrations are estimated through Eqs. 4.85 and 4.86 by using the isotherm parameters, a,- and by determined by the MCV method. This hybrid approach employs MCV only to estimate the mezzanine concentrations, so it is more accurate than MCV when the actual isotherm deviates from the Langmuir competitive isotherm. 4.2.2.2 Application of the Coherence Theory to Frontal Analysis Using this method, Jacobson et al. [14] showed that the competitive isotherms of pairs of simple solutes, like phenol and p-cresol, on alkyl-bonded silica stationary phases can be well fitted with a binary Langmuir isotherm. 1 The mezzanine is the intermediate plateau observed in multi-component frontal analysis, e.g., in Figure 4.21.
4.2 Determination of Competitive Isotherms
199
0.1
PHENOL
p-CR£SOL 0.4
2.1
20
AO
MJ
SO
ICO
Figure 4.25 Competitive isotherms of phenol and p-cresol on an octadecylsilica. In these two figures, the top isotherm is the single solute isotherm. Isotherms denoted A, B, and C correspond to solutions with 1:3, 1:1, and 3:1 mole ratios of phenol and p-cresol, respectively, (a) Competitive isotherms of p-cresol. (b) Competitive isotherms of phenol. Reproduced with permission from } . ]acobson, J. Frenz and Cs. Horvdth, Ind. Eng. Chan. (Res.), 26 (1987) 43 (Fig. 4), ©1987 American Chemical Society, (c) Competitive
isotherms of N-benzoyl D- (o) and L-alanine (<0>) in the racemic mixture; single component isotherms of N-benzoyl D- (+) and Lalanine (A). Reproduced with permission from S.C. Jacobson, A. Felinger and G. Guiochon, Biotechnol. Progr., 8 (1992) 533 (Fig. 2), ©1992 American Chemical Society.
10
zo
CfrngAJ
As an example, we show in Figure 4.25 the competitive isotherms of the mixture of p-cresol (Figure 4.25a) and phenol (Figure 4.25b) on octadecyl silica [14], and those of N-benzoyl-D- and L-alanine on BSA immobilized on silica [29]. The isotherms in Figure 4.25 were measured by binary frontal analysis (Section 4.2.1). Figures 4.26A and 4.26B compare the results of the experimental determination of isotherms using the traditional mass balance method (MMB) and those obtained with MMC. The adsorption isotherm predicted by MMC deviates significantly from the isotherm data obtained by MMB. This may be due to the limited applicability of the Langmuir competitive model for the modeling of the adsorption behavior even of such simple systems as p-cresol and phenol in reversedphase chromatography. Figures 4.26C and 4.26D compare the results obtained by MMB and HMMB for the same system. Over most of the concentration range, the agreement between the experimental data and the results of these two methods is
Competitive Equilibrium Isotherms
200
0.02
0.04
0.06
0.08
0.02
0.10
0.04
0.06
0.08
0.10
Total Mobile Phase Concentration, M
Total Mobile Phase Concentration, M
1.0
l.U
B
A a 3:1 -
0.8
•
0.6
-
a
= 0.1
u e 0
I 0.4
8
a -
B
0 a
a
a a a
0
0
0.4
0
B 13
a a
o 1:1
•
p
a
g
-
0.2
•
I," a
f
0.0 0.02
0.04
0.06
0.08
Total mobile phase concentration, M
0.10
ft n 0.12
S
a a e
0.02
a
s
0
a
e
8
0.04
D
8 0.06
1;3
a
a
Q
6
1:1
D O
o o
3.1
0.08
0.10
-
0.12
Total mobile phase concentration, M
Figure 4.26 Comparison of the competitive adsorption isotherm measured by FA and calculated by two different methods. p-Cresol (Left) and phenol (Right). Top Data from the mass balance method (MMB, binary frontal analysis) at molar ratios of 3:1 (Q), 1:1 (•) and 1:3 (A). Solid lines calculated by the method of composition velocity (MMC). Bottom Comparison of the competitive isotherms obtained by MMB (O) and HBBM (square symbol) (l~l) for p-cresol and phenol in three concentration regimes. Reproduced with permission from J. Jacobson and ], Frenz, J. Chromatogr., 499 (1990) 5 (Figs. 2 and 5).
excellent, even though, as shown earlier, this system deviates significantly from competitive Langmuir behavior. Accordingly, HMMB is a good approximation method for obtaining isotherm data. It should be emphasized that it is experimentally much simpler than MMB because there is no need for measuring the concentrations Q/W on the mezzanine plateaus (i.e., the intermediate plateaus).
4.2 Determination of Competitive Isotherms
201
4.2.2.3 The fe-root method (HRM) Chen et ah [109] proposed the use of a similar method for the determination of the best coefficients of the competitive Langmuir isotherm that model a set of experimental data. This method was called the /z-root method. In frontal analysis, the column is initially equilibrated with the pure mobile phase. At the initial time, the pure mobile phase is abruptly replaced with a solution that contains finite concentrations of different solutes. If there are n solutes, an equal number n of concentration shock waves is generated. Each shock migrates at a constant velocity that depends on the corresponding compound and on the amplitude of its shock. In other words, the resolution of a noncoherent into a coherent state is instantaneous. The elution order of the shocks will not change during the experiment and the plateaus regions between successive shocks are conserved. The /z-root waves can be determined by measuring the shock velocities and using Eq. 4.79 (Chen et al. [109]). 4.2.2.4 Determination of the coefficients of the Langmuir competitive isotherm From the injection of a small pulse of a highly dilute mixture of all the components involved, and the measurements of the retention times of each peak recorded (under linear conditions since this pulse is highly dilute), it is possible to calculate the parameters a^ for all the solutes
Frontal analysis gives the retention times of all the fronts that can be identified. These retention times can be used to calculate the velocities of the associated shocks and, therefore, the normalized velocities of the corresponding fronts. These velocities can be used to determine /z;-/fl using Eq. 4.79. In the first plateau zone, the least retained component (i.e. component n) is pure. In this zone, the concentrations of all the components, except that of the less retained one, are equal to zero. The concentration, Cn, of the least retained component n is given by Eq. 4.76 which simplifies to bnCn
—-1 = 0
(4.88)
The retention time of the first shock provides un, hence hnfl and we can calculate the Langmuir coefficient for the less retained solute (bn) from bn = ( ^ f l f l " M ) 1
(4.89)
For the second plateau, all concentrations are 0, except Cn_i and Cn. Therefore, Eq. 4.76 reduces to
202
Competitive Equilibrium Isotherms
Since hHia is also a nontrivial root in the second zone, it will satisfy Eq. 4.90 and, therefore, the second Langmuir coefficient, fon_i, can be calculated from "71—1
.-.
<-n-l
The iteration process affords the successive values of all the coefficients. So, for the coefficient b^, we have
The /i-root method provides an explicit method for determining the coefficients of a competitive Langmuir isotherm from experimental data. Two series of experimental measurements are required: (1) The determination of the retention times of the fronts, that are simple to carry out; (2) The determination of the concentrations of each solute on each intermediate plateau, which is more cumbersome to measure. Furthermore, the method relies on the assumption that the pair of compounds investigated follows Langmuir isotherm behavior in the range of concentrations studied and in the whole range of relative compositions. Any model error will result in the band profiles calculated with the isotherm model and the coefficients derived from the experimental data to deviate from the actual experimental band profiles. When there is a significant model error, the deviations will increase when the relative composition of the mixture differs more from the composition of the mixture used for frontal analysis. Since the relative composition of the eluate evolves progressively during the elution of a binary mixture (it decreases from C1/C2 — 00 to zero), model errors might prevent fruitful use of the isotherm data for computer-assisted modeling of the separation.
4.2.3 The Pulse Methods Two different implementations of these methods have been developed, the tracer pulse technique (or elution of an isotope on a plateau) and the concentration pulse technique (or elution on a plateau). They are very different in principle although they share much theoretical background. Only the second one has now any practical applications in liquid chromatography. The fundamentals of the phenomena involved in the pulse methods are discussed elsewhere in detail (Chapter 13). Suffice to say here that when a small sample of an n-component mixture is injected into a chromatographic system the mobile phase of which contains p additives, n + p peaks are formed and eluted, as illustrated in Figure 4.27. Although in pulse methods, tracers have the same isotherms as the components studied, their concentrations are different and, accordingly, the migration rates of the tracer peaks are different from the velocities of the component perturbations at the plateau concentrations. Thus, upon injection of one tracer (n — 1) in a the binary solution (p — 2) used as the mobile phase, we expect to see three peaks on the plateau of each component, as shown in Figure 4.27 in the case of a binary mixture, assuming that the two components
203
4.2 Determination of Competitive Isotherms
C mg/mL
a
6
b
Plateau-2
12
12
27 Total-1st
Plateau-1
3
10
10
24 Total
Tracer-1
0
8
8 0
5
21 0
10 c
6
5
10
Total-2nd Total-2nd
12
12
10
10
d 27
3
Total
24
Tracer-2
0
8 0
5
8
10
21 0
5
10
e
f
6 12
12
27
•dLr™" Total-1st
3 10
10
Tracer 1&2
0
8 0
5
24
Total
10
21
8 00
55
Time, min Time/ min
10 10
Figure 4.27 Illustration of the binary step and pulse method. Plateau concentrations: 45mg/mL for component 1, 50 mg/mL for component 2. Amount of tracer injected: 0.3 mg; tp = 0.143 min; Fv mL/min. L = 25cm; N= 2500; fc'ol = 6.17;fc'O2= 12.3; b\ = 0.0267; fo2 = 0.05; IQ = 60 s. (a) Elution of an isotopic tracer of component 1 in a mobile phase containing components 1 and 2; chromatograms shown by detectors selective for components 1, 2, and the tracer, (b) Same as (a), but chromatograms shown by a nonselective detector (Total) and by a detector selective for 1, but having the same response for 1 and its isotopic tracer, (c) Same as (a), but with injection of an isotopic tracer of component 2. (d) Same as (b), but with injection of an isotopic tracer of 2. (e) Selective chromatograms obtained upon injection of a mixture of isotopic tracers of 1 and 2. (f) Chromatograms obtained with a nonselective detector (Total), and with detectors selective for 1 or 2. studied are adsorbed but none of the other mobile phase components. Upon injection of a solution of the two tracers (n = 2), we should record four peaks on the plateau of each of the two components. Each of the peaks associated with the tracers (one in the first case, two in the second), moves at the tracer velocity, which is proportional to the slope of the chord of the corresponding isotherm (Chapters 7 and 13). With a selective detector, the recorded peaks are those of the pure trac-
204
Competitive Equilibrium Isotherms
ers (in the presence of the plateau mixture). The other two peaks or system peaks are coupled through the competitive isotherms and their velocities depend on this isotherm and on the composition of the mobile phase. Each one is the superimposition of a peak of each component of the mixture. Since the retention times of the two tracer peaks depend on the slopes of the chords, Aqj/ACj,i = 1,2, of the competitive isotherms, it is possible to determine competitive isotherms by the systematic measurement of the retention times of tracers as a function of the mobile phase composition. 4.2.3.1
The Tracer Pulse Method
As shown by Helfferich and Peterson [110], if a small pulse of one of the mixture components, isotopically labeled, is injected in a multicomponent solution, the labeled component moves at a velocity that is proportional to the slope of the corresponding chord of the isotherm, hi the same time, as many system peaks as there are components in the mixture arise and move at velocities related to the slopes of the corresponding isotherms (see Figure 4.27a,c,e). This assumes that equilibrium has previously been reached throughout the column. In the tracer pulse technique, only the labeled component is detected. Successive injections of a sample of each labeled component (or the simultaneous injection of all of them if their separate identification is possible) permit the direct determination of the competitive isotherms of all the components. From a theoretical point of view, the presence of large concentrations of the other components does not complicate the measurement nor its evaluation, since the retention time of each isotopic pulse is a linear function of the slope of the corresponding chord (Aq/AC, see Eq. 2.15). In practice, the method presents considerable difficulties. First, isotopically labeled samples of the studied compounds must be available or prepared, which is often difficult and costly for the complex components of the mixtures whose separations are performed in liquid chromatography. Radioactive isotopes are highly impractical to use, even at low concentrations, due to safety considerations, shielding, licensing, regulations, paperwork, and the high cost of personal protective equipment. Stable isotopes are easier to handle but the synthesis of labeled molecules remains expensive. They can be detected by mass spectrometry, but a functioning LC/MS interface is costly, making its use an expensive and hard-to-develop proposition. For all these reasons, the use of the tracer method seems to be limited to simple molecules, in gas chromatography [100-104]. 4.2.3.2
The Concentration Pulse Method
Concentration pulse chromatography (also called elution on a plateau, step and pulse method, system peak method, or perturbation chromatography) is experimentally much simpler [100,103,104]. The same experimental procedure is used as for the determination of single-component isotherms. First, the column is equilibrated with a solution of the multicomponent mixture of interest in the mobile phase. When the eluent has reached the composition of the feed to the column, a small pulse of the pure mobile phase (vacancy) or of a solution having a composition different from that of the plateau concentration (see end of this section) is
4.2 Determination of Competitive Isotherms
205
injected. This injection generates the elution of a series of system peaks, as many peaks as there are components in the solution that are retained by the stationary phase. Because of the coupling effect between the behavior of the feed components, due itself to the competitive behavior of their phase equilibrium, it is not possible to carry out a direct evaluation of the experimental results, as is done in the case of a single, pure compound. The perturbation signals obtained cannot be simply related to the equilibrium concentrations of the different components of the mixture in the stationary phase. It is not even possible to relate directly any of these perturbations or system peaks to any of the mixture components. Each perturbation peak is actually a perturbation of the concentrations of all the mixture components. As explained earlier (see also Chapter 13), the velocities of the system peaks are related to the slopes of the tangents to the multidimensional isotherm surface, in the directions of the N — 1 composition paths. These slopes can be derived from the retention times of the tracer peaks. The retention times of the peaks recorded on a chromatogram obtained with a non-selective detector (Figures 4.27b,d,f) give only the slopes of planar sections of the isotherm surface, as a function of the concentrations of the feed components. Integration is required to derive the isotherm parameters, which can be done only if an isotherm equation is available. An experimental isotherm, e.g., a set of experimental data points, cannot be derived from these measurements. The experimental data may be used only to determine the best values of the parameters of a selected isotherm model, by proper surface fitting. The determination of an isotherm surface requires the measurement of the pulse retention times for a number of solutions of the feed components in the mobile phase, in a wide range of absolute and relative concentrations of the components. Accordingly, it is not possible to derive adsorption data points as in FA. The method involves first the choice of an isotherm model. Then the sets of velocities measured for all the system peaks recorded is fitted to the theoretical expressions derived from the isotherm equation and the numerical values of the isotherm parameters that best fit the experimental data are calculated. The retention factors of the system peaks are given by the equation
with i = l,...,n. The fractions dqi/dCf are the total derivatives of the isotherm. The fractions dqi/dCj are the partial derivatives of the competitive isotherms. The fractions dCj/dC{ are the directional derivatives, which cannot be derived directly from the experimental data, i.e., from the retention factors k^ However, they can be derived from the coefficients of the isotherm model, by applying the coherence condition (see Chapter 12). In the case of a binary mixture, the retention factors of the perturbations on a plateau of concentrations CQ,I, CO,2 are solutions of the
206
Competitive Equilibrium Isotherms
eigenvalues of the matrix dq\ \ / dq-y dC L dC2 (4.94)
dq2
dqi
\
at the plateau composition. This equation becomes 0
+
r49
( dC2 321 M ' The two roots of Eq. 4.95 are inserted into the total derivative, Eq. 4.93, and this gives relationships between the isotherm parameters and the experimental data. The set of numerical values of the isotherm parameters is then derived from the fitting of the retention data measured to the isotherm model. Ching et ah [111] determined the competitive isotherms of fructose and dextran in water on silicagel, using the pulse method. They assumed for the two compounds the following competitive isotherm equations
= C^kx + A^+BnC^2)
qi
(4.96) (4.97)
By measuring the retention times of pulses of each solute on plateaus of series of their pure solutions at increasing concentrations (i.e., pulses of component 1 on plateaus of solutions containing only component 1 and pulses of component 2 on plateaus of solutions containing only component 2), the parameters k\, A\, m\, fc2, A2 and m2 can be determined. For solutions containing only component 1 (i.e., binary solutions of one component and the solvent), we have < 1
?
dC,
— i-- i A.(m. L
i •\\rmi
(4 98)
'
and for solutions containing only component 2, (4.99) Therefore, measurements carried out over a range of concentrations C\ and C2 with pure binary solutions, allow the determination of k\, A\,m\,k2, A2 and m2. From the retention times measured with pseudo-binary systems, i.e., for pulses of component 1 over concentration plateaus of solutions of component 2 alone (Q = 0) and for pulses of component 2 over plateaus of solutions of component 1 alone (C2 = 0), one can derive from Eq. 4.96: | |
(4.100) (4.101)
4.2 Determination of Competitive Isotherms
Figure 4.28 Retention times of the perturbations versus the concentration of the plateaus of Troger's base. Reproduced with permission from K. Mihlbachler, K. Kaczmarski, A. SeidelMorgenstern, G. Guiochon, J. Chromatogr. 955 (2002) 35 (Fig. 5).
207
1615-
/in
f a
14„ 13c: |i2-
'A
11109-
2
3
4
5
Concentration C [g/l]
Therefore, by measuring the retention times of pulses in pseudo-binary systems of various concentrations, the parameters B\2t ti\2, B2\, and n2\ can also be determined. This approach is valid for more complex isotherms only when the i.e., ternary interaction effects can be neglected and the adsorbate-adsorbate interactions in the ternary system are the same as these interactions between the same components in the corresponding binary systems. The validity of this assumption was confirmed for the mixtures of fructose and dextran T6, of fructose and dextran T9, and of dextran T6 and dextran T9, by comparing the predicted and the measured retention times obtained for the ternary system [111]. The tracer pulse method was also used by Bliimel et al. [112] to determine the binary isotherms of the enantiomers of l-phenoxy-2-propanol on Chiralcel OD, by Lindholm et al. [113] to determine the binary isotherms of methyl-mandelate on Chiral AGP, and by Mihlbachler et al. [1] to determine those of the enantiomers of Troger's base on Chiralpak AD. hi this last case, an unusual isotherm was obtained, illustrated in Figure 4.28. The adsorption of the more retained (+) enantiomer is not competitive: the amount adsorbed by the chiral stationary phase at equilibrium with a constant concentration of the (+) enantiomer is independent of the concentration of the (-) enantiomer. On the other hand, the adsorption of the less retained enantiomer is cooperative: the amount of this (-) enantiomer adsorbed by the CSP at equilibrium with a constant concentration of this enantiomer increases with increasing concentration of the (+) enantiomer. The isotherm data are best accounted for by an isotherm model derived assuming multilayer adsorption. h + 2bnCx + bl2C2 + b21C2 (b21
- b 2 C2 -2b22C2)
b2llC\C2
(4.102)
(4.103)
Competitive Equilibrium Isotherms
208
aI
54 52
bI
1775 Resp. [mV]
Resp. [mV]
56
1770 1765 1760
50 1.5
4 x 10
-3
6 8 Time [min.]
10
1.5
2
2.5 3 Time [min.]
3.5
0.02 aII
1
bII
c [mM]
c [mM]
0.01 0.5 0
0
-0.5 -0.01 -1 -1.5
4
6 8 Time [min.]
10 10
-0.02
1.5 1.5
2
2.5 3 Time [min.]
3.5
Figure 4.29 Perturbation signals recorded on a four-compound plateau. Top, experimental results. Bottom signal calculated with the equilibrium-dispersive model and the best coefficients of a competitive quaternary bi-Langmuir isotherm. Compounds: enantiomers of methyl- and ethyl-mandelate on a chiral phase. Perturbation as in Eq. 4.104 Reproduced with permission from J. Lindholm, P. Forssen, T. Fornstedt, Anal. Chem., 76 (2004) 5472 (Fig. 3). ©2004, American Chemical Society.
The solid lines in Figure 4.28 show the retention times of the perturbation pulses as functions of the mobile phase composition (concentration of each enantiomer, the solution being prepared from the racemic mixture) calculated from the isotherm model while the symbols show the experimental measurements. The principle of this pulse method and its general equations are easily extended to the case of several components in a mixture. The method was used by Lindholm et al. [24] to determine the quaternary isotherms of the enantiomers of methyl- and ethyl-mandelate on the chiral phase Chiral AGP. One of the serious roadblocks encountered in the use of the pulse tracer method is that the amplitudes of most of the system peaks decrease rapidly when the plateau concentration increases. Since the signal noise increases in the same time, it becomes rapidly impossible to make any accurate measurements of the retention time of these peaks. On the basis of fundamental work by Tondeur et al. [114], the origin of this variation of the relative intensity of the system peaks was explained by Forssen et al. [47], who then derived an effective rule to determine the composition of a perturbation pulse that generates system peaks that are detected easily. The concentrations of the components in the injected perturbation pulse should
4.2 Determination of Competitive Isotherms
0.2
0.4
0.6
209
0.8
1.2
Figure 4.30 Retention times of perturbation peaks versus the total concentration on quaternary concentration plateaus. Symbols are experimental data: peaks 1 (•), 2 (O), 3 (v), and 4 (A). The lines are calculated data using the parameters measured (see Table 1, ref. [24]: peaks 1 (dotted), 2 (dash-dotted), 3 (dashed), and 4 (solid). The main figure shows the high concentration range up to 1.250 mM. The inset in the upper right corner shows the medium-concentration range u p to 0.03 mM. Reproduced with permission from }. Lindholm, P. Forssin, T. Fornstedt, Anal. Chem., 76 (2004) 5472 (Fig. 4). ©2004, American Chemical Society.
be given by Cp,i = C0/i + (-l)'SC
(4.104)
whereCfy, Co,*, and SC are the concentration of compound i in the perturbation pulse, its concentration on the plateau, and a concentration increment that is small enough to avoid any concentration CS/I- being negative. The validity of this method was demonstrated for binary mixtures by Forssen et al. [47] and for quaternary mixtures by Lindholm et al. [24]. Figure 4.29 shows an example of the perturbation signal recorded during the determination of a quaternary isotherm [24]. The four competitive isotherms were all well accounted for using the competitive biLangmuir model. Figure 4.30 shows the dependence on the total concentration of the retention times of the four perturbations on a quaternary plateau of constant relative composition [24]. A significant drawback of the tracer pulse methods remains that they require large volumes of rather concentrated solutions, hence large amounts of the chemicals studied, larger even than those needed in FA (since the measurements to make on the plateau concentrations last longer). These solutions are more difficult to re-
Competitive Equilibrium Isotherms
210
TIME
Figure 4.31 Experimental determination of competitive isotherms by the method of the hodograph transform, (a) Individual elution profiles of the two components of a binary mixture, (b) Hodograph transform of the profiles in (a). Reproduced with permissionfromZ. Ma and G. Guiochon,}. Chromatogr. 603 (1992) 13 (Fig. 3).
cover than those used in the determination of single-component isotherms. For this reason, the use of microbore columns for the determination of isotherms by FA or perturbations has to be recommended. Unfortunately, the problems of column reproducibility, of the homogeneity of adsorbent lots, and of scale-up of the experimental data for applications in large-scale preparative chromatography are still almost unexplored.
4.2.4 The Simple Wave or Hodograph Method It can be shown that if a wide rectangular pulse of a binary mixture is injected and the pulse width is sufficient for a concentration plateau at the feed composition to appear in the elution profile of the mixed band, a plot of the local concentration of one component versus that of the other one is made of two lines which intersect at the point of the graph having for coordinates the composition of the feed (and that of the intermediate plateau) [106]. Figure 4.31a illustrates the elution profile of a large rectangular pulse of a twocomponent mixture. The symbols are the results of quantitative analysis of fractions collected. Figure 4.31b illustrates the data from Figure 4.31a replotted in the form of Q versus C%. The lines obtained depend on the isotherm and the isotherm coefficients can be obtained by curve fitting. In the case of competitive Langmuir isotherms the two lines are straight. We show in Figure 4.32 [115] the changes in the hodograph plot observed when the composition of the sample is changed, by increasing the concentration of either component in the feed sample at constant concentration of the other (Figures 4.32a or 4.32c), or by increasing both concentrations in the same ratio (Figure 4.32b) [115].
4.2 Determination of Competitive Isotherms
:"
:-'
-.'\
211
ta
Figure 4.32 Hodograph transformation obtained from a rectangular injection, (a) Increasing concentration of first component at constant second component, (b) Increasing concentration of both components, (e) Increasing concentration of second component at constant concentration of first component. Reproduced with permission from Z. Ma, A. Katti, B. Lin and G. Guiochon, } . Phys. Chem., 94 (1990) 6911 (Fig. 6a, 6b, and 6c), ©1990 American Chemical Society.
The method has been used for the determination of competitive isotherms in cases in which the deviation from the Langmuir model is moderate [115]. An HPLC chromatograph configured so as to permit injecting into the column a wide rectangular pulse, e.g., by pumping into it either the pure mobile phase or, for a known time, a solution of the compound of interest, was used to make the measurements described [115]. Excellent agreement was observed with other experimental data and with the experimental band profiles recorded in overloaded elution for binary samples of various compositions [48]. The most important advantages of the method are the rapidity with which a section of the isotherm surface can be determined and the small amount of material needed. From this viewpoint, the method has the same advantage over multicomponent FA as FACP with a rectangular pulse of sufficient width over single-component FA. However, the plot obtained depends little on the column efficiency, as long as this efficiency exceeds 1000 theoretical plates. The main drawback of the method is the need to collect and analyze several dozen fractions during the elution of the front of the second component and the rear of the first one (see Figure 4.31). It would rarely be possible to use this method with a selective detector, e.g., to find a wavelength at which one component can be detected and not the other one. Furthermore, the detector response at high solute concentrations is usually not linear and a deconvolution of the response would be inaccurate at best. Finally, an isotherm equation is needed. The Langmuir model has been used in the work described [115], but other equations could be used as well. It does not seem possible, however, to achieve sufficient precision on the points of the hodograph plot to differentiate readily between different isotherm equations when this isotherm is complex and depends on many parameters, or to determine precisely the value of these parameters.
212
Competitive Equilibrium Isotherms
4.2.5 The Inverse Method Competitive isotherms may be derived from overloaded chromatographic band profiles of mixtures, very much as single-component isotherms can be derived from single-component band profiles (see Chapter 3, Section 3.5.6). The principle consists in calculating the band profiles corresponding to an isotherm model, using e.g., the equilibrium-dispersive model of chromatography, and adjusting the isotherm parameters so as to minimize the difference between experimental and calculated profiles. A computer optimization routine allows the handling of either one or several chromatograms at the same time. The method has been described in detail [116]. It was implemented in the case of the enantiomers of 1-indanol on cellulose tribenzoate [117]. Excellent results were obtained as demonstrated in Figures 4.33 and 4.34. Experimental adsorption data acquired by FA are compared in Figures 4.33a and 4.34a with the best bi-Langmuir and bi-T6th isotherms obtained by the inverse method. The experimental band profiles are compared with the band profiles calculated using the equilibrium-dispersive model and the best coefficients of the bi-Langmuir isotherm model in Figure 4.33b and with the band profiles calculated using the best coefficients of the bi-T6th isotherm model in Figure 4.34b. The bi-T6th model seems to account better for the whole set of data, including band profiles obtained with smaller sample sizes [117]. This result is explained by the AED derived from the same set of experimental isotherm data (see Chapter 3, Figure 3.31). The AED is bimodal but the low-energy mode is relatively wide, suggesting that a certain degree of heterogeneity for the corresponding adsorption sites. Besides the problems encountered in the application of the method to singlecomponent profiles that were already discussed, the application of this method to binary isotherms encounters two new serious problems. First, it is most difficult to obtain the concentration profiles that are necessary for the calculation, except in the case of enantiomers. The reason is that, in most cases, high concentration profiles are recorded under such conditions that the detector response is not linear. Because the response factors of the detector for the two components are different, a proper calibration requires too much work to be practical. So, the only acceptable approach is to collect and analyze a hundred or so fractions, which is possible if long and tedious. In the case of enantiomers, however, the response of most detectors being achiral. the response factors of the two compounds are identical. A calibration curve is easily determined, using either enantiomer or the racemic mixture. The second problem is related to the dilution and separation that take place during the elution of an injection pulse. The range of mobile phase concentration sampled by the band profile during its migration extends from the feed concentration to 0. However, the band dilutes rapidly while it moves along the column and the amount of actual information contained in the elution profile does not extend much beyond the maximum of the elution band at the column outlet [116]. Further, if the injection band is wide, so as to decrease the extent of the dilution and to achieve a high concentration at the column exit, the degree of separation achieved is low and the recorded profile does not inform much on the rear part of the elu-
4.2 Determination of Competitive Isotherms
213
Figure 4.33 Top. Simultaneous fitting to a competitive bi-Langmuir model of the chromatograms obtained with a large (50.7 mg) and a moderate (10.14 mg) sample of the racemic mixture of 1-indanol. Bottom. Comparison of the FA adsorption data points (symbols) and the best competitive bi-Langmuir isotherms obtained by the inverse method (lines) for the racemic mixture. Reproduced with permission from A, Felinger, D. Zhou, G. Guiochon, } . Chromatogr. A, 1005 (2003) 35 (Figures 7 and 8).
tion profile of the first component [117]. In contrast, if the resolution between the two bands is good, then there is little information on the competition between the two compounds. In order to alleviate these obstacles, it is recommended to acquire several band profiles corresponding to different injection characteristics, injection concentration and duration of the pulse. These profiles can be treated simultaneously by the optimization program. Also, it is possible to use selective detectors (e.g., a polarimetric detector or a mass spectrometer) to acquire separately the elution profiles of the two compounds. In the case of mixtures of different compounds with different response factors, it is possible to collect fractions and analyze them separately. This procedure eliminates the calibration problem but may cause other problems by introducing errors on the time of the various data points and by causing some amount of back-mixing. Proper design of the fraction collection experiments allows minimization of this error [118]. Finally, the inverse method is fast and allows the rapid determination of the isotherm parameters corresponding to a systematic variation of the experimental conditions, e.g., temperature, mobile phase composition, pH, etc. It is particularly useful as a complement to the FA method, which is far slower but does not require the selection of an isotherm model in order to process the experimental data. Once a satisfactory isotherm model has been determined, it is unlikely that changing
214
Competitive Equilibrium Isotherms
Figure 4.34 Top. Simultaneous fitting to a competitive bi-T6th model of the chromatograms obtained with a large (50.7 mg) and a moderate (10.14 mg) sample of the racemic mixture of 1-indanol. Bottom. Comparison of the FA adsorption data points (symbols) and the best competitive bi-T6th isotherms obtained by the inverse method (lines) for the racemic mixture. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, } . Chromatogr. A, 1005 (2003) 35 (Figures 9 and 10).
the experimental conditions will affect the retention mechanism to the point of modifying the isotherm model itself [119].
4.2.6 Measurement of the Competitive Isotherms of Enantiomers Many reports in the literature deal with the determination of the competitive isotherms of particular systems. Many of them have already been referred to in this chapter. In this last section, we discuss the specific issues encountered in the measurement of these isotherms. This problem is particularly important from both practical and fundamental view points because these pairs of compounds have unusual properties arising from the identity of all their properties that are not chiral. The separation of enantiomers has become of particular importance in the production of pharmaceuticals. These separations are often difficult because the separation factors achieved on many chiral stationary phases (CSP) are low or moderate (often below 1.6) and the saturation capacities of most CSPs are low. The optimization of the experimental conditions of a separation is particularly important because most chromatographic separations of enantiomers are performed using the simulated moving bed process (SMB, see Chapter 17) and the complexity of this process, the long time that it needs to approach steady-state makes optimization by trial and error lengthy and costly. So, much attention has been devoted
4.2 Determination of Competitive Isotherms
215
to the improvement of the methods of determination of competitive isotherms for enantiomers. The inverse method, the method that calculates the set of isotherm coefficients that approximates as closely as possible the calculated elution profiles of one or several samples of different sizes and feed compositions and those calculated, is particularly suited in this case since the calibration of the detector response is not an issue [117]. Since pure enantiomers are usually rather expensive, the use of microbore columns for the measurement of their isotherm data has been developed [42,44]. Finally, it was shown that, in many cases, the competitive isotherms of a mixture of two enantiomers can be derived from experimental data acquired with the racemic mixture only [43]. This was done by using the converse of the method used to derive the competitive isotherm of a binary mixture from the single-component isotherms of the feed components. We can distinguish two types of chiral stationary phases (CSP): those that are obtained by chemically bonding ligands with a localized chiral center and those that are derived from natural products (e.g., cellulose) and in which a large proportion of the carbon atoms have a well defined chirality. In the latter type, the whole solution is a chiral environment, most molecular interactions are chiral, and, because of their huge number, they do not need to be strong to lead to a sufficient separation of the enantiomers. By contrast, chiral selective interactions are rare in the former type of CSP and chiral separations on them can be achieved only if these interactions are strong and if, when they take place, some strict conditions regarding the relative orientation between the enantioselective bonded group and the sample molecules are satisfied. These requirements are commonly referred to as involving the formation of a three-point complex interaction, although the actual condition may be more complex because adsorption is a dynamic process, not a static one as often implied wrongly [120-122]. Complex situations may be encountered with enantiomers, when conventional isotherm modeling is too simple an approach. For example, following earlier work by Cundy and Crooks [123] and Jung and Schurig [124], Baciocchi et at. showed how the enantiomers of l,l'-bi-2-naphthol can be separated on a nonchiral alkyl-amine bonded silica [125]. Because of the possible formation of the three possible dimers (two homogeneous, one heterogeneous) in solution and the possible separation of the corresponding two diastereoisomers, a pulse of a nonracemic mixture is split into two fractions, the first one of them containing almost all the enantiomer in excess, the second, almost only the racemic mixture. The model used in this first work [125] was later improved [126] leading to excellent agreement with experimental results. From the perspective of adsorption theory, chemically bonded CSPs are an unusual, important type of heterogeneous surfaces [31]. Besides the importance of their analytical applications, they offer the unique case of exhibiting tailored, well- defined active sites. For physico-chemical studies, they compare most favorably to conventional adsorbent surfaces, such as silicagel, the surface of which is poorly defined and most complex. Each type of sites on the surface of CSPs has a well-defined chemistry, a narrow adsorption energy distribution, and a specific kinetics. Admittedly, there might subsist some residual Si-OH groups, even after proper endcapping, and the adsorption energy distribution of the nonselective
216
REFERENCES
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Chapter 5 Transfer Phenomena in Chromatography Contents 5.1
Diffusion 5.1.1 Diffusivity or Diffusion Coefficients 5.1.2 Influence of the Concentration on the Bulk Diffusion Coefficients 5.1.3 Influence of the Pressure on the Bulk Diffusion Coefficients 5.1.4 Influence of the Temperature on the Bulk Diffusion Coefficients 5.1.5 Measurement of the Diffusion Coefficients 5.1.6 The Maxwell-Stefan Approach to Diffusion 5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media 5.2.1 Porous Media used in HPLC 5.2.2 Axial Dispersion in Porous Media 5.2.3 Influence of the Bed Heterogeneity 5.2.4 Kinetics of Mass Transfer in Porous Adsorbents 5.2.5 External Film Mass Transfer Resistance 5.2.6 Intraparticle Pore Diffusion 5.2.7 Surface Diffusion 5.2.8 Competitive Surface Diffusion 5.3 The Viscosity of Liquids 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5
The Viscosity of the Mobile Phase Importance of the Mobile Phase Viscosity in Preparative Chromatography . . . . Calculation of the Inlet Pressure in the Case of a Variable Viscosity Feed Concentration, Mobile Phase Viscosity and Inlet Pressure at Constant Flow Rate Flow Instability and Viscous Fingering
References
222 224 229 230 231 232 232 240 242 244 246 247 249 250 254 256 257 259 264 266 267 269
275
Introduction Chromatography is a powerful separation method because it can be carried out easily under experimental conditions such that the two phases of the system are always near equilibrium. This is because the kinetics of the mass transfers between these phases is usually fast. The separation power of a column, under a given set of experimental conditions, is directly a function of the rate of the mass transfer kinetics and of the axial dispersion coefficient. The scientists involved in the development of stationary phases for chromatography have produced excellent packing materials that permit the achievement of a very large number of equilibrium stages (i.e., theoretical plates) in a column. Thus, as we show later in Chapters 10 and 11, the thermodynamics of phase equilibria is often the main 221
222
Transfer Phenomena in Chromatography
controlling factor of the shape of individual band profiles in nonlinear chromatography. The effect of the finite column efficiency on the band profiles can be accounted for simply by lumping all the contributions to band broadening in an apparent axial dispersion coefficient, provided that the mass transfer resistances are relatively small. In this chapter the transfer phenomena at work in a preparative column are discussed. This includes diffusion, mass transfer resistances, the kinetics of adsorption-desorption, and the viscosity. The mobile phase viscosity is important in preparative chromatography. It influences strongly the molecular diffusion coefficients of the feed components, hence the mass transfer resistances. The mobile phase viscosity influences the cost of a separation because the inlet pressure at which the hardware, including the column, must be operated determines to a large extent the investment costs. The cost of the unit operation hardware, including the pump, the tubings, the valves, the monitoring system, and the column are a function of the operating pressure. Note that, if the equipment is properly optimized, the maximum production rate of a unit increases with increasing operating pressure. The sources of band broadening of kinetic origin in chromatography include molecular diffusion, eddy diffusion, and mass transfer resistances. Mass transfer in the general sense of the term, as it is used by chromatographers, includes (i) radial diffusion through the flowing mobile phase (from the center of the flow stream in the interparticle space to the particle surface); (ii) diffusion from the mobile phase percolating through the bed into the mobile phase stagnant inside the particles, across the stagnant film barrier; and (iii) diffusion through the stagnant mobile phase, inside the particle pores. It also includes the kinetics of the retention mechanism itself, e.g., adsorption-desorption, ion-exchange, complex formation/ dissociation.
5.1
Diffusion
Diffusion is the macroscopic result of the sum of all molecular motions involved in the sample studied. Molecular motions are described by the general equation of dynamics. However, because of the enormous difference in the orders of magnitude of the masses, sizes, and forces that characterize molecules and macroscopic solids, it can be shown [1] that, when a force field (e.g., an electric field to an ionic solution) is applied to a chemical system, the acceleration of the molecules or ions is nearly instantaneous, molecules drift at a constant velocity, and, in the absence of an external field and of internal forces acting on the feed components, which is the case in chromatography, the diffusional flux, /, of a chemical species i in a gradient of chemical potential is given by
'-7%
<">
where C is the concentration of the component considered, / the friction coefficient or molar drag constant (it is also the reverse of the mobility), and ji the
5.1 Diffusion
223
chemical potential
y. = f + RT In a
(5.2)
where ]t° is the reference potential, R the ideal gas constant, T the absolute column temperature, and a the activity of the component. Combination of Eqs. 5.1 and 5.2 gives
RTdln[a]dC In most applications of chromatography we can assume that the concentrations are small and that the activity coefficients do not vary much with the concentration. If we assume that the activity is equal to the concentration, Eq. 5.3 simplifies to the classical Fick's first law of diffusion [1,2] RTdC ]
dC DA
= -TTZ=- '«TZ
{5 4)
-
where dC/dz is the concentration gradient along the column, D^ g the diffusivity or diffusion coefficient of solute A in solvent B, and / the friction factor. In liquid chromatography, we are interested in reasonable estimates of D^g and in knowing the parameters that control or may influence the value of this coefficient. We note that, as shown by Eq. 5.1 and as pointed out by Ruthven [3], the true driving force for diffusion is the chemical potential gradient, not the concentration gradient. This has been demonstrated by the experiments of Haase and Siry [4]. In chromatographic practice, however, the distinction could be of importance only at high concentrations. In most cases, the chemical potential gradient will result essentially from the concentration gradient. As it is difficult to measure / , Fick's second law of diffusion is most widely used:
£ = "«£ For multicomponent systems, the diffusion flux of each component depends on the concentration gradient of all the system components. In order to account for these interactions, a matrix of diffusion coefficients is considered and the two laws of diffusion are written [5]:
and ^-TD.^l
(57)
where Da are the main coefficients and D;y, i ^ /, the cross-coefficients. This effect generally is not accounted for in chromatography. In nonlinear chromatography,
224
Transfer Phenomena in Chromatography
the kinetic contributions such as the dispersive effect of diffusion on the band profiles are small compared to the effects of thermodynamic origin. Furthermore, in practice, the solutions used in liquid chromatography, even in preparative applications, are rather dilute, and the effect of concentration on the diffusion coefficients seems rather small, except with proteins, for which the effect might be significant. This coupling of the molecular diffusion coefficients may contribute, however, to explaining some of the residual discrepancies between the theoretical predictions of band profiles and those determined experimentally (see Chapters 10 and 11). Recent work has shown the potential importance of the surface diffusion flux in accounting for mass transfer across the stationary phase. When proper models of this flux are used, excellent agreement is achieved between the chromatographic band profiles of multicomponent and those calculated using the general rate model (see Chapters 14 and 16).
5.1.1 Diffusivity or Diffusion Coefficients In general, we shall assume in this section that we are dealing with dilute solutions and the diffusion coefficient D^g corresponds to an infinitely dilute solution of A in B. In other words, each molecule A is in an environment of pure B solvent, there are A-B and B-B interactions in the system but no A-A interactions. In practice, we assume also that DA,B ls representative of the diffusion coefficient up to concentrations of the order of 5 to 10% [5], which happens to be the range within which preparative chromatography is usually carried out. There are several methods for obtaining an approximate value of the diffusion coefficient. These methods are preferred to direct determination because of the real and perceived difficulties of a direct measurement. The friction factor in Eqs. 5.1 and 5.4 can be estimated for quasispherical molecules by the Stokes-Einstein law of friction /
=
6nf]aN
(5.8a)
where N is Avogadro's number, r\ is the mobile phase viscosity, and a is the molecular radius. For elongated molecules, a different numerical coefficient should be used [2,6]. The Stokes-Einstein equation gives a reasonable estimate of the coefficient of self-diffusion (i.e., the diffusivity of an isotopic tracer) and a good firstorder estimate of the diffusion coefficient of dilute solutes, especially when the molecules of solute are larger than the solvent molecules [2,7]. However, when the average dimension of the solute molecules becomes less than five times that of the solvent molecules, Eq. 5.8a fails to predict properly the diffusion coefficients [8] and errors become large, especially in high viscosity solvents. A variety of empirical and semiempirical correlations have been proposed [5, 7,9-15]. These equations were reviewed by Li and Carr [16] for their particular application to liquid chromatography.
225
5.1 Diffusion
Table 5.1 Atomic Volumes Contributions to the Molar Volumeat the Normal Boiling Point
Group Bromine Carbon Benzene ring Naphthalene ring Chlorine Nitrogen Nitrogen, primary amines Nitrogen, secondary amines
n
27.0 14.8 -15.0 -30.0 -21.6 -15.6 -10.5 -12.0
Group Hydrogen Oxygen Oxygen, methyl esters Oxygen, higher esters Oxygen, acids Oxygen, methyl ethers Oxygen, higher ethers Sulfur
n
3.7 7.4 9.1 11.0 12.0 9.9 11.0 25.6
in mL/g atom 5.1.1.1
The Wilke and Chang Correlation
The equation derived by Wilke and Chang [9] is probably the most popular correlation for the molecular diffusivities of low molecular weight compounds in conventional solvents. It gives an excellent approximation for molecules having a moderate molecular weight: (5.9)
DArB = 7A x 10"
m
where V& is the molar volume (cm3 mole *) of the liquid solute at its normal boiling point, Mg the molecular weight of the solvent (g), TJ-Q its viscosity (cP), and I/>B is a constant which accounts for solute-solvent interactions. Recommended values of i/>g are 1 for all nonassociated solvents, 1.5 for ethanol, 1.9 for methanol, and 2.6 for water [9]. In Eq. 5.9, D^g is in cm 2 /s. V^ can be calculated from group contributions [5,15] (Table 5.1). The errors made when using Eq. 5.9 vary greatly from compound to compound, and this equation is not accurate. Diffusion coefficients can be predicted to within 10% when water is the solvent and 25% for organic solvents, but errors as high as 200% are possible when water is the solute [15]. A more accurate correlation method has been proposed, but it requires the use of the parachor, a parameter known for only a few hundred chemicals and which can be calculated from group contributions available for only a limited number of groups [5]. Equation 5.9 has been derived from a correlation of data obtained with compounds with molecular weights between 100 and 500, and it does not give good estimates of the diffusion coefficient of polymers, especially proteins. Nevertheless, as the specific volume of proteins is nearly constant, around 0.73 [17], their molecular volume is proportional to their molecular weight, and Eq. 5.9 predicts values of the diffusion coefficient between 0.5 and 1 x 10~6 cm2 s" 1 .
226
Transfer Phenomena in Chromatography
5.1.1.2 The Scheibel Equation Scheibel [10] has suggested the following equation: AT DA,B
=
(3VB\
2/3
1 + \VAJ
(5.10)
where the constant A is equal to 8.2 x 10 8 , except in the following cases: for water, when VA < VB, A = 25.2 x 10~8; for benzene, when VA < 2VB, A = 18.9 x 10~8; and for all other solvents, when VA < 2.5VB, A = 17.5 x 10~8. The molar volumes of the solvent and the solute are calculated as for the Wilke and Chang equation above. Note that this equation is very similar to one previously suggested by Wilke [9]. 5.1.1.3 The King Equation King et al. [18] have suggested the following equation (5.n)
This equation is more difficult to apply since it requires the heat of vaporization, AHV, of the solute and the solvent. 5.1.1.4 The Reddy-Doraiswamy Correlation This correlation [12] neglects the association factor
The numerical constant A is equal to 1.0 x 10~7 when VA < 1.5Vg and to 8.5 x 10~8 when VA > 1.5VB5.1.1.5 The Lusis-Ratcliff Correlation This correlation [13] was specifically derived for the estimation of diffusivities in organic solvents.
5.1.1.6 The Hayduk-Laudie Correlation In contrast with the previous one, this correlation [14] was derived to estimate the diffusivities of nonelectrolytes in water. 589 DA/B = 13.26 x 10" 0"5!/g !/ 1 - 44V^" V " aa589
(5.14)
5.1 Diffusion 5.1.1.7
227
Correlations for Macromolecules
For macromolecules, it is important to distinguish between random coils (most synthetic polymers) and globular molecules (most proteins). For the former, the molecular volume is approximately proportional to the square root of the molecular weight. The diffusion coefficient of polystyrene in toluene was estimated by Giddings et al. [19] DAiB = M / 5 5 3 e-3m29-
™
(5.15a)
In this equation, DAiB has the unit of cm2 s" 1 , M is in grams, and rjB is equal to 0.59 cP. Relations of this type are applicable to other synthetic polymers. In general, Eq. 5.8a can be rewritten for macromolecules as RT1
DA/B =
(5.15b) ^ 2 , where < R2 > 1/?2 is the root-mean-square radius of gyration of the polymer molecule in solution, the common estimate of molecular sizes. This parameter can be measured experimentally by various techniques, e.g., light scattering. 5.1.1.8
The Young et al. Correlation for Proteins
Proteins are globular macromolecules, with a radius smaller and much better defined than the average size of a random coil. Thus, Eq. 5.8a should give excellent results. It does not because these molecules are surrounded by a hydration shell, which increases their effective diameter. Young et al. [17] have derived a correlation for proteins DAB = 8.31 x 10- 8
1
—
(5.16)
in cm2s 1, MA in g, rjs in cP). Equation 5.16 gives excellent results because 75% of the 301 protein molecules studied have a diffusion coefficient within 20% of the value predicted by Eq. 5.16 [17]. Comparison between Eqs. 5.8a and 5.16 gives an estimate of the Stokes diameter which is 8% larger than the value derived from the molecular weight [20]. This apparent excess is related to the water shell traveling with protein molecules. 5.1.1.9
Comparison between these Correlations
We note that all the equations between Eqs. 5.8a and 5.16, except Eq. 5.14 assume the diffusion coefficient to be inversely proportional to the viscosity. This relationship does not seem to hold with viscous solvents (rj larger than ~ 20 cP), for which a proportionality to TJ~05 seems to give better results [5]. However, such highly viscous solutions are of little interest as mobile phases in liquid chromatography
228
Transfer Phenomena in Chromatography
due to the high pressure drop that would be necessary to ensure even a low flow velocity of the mobile phase. This issue will not be further considered here. The derivation of a correlation, the investigation of the accuracy of a correlation, and the comparison between the accuracies of different correlations require, first and foremost, the availability of accurate diffusivity data. Such data are not easy to obtain (Section 5.1.5). Accurate estimates of the viscosity of solvent mixtures and of their molecular volumes are also needed but they are easier to measure. Few systematic studies of the accuracy of the empirical correlations reported here have been made. There seems to be only one in HPLC, by Li and Carr [16,21]. The measurement of the diffusivities of compounds of homologous series of alkyl-benzenes and phenones were made in various water/methanol and water/acetonitrile solutions, using the Aris-Taylor method. The estimated precision of the diffusivities was 3%. Their conclusion was that, for the compounds studied, the Wilke-Chang, the Scheibel, and the Lusis-Ratcliff correlations all give results that could be expected to have errors below 20% in all aqueous solutions of methanol and acetonitrile. The Scheibel correlation seems to work better than the other two on the average, particularly in acetonitrile [16]. In the calculations of the results of the various correlations studied, Li and Carr recommended determining the molecular weights (MWB), the association factors (Tg), and the molar volumes (Vg) of the solvent mixtures using the following equations MWB
=
xOIgMWorg + xH2OMWHiO
(5.17a)
^B
=
x org f org + XH2o^H2o
(5.17b)
VB
= xorg Vorg + xH2oVH2o
(5.17C)
The data obtained in this work were further analyzed in a separate paper [21]. The proportional and the reversed dependences of the diffusivity on the square root of the solvent molecular weight and on its viscosity, respectively, were confirmed. So was its reversed dependence on Vg6, with exceptions for the higher molecular weight compounds in water-rich solvents for which the peaks obtained were unsymmetrical. The plot of DA,B versus TMWjp/r/V^6 (see Eq. 5.9) is not linear and a quadratic equation was suggested instead. This new equation gives values that have an absolute error less than 10% at the 95% confidence level. Note, however, that these results have been obtained with two similar homologous series and cannot be extrapolated to other compounds or groups of compounds without further detailed investigations. Finally, to illustrate the importance of the possible differences between the results predicted by different correlations, we give in Table 5.2 the values of the diffusion coefficient calculated for hypothetical compounds having increasing molecular weights from 100 to 300,000 Dalton, in a solvent with a viscosity of 1 cP, using Eqs. 5.8a to 5.16. These data offer convenient orders of magnitude for estimating the relative importance of various contributions to band broadening. The disagreement between the different correlations suggested becomes quite important at large values of the molecular weight, as expected because of the different assumptions made (e.g., globular molecule versus random coil). An accuracy better than 20% cannot be expected from any of the correlation methods reported. Better results would be merely coincidental.
5.1 Diffusion
229 Table 5.2 Diffusion Coefficient and Molecular Weight.
M (Dalton) 100 300 1,000 3,000 10,000 30,000 100,000 300,000
A
B
C
7.14 4.95 3.31 2.29 1.54 1.07 0.71 0.49
8.97 4.64 2.25 1.16 0.57 0.29 0.142 0.074
14.20 7.7 4.0 2.17 1.11 0.61 0.31 0.17
D 5.39 3.74 2.50 1.73 1.16 0.81 0.54 0.37
Diffusion coefficients (Dm in cm 2 /sec x 10 6 in a solvent with a viscosity of rj = 1 cP) after [A]: Eq. 5.8a, with a = [3M/(47rpN)]1/3, and T = 300 K; [B]: Eq. 5.9 with ipBMB = 41, p B = 1, and T = 300 K; [C]: Eq. 5.15a corrected for a solvent viscosity of 1 cP at T = 300 K; [D]: Eq. 5.16, at T = 300 K.
In an electrolyte solution, another problem is encountered. Ions of different sizes tend to diffuse at different rates, the smallest ones being the fastest. However, electrical neutrality must be conserved at all points in the solution. This requires that ions of opposite charges in a solution of two ions diffuse at the same rate. Nernst has derived the following equation, valid at infinite dilution [5,15]: U
AB
RT
F 2 A°+A°.
(5.18)
Z+Z_
where F is the Faraday constant, A? is the conductance of the ion of charge sign i, at infinite dilution, and Z,- is the charge of the ion. Equation 5.18 has been verified experimentally for dilute solutions.
5.1.2 Influence of the Concentration on the Bulk Diffusion Coefficients At low concentrations, the diffusion coefficient is independent of the component concentration. At high concentrations, the diffusion coefficient depends strongly on the concentration of the solute, especially if the molecular volume and viscosity of the pure solute differ strongly from those of the solvent. Since a concentrated solution is no longer ideal, we can write -
U
A,B
1+
3 In j /
(5.19)
where [a] A is the activity, 7^ the activity coefficient, and %A the mole fraction of the solute A [5]. Vignes [22] has shown that there is generally an excellent correlation between the composition of a solution and the diffusion coefficient, with D°AB = [D°A)B(xA
—s- 0)]-Tfl [D°BA(xB
—> 0)] x -*
(5.20)
230
Transfer Phenomena in Chromatography
where D°AB (XA —> 0) and DB A (Xg —> 0) are the diffusivities of A at infinite dilution in B, and of B at infinite dilution in A, respectively. Thus/ the plot of log DA,B versus the mole fraction of the solute is linear. Unfortunately, the determination of DA/B is not simple in most cases, and except when solutions are really dilute, we cannot assume that the activity remains proportional to the mole fraction. Experimental results suggest, however, that the concentration dependence of the diffusion coefficient is small in the concentration range 0-5% and that the expected variation is lower than the error probably made in using the relevant equation (Eqs. 5.8a to 5.16) to estimate the diffusion coefficient. Systematic studies of the dependence of the diffusion coefficient on the solution composition are certainly needed at this stage, in conjunction with the search for appropriate models of nonlinear chromatography.
5.1.3 Influence of the Pressure on the Bulk Diffusion Coefficients Secondary pressure effects are almost always neglected in liquid chromatography, because the pressure dependence of the density (i.e., the compressibility) and the viscosity of liquids are relatively small. Typical values are lxlO^ 4 atm^ 1 for the former and lxlO~ 3 arm" 1 for the latter. The pressure dependence of the liquid density tends to increase the retention volumes, compared to those expected with a noncompressible fluid under the same conditions [23,24]. The pressure dependence of the viscosity results in an increase of the retention time beyond the value that would be observed under the same conditions, with a solvent having a constant viscosity [23]. We discuss these effects in Section 5.3.1.2. The pressure can also affect the thermodynamic and kinetic parameters of the phase equilibrium. Retention factors and presumably the other coefficients of the isotherms change with increasing pressure [25-30]. As a first approximation, in a pressure range wider than the one within which preparative chromatography is carried out (i.e., 0-200 atm), the retention factors of most compounds increase linearly with increasing pressure. The slope of the variation is proportional to the difference between the partial molar volumes in the mobile and stationary phases [26,29,30]. The effect of the pressure on the separation factor in homologous series has been measured [26,29,30]. It is quite negligible in practice. From Eqs. 5.8a to 5.16, it is clear that the diffusion coefficient is a function of two parameters, the solvent viscosity and the solute molar volume, both of which depend on the pressure. Thus, we can write =
DAB
dP
constant - ^ r
VB
dP
(5.21) U
VA dP
where DA B is the diffusion coefficient of solute A in solvent B, r/g the solvent viscosity, VA the solute molar volume, and n an exponent between 0.33 and 0.67. Thus, the pressure dependence of the diffusion coefficient can be derived from the pressure dependences of the viscosity and density. As the former is an order of magnitude larger than the latter, we may even neglect the compressibility effect in
5.1 Diffusion
231
most cases. As we see later (Section 5.3.1.2), the viscosity increases with increasing pressure Vs = riBfli1 + aAp]
(5-23)
where t]s,o is the reference viscosity (i.e., under atmospheric pressure) and a a numerical coefficient, depending on the solvent and of the order of 1 x 10~3 atm^ 1 . Since solvent compressibility is small, we have
«
DAtBJ0[l-(a-nXA)AP]
(5.24b)
«
DA/Bi0[l-clAP]
(5.24C)
where XA is the solute compressibility and DArBro the reference diffusion coefficient. Linear relationships of this type have been reported by Jonas et al. [31] in the case of cyclohexane, with a value of oc between 0.9 and 0.95 times the value of cc, in agreement with the small value of the compressibility contribution. The coefficient a increases with increasing temperature, from 1.45 xlO~ 3 atm^ 1 at 40°C to 2.95 x 10~3 atm" 1 at 110°C. Comparable results, with values of the proportionality coefficient cc of the same order of magnitude, have been reported in a number of experimental investigations [32]. For example, the pressure dependence of the diffusion coefficients of several solutes in conventional LC solvents satisfies Eq. 5.24, with cc between 0.9 and l.lxlO" 3 {i.e., the diffusion coefficient decreases by 1% every time the pressure is raised by 10 atm) [33]. This has limited consequences in preparative chromatography, a field in which the column head pressure rarely exceeds 100 to 150 bar. The relatively considerable influence of the pressure on the diffusion coefficients has a major consequence when chromatography is carried out under very high pressures, in the range of 0.5 to several kbar [24]. The mass transfer resistances are all direct functions of the diffusivity, except the adsorption/desorption kinetics (see Sections 5.2 and 5.2.7 for the case of surface diffusion). Accordingly, the efficiency of columns packed with very fine particles (dp of the order of 1 ^m) decreases rapidly with increasing velocity, far more rapidly than anticipated from the small particle diameter. Since it seems highly improbable that this range of pressure will ever be used in preparative HPLC, there will be no elaboration here on the reasons for this fact.
5.1.4 Influence of the Temperature on the Bulk Diffusion Coefficients Equations 5.8ab, 5.9, and 5.16 assume that D^grj/T is constant. Since the viscosity decreases exponentially with increasing temperature (see Section 5.3.1.1), it seems realistic to assume that the diffusion coefficient increases as DA/B = Ae-B/T
(5.25)
232
Transfer Phenomena in Chromatography
For most solvents, viscosity data are usually available in the whole temperature range of interest in chromatography. The most practical method is to use these data and apply the relevant equation to the calculation of the diffusion coefficient, although these correlations have been derived using data collected in a narrow temperature range [5].
5.1.5 Measurement of the Diffusion Coefficients Many methods have been proposed for the measurement of diffusion coefficients. The three methods most frequently used are the diaphragm cell method, the capillary method, and the Taylor dispersion method. This last method is the most practical for chromatographic applications. It uses a slightly modified chromatograph and is quite accurate. The Taylor dispersion method was introduced by Ouano [34], Pratt and Wakeham [35], and Grushka and Kikta [36]. It has been investigated and discussed in detail by Alizadeh et ol. [37] and by Atwood and Goldstein [33], who made a very thorough study of the instrumental design and of the sources of errors. The method uses a long, narrow, empty tube whose walls are smooth and inert to prevent the retention of the compounds studied. The diffusion coefficient is derived from the flow rate dependence of the broadening of a narrow plug of dilute solution of the compound studied. Fused-silica capillary tubes, 0.21 mm i.d., 27 m long, coiled on a 25-cm spool, were used. The retention volumes of all the compounds were equal to the hold-up volume, with an error of ca 0.2%, showing the lack of retention [33]. The same method was used by Li and Carr to measure the diffusivities of series of homologous alkyl-benzene and alkyl-phenones in aqueous solutions of methanol and acetonitrile [16]. These authors used 16 m long, 0.5 mm i.d. tubes, coiled into diameters of 12 and 13 cm, which was necessary to fit them into a temperature-controlled oven but was probably the main source of systematic error because of the secondary circulation of the stream that is generated by Coriolis acceleration in coiled pipes. The diffusion coefficient is given by the Aris-Taylor equation, from the variance of the peak observed (corrected for the contributions of the extra-column volumes) al = {atuf = 2DAtBtR + ^L))
(5.26)
where c z and o"t are the standard deviation of the peak in length and time units, respectively, u is the solvent velocity, tR = L/u is the retention time of the peak, dc and L are the tube diameter and length, respectively. This method permits the rapid and accurate determination of many diffusion coefficients, as well as detailed studies of the influence of the chemical structure of the molecule, the nature of the solvent, its composition, the solute concentration, the temperature, and the pressure on these coefficients.
5.1.6 The Maxwell-Stefan Approach to Diffusion The correct description of the diffusion of multicomponent mixtures within fluid phases, across the boundary between two phases or within porous materials is
5.1 Diffusion
233
essential in the calculation of the design parameters of chromatographic columns as well as in many other processes in chemical engineering. A variety of solute transport mechanisms are involved in chromatography, such as bulk liquid phase diffusion, solid-phase diffusion and diffusion inside the pores of the particles of packing materials. Traditionally, Fick's law of diffusion is used by chromatographers and by chemical engineers for the design of separation and reaction equipments. Fick's law postulates a linear dependence of the diffusive flux, /; of component i, with respect to the molar average velocity of the mixture, u and to the composition gradient Axf. Ji = Ci(ui -u) = -QDiVxi
(5.27a)
and the molar flux, N,, of component i is given by Nt = Qui = CtXiUi = -CtDiVxi + XiNt
(5.27b)
with
Nt = t,Ni
(5.27c)
However, Eq. 5.27a is valid only under the following set of conditions 1. The system considered is a binary mixture. 2. Or a dilute solute i diffuses in a multicomponent mixture. 3. And there are no electric, electrostatic, or inertial forces involved. When uphill diffusion can take place (because the directions of the gradients of chemical potential and of concentrations are opposed), the Fick approach fails, even at the qualitative level, to describe the mass transfer phenomena observed experimentally. Several known examples underly the shortcoming of the Fick's formulation [38]. A more general approach to the diffusion problem is needed. The essential concepts behind the development of general relationships regarding diffusion were given more than a century ago, by Maxwell [39] and Stefan [40]. The Maxwell-Stefan approach is an approximation of Boltzmann's equation that was developed for dilute gas mixtures. Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in this theory. Krishna et al. [38] discussed the Maxwell-Stefan diffusion formulation and illustrated its superiority over the Fick's formulation with the aid of several examples. The Maxwell-Stefan formulation, which provides a useful tool for solving practical problems in intraparticle diffusion, is described in several textbooks and in numerous publications [7,41-44]. 5.1.6.1 Maxwell-Stefan Isothermal Diffusion for Binary mixtures For a b i n a r y fluid mixture, t h e force balance o n c o m p o n e n t 1 expresses t h e equality b e t w e e n the gradient of its chemical potential a n d the d r a g force, so M a x w e l l Stefan diffusion is given b y [38] M2)
(5.28)
234
Transfer Phenomena in Chromatography
where V is Maxwell-Stefan diffusivity and U\ — u2 is the molar velocity difference of the two components. Multiplying both sides of this equation with X\ /RT gives _Xj_cLjii
_ xlx2u1
RT dz ~
- x1x2u2
V
.
(
(i) y j
-
Since the fluxes are defined as IV,- — Qx,-M;, this equation can be generalized in the three dimensional space as the vectorial equation - xtN2 QV
(5
"30)
In a nonideal fluid, the chemical potential may be written as a function of the activity and we have
where F is the thermodynamic correction factor that accounts for the nonideal behavior of the system. Finally, combining Eqs. 5.27c, 5.30, and 5.31 with x2 = 1 — x\, we obtain Ji = Ni - x-iNt = -QVFVx!
(5.32)
Comparing Eqs. 5.28 and 5.32 with those stating Fick's law (Eqs. 5.27a to 5.27c) for a binary system yields the following relationship between Fick the diffusivity, D and the Maxwell-Stefan diffusivity T>. D = VF
(5.33)
For gas mixtures at low to moderate pressures and for ideal liquid mixtures, the thermodynamic correction factor is equal to 1 and the Maxwell-Stefan diffusivity is independent of the composition. In this limit case, Fick and Maxwell-Stefan diffusivities are identical. The Maxwell-Stefan diffusivity has the physical meaning of the inverse of a drag coefficient. The Fick diffusivity is the combination of two separate effects, a drag effect and an effect due to a thermodynamic nonideal behavior. Maxwell-Stefan is more easily predictable and interpretable than the Fick diffusivity. For nonideal liquid mixtures, the thermodynamic factor F depends strongly on the composition. Accordingly, we should expect the Fick diffusivity to exhibit a similarly strong composition dependence. This statement is supported by experimental data available in the literature and is illustrated in Figure 5.1 for the solution of methanol and n-hexane in which the Fick diffusivity tends toward zero in the region of the phase transition point near X\ ss 0.5 [45]. On the other hand, the Maxwell-Stefan diffusivity, calculated from Eq. 5.33, using the Fick diffusivity and thermodynamic data, shows only a mild composition dependence. Vignes [22] proposed an empirical relationship for the composition dependence of Maxwell-Stefan diffusivity. V = {VXl^{VXl^)1-^
(5.34)
where DXl^\ and T>XI^Q are the values of T> at infinite dilution of either component in the other one. Equation 5.34 indicates that lnX> is a linear function of x\.
5.1 Diffusion
235
Figure 5.1 Plot of the experimental data for the Fick and Maxwell-Stefan diffusivities in solutions of methanol and nhexane. NB (14) should mean Ref. [22] and Eq. 5.34. The plot is reprinted from
Vlgnes relation (14)
mn
Maxwell-Stefan D (o)
:
°°°
R. Krishna and ]. A. Wesselingh, Chem. Eng. Set, 52 (1997) 861 (Fig. 11). It
° FickD{o)
" methanol* nhexane 1
o °
B
was drawn from data from W. M. Clark, R. L. Rowley, AIChE ) . , 32 (1986) 1125. Reproduced by permission of the American Institute of Chemical Engineers. ©1986 AIChE. All rights reserved.
o
•
Diffusivity 10 i n - " m* s-' J :
i
i
i
i
i
i
i
i
i
mole fraction of methanol
From the data in Figure 5.1, we see that this empirical correlation gives excellent results, particularly if we consider the large amplitude of the change in the Fick diffusivity. The generalization of Eq. 5.28 for a multicomponent system is
fi
U
±p>
U
lp*
^ l p i +• • •
(5.35)
V
By following the same method as the one used to derive Eq. 5.30 from Eq. 5.28, we can derive from this equation the following: X;
(5.36)
j\ J.
If we introduce the (n — 1)-dimensional matrix [F] of thermodynamic factors, we can recast the LHS of Eq. 5.35 in terms of the mole fraction gradients. The Maxwell-Stefan diffusion for multicomponent systems is thus [38] X{
(5.37)
RT x
(5.38) (5.39)
Combination of this equation and of Eq. 5.35 gives (5.40)
where (J) is the (n — 1) column vector of the fluxes in the column and the elements of matrix [B] are given in term of the Maxwell-Stefan diffusivities as: (5.41) (5.42)
236
Transfer Phenomena in Chromatography
with i,j = 1,2, • • • ,n — 1. A matrix, [D] of Fick diffusivities similar to the one introduced in the binary case can be defined as [D] = [B]- 1 ^]
(5.43)
The coefficients of the matrix [B] have no clear physical interpretation. The advantage of the Maxwell-Stefan formulation is that it decouples the drag effect ([£>]) from the thermodynamic effects ([F]). Comparing Eqs. 5.37 and 5.43 gives the diffusion fluxes as J,- = -C f [D](Vx)
(5.44)
Equation 5.44 is the proper generalization of Fick's formulation to the case of multicomponent mixtures. There are only a few particular cases in which the simplification of Fick's formulation for a binary mixture with the assumption of a constant effective Fick diffusivity Dt- is justified: 1. When the binary Maxwell-Stefan diffusivities T>ij are equal and the mixture is ideal (F^ = 1). With these assumptions, Eq. 5.44 yields Ji = -CtVVxi(-Dirj = V, I y = 1)
(5.45)
2. When all the components are present at small concentrations (i.e., = 0), except for the major component of the mobile phase for which xn = 1. Then, with this assumption, we derive from Eq. 5.37 that F^j = 1 and from Eq. 5.41 that Bjj = 1/T>in. As a result, Eq. 5.44 reduces to Ji = -CtViinVxi(xi
-^0,xn^
1)
(5.46)
So, each dilute component interacts only with the solvent. In all the other cases, the diffusion coefficient is not constant but it is concentration dependent. 5.1.6.2 Diffusion Mechanism Inside Porous Structure Most commercial adsorbents consist of small microporous or nonporous crystals formed into macroporous pellets or particles. The solutes carried along the column by the fluid mobile phase must first be transported from the bulk fluid phase to the external surface of the adsorbent and then they must diffuse inside the particles. Within a particle there are two distinct kinds of diffusion phenomena that contribute to the resistances to mass transfer, the macropore (or inter-crystalline) diffusion through the pellet and the micropore (or intra-crystalline) diffusion resistance. The relative importance of macropore and micropore diffusion resistances depends on the pore size distribution within an adsorbent particle. Micropores have diameters smaller than 2 nm, macropores diameters greater than 50 nm while mesopores are in the range of 2 to 50 nm. Keil et al. [46,47] discussed the modeling of diffusion and reaction in porous media. For a successful modeling of these phenomena, a variety of difficult problems must be solved, such as multicomponent diffusion in porous media, surface
5.1 Diffusion
237
diffusion, the description of the porous structure, the modeling of the active centers for adsorption and for chemical reactions. Within a pore we may distinguish three different types of diffusion mechanisms: 1. Molecular diffusion is significant in large pores and under high pressures. In these cases, the effects of molecule-molecule collisions dominate over those of molecule-wall collisions. However, compared to molecular diffusion in the free bulk phase, molecular diffusion in pores is hindered by the pore walls and may be slower or even much slower. 2. Knudsen diffusion becomes predominant when the mean-free path of the molecular species considered is larger than the pore diameter and hence, when the effects of the molecule-wall collisions become important. In liquid chromatography, Knudsen diffusion is negligible. 3. Surface diffusion accounts for the movements of adsorbed molecules along the pore wall surface. This mechanism of transport becomes dominant for micropores and for strongly adsorbed species. The total diffusion flux of any species within the pores of the particles of packing material is obtained by combining these different contributions. 5.1.6.3 Multicomponent Diffusion in Porous Media Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46-48]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the MaxwellStefan approach for dilute gases, itself an approximation of Boltzmann's equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. In the DGM model, porous media are considered as arrays of heavy molecules {i.e., dust) that are motionless and uniformly distributed in space. By treating the dust particles as giant molecules it is possible to use the Chapman-Enskog kinetic theory. The dust molecules are treated as an (n + l)th pseudo-species added to the n-component gas mixture. The dust particles are kept fixed in space (i.e., motionless) and are considered like another gas component in the Maxwell-Stefan equations. The DGM has two independent parts, a diffusion part, consisting of a set of Maxwell-Stefan diffusion equations and a viscous-flow part consisting of an equation of motion. The structure of the porous medium is treated as a completely independent problem. Although the DGM model is not an exact approach and it
238
Transfer Phenomena in Chromatography
takes into account only the classical molecular cross-sections and binary collisions, it has proved to be fairly accurate in engineering applications. The effective binary pair diffusion coefficient CDf.) in a porous medium is given by v
i,j = ~vi,j
(5-47)
The porosity to tortuosity ratio, e/r, characterizes the pore matrix and is best determined by experimental measurements. 5.1.6.4 Diffusion within Micropores and Surface Diffusion Within micropores, surface forces are dominant and an adsorbed molecule never escapes completely from the force field of the surface. Diffusion within this regime has been called configurational diffusion, intra-crystalline diffusion, micropore diffusion, or simply surface diffusion. The Maxwell-Stefan formulation, which is generally accepted for diffusion in the bulk fluid phase, can be extended to describe surface diffusion by considering the vacant sites to be a (n + l)-th pseudospecies on the surface [38,47,49-52]. Using the Maxwell-Stefan diffusion formulation , the following relationship was obtained for surface diffusion. 1
"•'•-
v -
.
=
> «
^ pyeqsatVfj
) ,
i
(5.48)
ppeqsatVf
for i = 1,2, • • • ,n. In this equation, 6 is the fractional coverage (6 = q/qsat, with q concentration in the solid phase and qsat the saturation capacity of the adsorbent, for a Langmuir isotherm) pp is the particle density, and e the porosity of the material. In matrix notation, this equation is written -p e qsat[r](Ve) = [BS](NS)
(5.49)
where the matrix of surface diffusion has as elements
*y = w*+ ^
vr
(5 50)
'
Ki - " J r
(5-51)
(Ns) =-pp e qsat[Ds](Ve)
(5.52)
hi (with i, j , = 1,2, • • •, n). A matrix of Fick surface diffusivities would be defined as
and we have the following expression for [Ds]: [Ds] = [Bs]-1[r]
(5.53)
5.1 Diffusion
239
For single file diffusion mechanism, with no possible counter-exchange between adsorbed species, these equations simplify to ( V\v
o' o
0
0
0
v%w o o o
•••
(5.54)
o
In this equation, D,-y is the Maxwell-Stefan surface diffusivity defined as Hv =
V
^
(5-55)
The (n + l)-th component accounts for the unoccupied fraction of the surface, with 6n+i = l-6t and 9t = 9\ + 82 H h 9n, given by Eq. 5.52. This Fick surface diffusivity matrix includes both the effects of surface mobility of the adsorbates and of the equilibrium thermodynamics accounted for by [f]. For single component, Eq. 5.52 reduces to Nf = -pv e qsat^lVOx
(5.56)
with D| Fick diffusivity given by
D\ = vyr
(5.57)
For the Langmuir isotherm, we have T=T-L-
(5.58)
Inserting Eq. 5.57 into 5.58 gives:
(5,9)
D5 . £ £
The use of other adsorption isotherms such as the Freundlich, Toth, or LangmuirFreundlich isotherms results in different thermodynamic correction factors [53]. For single file diffusion involving two components, Eqs 5.52, 5.53, and 5.54 reduce to: (Ns) = -pp e qsat[Ds}{W9)
(5.60)
Using the Langmuir isotherm to calculate [F] gives
02 - 1 - 0i - 02
(5.61)
Kapoor and Yang [54,55] modified Eq. 5.54 to account for surface heterogeneity. Multilayer surface adsorption was considered by Chen and Yang [56], Chen et al. [57], and Sikavitsas and Yang [58] who derived an important extension of
240
Transfer Phenomena in Chromatography
this approach by writing an expression for [F] that takes into account adsorbateadsorbate interactions. When these interactions are negligible, their model reduces to Eq. 5.54. The effective Fick surface diffusivities -N? D
/=
(5-62)
W"
1
peqV9
can be derived from Eqs. 5.52 and 5.62: 1
(5.63)
1-01-02 -0!-02 V
'
|V0 2
(5.64)
Equation 5.63 can also be derived assuming Langmuir kinetics for adsorption processes [59]. This equation shows that the effective surface diffusivities depend strongly both on the surface concentration and on the surface concentration gradient. In addition the effective surface diffusivity of each component is affected by the surface concentration gradient of the other component.
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of adsorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. As discussed in more detail later, in Chapter 6, the broadening of a band during its migration along a chromatographic column of length L is best characterized by a coefficient related to the first moment (p{) and to the second centered moment (^2) of the elution profile of a narrow pulse of a probe compound, the column height equivalent to a theoretical plate, HETP or H, through H = ^
n
(5.65)
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
241
This coefficient combines the broadening effects of axial dispersion and the mass transfer resistances. The former effects decrease with increasing mobile phase velocity while the latter increase, hence there is an optimum velocity for which H is minimum. The solution of the general rate model shows that H is related to the column parameters through the equation (5.66a) with (5.66b) (5.66c) (5.66d) h
= (l-ee)^-(ei
+ ppK)2
(5.66e)
where UQ is the superficial velocity of the mobile phase, ee the interparticle void fraction, £j and pp the porosity and the density of the packing material, respectively, K the adsorption equilibrium constant, Di the axial dispersion coefficient, dp the particle diameter, kr the external mass transfer coefficient, and De the intraparticle diffusivity. The terms Sax, Sf, and 8^ are the contributions to ji2 of axial dispersion, the external (or fluid-to-particle) mass transfer, and intraparticle diffusion, respectively. The difficult problem in investigations of mass transfer kinetics in chromatography is to find methods to determine or estimate the coefficients in Eqs. 5.66b that are independent of the chromatographic process, that do not use parameter identification. Although legitimate in the study of a specific application, this last method would lead to a circular argument in fundamental investigations. The present section is mainly devoted to the description of the main methods used to derive values of the rate and diffusion coefficients. The derivation of these parameters from systematic series of chromatographic measurements is discussed in Chapter 6, Section 6.4.7. As pointed out by Ruthven [3], the rates of adsorption and desorption in porous adsorbents are usually controlled by the rate of diffusion within the pore network, more than by the kinetics of adsorption-desorption. This is especially true in chromatography, where adsorbents are carefully prepared to exhibit only moderately strong energy of physisorption and no chemisorption. Thus, it is important to consider diffusion within the pore networks existing in the columns. A packed column as used in chromatography is a porous medium with a multimodal pore distribution. There are usually two modes in this distribution, but three-mode distributions may also be encountered, as we see later. In a classical column made by packing the porous particles of an adsorbent, the first mode is made of the interparticle pores, the fraction of the column volume through which the mobile phase flows. The second one is made of the intraparticle pores, within
Transfer Phenomena in Chromatography
242
Logarithm of Pore Size: angstrom
Logarithm of POre Sizo: angstrom
Figure 5.2 Pore size distribution obtained by Mercury intrusion Porosimetry for four packing materials, all with a nominal average particle size of 10 ]im.. (a) High pore size mode, above 1000A. (b) Low pore size mode, below 1000A. Reprinted with-permissionfrom H. Guan, G. Guiochon, E. Davis, K. Gulakowski, D. W. Smith, J. Chromatogr. A, 773 (1997) 33, Fig. 1.
each adsorbent particle. Those are usually several orders of magnitude smaller than the particles (e.g., most silica adsorbents are available in particle sizes ranging from 3 to 60 ^m, with pores between 30 and 1000 A). Dispersion and diffusion behaviors are quite different in these two types of pore volumes.
5.2.1 Porous Media used in HPLC Two types of porous media are currently used in liquid chromatography, beds of particles packed in a cylindrical tube and monolithic rods encapsulated in a cylindrical tube. The former have been used since Tswett. The typical average particle size used has decreased over the years, the particle size distribution has become narrower, and the quality of the material itself, particularly the homogeneity of the adsorbent surface has markedly improved. The use of small particles in preparative HPLC is limited by mechanical constraints. The operating pressure of a column increases rapidly with decreasing average particle size, in proportion to dp3 at constant column length, to dp2 at constant column efficiency (because the column HETP decreases in proportion to dz1 and the optimum mobile phase velocity increases in proportion to dp). Accordingly, while analysts use typically 0.46 mm i.d. columns 15 to 30 cm long, packed with 3 or 5 jim particles (and 1 mm i.d. columns packed with 1 or 1.7 ^m particles are now available, to be used at pressures in the 1 to 2 kbar range), separation engineers use 5 to 80 cm i.d. columns, 20 to 100 cm long, packed with 10 to 20 ]ivn particles, at pressures rarely exceeding 150 bar. Such beds have obviously a bimodal pore size distribution (see Figure 5.2), with a mode of large pores that have an average size of the order of a quarter to a third of the particle diameter and extending between 10 and 50% of this diameter (see Figure 5.2a) and which corresponds to the interparticle space or external porosity of the bed [61]. The small pores are all smaller than 100 ran for the pack-
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
243
ing materials studied and have an average size of the order of 40 to 80 nm (see Figure 5.2b). These pores are the internal pores, defining the internal porosity of the packing material and, indirectly, controlling the specific surface area of the adsorbent, the area that is responsible for the retention of the feed components and the saturation capacity of the adsorbent. The idea of replacing the packed bed with a monolithic rod column is ancient. However, the first implementations of this approach pioneered by Nakanishi for silica-based monoliths [62,63] and by Svec for polymeric monoliths [64,65] are relatively recent and the first successful applications are less than ten years old [66]. The technique has not yet reached maturity and further important progress should be expected. Currently, the silica monolithic rods are derived from sol-gel and then encapsulated. Polymeric monoliths are obtained by cross-linking polymeric solutions inside the column tube. Although preparing analytical columns of conventional dimensions does not raise important problems any longer, the scaling up of these processes to prepare the large diameter rods needed for preparative HPLC seems to raise serious difficulties. Nevertheless, because of their considerable potential advantages, it is highly probable that large diameter monolithic columns will be developed in the near future. The advantages of monolithic columns over packed beds stem from their different structures. Monolithic rods have a bimodal pore size distribution, as do packed columns. They have a distribution of mesopores that is much like that of the porous particles packed in conventional columns. These pores have an average size and a size distribution that are similar to that of the particles. They give to these rods a specific surface area and an "internal porosity" similar to those of packed columns [67]. Monolithic rods, however, also have many large macropores or through-pores. In typical silica-based monoliths, these pores have an average size that is slightly larger than that of the external pores of columns packed with 10 jim particles. These macropores are much less tortuous and constricted than the interparticle pores of packed columns. The consequence is that the monolithic columns have a much higher porosity than conventional packed columns for a given HETP. In other words, the achievement of a given column efficiency at a given flow velocity can be obtained with an inlet pressure that is several times lower [68,69]. Marginally useful in analytical applications, except when extremely high efficiencies (e.g., 50,000 theoretical plates or more) are required, this advantage would be critical in preparative applications since replacing a packed column with a rod column would allow the achievement of a markedly higher column efficiency for the same operational pressure or of the same efficiency while running the column with a much lower inlet pressure. Either way, the production costs could be significantly reduced. Numerous studies have been made characterizing the performance of silicabased rod columns [69], demonstrating the high level of reproducibility of analytical retention data [68] and of isotherm data [70], showing that the thermodynamic properties of the interactions between various solutes and chemically bonded Qg silica were very similar whether the silica support was made of particulate or monolithic material [71]. It has also been shown that the mass transfer kinetics was very similar for particulate and monolithic columns [72-74]. Even the satu-
244
Transfer Phenomena in
Chromatography
Overall Flow Direction Figure 5.3 Schematic diagram showing the flow pattern in a small region of a column. Reprinted with permission from J.C. Giddings, "Dynamics of Chromatography" (Figure 2.9- 1), by courtesy of Marcel Dekker Inc.
ration capacities of the two forms of this adsorbent are close. Thus, if monolithic columns of large diameter and of a high degree of radial homogeneity can be prepared, they will afford significant improvement in performance while raising no significant difficulties from the thermodynamic point of view.
5.2.2 Axial Dispersion in Porous Media The average dimension of the interparticle pores in a packed bed is of the order of a fraction of the particle diameter. As the particles are generally convex and the packed bed is not consolidated (i.e., the particles are not fused but remain independent), the structure of the extraparticle space is relatively simple, the porosity distribution is rather narrow, and the channel anastomosis that is illustrated in Figure 5.3 does not leave any significant part of the bed isolated over more than a few particle diameters [75]. The diffusion coefficient in this medium is practically the same as in the bulk, with no significant contribution of the structure of the packing bed to axial dispersion other than that due to the influence of the tortuosity and the constriction of the channels. In a packed bed, it is impossible to move very far along a straight line without hitting the surface of a particle. The channels follow tortuous paths around the particles. The tortuosity of the packing is the ratio of the average actual path length between the particles, along the packed column to the column length [75]. Channels are alternately narrow, as they squeeze between closely placed particles, and wide, as they expand between two successive narrows. The narrow constrictions can slow diffusion markedly [75]. In practice, however, all chromatographic beds seem to have very similar tortuosities and degrees of constriction, although data on this topic are lacking, in large part because axial dispersion has rarely been studied at low mobile phase velocity. Differences in these factors could explain the perceived, but still largely unquantified and hardly documented, differences in performance between columns packed with spherical and with irregular shaped particles. Tallarek et al. applied Pulsed Field Gradient (PFGNMR) to study mass transfer, flow, and dispersion in packed chromatographic columns, using *H NMR [76]. The measurements allow the determination of the probability distribution of all
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
245
the displacements of the fluid located in the measurement volume by using a short PFG pulse to give an address to the -"-H atoms of the solvent molecules and determining their position distribution after a certain time. This technique allows the quantitative determination of the relative amounts of the stagnant (i.e., purely diffusive) fluid pools in a bed of porous particles and of the convective fluid stream. It permits a detailed investigation of the kinetics of the mass transfer between these two fluid fractions. It provides a direct derivation of accurate estimates of the average velocity of the mobile phase, of the total porosity of the packing, and of the volume fraction of the unexchanged fraction of the stagnant mobile phase at different mobile phase velocities. This noninvasive method could allow the differentiation between the various packing materials used in chromatography, a correlation between the chromatographic properties of these materials that are controlled by the mass transfer kinetics (e.g., the column efficiency) and the internal tortuosity and pore connectivity of their particles. It could also provide an original, accurate, and independent method of determination of the mass transfer resistances, especially at high mobile phase velocities, and of the dependence of these properties on the internal and external porosities, on the average pore size and on the parameters of the pore size distributions. It could be possible to determine local fluctuations of the column external porosity, of its external tortuosity, of the mobile phase velocity, of the axial and transverse dispersion coefficients, and of the parameters of the mass transfer kinetics discussed in the present work. Further studies along these lines are certainly warranted. In the modeling of chromatography, the contributions of all the phenomena that contribute to axial mixing are lumped into a single axial dispersion coefficient. Two main mechanisms contribute to axial dispersion: molecular diffusion in the interparticle pores and eddy diffusion. In a first approximation, their contributions are additive, and the axial dispersion coefficient, Dj,, is given by DL = 7lDm
+ 72dpU
(5.67)
where dp is the particle diameter and 71 and 72 are geometrical constants, whose values are usually around 0.7 and 0.5, respectively. Chung and Wen [11] have proposed the following equation for the evaluation of axial dispersion in packed
beds: Pe = - ^ - [o.2 + 0.011Rea48l ejdp L
(5.68)
J
where Re = (upe^dp)/^ is the Reynolds number, Pe = uL/Di is the column Peclet number, L is the column length, dp is the particle size, ej, is the interparticle void fraction (i.e., the external porosity), p is the density of the mobile phase, and r\ is its viscosity. Gunn correlation The axial dispersion coefficient may be estimated using a correlation given by Gunn [77] rReSc
246
Transfer Phenomena in Chromatography A
=
B
=
ReSc —2-
,„ ,, r(l-p)2
(ReSc)*
where x\ is the first root of the zero-order Bessel function (2.405), Sc = t]/(pDm) is the Schmidt number, T is the bed package tortuosity factor (assumed to be equal to 1.4), and o% is the dimensionless variance of the distribution of the ratio between the local fluid linear velocity and the average velocity over the column cross-section. Usually, this variance is assumed to be equal to zero, p is a parameter defined by p = 0.17 + 0.33exp(-24/Re)
(5.69b)
There is often a good agreement between the values of Di measured experimentally and those calculated from the Gunn correlation [78].
5.2.3 Influence of the Bed Heterogeneity In practically all the theoretical developments published regarding gas or liquid chromatography, the column bed is considered as radially homogeneous and the feed as distributed in an homogeneous fashion across the whole column crosssection, at its entrance into the column. With such assumptions, the chromatographic column can be considered as linear and one space dimension is sufficient to study the migration and the separation of the bands of the feed components. In spite of all the efforts paid to the preparation of homogeneous column beds, however, columns have always some residual degree of heterogeneity. The essential causes of this heterogeneity are the frictions forces between the particles in the bed and between the bed and the column and the insignificance of the thermal energy at the scale of a particle of packing, even the smallest [79]. Friction between the wall and the particles causes local stress that results in differences between the packing density close to the column wall and in its center [80,81]. Sarker, Stanley, and Guiochon have demonstrated the importance of the stress applied to a bed during packing by axial compression or by the percolation of a packing solvent under high flow rate on the properties of a packed bed, particularly its porosity, its permeability, and its efficiency [82-84]. Cherrak, Drumm, and Guiochon demonstrated the importance of friction in the performance of a bed of particles [85-87]. Following earlier results by Knox [88], Eon [89] and Baur et cd. [90], Farkas et al. showed that columns are not radially heterogeneous, even when they are carefully prepared by professionals [91-93]. The mobile phase velocity is significantly higher along the column axis than along its wall. The difference may be a few percents. Furthermore, the column efficiency is markedly lower in the wall region than in its center. The phenomenon is the same for analytical and preparative size columns [92,93]. This radial heterogeneity of the column has serious detrimental effects on the separation power of the column. The only significant difference between the earlier results [88,89] and the more recent ones of Farkas
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
247
et al. is that the former authors were dry-packing their columns and observed a higher mobile phase velocity in the region along the wall while later authors used slurry-packing methods that maximize the effects of the friction stress at the wall, causing the packing density to be higher there, hence the mobile phase velocity lower [79]. The consequences of the radial heterogeneity of the column are that, although the sample may be injected as a thin flat zone, the bands of feed components become warped during their elution, under the influence of the heterogeneous radial distribution of the velocity. Because the local permeability is also a function of the radial position, the streamlines are not parallel, and the velocity has a radial component in certain regions. The results of Sarker and Cherrak give a physical justification to the considerations of Giddings regarding the different scales at which axial dispersion must be considered [75]. This author suggested that the axial dispersion in term in the HETP equation was the sum of four contributions. The first, or trans-channel contribution, arises from the radial distribution of the velocities inside each individual channel between particles, which is similar to the HagenPoiseuille flow profile in a cylindrical tube, except that channels in a packed column have a much more complicated structure and flow velocity distribution. The second, or short-range interchannel contribution, is caused by small groups of particles being more closely packed than average and separated by more loosely packed regions. The third, or long-range interchannel contribution, is a consequence of the fluctuations of the density of these loose agglomerates. Finally, the last, or trans-column contribution, originates from systematic variations of the packing density across the entire column, in the transverse and axial directions. The combination of these different contributions explains the importance of the axial dispersion contribution to the HETP of poorly packed columns. It also explains the difference between the HETP curves determined by chromatography, from the profile of the concentration history at the outlet of a column, and by NMR from the dispersion of a band inside the column, illustrated later (Chapter 6 and Figure 6.13 [76]). Miyabe and Guiochon have studied the influence of the heterogeneity of the column bed on the band profile, assuming a parabolic velocity profile in the radial direction, with a velocity slightly larger in the center than along the wall [94,95]. They showed that the peak becomes unsymmetrical and exhibits a degree of tailing that is related to the difference between the extreme values of the velocity [95]. Since most columns used in HPLC are well or reasonably well packed, the effect of this radial heterogeneity is not large and it is usually ignored. The result is that most interpretations of band profiles suffer from a model error which propagates into errors in the determination of kinetic coefficients.
5.2.4 Kinetics of Mass Transfer in Porous Adsorbents Several phenomena affect the overall rate of the mass transfers in chromatography: the mass transfer resistances from the percolating stream of the mobile phase to the mobile phase stagnant inside the pores of the particles, the dispersion through the porous particles, and the kinetics of adsorption-desorption. In
248
Transfer Phenomena in Chromatography
Interstitial mobile phase
TRANSPORT AT THE PARTiCliE BOUNDARY
Figure 5.4 Schematic illustration of the eluite diffusion into a stationary phase particle. ka and kd, rate constants for adsorption and desorption; ke, mass transfer coefficient at the particle boundary; 9, tortuosity factor; e,-, internal porosity. Reproduced with permission from Cs. Horvdth and H.J. Lin, J. Chromatogr., 149 (1978) 43 (Fig. 1).
most practical cases, the kinetics of adsorption-desorption is fast, and the kinetics of equilibration is controlled by the mass transfer resistances. In affinity chromatography, by contrast, the rate of dissociation of the complex is often slow and controls the rate at which equilibrium is reached. There are at least two main sources of resistance to mass transfer (Figure 5.4 [96]): external film mass transfer resistance and intraparticle diffusion that is composed of pore and surface diffusion. The latter diffusion is insignificant in numerous adsorbents but plays an important role in most adsorbents used in RPLC. For particles having micropores, there is an additional mass transfer resistance, the resistance to diffusion through micropores which is often important. This explains why considerable attention is paid in the preparation of stationary phases for HPLC to avoid the formation of micropores. This explains also why graphitized carbon black, which tends to be plagued by a profusion of micropores, has not been a successful stationary phase for HPLC. The relationships between band broadening in chromatography and the various sources of mass transfer resistances are discussed later in this book, to avoid replication and for the sake of clarity. These relationships are discussed in Chapter 6, in the case of linear chromatography (the band width depends only on this kinetics), in Chapter 14 in the case of the bands of single components in nonlinear chromatography (the influence of the mass transfer kinetics on the band width decreases with increasing nonlinear behavior of the isotherm), and in Chapter 15, in the case of multicomponent chromatography under nonlinear conditions (the mass transfer kinetics being also nonlinear, the presence of other components may
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
249
affect the mass transfer kinetics of each of them [97]). We discuss here only the fundamentals of mass transfer.
5.2.5 External Film Mass Transfer Resistance Each particle in a bed of porous particles is surrounded by a laminar sublayer (Figure 5.4), through which mass transfer takes place only by molecular diffusion. On one side, this layer is exposed to the flowing mobile phase and is entirely accessible. On the other side, it wraps the particle wall and is accessible from the particle inside only at the pore openings. The thickness of this layer, hence the mass transfer coefficient, is determined by hydrodynamic conditions and depends on the flow velocity. The mass transfer rates can be correlated in terms of the effective mass transfer coefficient, kf, defined according to a linear driving force equation: a
"
6k f
= -jl-(C-C*) (5.70) dp where a is the external surface area of the particles per unit volume (i.e., 6/dp for spherical particles), q is the adsorbed phase concentration averaged over a particle, C is the concentration of solute in the mobile phase, and C* is the solute concentration in the mobile phase which would be in equilibrium with q. Several empirical correlations are available for the evaluation of external mass transfer coefficients [74,98]. 5.2.5.1
The Wilson-Geankoplis Correlation
The following relationships were proposed by Wilson and Geankopolis [99] —
Re033 Sc033
for 0.0015 < Re < 55 (5.71)
Sh = I —
Re
069
Sc
033
for 55 < Re < 1050
where Sh = (kfdp)/Dm is the Sherwood number and Sc = i]/(pDm) is the Schmidt number. In liquid chromatography, the Reynolds number is almost always less than 0.1 and frequently less than 0.01. So, the first equation is certainly the proper one to use. 5.2.5.2
The Kataoka Correlation
Kataoka et al. [100] found that, when only the data for Re < 100 are considered, most experimental data published earlier fit the following equation using the Sherwood, Reynolds, and Schmidt dimensionless numbers Sh
= 1.85 ( ^ )
Re1/3Sc^3
(5.72)
250 5.2.5.3
Transfer Phenomena in Chromatography The Penetration Correlation
The penetration theory [7] provides the value of the film mass transfer coefficient as (5.73)
Recent work on the mass transfer kinetics in monolithic columns [74] has suggested that the Kataoka and the Penetration correlations give similar results. These results are better than those given by the Wilson-Geankoplis correlation.
5.2.6 Intraparticle Pore Diffusion Adsorption of molecules proceeds by successive steps: (1) penetration inside a particle; (2) diffusion inside the particle; (3) adsorption; (4) desorption; and (5) diffusion out of the particle. In general, the rates of adsorption and desorption in porous adsorbents are controlled by the rate of transport within the pore network rather than by the intrinsic kinetics of sorption at the surface of the adsorbent. Pore diffusion may take place through several different mechanisms that usually coexist. The rates of these mechanisms depend on the pore size, the pore tortuosity and constriction, the connectivity of the pore network, the solute concentration, and other conditions. Four main, distinct mechanisms have been identified: molecular diffusion, Knudsen diffusion, Poiseuille flow, and surface diffusion. The effective pore diffusivity measured experimentally often includes contributions for more than one mechanism. It is often difficult to predict accurately the effective diffusivity since it depends so strongly on the details of the pore structure. The structure of the intraparticle pores is more complex than that of the extraparticle volume of a packed column (Figure 5.4 [96]). The pore distribution is much wider than the particle size distribution. The pore network has been compared to the network of roads in a city, with its highways, streets, alleys, and staircases. Local obstructions make the lengths of channels very different. Some local pore networks can be isolated from close-by channels. In connection with this issue, the concept of pore connectivity [101] is important for the description of the properties of adsorbents and their behavior in preparative chromatography. Generally, however, there is no flow through the channels. Their average diameter is so small that only an insignificant flow velocity can be sustained by the pressure drop between the opposite ends of a particle. The mass transfer mechanism is further complicated because surface diffusion may take part in the process, to a degree that depends on the nature of the material, but may also vary locally. Finally, although with silica it is reasonable to assume that no transfer can take place across the walls between pores, this is not necessarily true for all the resins used as packing materials. It is thus very difficult to make accurate predictions of the effective diffusion coefficient inside particles, since it depends so much on the pore structure and on the properties of the walls between pores.
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
251
In analyzing macropore diffusion it is usually assumed that the transport occurs only through the pores and that the mass flux through the solid can be neglected. This is certainly true when the packing material is silica-based. When it is made of a polymeric resin, this is less obvious as small molecules can diffuse unexpectedly fast through the network of certain types of coiled polymeric chains which are highly flexible (e.g., silicons). It is convenient to define a pore diffusivity, Dp, based on the pore cross-section area. / = -tvDp—
(5.74)
Dp is smaller than the diffusivity in a straight cylindrical pore as the combined result of two effects: the random orientation of the pores which increases the actual length of the diffusion path and the rapid, random variation of the local pore diameter. Both effects are usually accounted for by a tortuosity factor, T: Dp = -
(5.75)
Where D is the diffusivity observed under the same conditions in a straight cylindrical pore. 5.2.6.1
The Correlation of Mackie and Meares
The concepts of tortuosity and constriction of the packing material can be extended to all networks available to the molecules. There is little reliable data in this area. For small ions, and assuming that the pores inside the particles are of uniform size, the following equation may be used to estimate the pore diffusion, Dp [102]:
r e i2 Dp = —?—
Dm
(5.76)
where £p is the intraparticle porosity and Dm the diffusivity. For typical chromatography packings, tp is between 0.30 and 0.70, hence Dp is between 3 and 30 times smaller than Dm. 5.2.6.2
The Correlations of Satterfield et al. and of Brenner and Gaydos
Two correlations were suggested by Satterfield et al. [103] and by Brenner and Gaydos [104]. Both rely on the following equation
Dp = £ p J g , P w
(5.77)
where Kp is the hindrance parameter and k is the tortuosity of the pores inside the particles. While k is derived from the band width of an unretained tracer, we have two different equations to estimate the hindrance parameter. Satterfield et al. suggested that it is simply given by = -2.0Aw
(5.78)
252
Transfer Phenomena in Chromatography
where Xm is the ratio of the diameters of the sample molecule and the pores (hence Km < 1.0). The diameter of the molecule is usually calculated from its molar volume at its boiling point, assuming that the molecule is spherical. Brenner and Gaydos wrote that l + lA m lnA m -1.593A m (1-A W ) 2
KnP — —
Miyabe et ol. reported that both equations give values of Kp that are very close [74]. 5.2.6.3 Diffusion in Particles with a Two-mode Internal Pore Size Distribution Almost all stationary phases used in chromatography have a bimodal pore size distribution. The first mode corresponds to the macropores or throughpores that allow the percolation of the column by the stream of mobile phase. The second distribution corresponds to the mesopores that combine to give the conventional internal porosity distribution described in the previous section. The mesopores are responsible for most of the specific surface area necessary to provide the retention and the saturation capacity that are needed to permit the retention of the mixture components in a good solvent, a condition for chromatographic separation. Nonporous particles have been used with only moderate success because very weak solvents must be used to achieve sufficient retention, which often causes solubility problems, and the saturation capacity of these particles is small. The terms of macro- and meso-pores apply as well to columns made of packed particles and to monolithic columns. Some particles, however, may have a bimodal distribution for their internal pores. Besides the mode corresponding to the mesopores (typical dimensions between ca 6 and 30 nm), the second mode corresponds either to micropores (dimensions below ca 2 nm) that have properties that are rarely acceptable or to macro- or through-pores (dimensions above 500 ran) that are intermediate in size between the mesopores of an adsorbent and the extraparticle pores of a packed column. When the micropores are in large number on the adsorbent surface, the column efficiency is often poor and bands may exhibit a strong tailing under linear conditions. Molecules move too slowly through the micropores and are strongly adsorbed in them, as they find themselves surrounded in the electrical field generated by the pore walls. Graphitized carbon blacks tend to have many micropores which adsorb strongly straight alkyl chains longer than about three carbon atoms. This causes a very slow kinetics of desorption, a poor column efficiency at low sample concentrations, and unusual isotherm effects [105]. Although they contribute much to the specific surface area, micropores contribute little to the saturation capacity and are undesirable. There are few micropores in common stationary phases. One exception is molecular sieves (e.g., zeolites) which have found little practical use in HPLC. Another exception is the stationary phases used for separation of small molecular weight compounds by size exclusion chromatography. Provided the pore size is suitable to the size of the components separated, a high
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
253
degree of size selectivity can be achieved, but it is difficult to obtain good or even reasonable column efficiency. In the mid 1980s, a new packing material was patented and commercialized. It was claimed that this material exhibited exceptionally fast mass transfer kinetics, particularly for protein separations. These properties were rationalized on this stationary phase having a bimodal pore size distribution with large through-pores having an average size of the order of 0.10 to 0.02 times the particle diameter. It was also claimed that these pores went through the entire particle [101,106-108]. A significant flow velocity through these macropores was to cause a convective mass transfer with a rate exceeding that of mass transfer by diffusion. This convective flow assisted in the mass transfer of the feed components across the particles, and, thus allowed high column efficiencies at high flow velocities. The claim was never demonstrated and, eventually, the material and the company vanished after a patent infringement trial lost by the inventors. Those scientists interested in patent law dynamics will enjoy reading the decision of the judge in the case [109]. Particles of this type had been described by Kirkland [110,111] as early as 1976 (cf. the micrographs in Figures 2 of refs [106] and [110,111]). In the same year, Van Kreveld and Van den Hoed [112] had demonstrated that mass transfer through large size conventional silica particles used in size exclusion chromatography involved conventional diffusion and a contribution of convective origin. However, they did not identify the origin of this contribution. Frey et cd. [113] studied theoretically and experimentally the effect of intraparticle convection on the elution and separation of biomacromolecules and developed a model relating the characteristics of the particles and an apparent dispersion coefficient. The results were in agreement with experimental data measured with proteins under conditions of no retention. Obviously, such an agreement does not demonstrate the validity of the model. Frey et cd. concluded that bimodal particles with gigapores in which convection enhances the rate of mass transfer have some minor advantages over classical particles for applications that require fast separations but not high resolving power. However, pellicular particles are more effective for rapid, high-resolution separations where the saturation capacity and the specific surface area are not issues of major importance. The practical advantages of packing materials with large through-pores seem to be marginal and the interest in this type of material is now dormant. The measurement of the pore size distribution of a material is complex and the results must be interpreted with great caution. Most of the data that are available in the literature (including pamphlets on commercial products) are to be understood as semi-quantitative at best. The pore size distribution of adsorbents is measured by mercury porosimetry or derived from the adsorption of nitrogen [114]. Both methods are precise. The accuracy of the data given by mercury porosimetry relies on the accuracy of the contact angle of mercury with the material studied. This parameter can be measured only for planar surfaces, so it has to be estimated for the surface of a porous adsorbent. It depends to some extent on the chemistry of the surface, hence is prone to change from brand to brand, if not from batch to batch within a brand. The adsorption of nitrogen is measured in the gas phase, under cryogenic conditions. Accordingly, the conformations of a bonded layer
254
Transfer Phenomena in Chromatography
during these measurements and during the use of the adsorbent in HPLC may not be closely related. Additionally, the small size of nitrogen molecules allows their access to pores inaccessible to eluites in HPLC.
5.2.7 Surface Diffusion Mass transfer phenomena in chromatography can be separated into two kinds, those that take place in the fluid impregnating the bed of porous particles and those that involve the interactions between the solutes and the stationary phase. The former include axial dispersion, fluid-to-particle mass transfer and pore diffusion. They are all related to bulk diffusion. The latter include the kinetics of adsorption-desorption and surface diffusion. So, diffusion through porous particles depends on two different kinds of diffusion phenomena: pore and surface diffusion. Pore diffusion takes place within the pore space, in the free solution, away from the adsorbed layer which is under the influence of the electromagnetic field that exists above the solid surface. Surface diffusion is the direct contribution of transport that takes place through the adsorbed layer along the surface of the macropores. Although the mobility of the solute molecules in the adsorbed phase is much smaller than its mobility in the liquid phase, this can be compensated by the effect of the concentration of the solute in the adsorbed layer being much higher than in the solution when the thickness of the adsorbed layer is significant. Then, the contribution of surface diffusion can be quite significant. This is especially true in the case of RPLC when the stationary phase is made of a relatively thick layer of bonded alkyl chains. Although most models of adsorption assume that adsorbate molecules stay immobile on the adsorption site where they landed until they are desorbed (one of the basic assumptions of the Langmuir model, see Chapter 3, Section 3.2.1.1), this is not necessarily always true. Depending on the surface energy and, to a degree, on the nature of the compound studied, adsorbate molecules may remain immobilized in the adsorbed monolayer or they may migrate along and close to the surface, within the electric field of the adsorbent surface. Under certain conditions, the activation energy needed for such a lateral move may be lower than the desorption energy. Then adsorbate molecules may move laterally without having to desorb and surface diffusion is possible. Although this phenomenon has been shown to take place in a variety of gas-solid and liquid-solid systems, it is often important in the case of RPLC, because the adsorption energy is often rather low in such systems. Suzuki [115] has shown that, when surface diffusion takes place, the coefficient of diffusion across porous particles is the result of two parallel contributions due to pore and to surface diffusion De = Dp + pkKDs
(5.80)
where De is the interparticle diffusion coefficient, Dp is the pore diffusion coefficient, Ds is the surface diffusion coefficient, p^ is the particle density, and K is the Henry constant of adsorption (i.e., under linear conditions). Similar to the bulk diffusivity, the surface diffusion coefficient is the proportionality coefficient
5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media
255
between the surface diffusive flux and the gradient of chemical potential of the adsorbate along the surface. In principle, a tortuosity coefficient of the surface should also be used. However, surface diffusion is still poorly known. The surface diffusion coefficient depends much on the adsorption energy, it might depend on the local adsorption energy, hence vary from place to place on the surface, and it is likely impossible to separate the influence of the local surface chemistry and that of its tortuosity. The mechanism of surface diffusion has been much studied by Miyabe and Guiochon [116-122]. These authors showed that the activation energy of surface diffusion can be considered as the sum of two terms. The first one is the energy needed to make a hole in the mobile phase. It is independent of the adsorption energy of the solute considered but depends only on the nature of the mobile phase. The second contribution is the energy needed for the molecule of adsorbate to jump from the monolayer into this hole. This activation energy is proportional to the isosteric heat of adsorption. Experimental results confirmed that the values of the surface diffusion coefficients of several series of compounds are related to those of their bulk diffusivities through the equation: Ds = Dme~^r
(5.81)
Er = 5 + X(-Qst) = 1.5 - 0.30Qst
(5.82)
with
where Er is the additional increment of activation energy required for molecular diffusion when molecules migrate in the potential field of adsorption, Qst is the isosteric heat of adsorption of the compound, 5 and A are proportionality constants. A is positive and smaller than unity [119]. The surface diffusion coefficient of weakly retained compounds is slightly lower than their bulk diffusivity, close to half [118]. This might be explained by a steric hindrance to diffusion due to the swollen layer of bonded ligands through which surface diffusion must take place. The surface diffusion of strongly retained compounds can be so small as to become practically negligible. Thus, bulk and surface diffusion are not two different kinds of mass transfer mechanisms as is often assumed. On the contrary, surface diffusion should be considered as molecular diffusion proceeding in the potential of adsorption energy and restricted by it. The two phenomena are not fundamentally different from each other. In principle, there should not be surface diffusion for nonretained compounds. Qn the other hand, the mobile phase is adsorbed at the liquid-solid interface and its properties in this monolayer cannot be the same as those of the bulk solvent. This explains why the limit of the surface diffusion coefficient when the equilibrium constant tends toward zero is smaller than Dm (Eq. 5.81). Finally, Miyabe has shown that there is a linear free energy relationship between the equilibrium constant of retention and the surface diffusion coefficient [119]. This has led to the development of a procedure for the derivation of a first approximation of the surface diffusion coefficient. However, data are available only for alkyl-benzenes and alkyl-phenols at this time.
256
Transfer Phenomena in Chromatography
An interesting feature of surface diffusion is that the surface diffusion coefficient increases with increasing concentration of the adsorbate in the adsorbed phase. It has been shown that (5.83)
where q(C) is the equation of the equilibrium isotherm. This dependence may contribute to explaining why, as was often reported, the mass transfer kinetics seems to be faster at higher concentrations. Two other dependences were suggested by Kaczmarski et al. [123]. First, they used a model borrowed from gas-solid adsorption, the Maxwell-Stefan model, that is valid if the mobile phase is much less strongly adsorbed than the compound studied, which is generally true in RPLC. This model gives Ds = D s , o ( l - 0 ) n
(5.84)
where n is the number of nearest neighbor sites on the surface. Second, neglecting A in Eq. 5.82, and assuming that the isosteric heat of adsorption is a decreasing function of the surface coverage, these authors used the relationship Ds = Dsfie-^
(5.85)
where a. is an empirical coefficient. Both methods were used to account for the mass transfer of either enantiomer of 1-indanol on cellulose tribenzoate. They lead to estimates of D s 0 equal to 1.47 and 1.65 xlO~ 5 , respectively (with n = 3 and a = 0.029). An excellent agreement was obtained between the experimental band profiles and those calculated with the general rate model and a mass transfer kinetics involving surface diffusion. On the basis of Eq. 5.83 and using the POR model, Gubernak et al. [124] have shown that the profiles of the overloaded elution bands of insulin on an ODS column are well accounted for using a single value of the surface diffusivity for a wide range of sample sizes, hence concentrations.
5.2.8 Competitive Surface Diffusion When two components are partially separated and their mass transfer kinetics depends on their concentration, which seems to be the general case, it is reasonable to expect that the egress of the molecules of the first eluted component from a particle will interfere with the migration of the molecules of the second component into the same particle. Both mass transfer kinetics will be slowed down. This issue has been discussed mostly by Kaczmarski et al. [123,125]. Previous work [126] had shown that the POR model accounted well for the overloaded band profiles of samples of mixtures of various compositions of the enantiomers of indanol on cellulose tribenzoate, in a wide range of sizes. However, this was possible only by letting the effective diffusion coefficients of the two enantiomers float. The best values of these coefficients depended on the sample size. This suggested an important model error, the influence of surface diffusion
5.3 The Viscosity of Liquids
257
on the intraparticle mass transfer being neglected in this model. Kaczmarski et al. [123] used the dusty gas-like model to describe the surface diffusion of the two adsorbed molecules and the generalized Maxwell-Stefan model of surface diffusion. This model introduces coefficients of counter-sorption diffusivity that accounts for the interactions between the opposite flux of the two compounds during their elution. These coefficients are functions of the fractional surface coverages of the two compounds. With this model, there was an excellent agreement between the experimental band profiles and those calculated with the combination of the general rate model and this generalized Maxwell-Stefan model of surface diffusion, using Eq. 5.84 to estimate the single-component surface diffusivities. Only one value of DS/Q was needed. This confirms both the importance of surface diffusion in the mass transfer kinetics in RPLC and the complexity of this kinetics in the case of mixtures.
5.3 The Viscosity of Liquids The mobile phase viscosity is an important characteristic of the mobile phase. It relates the mobile phase velocity and the inlet pressure required to achieve this velocity. After Darcy law [127], the mobile phase velocity, u, is proportional to the local pressure gradient: u = -
~ tj dz
(5.86)
where rj is the mobile phase viscosity and B the column permeability, which is proportional to the square of the average particle size and denoted B = kgdl or B = dp/<J>, depending on the authors. The dynamic viscosity, rj, is nearly constant under typical conditions of preparative chromatography, although, as discussed below, rj varies with temperature and depends slightly on the pressure. The permeability, ko, depends somewhat on the width of the particle size distribution and on the particle shape. It has been claimed that spherical particles have a larger permeability than irregular ones; however, definitive data are not available. The effect, as far as it exists, may in part result from uncertainties in the definition and measurement of the average size of irregular particles. Since under steady-state conditions B and rj are constant, integration of Eq. 5.86 shows that u is proportional to the column pressure drop, AP k0d2p AP u = —p-— rj L
(5.87a)
AP = i ^ -
(5.87b)
or
where L is the column length. Equation 5.87a is often referred to as the Darcy equation, and the differential Eq. 5.86 is generally ignored. In the MKSA system
258
Transfer Phenomena in Chromatography
of units, L and dp are in m, u in m/s, t] in newton s/m 2 , and AP in N/m 2 or pascal. B is equal to k^d^,, withfcobetween 0.5 x 10~3 and 2 x 10~3, depending on the compactness of the packed bed, i.e., on its external porosity. Traditionally, a value of kg = 1 x 10~3 is used for irregular particles, and 1.2 x 10~3 for spherical ones. In practice, however, the CGS system of units is more commonly used in chromatography. The viscosity is given in cP (the Poise is the CGS unit of viscosity), the lengths are in cm, and AP is in barye (the CGS unit of pressure, 1 atm = 106 barye). If the column length is in cm, the particle diameter in ,um, the viscosity in cP, and the pressure drop in psi, Eq. 5.87b becomes =
15u(an/s) ? (cP)L(cm) kodjbim2)
As an example of calculation, assume the following values: linear velocity 0.10 cm/s, viscosity, 1.0 cP, column length, 15 cm, particle size 10 fim, and permeability constant, 1 x 10~3. The pressure drop is _ 15[0.10cm/S][1.0cP][25cm] Ai [ixl0- 3 ][10 2 ] Ai
It has also been shown that the diffusion coefficients in a solvent are inversely proportional to its viscosity. The viscosity changes with temperature, composition, and the concentration of the feed. When the column is operated at a constant reduced velocity, v = (udp)/Dj\^, the efficiency constant. The effect of a change of viscosity due to an adjustment in any of the parameters just listed will have little effect on the pressure required to keep the reduced velocity constant (since the product of the viscosity and the diffusion coefficient remains constant) but it will markedly affect the retention times (which will increase with increasing viscosity) and, in preparative applications, the production rate. Thus, conditions under which the viscosity is low should be preferred. The effect of a change of viscosity due to the use of high concentration feeds, organic modifier gradients or gradients in hydrophobic interaction chromatography (HIC) can be significant. In order to maximize the production rate, the feed flow rate and the elution flow rate may be different. In order to maximize the production rate, operation at constant column efficiency may be sacrificed for flow at constant pressure. This allows for the use of process control systems to adjust easily to changes in temperature, composition and concentration. Changes in temperature not only affect the viscosity, but also the retention time. The use of heat exchangers and other temperature control equipment can lead to improved reproducibility and robustness. We review here the viscosity of the most common mobile phases, the factors that influence this viscosity, the temperature, the pressure, and the mobile phase composition, and we discuss two phenomena of practical importance in the preparative applications of chromatography: (i) the dependence of the mobile phase viscosity on the concentration in feed components and the pressure excursion generated by the elution of high concentration bands of viscous feed; and (ii) the occurrence of flow instabilities and fingerings due to the rapidly varying viscosity of the eluent.
5.3 The Viscosity of Liquids
259
Table 5.3 Viscosity of Solvents Used in Mobile Phases1
Formula H2O
Name Water
CH 4 O C2H6O C3H8O C3H8O C2H3N C4H10O C3H6O C3H6O2 C4H8O2 QHi2 C5H12 CeH 14 C7H16
Methanol Ethanol 1-Propanol 2-Propanol Acetonitrile Diethyl ether Acetone Methyl Acetate Ethyl Acetate Cyclohexane n-pentane n-hexane n-heptane n-octane Benzene Toluene m-Xylene Carbon Tetrachloride Chloroform Methylene Chloride Methyl Chloride
QH6 C7H8 C 8 H 10 CCI4 CHCI3 CH 2 C1 2 CH 3 C1
»/(cP) 1.0 0.90 0.55 1.04 1.94 0.98 0.35 0.23 0.32 0.38 0.46 0.88 0.22 0.30 0.40 0.51 0.61 0.55 0.60 0.86 0.52 0.41 0.18
T(°C) 20 25 25 25 25 52 25 20 25 20 20 25 25 25 25 25 25 25 25 25 25 25 20
1
Data from R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 4th ed., 1987.
5.3.1 The Viscosity of the Mobile Phase A considerable number of studies have been devoted to the viscosity of liquids. We summarize here the properties that are relevant to chromatographic applications. We give in Table 5.3 the viscosities of the most important solvents used as mobile phases. 5.3.1.1 Influence of the Column Temperature on the Viscosity
The viscosity of liquids decreases with increasing temperature. The phenomenon has an exponential character, with a relatively low activation energy, of the order of a few kcal/mole. The dependence of the viscosity on the temperature is represented by an Antoine equation log t] = A +
B T+C
(5.88)
Transfer Phenomena in Chromatography
260 Viscosity, Cp
2 Toluene
Pentane
\ Methanol Ol
a a Water Water
\
Chlorofom Chloroform \ % Isopropanol Isopropanol V— Acetone
1.5
\
\
\
\
V
\
. \
^\
\ \
' \
\ \
1 \
Ethyl Acetate Ethyl \ \
\ \
:
\
0.5 \ ••
Acetonitrile Acetonitrile
- Methylene Chloride Chloride
0 150
200
250
300
350
400
T oK
450
Figure 5.5 Effect of the temperature on the viscosity of selected solvents.
In many cases, the constant C can be neglected and the viscosity data accounted for by a classical exponential dependence, as long as the temperature is not too close to the boiling point. If C = 0 and T is in °K, B is of the order of 1 to 3 x 103. For example, the viscosity of benzene decreases by one-half when the temperature increases by 120°C. Figure 5.5 illustrates the temperature dependence of a few typical solvents and Figure 5.6 that of various organic solutes [5]. By definition, measurements of adsorption isotherm data are made at constant temperature. So is that of other thermodynamic data. Often, such data are measured at different temperatures. Because the viscosity of liquids decreases with increasing temperature, it is strongly recommended that these data be measured at constant column inlet pressure, not at constant flow rate. At constant pressure, the effects of the temperature and of the pressure on the equilibrium constant can be easily deconvoluted. At constant flow rate, a change in the column temperature causes a change in the mobile phase viscosity, hence in the inlet pressure of the column. With large molecular weight compounds, {e.g., proteins), this pressure change may affect markedly the equilibrium constant [128] and result in large errors being made in the thermodynamic functions of the equilibrium [129]. It is useful to know the viscosity of the solvents that can be used as mobile phases in liquid chromatography. The number of such solvents is limited, and accurate viscosity data are available for all of them. Thus, this discussion is limited to the viscosity of mobile phases used in HPLC and the predictive equations that can be used to estimate the viscosities of liquids when accurate data are not available [5] will not be covered. However, note that any liquid, whether it is a conventional solvent or not, can be used as the mobile phase provided it affords
5.3 The Viscosity of Liquids
261
the required performance. For example, Kele [130] used methyl-1-naphthalene for the separation of C^Q and C/o on phenylporphyrin-bonded silica. 5.3.1.2 Influence of the Pressure on the Viscosity The viscosity of liquids increases slowly with increasing pressure (water is an exception below about 20° C). For pressures below about 2000 atm, and for temperatures below 0.85Tc (Tc, critical temperature), the increase is practically linear [5,24]: (5.89)
tj = TjO[l + aAP]
where t]Q is the viscosity under atmospheric pressure. Values of the coefficient a are reported in Table 5.4. The deviation from linear behavior is less than 5% below 1000 atm, hence in the entire operating pressure range in preparative liquid chromatography. When the pressure increases from 0 to 200 atm, the viscosity increases by 10% for methanol and 25% for benzene. These changes cannot be neglected in all cases. An increase in the pressure, hence in the viscosity, results in an increase of the retention times compared to the times predicted for a mobile phase with a constant viscosity [23]. However, there is no change in the retention volumes. Viscosity, Cp
b p-Cresol
Ethanol
12 Octadecenoic Acid m-chlorophenol
Decylbenzene
8
2-phenylethanol
Propylene Glycol
4 Furfural n-Heptane
0 200
250
300
350
400
OK T oK
Figure 5.6 Effect of the temperature on the viscosity of selected organic compounds. Data in these two figures are from R. C. Reid, ]. M. Prausnitz, and B. E. Poling, "The Properties of Gases and Liquids," McGraw-Hill, New York, 4th Ed., 1987.
Transfer Phenomena in Chromatography
262
Table 5.4 Pressure Dependence of Viscosity at 30° C
Compound
Viscosity1 (cP) under 1 atm 0.220 0.296 0.355 0.566 0.519 0.212 0.285 0.390 0.520 1.003 1.79 1.29 0.80 0.38 0.324
n-Pentane n-Hexane n-Heptane Benzene Chloroform Diethyl ether Acetone Ethyl acetate Methanol Ethanol Water (at 0°C) Water (at 10°C) Water (at 30°C) Water (at 75°C) Acetonitrile 1 2
(10"3 atm" 1 ) 1.06 1.15 1.09 1.22 0.62 1.11 0.684 0.810 0.470 0.585 -0.080 -0.046 0.053 0.076 0.624
Compressibility2 (10"4 atm" 1 ) 3.14 1.61 1.42 0.96 0.97 1.87 1.24 na
1.23 1.11 na na na
0.46 1.10
Viscosity at 30°C unless otherwise indicated. Compressibility at 25°C, except for diethyl ether, methanol and ethanol, 20°C [23].
5.3.1.3 Influence of the Mobile Phase Composition on Its Viscosity The viscosity of concentrated solutions can be quite different from that of the pure solvent. Many models have been proposed to calculate the viscosity of liquid mixtures from the viscosity of the pure components. These models rely on interpolation. As is often the case, mixtures of components that can exhibit intermolecular hydrogen bonding and especially aqueous solutions must be considered separately. The most recommended method for the calculation of the viscosity of mixtures is the Grunberg-Nissan equation [131]. Although it has only one constant, this equation is accurate for nonaqueous solutions. It gives the viscosity of lowtemperature liquid mixtures as \nt]m =
(5.90) '
l
where jy,- and rjm are the viscosities of the pure compounds i and of the mixture, respectively, and x,- is the mole fraction of component i. The interaction parameter, Gij, is a mild function of temperature (i.e., a linear, not an exponential, function of T). It depends on the nature of the pure components i and j (G;,- = 0). The constant G{j can be determined from experimental data. In the case of binary mixtures, it
5.3 The Viscosity of Liquids
263
Viscosity, C cp
a
p
T in oC
T=15
2
20 25 30
1.5
35 40 50
1
60
0.5 {'.
0 25
0
5 500
Pure H H2;0
755 7
MaOH/ H2;00 %v/v %v/v MeOH/ H
100 Pure MaOH Pure MeOH
b D
15 15
a 20
1.2
•.D ^ ^ ~ ^ ~
25 25
30 30 " " * — - .
• --•
35 35 40 40
0.8 «•
A
(1
*X««.
'
.
—
50 0 60
* " "
""" "-F
•-¥
0.4
"•""i--^7
0 0 0
25
Pure H220 re H
50 0 Acetonitrile/H220 0 %v/v AcatOnitrila/H
75
100 100 Pure ACN
Figure 5.7 Effect of the temperature on the viscosity of methanol-water ((a)) and acetonitrile-water ((b)) solutions. Data from H. Colin, } . C. Diez-Masa, T. Czaychowska, I. Miedziak and G. Guiochon, J. Chromatogr., 167 (1978) 41.
can be calculated using a group contribution method [5]. In many cases, however, the excess term in Eq. 5.90 is ignored, and this equation becomes identical to the one proposed by Arrhenius in 1887 for binary mixtures: = X\ In j/i + x2 \ntj2
(5.91)
The results obtained with this equation might not account exactly for the behavior of a particular mixture, but they seem to give an excellent approximation of the dependence of the mobile phase viscosity on its composition, provided the reduced temperature (T/Tc) is below about 0.7 [5]. Li and Carr [16] have shown that the viscosity data of mixtures of water and methanol or acetonitrile [132] are not
264
Transfer Phenomena in Chromatography
well accounted for by the Grunberg-Nissan equation [131]. Far better results were obtained with the following empirical equation where (p is the volume fraction of the corresponding component Vm =
^HZOVHZO
exp((pOIgaOIS)
(5.92)
hi this equation, aOrg and K.H2Oa r e two empirical constants. Both are functions of the temperature and are determined by fitting experimental data to this equation. Using this empirical equation, Li and Carr [16] obtained relative errors less than 10%. Note that the progressive changes in the profiles of the t](T) curves with decreasing temperature for acetonitrile/water mixtures is probably explained by the formation of water-rich and ACN-rich clusters of increasingly large size with decreasing temperature [133]. Teja and Rice [134] have proposed another relationship to calculate the viscosity of mixtures of polar liquids. It is based on a corresponding states treatment for the mixture compressibility factors. This method is more accurate than the previous one for polar-polar mixtures, particularly for aqueous solutions. It uses two reference-solution models for nonspherical molecules and a single interaction parameter. For methanol-water systems, the accuracy was within 9% [5]. Figure 5.7 shows the temperature dependence of the viscosity of water-methanol and water-acetonitrile mixtures in the entire composition range [132]. The existence of a viscosity maximum for a methanol concentration close to 45% is conspicuous. For water-acetonitrile, there is also a viscosity maximum, but it is less pronounced and takes place at a lower organic solvent concentration (~ 20%). We also show in Figure 5.8 the variation of the viscosity of various aqueous solutions with their composition [135].
5.3.2 Importance of the Mobile Phase Viscosity in Preparative Chromatography The viscosity data in Figures 5.5 to 5.8 have interesting and important implications for the pressure requirements in liquid chromatography, because of the relationship between the inlet pressure, the temperature, and the flow rate (Eq. 5.87). Ultimately, these parameters control the retention and the separation. The column inlet pressure required to achieve a certain flow rate is proportional to the mobile phase viscosity, which in turn depends considerably on the nature and the composition of this phase (see Figs. 5.5 to 5.8) and on the nature and concentration of the feed components when this concentration is not small (see Section 5.3.3). The consequences are quite different in normal phase, reversed phase, ion exchange, and hydrophobic interaction chromatography. For example, at 300 K, the viscosity of ethyl acetate is 0.4 cP, that of a 70% methanol-water solution is 1.2 cP, that of a 20% acetonitrile-water solution is 0.9 cP; while that of a 70% acetonitrile-water solution is only 0.5 cP. The viscosity of a 20% 2-propanol-water solution is 2.5 cP, and that of a 5% sodium acetate solution is 1.2 cP. The relative importance of these viscosities is not intuitive, and these tables, graphs, and charts are critical to understanding and trouble-shooting the process.
5.3 The Viscosity of Liquids
265
Relative Viscosity 4.5 Sodium Acetate Acetate Sodium
1 20 OC oC
1 Glycerol Glycerol
^
3.5
Sodium .Sulfate Sodium Sulfate
1
\r
^ ilfate Ammonium Sulfate
J
2.5
1.5
Isopropanol Isopropanol
Acetic Acid Acetic Acid
^^^^^
\
Methanol Methanol
Sodium Nitrate Nitrate Sodium
0.5 0
25
50 50
75
wt % % AA wt
100
Figure 5.8 Effect of the concentration of various compounds on the viscosity of their aqueous solutions. Data from CRC Handbook of Chemistry and Physics, D.R. Lidde, Ed., Boca Raton, FL, 71st ed., 1990.
In normal phase chromatography (NPC), the significantly lower viscosity allows higher flow rates at a given operating pressure. The concentrated buffer solutions used in hydrophobic interaction chromatography (HIC) are often more viscous than the methanol-water mixtures used in reversed phase liquid chromatography (RPLC). Concentration gradients of methanol-water in RPLC or of buffer in HIC can result in significant variations in the column inlet pressure at constant flow rate or in changes in the flow rate at constant pressure because of the viscosity maximum at about 45% methanol (Figure 5.7a). The effect is less significant with acetonitrile-water gradients (Figure 5.7b). Working at constant flow rate can have a significant impact on the cycle time. Similarly, working at constant pressure may have an impact on the separation, due to changes in the column efficiency. These issues need to be evaluated in developing a preparative or process separation. Accelerating mass transfers through porous particles in chromatography requires the use of packing materials made of smaller particles (as long as widebore monolithic columns are unavailable). Because the optimum reduced velocity of packed columns is about constant, their optimum operating velocity is proportional to the reverse of the average particle diameter. Accordingly, a large pressure drop is the price to pay to improve any separation on a given phase system, as stated by Eq. 5.87 and by the dependence of the permeability on di. Although
266
Transfer Phenomena in Chromatography
high-pressure catalytic reactors are used in the petroleum and the chemical industries, chemical engineers in the fine chemicals, pharmaceutical, and biotechnology industries are less comfortable with working at high pressures [136]. Chromatographers have been more willing to accept it because a high inlet pressure has always been a characteristic feature of the method. The chromatographic performance of columns made of bundles of parallel fibers [137] and of packings made of randomly oriented fibers [138,139] has been studied, but the results obtained so far have been rather disappointing. Packing materials for which the internal porosity has a bimodal size distribution with macropores large compared to the particle diameter could permit the achievement of efficient chromatographic separations while using low inlet pressures. Intraparticle mass transfer is enhanced if the macropores are large enough to allow convection of the mobile phase at a significant velocity (see Section 5.2.6) [101,106-108,110-113].
5.3.3 Calculation of the Inlet Pressure in the Case of a Variable Viscosity In liquid chromatography, the compressibility of the mobile phase is small, and in practice it is often negligible, unless high accuracy measurements are made for a specific purpose [24]. Under steady-state conditions, the equation of continuity of the mobile phase can be integrated to give u = UQ = constant, where UQ is the outlet velocity. When the viscosity of the sample and that of the mobile phase do not differ significantly, Eq. 5.86 can be integrated, and the total pressure drop can be calculated as _ urjL _ t,LFv
Since the compressibility of liquids is not entirely negligible, we may derive the velocity profile along the column by integration of the compressibility definition [23] X=
\dV
ldu
(5 94)
-vlP=-udP
"
Assuming a constant compressibility equal to xo (it actually decreases slowly with increasing pressure), gives the following equation [23] u(P) = uoe-x°(pz-p^
(5.95)
where u(P) is the local velocity, Pz the local pressure, and u$ the outlet velocity. Combination of Eqs. 5.89,5.86, and 5.94 gives a differential equation dz
(5.96)
Numerical integration of Eq. 5.96 between the column inlet and the point of abscissa z permits calculation of the pressure profile along the column. Because the mass flow rate of the mobile phase along the column is constant and if the mobile phase compressibility is assumed to be negligible, the mobile
5.3 The Viscosity of Liquids
267
phase velocity is constant. When the dependence of the viscosity on the pressure and the influence of the pressure gradient must be taken into account, the pressure drop is obtained by integration of the following equation
M5 = / P °JL kodj JPe 1 + ccP
(597) { '
Integration between the column inlet and the local point permits calculation of the pressure profile. It is useful to keep in mind that the orders of magnitude of the influence of the pressure on the specific volume and on the viscosity of a fluid are quite different. For all common solvents, the compressibility (x) is between 1 and 1.5 x 10~4 bar" 1 at room temperature. The pressure coefficient of the viscosity is between 0.5 and 1.2 x 1(T3 bar" 1 , except for water for which a = 0 [24].
5.3.4 Feed Concentration, Mobile Phase Viscosity and Inlet Pressure at Constant Flow Rate During a separation, the eluent is a solution of the feed components in the mobile phase that are becoming progressively better separated and more dilute in the mobile phase. At most intermediate points, these components are eluting as a series of partially resolved bands. When the viscosity of the feed and the mobile phase are different, the viscosity of the eluent depends on its local composition. As the feed is diluted in the column by the mobile phase, the viscosity of the eluent is a function of time and location. Thus, the pressure drop at constant flow rate will vary during the elution of the sample band. Conversely, at constant pressure, the flow rate will vary during elution. At constant flow rate, the combination of a highly viscous feed, a gradient of the mobile phase composition, and/or the change in eluent composition during column regeneration can lead to significant variations in the viscosity throughout the column length. This will result in significant increases and/or decreases in the operating pressure. This phenomenon increases the complexity of preparative chromatography and requires a careful investigation of how to load the column and to manage the flow rates, the pressure control systems and the settings of the proper pressure safety devices. Since the viscosity of a mixture is a nonlinear function of its composition and of the viscosities of the pure components of this mixture (Eqs. 5.90 or 5.91), the pressure drop in the column depends both on the amount of viscous solutes in the column and on their concentration profiles. Assuming that the flow rate is kept constant through the column, the numerical integration of the mass balance equation (Eq. 2.2 in Chapter 2) gives the concentration profile of the feed components in the mobile phase along the column length as a function of time (Chapter 10). From this composition profile, the viscosity profile is derived through Eq. 5.90 or 5.91. Knowing the local viscosity of the eluent solution (mobile phase plus feed components), we can calculate the local pressure drop across the column slice of length Az AP{ = ^Vmj Kat
(5-98)
268
Transfer Phenomena in Chromatography
Figure 5.9 Variation of the column inlet pressure during elution at constant flow rate of a viscous sample. (Top) Plot of the ratio AP/APQ during the migration and elution of the band. Abscissa, time (min). AP, inlet pressure at constant flow rate during elution. APQ, inlet pressure for the pure mobile phase. (Bottom) Concentration profiles of the elution bands. Sample size: 1 mL, concentrations: 200, 400, and 600 mg/mL. Solvent viscosity: 0.35 cP; solute viscosity: 50 cP. Langmuir isotherm. Reproduced with permission from A. Felinger and G. Guiochon, Biotechnol. Progr., 41 (1993) 450 (Fig. 3). ©1993, American Chemical Society. time (min)
Because in most cases the pressure achieved in preparative chromatography does not exceed 150 atm (~ 2 000 psi), the mobile phase viscosity in Eq. 5.96 can be assumed to be independent of the pressure during a separation (within a few percent). We will also neglect the influence of the pressure on the other chromatographic parameters {e.g., the hold-up volume and the mass transfer resistances). The integration of the local pressure drop along the column gives the total pressure drop through the column. Since the concentration profiles are calculated numerically, using a time/space grid {At, Az), for each time to + At, the pressure drop is (5.99) where N — L/Az — L/H is the number of theoretical plates of the column (Chapter 10, Sections 3.4b and 3.5) and L is the column length. The profile of the pressure drop versus time during the elution can be obtained as we repeat the preceding calculations with At time increments. We note that the pressure profile depends essentially on the concentration profile and on the ratio, J/m/j/o, of the viscosity of the solution of highest concentration involved in the profile to the viscosity of the pure mobile phase. The pressure profile can be scaled by the pressure drop obtained with the pure mobile phase, and the relative pressure surge, AP/APQ where APQ is the pressure drop when there are no feed components in the column, depends on the column length and particle size only through the influence of these parameters on the concentration profile (besides the amount of feed injected). Calculations show that there is a significant pressure excursion above the value
5.3 The Viscosity of Liquids
269
corresponding to steady state only if (i) the feed is very viscous; (ii) the feed is very soluble in the mobile phase; and (iii) a very large sample is injected. Figure 5.9 shows the variations of the column inlet pressure during the elution at constant mobile phase flow rate of increasingly large samples of a viscous pure compound (top) [140]. For simplicity, the figure shows the plot versus time of the ratio of the actual inlet pressure to the inlet pressure when the pure mobile phase is pumped through the column. The elution band profiles are calculated with the equilibrium-dispersive model (Chapter 10) and shown in the bottom of the figure. The pressure rises linearly during the injection of the sample and then begins to decay shortly after switching from the feed to the eluent since dilution is rapid at the beginning of the separation. The pressure decreases to the value of the initial inlet pressure in isocratic elution during the elution and dilution of the band. Note that it is only when the solute is highly viscous, when its concentration in the feed is very high, and when the amount of feed injected is large that a significant pressure excursion takes place during the injection and the separation. In gradient elution, the variation of the pressure post loading is more complex. It depends on the gradient characteristics and on the variation of the mobile phase viscosity and dilution is slower in gradient separations. During scale-up of the separation from laboratory to pilot to industrial scale, the pressure profile remains similar if the linear velocity, the column length, the particle size, the void fraction, the gradient characteristics, the load volume of feed, the load concentration, pressure drop in the extra-column tubings, the temperature of the feed and that of the mobile phase are all held constant. In practice, the pressure profile remains similar on scale-up when these details are well matched in the laboratory and in the plant. Often, however, the challenge is that, for operational reasons, these parameters cannot be maintained constant during the scale-up. Thus, additional experiments may be required in the laboratory to match the conditions in the pilot plant or in the plant.
5.3.5 Flow Instability and Viscous Fingering Fluid mechanics shows that the behavior of the interface between a plug of a viscous liquid that percolates through a packed bed and a less viscous liquid that pushes the viscous plug is unstable. The pressure drop along a streamline is less if a longer fraction of the length of this streamline is occupied by the less viscous fluid. Accordingly, the velocity along this streamline is faster and any small fluctuation amplifies quickly into a finger [141]. This instability is referred to as the Saffman-Taylor instability and the formation of fingers of the less viscous fluid into the more viscous one is called viscous fingering (VF). The phenomenon is general. It requires only that the less viscous fluid pushes the more viscous one. Its intensity depends on the difference between the viscosity of these two fluids or viscosity contrast. Chromatography does not enjoy a waiver from this instability, which may take place easily under experimental conditions that are typical of those used in preparative chromatography. The solutions of most feeds in the low viscosity solvents that are used to make chromatographic mobile phases are more viscous than the initial solvent. Thus, VF will tend to take place at the rear of
270
Transfer Phenomena in Chromatography
most elution bands, particularly when large samples of concentrated solutions are injected rapidly. This phenomenon should prevent or severely limit the achievement of separations under a high degree of column overloading. Yet, there are surprisingly few reports regarding VF and its effects on chromatographic separations. This apparent contradiction remains somewhat of a mystery. The reason is probably that the rear part of band profiles (the side along which the viscosity decreases with decreasing compound concentration) are often diffuse boundary. Under such conditions, it takes a certain time for VF to develop and fingers form only under conditions that are rarely met in practice. Although never analyzed in detail, the effects of VF in chromatography have already been noticed and the differences in viscosity between the eluent and the sample bands have justly been blamed for excessive band broadening in certain attempts at performing high concentration separations or purifications of high molecular weight polymers or of proteins [142-144]. For example, Moore [143] has reported that VF appears at concentrations between 1 and 10 mg/mL for polystyrene samples of molecular weights above 150,000 Daltons. He suggested eliminating VF entirely by doing SEC analyses in frontal analysis. VF takes place only when the column is cleaned for the next analysis. This was shown to be most efficient for analyses but cannot be extended to preparative separations. The viscosity effect has also been reported by Flodin [145] who observed increasingly tailing bands and a marked decrease of the resolution with increasing dextran concentration in a dilute sample of sodium chloride and hemoglobin. The resolution was barely improved by a decrease in the flow rate. However, the effect in this case was incorrectly blamed on bed compression [145]. Very similar results were reported later by Czok et al. [146], who demonstrated the implication of VF in its mechanism and suggested eliminating VF by using a mobile phase having a viscosity close to that of the sample (at the cost of operating the column with a high inlet pressure). VF seems to be of particular importance in size exclusion chromatography (SEC), because there is little retention with this method, hence little sample dilution, and the viscosity of a polymer solution increases rapidly with the polymer concentration. The phenomenon has also been observed in the separation of concentrated protein solutions in SEC, in ion-exchange chromatography, when the retention is low, and in gradient elution, a method which may generate excessively concentrated solutions. James and Ouano studied the errors that are made in the measurements by SEC of the properties of polymers and are due to column overloading. They found that these errors originate in the sample viscosity [147]. Emneus avoided the band broadening effect due to fingering in the desalting of proteins by carrying out size exclusion separations in a batch mode [148]. The sample is mixed with the dry packing and the highly viscous sample solution containing the proteins is separated by centrifugation from the particles of the packing material that contain only the salt solution. Such a procedure is not chromatographic, however, and is unacceptable for difficult separations. In order to avoid VF, it has been recommended that the sample viscosity be kept below twice the mobile phase viscosity [149]. For proteins, it was suggested also that the sample concentration should not exceed 0.2% (w/w) or about 0.04
5.3 The Viscosity of Liquids
aa
271
b
c
Figure 5.10 Photograph of VF pattern on the front of a breakthrough curve of acetonitrile in carbon tetrachloride that has the same refraction index as the silica used as packing material. Packing, 10 \vn\ particles of Nucleosil C^g, u = 0.030 cm/sec. Patterns after migrations of ca (a) 2 cm, (b) 5 cm, and (c) 7 cm along the column. Reproduced with permissionfromR. A. Shalliker, V. Wong, G. Guiochon,}. Chromatogr. A, (2005) (Fig. 1).
mg in a 20-fiL sample loop for conventional analytical columns [149]. This figure is conservative and aims at avoiding other sources of difficulties as well. Obviously, a general method of reducing the importance of the viscosity effects in analytical or preparative applications of SEC is to inject a larger, more dilute sample. However, there is a limit to the extent of volume overload that can be achieved before the resolution becomes too low for a useful purification. Similarly, in preparative applications, we need to keep feed component dilution at a minimum to reduce the costs of the post-separation solvent reduction. Since the separation mechanism of SEC is essentially entropic, it is affected by temperature only insofar as the molecular size of the feed components is temperature dependent. On the other hand, the sample viscosity decreases with increasing temperature. If it does so faster than the solvent viscosity, SEC separations will be advantageously carried out near the upper temperature limit of the sample, the column packing,
272
Transfer Phenomena in Chromatography
Figure 5.11 Illustration of the effects of viscous fingering on the band profiles in size exclusion chromatography. (a) Elution profiles of chicken ovalbumin at increasing sample concentration (0.02 to 1.2 mg sample). Flow rate of phosphate buffer solution 0.5 ml/min. (b) Elution profiles of uracil solutions containing increasing concentrations of glycerol. Flow rate 0.1 ml/min. Sample volume 20 \i\. Glycerol concentrations: 1 = 0%; 2 = 10%; 3 = 17%; 4 = 33%; 5 = 50%; 6 = 67%. (c) Elution profiles of turkey ovalbumin solutions. Injections of a low-concentration sample expanded 15fold (dotted line — 3), and of two high concentration samples (2.8 mg in 20f*L). The first concentrated sample (dashed line — 1) is injected conventionally; the injection of the second one is followed by the injection of a 1.4-mL viscous plug of 15% glycerol (solid line — 2). Reproduced with permission from M. Czok, A. M, Katti and G. Guiochon, J. Chromatogr., 5501991 705 (Figs. 1,2 and 7).
or the equipment. For other separation mechanisms, operation at temperatures above ambient should be considered in order to reduce the extent of VF. That may require appropriate changes in the mobile phase composition. A detailed study of the influence of VF in SEC was published by Czok et al. [146]. Other more fundamental studies were made by Czok and Cherrak [150], Fernandez et al. [151-154], and Shalliker et al. [155-158]. These last two groups made important investigations of VF based on the visualization of the phenomenon, the first group by NMR imaging and the second by optical means, using a balanced refractive index method. This latter method consists merely in packing a RPLC adsorbent in a glass column, using a mobile phase that has the same refractive index as the packing material, and placing the column in a rectangular shaped box with four transparent windows. The injection of colored bands is easily followed visually and can be analyzed in detail from digital photographs [157].
5.3 The Viscosity of Liquids
273
Figure 5.10 is an example of the results obtained. While most investigators of VF used Hele-Shaw cells which approximate the three-dimensional bed with a pseudo-two-dimensional one, these last two groups have visualized the VF phenomenon in actual columns. Czok et al. [146] showed that bands of ovalbumin of increasing concentration have increasing width and complexity (Figure 5.11a). The mobile phase was water and the samples were mixtures of glycerol and uracil. These two compounds are not separated by SEC (they are not retained), so glycerol provides viscosity to the band while uracil provides UV absorbance, permitting the detection of the band. When samples of increasing concentration, hence viscosity, were injected into the pure mobile phase (Figure 5.11b), increasingly wide elution band profiles of increasing complexity were recorded. However, glycerol is not retained and the viscosity of the solutions is not very high. A 33% glycerol solution in water has a viscosity only 2.5 times as large as pure water. The bands in Figure 5.11a and b have nearly the same front, suggesting that the pure water mobile phase fingers into the viscous sample plug that it pushes and causes the formation of VF which deforms the band. Samples of increasing viscosity were injected in a mobile phase containing 33% glycerol, with a reduced flow rate to keep the Peclet number approximately constant (Figure 5.11b). The elution bands are somewhat similar to the previous ones when the sample viscosity is higher than the mobile phase viscosity. They exhibit the same kind of wide band, having the same front as the band of a 33% glycerol sample, but a delayed and wavy rear boundary. The band of a 33% glycerol sample is very similar to that of a nonviscous sample injected in a nonviscous mobile phase. The bands of low-viscosity samples, on the other hand, exhibit the opposite deformation, with a constant rear boundary but a front appearing at a time that decreases with decreasing viscosity. Thus, the band front of a low-viscosity sample tends to finger into the high-viscosity mobile phase. Finally, if a wide enough plug of a solution of glycerol having the same viscosity as the sample is injected immediately after this sample, the effects due to VF disappear (see Figure 5.11c and [146]). A size exclusion separation of a concentrated solution of hemoglobin and bovine serum albumin (1.4 mg each in 20 fiL of solution) could be achieved if the sample injection is followed by the injection of 2 mL of a 15% solution of glycerol which has the same viscosity as the sample. The high viscosity of this rear plug prevents VF. Magnetic resonance imaging (MRI) provides a means of direct analysis of the three-dimensional flow pattern of a liquid percolating through a porous medium, although its spatial resolution is still limited to ax 1 mm 3 [151-154]. This approach allows a comparison between the results of actual observations and theoretical descriptions of the behavior of fingers [159]. The effects of the sample molecular weight (i.e., its diffusivity), of the volume of sample solution injected, of the heterogeneity of the packing, and of the reduced velocity were investigated. The numerical calculation of the behavior of a single finger showed that the faster flow velocity associated with the progression of the finger of less viscous fluid inside the more viscous one generates vortices that tend to enlarge the tip of the finger. Fingers cannot penetrate through the whole band of a finite sample and pass its leading edge because the growth of fingers is controlled by the viscosity
274
Transfer Phenomena in Chromatography
contrast. Accordingly, an isolated finger tends to spread and grow laterally when it nears the leading edge of an unretained zone [153,159]. Although, in practice, numerous fingers coexist and the phenomenon is more complex, this result agrees with experimental observations [152]. The fingers distort the leading edge of the band but do not break through it. The growth rate of the fingers increases rapidly with increasing viscosity ratio (relatively small in chromatography), hence with the sample concentration (glycerol and bovine serum albumin in Ref. [152]). Their wavelengths, on which depend their diameters, decrease rapidly with increasing viscosity contrast and are proportional to the sample diffusivity [159]. It is predicted to be 2 mm for glycerol and 0.5 mm for BSA but the resolution of MRI does not allow a significant estimate of this difference [152,159]. The experimental results are consistent with those of Czok et al. [146] but provide far more detailed explanations of the VF phenomenon. Although considerable perturbations of the band profiles had been observed when the viscosities of the feed and of the mobile phase differed markedly and circumstantial evidence made it plausible to attribute these effects to VF, actual demonstrations of the formation of VF were made later. Broyles and Shalliker showed that numerous phenomena that take place inside a column, among which is VF, can be visualized by packing the column in a glass tube, using a mobile phase having the same refraction index as the packing material, and placing the cylindrical column inside a rectangular cuve filled with a liquid of the same refraction index [155-158]. The column seems transparent and the cylindrical lens effect is canceled. The intensity of VF was correlated with the extent of band distortion [155]. Although VF is a chaotic phenomenon, at the low Reynolds numbers characterizing the flow of the mobile phase in chromatographic columns (Re < 0.01), both the VF pattern [158] and the band distortions [155] are well reproducible, as illustrated in Figure 5.10. Thus, VF takes place in chromatography when concentrated solutions are injected. The phenomenon is observed essentially in SEC, as retention is short and the extent of dilution is limited. It is certainly prone to take place in other modes of chromatography as well. With them, the remediation method suggested by Czok et al. [146] for SEC would be more difficult to apply, as the concentration, hence the viscosity, of bands varies in a broad range during their elution. Since most isotherms are convex upward, chromatographic bands tend to have a sharp front and a diffuse rear boundary (see Chapters 7 and 10). On the other hand, the viscosity of the sample solution is almost always higher than that of the pure mobile phase. Hence, the sharp front, or shock layer, is hydrodynamically stable, since the viscosity increases across it. The diffuse rear boundary is not stable, but the concentration, hence the viscosity, decreases only slowly with increasing time when the rear of the band passes. Accordingly, VF develops rather slowly. This may explain why the phenomenon has rarely been mentioned in the literature and why users never complain about it, even in private discussions, except in connection with size exclusion chromatography. Yet, VF has been observed by us under such experimental conditions that the band profile is still barely broadened. There is more to this phenomenon than we know yet. Finally, the influence of the band retention on the development of VF has not yet been studied. It is probably not
REFERENCES
275
negligible. Finally, VF develops quite differently when the fluid studied has a nonmonotonic viscosity profile, as was shown by Manickam and Homsy [160,161]. In this situation, the composition of the fluid that is pushed is not constant but has a concentration gradient away from the interface. Aqueous solutions of 1- and 2propanol, that were taken as cases in point in the simulations [160], or of methanol, that is often used as an organic modifier in HPLC, have a nonmonotonic viscosity. The viscosity of these solutions is maximum for concentrations of alcohol around 60% (nearly the same for all three alcohols). These authors showed that while an unfavorable viscosity contrast always leads to VF instability for monotonic viscosity fluids, this is not true for nonmonotonic fluids. Under certain conditions, and for a certain time, a stable displacement can take place when the viscosity contrast is unfavorable and conversely, VF instabilities may eventually develop with an initially favorable viscosity contrast. This is due to the effects of diffusion (and of dispersion which is even faster in HPLC) that slowly affects the viscosity profile. The condition for instability in a nonmonotonic viscosity profile is ((3/9C)c=o + (9/3C)c=i < 0. The viscosity profile of a solution of feed components in a water/methanol mobile phase, at relatively high concentrations, may very well satisfy the stability condition of Manickam and Homsy [160,161] for a time, at least. This may provide an answer to our question. A high-concentration band profile generally constitutes a nonmonotonic fluid, with a maximum viscosity at the band apex. In most cases, the stable region is downstream. Under such conditions, the band propagation is stable initially and instability might very well take place too late to cause serious problems.
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Chapter 6 Linear Chromatography Contents 6.1 The 6.1.1 6.1.2 6.1.3 6.1.4 6.2 The 6.2.1 6.2.2 6.2.3 6.3 The 6.3.1 6.3.2 6.3.3
Plate Models 283 Overview of the Approach 283 The Martin and Synge Plate Model 284 The Craig Plate Model 286 Comparison of the Two Plate Models 288 Solution of the Mass Balance Equation 290 The Equilibrium-Dispersive Model 290 Solution of the Lumped Kinetic Model 295 From the Lumped Kinetic Model back to the Equilibrium-Dispersive Model . . 300 General Rate Model of Chromatography 301 Inverse Laplace Transform of the Solution 304 Analytical Solution in the Case of Periodic Rectangular Injections 305 Analytical Solution for Chromatography with a Packing Material having Nonuniform Particles 308 6.4 Moment Analysis and Plate Height Equations 310 6.4.1 Moments of a Chromatographic Peak 310 6.4.2 Plate Height Equation in the General Rate Model 313 6.4.3 Coupling of Molecular and Eddy Diffusion 315 6.4.4 Plate Height in Perfusion Chromatography 320 6.4.5 The Golay Plate Height Equation 324 6.4.6 Dispersion and Partitioning in Short Coated Tubes 325 6.4.7 Evaluation of Transport Parameters from Chromatographic Peaks 326 6.5 The Statistic Approach 328 6.5.1 The Use of Characteristic Functions 329 6.5.2 Transport Equation in Chromatography with a Finite Speed of Signal Propagation 334 6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography . . . . 335 6.6.1 Extra-column Source of Band Tailing 336 6.6.2 Surface Heterogeneity and Complex Isotherm 338 6.6.3 Heterogeneous Kinetics 339 6.7 Extension of Linear to Nonlinear Chromatography Models 341 References 342
Introduction Linear chromatography is a specific, limiting case of nonlinear chromatography. It is an important one for our purpose, however, because in linear chromatography the influence of thermodynamics on the shape of the band profiles vanishes. Thermodynamics controls only the band positions; kinetics controls their profiles. Thus, the influence of the parameters of kinetic origin remains isolated and can
281
282
Linear Chromatography
be studied. Linear chromatography means that the equilibrium isotherm is linear and that the amount adsorbed on the stationary phase at equilibrium is proportional to the solute concentration in the mobile phase. Thus, the isotherm is summarized by a proportionality coefficient, the equilibrium or Henry constant. Under linear conditions, the band width depends on the kinetics of mass transfer in the column and on axial dispersion but is independent of the equilibrium constant. The retention time, on the other hand, will be independent of the kinetics in most cases (unless the mass transfer kinetics is very slow). Thus, linear chromatography permits a detailed study of the mass transfer kinetics and of axial dispersion in chromatographic columns, while ideal chromatography (Chapters 7 to 9) permits the study of the influence of thermodynamics on the band profiles. The importance of linear chromatography comes from the fact that almost all analytical applications of chromatography are carried out under such experimental conditions that the sample size is small, the mobile phase concentrations low, and thus, the equilibrium isotherm linear. The development in the late 1960s and early 1970s of highly sensitive, on-line detectors, with detection limits in the low ppb range or lower, permits the use of very small samples in most analyses. In such cases the concentrations of the sample components are very low, the equilibrium isotherms are practically linear, the band profiles are symmetrical (phenomena other than nonlinear equilibrium behavior may take place; see Section 6.6), and the bands of the different sample components are independent of each other. Qualitative and quantitative analyses are based on this linear model. We must note, however, that the assumption of a linear isotherm is nearly always approximate. It may often be a reasonable approximation, but the cases in which the isotherm is truly linear remain exceptional. Most often, when the sample size is small, the effects of a nonlinear isotherm (e.g., the dependence of the retention time on the sample size, the peak asymmetry) are only smaller than what the precision of the experiments permits us to detect, or simply smaller than what we are ready to tolerate in order to benefit from entertaining a simple model. The hypothesis of linear behavior of the equilibrium isotherm in analytical chromatography has three important consequences. First, the different components contained in a sample of a mixture behave independently of each other. They do not compete for interaction with the stationary phase because the sample size is small and the solutions are dilute. Therefore, the elution profiles and the retention times of the various components of a mixture are independent of the presence of other solutes and of their relative concentrations. Each band profile is the same as if the corresponding solute were alone, pure. As a consequence and in contrast with nonlinear chromatography, there is only one problem to solve in linear chromatography, the determination of the peak profile of a single component. The second consequence of the assumption of a linear isotherm is to make simple the mathematics of describing the migration of these independent, individual bands and of calculating their retention times and profiles. As we show later in this chapter, an analytical solution or, at least, a closed-form solution in the Laplace domain can be obtained with any model of linear chromatography. This is certainly not the case in nonlinear chromatography.
6.1 The Plate Models
283
Finally, considerable attention has been paid to the kinetics of mass transfer in chromatographic columns under linear conditions and to all the sources of band broadening. Much of that knowledge can be put to good use in the study of nonlinear chromatography because the fundamental phenomena remain the same. Moreover, the study of the kinetics contributions to band broadening in nonlinear chromatography is simplified by the application of the results obtained in linear chromatography. In particular, this hybrid combination is the basis of the equilibrium-dispersive model (Chapters 10 to 13). The theory of linear chromatography has developed along three axes, leading to the three broad classes of models that are used to describe and predict elution profiles. These classes are (i) the plate or "tank in series" models of Martin and Synge [1] and Craig [2]; (ii) the solutions of systems of differential equations that describe the mass balance and the mass transfer kinetics and which have been the basis of the fundamental work of Lapidus and Amundson [3] and of Van Deemter et ol. [4]; and (iii) the statistical models first developed by Giddings and Eyring [5] and refined later by Dondi [6,7] and Felinger [8,9].
6.1 The Plate Models The plate models assume that the column is divided into a series of an arbitrary number of identical equilibrium stages, or theoretical plates, and that the mobile and the stationary phases in each of these successive plates are in equilibrium. The plate models are in essence approximate, empirical models because they depict a continuous column of length L by a discrete number of well-mixed cells. Although any mixing mechanism is clearly absent from the actual physical system, plate models have been used successfully to characterize the column operation physically and mathematically. Therefore, by nature, plate models are empirical ones, which cannot be related to first principles. Note that, in the following, we consider alternately two types of concentration profiles, the spatial profiles and the history profiles. Spatial profiles give the distribution of the concentrations of a compound along the column at a given time. Historical profiles give the variation of the concentration during the migration of the band past a given point of the column. Of special concern among historical profiles is the one recorded at the column outlet, or elution profile. These two types of profiles are different because apparent axial dispersion takes place constantly, hence disperses more the rear of a band than its front which passes first at the point considered. When comparing historical and spatial profiles, one should remember that the part of a spatial profile to be eluted first is the one that has achieved the longer migration distance, hence is closer to the column outlet.
6.1.1 Overview of the Approach Martin and Synge published their first description of the plate theory in 1941 [1], at the time when Wicke [10], Wilson [11], and Devault [12] were beginning to study the solution of the mass balance equation of chromatography (Section 6.2).
284
Linear Chromatography
The plate theory assumes a linear isotherm and divides the column into a series of theoretical plates. In each plate there is equilibrium between the mobile and the stationary phases and an integral mass balance is written for each plate. The length of the plate is called the height equivalent to a theoretical plate (HETP). Thus, the two theoretical approaches to the theory of chromatography that illustrate the opposite viewpoints of analytical chemists and chemical engineers began at the same time. Wilson [11] and DeVault [12] neglect the influence of the mass transfer kinetics and the axial dispersion (band broadening contributions) on the band profile, to focus on the thermodynamic influence of a nonlinear isotherm (which is most important at high concentrations), whereas Martin and Synge [1] do the opposite and assume a linear isotherm in order to study nonideal effects. A few years later, in the mid-1940s, Craig [2] developed an apparatus to perform liquid-liquid extraction with a large number of stages (up to several hundreds). The theory of the Craig machine can easily be extended to a method to describe the distribution of solutes in linear liquid-liquid chromatography (LLC) based on a multiple step extraction process, using a finite number of vessels, where each stage is considered as an ideal mixer. The Craig machine gives the binomial distribution as the response to a pulse injection in linear chromatography, and the plate model of Martin and Synge [1] gives a Poisson distribution. Both distributions, however, tend toward the Gaussian one when the number of transfer stages increases and, in practice, they are equivalent.
6.1.2 The Martin and Synge Plate Model The Martin and Synge plate model [1] is a continuous plate model. It assumes that the column is equivalent to a series of continuous flow mixers. Mobile phase is transferred from one vessel to the next one as new mobile phase is added to the first vessel. Hence, the mobile phase flows continuously, and in each mixer the volumes of the mobile phase, vm, and of the stationary phase, vs, remain constant. The model is also based upon the assumption that, at the beginning of the experiment, only the first plate (rank I = 0) is loaded with the sample and that there are no sample components in the other plates. Said [13] has extended the theory of the continuous plate model to the case in which the solute is initially distributed on several plates, according to a certain distribution function. The mass balance for the mixer of rank I, when a volume dv of mobile phase is moved through the series of plates, is given by Amount of solute entering the mixer / = Amount of solute exiting from the mixer I + Accumulation due to concentration change or c
m,l-idv = CMridv + vmdCmri + vsdCSii
(6.1)
Since we assume that complete (linear) equilibrium is reached in each plate, we have dCS/i = adCm/i
(6.2)
6.1 The Plate Models
285
where a is the slope of the adsorption isotherm. Combination of Eqs. 6.1 and 6.2 gives a linear, first-order, differential equation, which is easily solved. If we assume, in agreement with the model, that the solute exists only in the first plate at the beginning of the experiment, where its concentration is Cm/o in the mobile phase and CS/o in the stationary phase, we obtain e xxl Cw,0
(6.3)
l\
Equation 6.3 gives a Poisson distribution function, where the variable is the plate rank, I, and x is a constant given by ^
^
_ ^ _ = 1 (6.4) T vvm + + «»s «» T where x = (vm + avs)/Fv is the time needed to pass through the column a volume equal to vm + avs, or retention time per plate, V is the total volume of mobile phase passed through the plate / since the beginning of the experiment, vm + avs is the equivalent volume of each plate, and Fv is the volumetric flow rate of the mobile phase. For large values of x, this Poisson distribution is well approximated by a Gaussian distribution: vm + avs
eX1p
\-iL2Ff"
(6.5)
with a standard deviation crs and an average }is. According to the laws of statistics, the mean of the distribution in Eq. 6.3 is given by }is = x and its variance is given by of = x. The maximum concentration of the solute is eluted from the column when it leaves the last stage (Nth stage, of rank I = N — 1), i.e., only when the mean location of the solute distribution is in this stage, so the number of operations needed to elute the band maximum is given by }is = x = N
(6.6a)
hence:
^f = x = N
(6.6b)
S
It is important to note that Eq. 6.3 is not the elution profile of the peak but the spatial distribution profile of the solute along the column. The elution profile is the distribution of the amount of the solute that leaves the last stage [rank (N — 1)]. This distribution is given by Cm^N_y/(x Cmjo). We notice that, in contrast to Eq. 6.3, the number I is now constant and equal to (N — 1), while x is a variable. Finally, the elution profile in the case of the continuous plate model is given by
286
Linear Chromatography
This equation with a fixed value of N and a variable value of t is not a Poisson distribution function (in contrast to Eq. 6.3), but a gamma density function [14], with a first moment1 given by P = TN
(6.8a)
and a variance equal to o2 = TZN
(6.8b)
Since the first moment of the distribution is equal to the retention time, we can derive from Eq. 6.8a that
T-§
(6.8C)
Combination of Eqs. 6.7 and 6.8c gives
For large values of the plate number, N, the gamma density function approaches the Gaussian function [15], as given by Eq. 6.5.
6.1.3 The Craig Plate Model The Craig model [2] is an exact description of the separation process implemented in the Craig machine. It makes an approximate model of chromatography. In this model, we divide the column into a series of Nc discrete stages of equal length and number these stages in the direction of flow, with rank I = 0,1, 2, • • •, Nc — 1. Each stage contains a volume vm of mobile phase and a volume vs of stationary phase. We assume that initially the first stage (stage number I — 0) contains all the solute used, that the other stages are entirely free of solute, and that all further portions of the mobile phase introduced in the first stage (I = 0) are pure mobile phase. The distribution coefficient of the solute is a = Cs/Cm. After equilibrium has been reached in the first stage, the amount of mobile phase contained in the last stage (I — Nc — 1) is withdrawn from that stage and collected. Then the volume of mobile phase contained in each stage is moved from that stage to the next, and the same amount of pure mobile phase is added to the initial stage (I = 0). After equilibrium is reached again in all the stages containing some solute, the mobile phase contained in one stage is moved again to the next stage, the volume contained in the last stage is collected, and the first stage is replenished. The process is repeated as often as necessary to sweep all the sample components from the column. Thus, the Craig model is a discontinuous plate model. At each step, the fraction of solute R = (vmCm)/(vmCm + vsCs) = 1/(1 + Fa) = 1/(1 + k'o) moves from one stage to the next. As the mobile phase is transferred, the fraction 1 — R remains in the stage. After r such operations have been ]
See later, Eq. 6.77 for the definition of the moments of a distribution.
6.1 The Plate Models
287
performed, the composition of the first r + 1 stages is given by the terms of the binomial distribution of the expression: [(1 - R) + R]r
(6.10)
Thus, the probability of finding a molecule in the stage of rank / [i.e., the (l+l)th stage] after r operations have been completed (with / < r) is given by
At this stage in the process, after r transfer operations have been completed, the mean location of the amount of solute introduced initially in the first stage is the stage of rank fis, with yis = rR
(6.12)
The variance of the distribution of the solute in the separation column is *
R) = Ml-R)
(6.13)
According to the central limit theorem of statistics, if r is sufficiently large, the binomial distribution (Eq. 6.11) can be considered identical to a Gaussian distribution with the same average, fis, and standard deviation, crs. The distribution of the solute in the Craig model is now given by Eq. 6.5. The maximum solute concentration is eluted from the column when it leaves the last stage (Ncth stage, of rank I = Nc — 1), i.e., only when the mean location of the solute distribution is in this stage, so the number of operations needed to elute the band maximum is given by ¥s
= rR = Nc
(6.14)
an equation very similar to Eq. 6.6a. The total volume of mobile phase that has passed through the last stage during the entire process is VR = rvm. From Eq. 6.14, we obtain: Nr
VR = rvm = — vm
(6.15)
We note that Ncvm is equal to the total volume of mobile phase, Vm, contained in the column. By substituting Vm and inserting the value of R in Eq. 6.15, we obtain VR = Vm(l + k'o)
(6.16)
Since the mobile phase moves along the column at a velocity that is 1/R times faster than the solute, the standard deviation of the distribution at the end of the column, expressed in volume units, is given by ^
= vm\JrR(l - R)R
(6.17a)
A combination of Eqs. 6.15 and 6.17a results in (6.17b)
288
Linear Chromatography
Finally, by inserting the value of R in Eq. 6.17b, we obtain the number of stages in the Craig model corresponding to a Gaussian distribution of the solute in a column, given by
As already pointed out in the case of the Martin and Synge model, it is important to understand that the distribution given by Eq. 6.11 (like the distributions given by Eqs. 6.3 and 6.5) is not an elution profile but rather the spatial profile of the solute concentration along the column, after a given number of transfers, r, has been performed, and for the stages with a rank I between 0 and r — 1. The elution profile is determined by the amount of solute in the mobile phase that leaves the last or Ncth stage (rank Nc — 1) during each operation. This amount is equal to R PI=NC-IS- Unlike the case of Eq. 6.11, here I is constant and equal to Nc — 1, since we consider the last stage, and r varies from Nc — 1 (corresponding to the holdup volume) to infinity. The elution profile is given by the distribution f(r) = R
J{)
( N l ) ! ( ( N l )
6.1.4 Comparison of the Two Plate Models The difference between the two plate models has been discussed by Klinkenberg and Sjenitzer [16]. In both models, we have a series of identical mixers, each containing the same amount of mobile and stationary phases. In the Martin and Synge model, by contrast with the Craig model, the mobile phase flows continuously. The result is that we have two different distributions, but both of them tends toward the same limit (Eq. 6.5) when the number of stages increases indefinitely. Comparison of Eqs. 6.6b and 6.18 shows that the peak dispersion in the Craig model is less than in the Martin-Synge model if we use the same number of stages. In order for the two results to become equal, the number of stages used in the Craig model should be smaller than in the Martin-Synge model. We should have NC = ^
N
(6.20)
This relationship is important when comparing the performance of these models. In order to compare the band profiles predicted by both models and the Gaussian distribution, we use very small plate numbers, which magnifies the differences. With high plate numbers, the differences would be insignificant. In Figures 6.1a and 6.1b, we compare the profiles of the three distributions in the cases in which the plate numbers (Martin and Synge definition) are 25 and 100, respectively. The Gaussian profile (solid line) is derived from Eq. 6.5. The solution of the Craig model (dashed line) is calculated from Eq. 6.19. For the Craig model, the number of stages has been calculated according to Eq. 6.20. As can be seen in these figures, the difference between the three profiles is practically negligible
6.1 The Plate Models
289
Figure 6.1 Comparison of the chromatograms given by the two plate models of chromatography and a Gaussian peak. Solid line: Gaussian profile (Eq. 6.5). Dotted line: Continuous (Martin and Synge) plate model (plot of f{t), Eq. 6.9, vs. t/tR). Dashed line: Discrete or Craig plate model (Eq. 6.19). (a) Plate number: N = 25. (b) Plate number: N = 100. Reprinted by permission ofKluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 1), with kind permission of Springer Science and Business Media.
when the column has 100 theoretical plates. This difference will be insignificant for columns having an efficiency in excess of 100 theoretical plates. Even for 25 stages (Figure 6.1a), the difference is quite small. In this last case, the experimental profiles would be modeled as well with the profiles calculated from the Martin and Synge model and with those calculated from the Craig model (Figure 6.1a). Although the plate models are hypothetical models of band retention and band broadening, these models have the advantage of introducing a simple but powerful concept, the column HETP. Measuring the HETP of different columns or of a column under different experimental conditions permits an easy assessment of the parameters which influence the separation power of a column. This method has been put to considerable use to optimize the experimental conditions in analytical chromatography. However, the plate theory does not supply any a priori information on the relationships between column efficiency and the various parameters that control it or may influence it. These fruitful relationships originate from the conclusions of a more powerful method, closer to the fundamental principles of physics and chemistry, which we discuss now. Because all theories of linear chromatography, plate theory, statistical theory, and mass balance solution give the same practical result, a Gaussian profile, their results have been pooled into a set of conventions whose different origins have now been forgotten.
290
Linear Chromatography
6.2 The Solution of the Mass Balance Equation The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. The ideal model of chromatography, which has great importance in nonlinear chromatography, has little interest in linear chromatography. Along an infinitely efficient column, with a linear isotherm, the injection profile travels unaltered and the elution profile is the same as the injection profile. We also note here that, because of the profound difference in the formulation of the two models, the solutions of the mass balance equation of chromatography for the ideal, nonlinear model and the nonideal, linear model rely on entirely different mathematical techniques.
6.2.1 The Equilibrium-Dispersive Model In the equilibrium-dispersive model, we assume that the mobile and the stationary phases are constantly in equilibrium. We recognize, however, that band dispersion takes place in the column through axial dispersion and nonequilibrium effects (e.g., mass transfer resistances, finite kinetics of adsorption-desorption). We assume that their contributions can be lumped together in an apparent dispersion coefficient. This coefficient is related to the experimental parameters by
where H is the column height equivalent to a theoretical plate (HETP), N is the column plate number or column efficiency, u — L/to is the mobile phase linear velocity with t$ the hold-up time, L is the column length, and Da is the apparent axial dispersion coefficient. From Eq. 6.21, we derive that N = Pez/2, where Pez is the axial Peclet number (uL/Da). With these assumptions, the differential mass balance of the solute in a slice of column (Chapter 2, Eq. 2.2) is given by
where F — Vs/Vm — (1 — e)/e is the phase ratio, e is the total porosity or void volume fraction of the column packing, C is the solute concentration in the mobile phase, and q is the solute concentration in the stationary phase in equilibrium with
6.2 The Solution of the Mass Balance Equation
291
the mobile phase at concentration C. Since we assume that the isotherm is linear, q = aC, k'o = Fa, and Eq. 6.22 can be rewritten as (6 23)
-
There is a simple dimensionless form of this equation 3Q dCd=J_&Cd dtd ^ dzd 2N dzj
K
'
The dimensionless distance, time, and concentration, zd, td, and Q , respectively, are given by zd
=
*d = =
j
(6.25a)
lt^\-us
a, Ap
=c
=^
^
= 1= c£SL0
n
(6.25b)
+g n
where tg is the retention time of the band, Ap = n/Fv is the area of the injected pulse (which should be conserved during the band migration), Fv is the volume flow rate of the mobile phase, n is the amount of component (mole) injected in the column, and S is the column cross-sectional area. As always with differential equations, the solution of Eq. 6.24 depends on the choice of the initial and boundary conditions. The initial condition in linear chromatography is almost always a column empty of sample, with the two phases in equilibrium. The problem of the selection of appropriate boundary conditions for the solution of a partial differential equation is mathematically subtle and full of pitfalls. Boundary conditions which, for a chromatographer, seem to describe nearly identical experiments may result in different solutions (e.g., the next three sub-sections). We give here an example of the problems found and show how their solution is important for a proper understanding of the chromatographic process. 6.2.1.1
Open-Open Boundary Condition
A closed-form solution of Eq. 6.24 has been derived by Lapidus and Amundson [3], Levenspiel and Smith [17], Carberry and Bretton [18], Reilley el al. [19], and Wicke [20]. All these authors used an "open-open" boundary condition, i.e., conditions assuming that the column stretches to infinity in both directions (z —• — oo, dC/dz = 0; z —• oo, dC/dz = 0), and that a Dirac <S(z) pulse of solute is injected at z = 0. With these boundary conditions, the solution of Eq. 6.24 is given by
J^%(Zt)2
(6.26)
292
Linear Chromatography
Like Eq. 6.5, Eq. 6.26 gives the concentration profile of the solute along the column. In this equation the time, td, is a constant parameter, and the position, z<j, is the variable. The solution is in the form of a Gaussian profile. On the other hand, the elution profile, which is given by writing zd = 1 in Eq. 6.26 and taking the parameter td as the variable, is not a Gaussian distribution. It is written: (6.27) We see in Eq. 6.27 that the dimensionless plot of Q versus td depends only on the plate number, N. The first and second moments (see Section 6.4.1) of the profile are given [17] by id =
1+ ^
(6.28a)
oj
1 ±
(6.28b)
=
+
? J-
Equation 6.27 is not a Gaussian profile. However, if we use the approximation that N/tj ~ N = 1/Vf (see Eq. 6.28ab), we obtain a Gaussian profile: Q =
i=e
^
(6-29)
V2 Unless cj is very small (i.e., unless N is sufficiently large), the elution profile is not Gaussian (we show later that the profile is very nearly Gaussian for N = 100). When the boundary condition is a Dirac S(t) function at z = 0 [which is different from the Dirac S(z) pulse at z = 0 of the open-open boundary condition], the solution obtained is different. It can be derived using the inverse Laplace transform [21,22]:
The first two moments of this distribution are td
=
1
(6.31a)
°d
=
Jj
( 6 ' 31b )
Comparing Eqs. 6.27 and 6.30 shows that the ratio of the two solutions is 1/tj: Crf (pulse boundary condition) = —(open
open boundary condition) (6.32)
In Figures 6.2a and 6.2b, we compare the profile of a Gaussian distribution and the exact solutions of the equilibrium-dispersive model with the two different boundary conditions, for columns having 100 (Figure 6.2a) and 1000 (Figure 6.2b) theoretical plates, respectively. In each figure, the dotted lines are the band profiles
6.2 The Solution of the Mass Balance Equation
293
Figure 6.2 Comparison of the chxomatogram given by the equilibrium-dispersive model of chromatography with a Gaussian profile. Dimensionless plot of Q = CtR/Ap versus frf = t/tR- Solid line: Gaussian profile with N theoretical plates. Dotted line: equilibriumdispersive model with an "open-open" boundary condition and N disp = Lu/(2Da). Dashed line: equilibrium-dispersive model with a S(t) pulse boundary condition and Ndisp = Lu/(2Da). (a) N = N disp = 100 theoretical plates, (b) N = Ndisp = 1000 theoretical plates. Reprinted by permission ofKluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 2), with kind permission of Springer Science and Business Media.
calculated as the solution of the equilibrium-dispersive model with the openopen boundary condition (Eq. 6.27), the dashed line is the solution of the same model with the 5 pulse boundary condition (Eq. 6.30), and the solid line is the Gaussian approximation (Eq. 6.29). We see that there is a slight difference between the two band profiles for N = 100, especially long before or after the retention time of the band maximum, tR. For N = 1000, the difference between the two solutions is entirely negligible. 6.2.1.2 Open-Closed Boundary Condition Analytical solutions of Eq. 6.24 can be obtained with another boundary condition. We now assume a semi-infinite column that stretches from z = 0 to infinity in the positive direction. This is the "closed-open" boundary condition, which
Linear Chromatography
294 corresponds to At
z=0
eS for
and
C = 0
for
C —> 0
when
0 < t < tp
t>u
(6.33a)
This boundary condition is equivalent to the Danckwerts [23] boundary condition:
UCQ =
uC-Da
0 =
uC-Da
and
C —> 0
ac dC
for 0 < t < tp
(6.33b)
for
p
(6.33c)
z —> oo
(6.33d)
when
t > ih
These equations can be written in dimensionless form =
Q-
Q->o
J_dCj 2N dzd
(6.33e) •*o
oo
(6.33f)
Villermaux and Van Swaay [24] have derived the solution of Eq. 6.24 with the boundary condition described in Eqs. 6.33a7 using the inverse Laplace transform. The solution is (6.34) where erfc is the complementary error function. The first two moments of the band profile are tA =
1+
XT^
(6.35a)
(6.35b) N 4N2 From the mathematical viewpoint, this solution is quite different, from the solutions obtained with the open-open boundary conditions, although in the latter case we do not make use of the part of the column for which z^ < 0. 6.2.1.3 Closed-Closed Boundary Condition For a finite column of length L, the boundaries are closed for dispersion at both the column inlet and its outlet, hi dimensionless form, this "closed-closed" boundary
6.2 The Solution of the Mass Balance Equation
295
condition is written At z = 0,
6{td)
=
Cd-
1 3Q 2Ndzd (6.36)
At
z = L,
^
=
0
The solution of Eq. 6.24 for these boundary conditions has been derived by Otaka and Kunigita [25]:
where 8n is the nth root of the transcendental equation:
coi5
« =^-wn
(6 37b)
-
The first two moments of the concentration distribution given by Eq. 6.37a are h
= 1
(6.38)
Numerical calculations are easily made to compare in a real case the four different distributions derived as solutions of Eq. 6.24 with as many different boundary conditions. Then we observe that the peak profiles obtained are very close, unless the plate number is small, much smaller than the values encountered in practical applications. These differences can be completely neglected in practice.
6.2.2 Solution of the Lumped Kinetic Model In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermodynamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written
where Cs is the concentration of the solute in the stationary phase and Di accounts for the axial dispersion. In the lumped kinetic model, the contributions of all the mechanisms involved in band broadening, due to their relatively slow kinetics, are lumped in a single rate coefficient.
296
Linear Chromatography
Depending on the main cause of sluggishness in reaching equilibrium in the column, we can distinguish several kinetic models. If the kinetics of the retention mechanism (e.g., the kinetics of adsorption-desorption) is slower than the other steps of the chromatographic process, we use the reaction-dispersive model. If the slowest step in the chromatographic process is the mass transfer kinetics, we have the transport-dispersive model.
6.2.2.1 The Reaction-Dispersive Model In this model we assume that the contributions of the mass transfer resistances to band broadening are negligible, but that the kinetics of adsorption-desorption is slow. So the behavior of the chromatographic system is described by the mass balance Eq. 6.40. If we assume now that the kinetics of adsorption-desorption is of first order, we have the kinetic equation: —^ = kaCat
kdCs
(6.41)
where ka and k^ are the rate constants for adsorption and desorption, respectively. Lapidus and Amundson [3] have derived the analytical solution of the reactiondispersive model of chromatography, where the reaction kinetics is linear (Eqs. 6.40 and 6.41), for any initial condition, any injection profile, and for a column of infinite length. This solution is too complicated to be reproduced here. It is more general than the one derived by Giddings and Eyring [5] (see Section 6.5) from a completely different approach (equation of motion versus probability distribution). In the solution of Lapidus and Amundson, the contribution of axial dispersion is also included, besides that of the adsorption-desorption kinetics, while axial dispersion is ignored in the treatment of Giddings and Eyring [5]. The simplified Lapidus-Amundson solution corresponding to a Dirac pulse injection into an empty column was used by Van Deemter et dl. as the starting point of their landmark work [4]. It is discussed later (Eqs. 6.47 and 6.48). 6.2.2.2 The Transport-Dispersive Model In this second lumped kinetic model and in contrast to the first one, we assume that the kinetics of adsorption-desorption is infinitely fast but that the mass transfer kinetics is not. More specifically, the mass transfer kinetics of the solute to the surface of the adsorbent is given by either the liquid film linear driving force model or the solid film linear driving force model. In the former case, instead of Eq. 6.41, we have for the kinetic equation: d
-^=k'm(C-n
(6.42)
where C* is the solute concentration in the mobile phase which is in equilibrium with the solid phase concentration Cs. Thus, C* = Cs /a where a is the slope of the linear isotherm (a = k'0/F, F, phase ratio), and k'm is the apparent mass transfer coefficient.
6.2 The Solution of the Mass Balance Equation
297
If we instead use the solid film linear driving force model of mass transfer kinetics, we have ^•=km(q-Cs)
(6.43)
where q is the solute concentration in the stationary phase which is in equilibrium with the mobile phase concentration C. Thus, q = aC, and km is the apparent mass transfer coefficient. In linear chromatography, the solid film driving force model (Eq. 6.43), and the liquid film driving force model (Eq. 6.42) are special cases of the first-order linear kinetics (Eq. 6.41), which can be rewritten as )
(6.44)
Equation 6.44 is equivalent to Eq. 6.43 provided that kd = km and ka = akm. So, in linear chromatography, the solid film driving force model is a special case of the linear kinetic model, with kd = km and ka = akm. On the other hand, Eq. 6.42 can be rewritten as
Thus, in linear chromatography the liquid film driving force model (Eq. 6.42) is equivalent to the solid film driving force model (Eq. 6.43) with k'm = akm and is a special case of the linear kinetic model, with kd = k'm/a and ka = k'm. The solution of the kinetic model derived by Lapidus and Amundson [3], which was obtained for a first-order linear kinetic model, is directly applicable to both the solid film and the liquid film driving force models. We now discuss these equivalent solutions. The solution of the solid film driving model is the same as the analytical solution of the linear kinetic model derived by Lapidus and Amundson [3], in which we replace ka and kj by akm and km, respectively. The solution of the liquid film driving force model is the same as the analytical solution of the linear kinetic model derived by Lapidus and Amundson [3], in which we replace ka and kd by k'm and k'm/a, respectively. This solution is valid for any initial and boundary conditions. As an example of the experimental validity of the solution derived by Lapidus and Amundson, Figure 6.3 shows the breakthrough curve measured by Rixey and King [26] (symbols) for succinic acid on pre-wet Porapak Q (Waters-Millipore). The experimental results are in excellent agreement with the analytical solution (solid line) derived by Lapidus and Amundson [3]. The Peclet number and the external mass transfer coefficient were estimated using conventional relationships taken from the literature [26]. The analytical solution of Lapidus and Amundson [3] for an impulse injection, using the solid film driving force model is written [4]: = C0
(6.46)
Linear Chromatography
298 1 0,9 -
f
o.e 0.7 -
7m -VZ/D
/ /
L
O.S -
y
o.s -
A
-
Exp*rlm*ntil Data
4
Eq. 1:
o / 0.4 -
/
0.3 -
J
o.a 0.1 Q _
110
P« - 970
and N - 3300
fl 4 - - m - 0 " 1 ^ 130
ISO
17
Volum* of Effltjvnt. ml
Figure 6.3 Breakthrough curve of succinic acid on pre-wet Porapak Q. Flow rate, 0.25 mL/min; column diameter, 1.5 cm; particle size, 80-100 (US sieve series); bed volume, 80 mL; interstitial volume, 35 mL; L = 45 cm; DL = 0.00027 cm 2 /sec; kf = 0.0020 sec" 1 . W. Rixey and C.J. King, AIChE ]., 35 (1989) 69 (Fig. 2). Reproduced by permission of the
American Institute of Chemical Engineers. ©1989 AIChE. All rights reserved.
with (6.47) Equations 6.46 and 6.47 supply the concentration profile along the column (t constant, z variable). As shown previously in the case of the plate models, the elution profile can be obtained by letting z — L, which, after some rearrangements, can be written in dimensionless form as 1.5
(6.48) with (6.49a) J
\
In these equations, Jj (s) is Bessel's modified integral of the first kind, order 1, and N(jiSp and Nm are the numbers of dispersion units and of mass transfer units,
6.2 The Solution of the Mass Balance Equation
299
respectively. The other parameters are dimensionless quantities: h,d =
=
Y
(6.49b)
t> t'oj
=
td
=
t'd
=
(6.49c)
k t
(6.49d)
f tR
(6.49e)
The number of dispersion units or number of mixing stages is given by _ Pez _ Lu disp - - y - 2Dl -
L2 D 2 Lt0
(6.50)
where Pez = uL/Di is the column Peclet number. The number of mass transfer units is given by kmk'L
.
(6.51)
6.2.2.3 Origin of the Van Deemter Equation Using the results of Lapidus and Amundson [3], Van Deemter et al. [4] demonstrated that a simplification of considerable importance can be made to the solution derived by these authors if we assume that the mass transfer kinetics is not very slow, which is almost always the case in analytical or preparative applications of chromatography. Then, Eqs. 6.46 and 6.47 can be reduced to a Gaussian profile: C
A,
=
: exp
: exp
(t-TTF)' + k'Q)z-ut)2
(6.52)
with = 2
DLz 2k>0z
+ k'0)2kmu
(6.53) (6.54)
From the concentration profile along the column, given by Eq. 6.52, we can derive the corresponding elution profile by inserting z = L. Using Eqs. 6.50 and 6.51, we
300
Linear Chromatography
can write the elution profile in a dimensionless form: „
CtR
1
exp
Ar,
(6.55)
Comparing Eqs. 6.29 (classical Gaussian profile) and 6.55 shows that ^p
N disp
(6-56)
'w
or, in terms of the column HETP 2
According to Glueckauf [27,28], the lumped mass transfer coefficient, km, is related to the film mass transfer coefficient and to the pore diffusion by the equation: 1 k'okm
" +' J^ 6QFD
6Fkf
p
(&S7b)
where kt is the mass transfer coefficient across the stagnant liquid film that surrounds the particles, Dp is the pore diffusion coefficient, F is the phase ratio, and dp is the diameter of the particle.
6.2.3 From the Lumped Kinetic Model back to the EquilibriumDispersive Model The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deentter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibriumdispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k' = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. In order to give a quantitative assessment of the assumption we have made, that the rates of the kinetic phenomena involved in the achievement of phase equilibrium in the column are not very slow, we compare the band profiles in Figure 6.4, obtained as exact solutions of Eqs. 6.48 and 6.49a (dotted line), and a Gaussian profile (Eqs. 6.29 or 6.55). For the calculation of the exact solution of Eqs. 6.48 and
6.3 The General Rate Model of Chromatography
301
Figure 6.4 Comparison of the chromatogram given by the Lapidus and Amundson model of chromatography with a Gaussian profile. Dimensionless plot of Q versus tj. Solid line: Gaussian profile with N theoretical plates. Dotted line: Lapidus and Amundson model with Nap plates, (a) N ap = N = 25. (b) N ap = N = 100. Reprinted by permission ofKluwer Academic Publishing, from S. Golshan-Shimzi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 3), with kind permission of Springer Science and Business Media.
6.49a, we have selected the following values: N^p = 200 and [(1 + k'o) /k'0]2Nm = 400. According to Eq. 6.51, iVap is then equal to 100. Thus, for the Gaussian profile a value of N = 100 was also used. As can be seen in Figure 6.4b, there is good agreement between the two band profiles, although the column efficiency is only 100 theoretical plates. The difference between the exact solution of the lumped kinetic model and a Gaussian profile becomes rapidly insignificant for higher efficiencies. Even in Figure 6.4a, in the case N = 25, there is a rather small difference. In practice, as experimental signals are always blurred by some amount of noise and drift, it would be difficult to derive with any accuracy a rate constant from a parameter related to the asymmetry of band profiles corresponding to an efficiency higher than 25 theoretical plates.
6.3 The General Rate Model of Chromatography The chromatographic process involves an intricate combination of complex phenomena of hydrodynamic, thermodynamic, and kinetic origins, which often interact. In all the models we have considered until now, we have made considerable simplifications by focusing on the phenomena we thought were the most important and neglecting the other ones. Unfortunately, in the case of the band broadening mechanisms, many steps take place during the migration of the solute
302
Linear Chromatography
molecules along the column and it is often impossible to identify a single step as the rate-controlling factor. The study of the lumped kinetic models shows that, as long as the equilibration kinetics is not very slow and the column efficiency exceeds 25 theoretical plates (a condition that is satisfied in all the cases of practical importance), the band profile is a Gaussian distribution. We can thus identify all independent sources of band broadening, calculate their individual contributions to the variance of the Gaussian distribution, and relate the column HETP to the sum of these variances. The method is simple and efficient. It has been used successfully for over 40 years [29]. We may want a more rigorous approach. The general rate model attempts to consider simultaneously all the possible contributions to the mass transfer kinetics and to deal with them at the basic stage of the model by including their contributions in the system of partial differential equations that states mass conservation and transport. The first problem is to make a complete census of all the contributions of significance. The general rate models usually consider the axial dispersion as defined above (i.e., as the sum of axial and eddy dispersion), the external film mass transfer resistance, the intraparticle diffusion (the sum of the contributions of pore and surface diffusion), and the rate of adsorption-desorption. Because in this model we consider separately what is taking place inside and outside the particle, we need to write two mass balance equations for the solute, one for the mobile phase flowing between the particles, the other one for the stagnant mobile phase inside the particles. The mass balance equation in the mobile phase is written
where u is the linear velocity (average interstitial velocity, L/to), F = (1 — e)/e is the phase ratio (e, total porosity of the column), and D^ is the axial dispersion coefficient, which is the sum of the molecular and the eddy diffusion coefficients. q is the value of the stationary phase concentration, q, averaged over the entire particle. Calculated for a spherical particle, q is
The term dq/dt in Eq. 6.58 is the rate of adsorption averaged over the particle. For a spherical particle it is given by dq/dt = 3 / (RpMp) where Mp is the mass flux of solute from the bulk solution to the external surface of the particle. The boundary condition at the outer surface of the particle is given by MF = D p ^ | r = R p = kf(C-Cp\r=Rp) e kf is the external mass transfer coefficient and Cp is solute within the pores inside the particle, a function of r.
(6.60)
6.3 The General Rate Model of Chromatography
303
The differential mass balance of the solute inside the pores of an adsorbent particle is given by another partial differential equation: ( +
2dCp\ )
(661)
where £p is the internal porosity of the particle, Dp is the diffusion coefficient of the solute in the particle pores, Cp is the concentration of the solute inside the pores, and Cs is the concentration of the solute adsorbed by the stationary phase. Furthermore, due to the symmetry condition, we have: ^|r=o = O
(6.62)
The concentrations of the solute in the stationary phase, CS/ and in the mobile phase, Cp, are related. If we assume that the kinetics of adsorption-desorption is infinitely fast, these concentrations are related through the adsorption isotherm equation (here, linear), Cs = KaCp. If the kinetics of adsorption-desorption is too slow, then they are related through the kinetic equation of the adsorptiondesorption. Assuming first-order slow adsorption-desorption kinetics, we have ^
- C*p) = kads (cp - | )
(6.63)
where kads and Ka are the adsorption rate constant and the adsorption equilibrium constant, respectively. Finally, for a pulse injection, the initial and boundary conditions are: C(z,O) Cp{r, z,0)
= =
(6.64a)
0 0
C(O,t) = Co for C(O,t) = 0 for
(6.64b) 0 < t
t > tp
<tp
(6.64c) (6.64d)
A closed-form analytical solution of this system of partial differential equations and relations (Eqs. 6.58 to 6.64a) is impossible to derive in the time domain. This is due to the extreme complexity of the general rate model, which accounts for the axial dispersion, the film mass transfer resistance, the pore diffusion and a first-order, slow kinetics of adsorption-desorption. However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis.
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6.3.1 Inverse Laplace Transform of the Solution The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected (i.e., Di = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast (i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen's model is equivalent to Carta's [34]. Pellet [35] and Rasmuson and Neretnieks [36] extended the solution of Rosen by including axial dispersion, but still assuming that the kinetics of adsorptiondesorption is infinitely fast. Later, Rasmuson [37] extended the earlier solution and calculated the profile of a breakthrough curve (step boundary condition, or frontal analysis) in the framework of the general rate model (Eqs. 6.58 to 6.64a), which includes axial dispersion, the film mass transfer resistance, the pore diffusion, and a first-order slow kinetics of adsorption-desorption. All the solutions discussed above are given in the form of infinite integrals. In the case of a pulse injection, a similar analytical solution has not been derived yet, except for Carta's solution of Rosen's model. However, the numerical evaluation of the inverse Laplace transform is possible. It has been calculated in the case of the general rate model (i.e., Eqs. 6.58 to 6.64a) by Lenhoff [38]. The numerical integral derived by Lenhoff is given by: (6.65a)
and m\ — m-2
(6.65b)
disp &
where Pez
uL
(6.65c)
1+b 2 1-b
b =
/1-4
(6.65d) (6.65e)
w
15Fs/Np
= \
1 + Np(w
(6.65f)
5N cothw—1)
(6.65g)
(6.65h)
6.3 The General Rate Model of Chromatography
305
Figure 6.5 Comparison between the chromatogram given by the general rate model of chromatography and a Gaussian profile. Dimensionless plot of Q versus t^. Solid line: Gaussian profile. Dotted line: General rate model solution, (a) N ap = N = 25. (b) N ap = N = 100. Reprinted by permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 5), with kind permission of Springer Science and Business Media. Nf and Np are defined in Eqs. 6.68 and 6.69. D = (fcads<^)/M
is
tb-e Damkohler
number, tQ^ = t/to, and z^ = z/L. In Figure 6.5, we compare a Gaussian profile (Eq. 6.29, solid line) corresponding to a column efficiency equal to 25 (Figure 6.5a) and 100 (Figure 6.5b) theoretical plates with the band profile calculated using the numerical inversion of the Laplace transform of the general rate model (Eqs. 6.65a (dotted line). For the solution of the general rate model, we have chosen the values of Ndisp/ Nf, and Np so that Nap = 25 (Figure 6.5a) or 100 (Figure 6.5b). As shown in these figures, the two band profiles are still very close for a column efficiency of 25 theoretical plates.
6.3.2 Analytical Solution in the Case of Periodic Rectangular Injections If we assume that the axial dispersion is negligible (i.e., DL = 0 in Eq. 6.58), and that the kinetics of adsorption-desorption is infinitely fast, but that the rate of mass transfer is finite, we can replace Eq. 6.63 by Cs = KaCp
(6.66)
These assumptions give a simplified general rate model, which considers only film mass transfer and pore diffusion. The advantage of this simplified model is that it is possible to calculate the inverse Laplace transform of its solution in the Laplace domain, and obtain a time-domain solution provided that we may
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306
assume the injection to be periodic. In this case, Carta [34] has obtained an analytical solution in the form of a rapidly convergent infinite series. The solution in the time domain is derived by applying the residue theorem to the inversion integral. Although the solution is strictly valid only for periodical injections, it can be used as a very good approximation to calculate the response of the column to a single injection by choosing the injection period long enough so that the effluent concentration has reduced to zero at the end of the first cycle. The elution profile, at the end of a column of length L, for an injection pulse of width tp and a period tc can be written as tp
c
Co
(6.67a)
tc + h T
OO
nkt
+ 71
sin k=l
p F
* (2nk(t-to-tp/2) 'cosI — - - Nfbk
where k is an integer, and the coefficients ak and bk are given by
_ (7k " h) [7k " (1 " 5Nf/Np)Ak) + 4 (6 67b)
"
and
h=
(7k ~
[7k-(l-5Nf/Np)Ak}Vk
and where: (6.67d)
rjk = Vk/2r(sinh y/2k/r - sin \/2k/r) Ak = (cosh \Jiklr - cos y/2k/r)
(6.67e)
{l-ep)Ka]
(6.67h)
(6.67f)
Nf is the number of mass transfer units corresponding to diffusion across the film that surrounds the particles: Nf = 1
6FkfL dpu
6Fkft0 dp
(6.68)
and Np is the number of mass transfer units in the pores, which is given by
As shown by Glueckauf [27,28], the effects of the different phenomena contributing to band broadening are additive. Since in this simplified model axial dispersion is neglected and the kinetics of adsorption-desorption is also ignored, the
6.3 The General Rate Model of Chromatography
307
Figure 6.6 Comparison of the chromatogram given by the film mass transfer-pore diffusion model of chromatography with a Gaussian Profile. Dimensionless plot of Qj versus t^. Solid line: Gaussian profile. Dotted line: Carta's solution [34]. (a) N ap = N -25 theoretical plates, (b) Nap = N = 100. Reprinted by permission of Kluwer Academic Publishing, from S. Golshan-SMmzi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 4), with kind permission of Springer Science and Business Media.
apparent number of transfer units can be evaluated through Eqs. 6.57a and 6.57b, and we obtain v
(6.70) ap
In Figure 6.6, we compare the solution derived by Carta [34], calculated from Eqs. 6.67a (dotted line) and a Gaussian profile (Eq. 6.29, solid line) for column efficiencies N = 100 (Figure 6.6b) and N = 25 (Figure 6.6a). To calculate the Carta solution, we have chosen for Figure 6.6b the following values of the expressions: [(fci +1) /h]2Nf = 400 and [(fcx +1) /h]2Np = 400. With these values, and according to Eq. 6.70, JVap becomes equal to 100. Figure 6.6a shows clearly that, although the column efficiency is only 25 plates, there is a very good agreement between the two band profiles. With 100 plates (Figure 6.6b), the difference would be difficult to establish through experimental measurements. This demonstrates that this solution of the general rate model, as well as the Lapidus and Amundson solution, can be approximated very well with a Gaussian band profile for any column efficiency of practical interest.
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6.3.3 Analytical Solution for Chromatography with a Packing Material having Nonuniform Particles Most discussions of the properties of the General Rate Model (GRM) of chromatography and of its solutions assume that the particles of the packing material have all the same dimensions. Actual packing materials, however, are made of particles with significantly different sizes, if not shape. The effects of a finite width of the particle-size distribution on the profiles obtained in linear chromatography were studied using the moment analysis method [39,40]. Carta et ah [41] extended their previous analytical solution for homogenous particles [34] to the case of a bed packed with particles having an arbitrary size distribution when intraparticle diffusion controls the mass transfer kinetics in the chromatographic system. The solution is derived in the Laplace domain. Then, the time domain solution is obtained by applying the residue theorem to the inversion integral [34,41]. Unlike previous treatments made by other authors, this solution does not require the numerical evaluation of infinite integrals and is valid for both pulse (elution chromatography) and step change (frontal analysis) of the inlet sample concentration. This solution also permits the prediction of the cyclic performance of periodic operations. The analytical solution for an arbitrary but discrete distribution of particle size, such as might be obtained from sieve fractions or a particle size analyzer, is given by: M
C
2r
Co
k=l
K
exp ;=1
kt sin
2r
cos
T ~ ~m
2~r
~u H\
V2raA^Rj
(6.71)
where M is the number of sieve fractions considered, Rj the average radius of the particles in the fraction of rank ; and /y the volume fraction of this cut, r? = tp/n and 2nr = tp + t£, with Co and tp the height and time width of the rectangular pulse injection, tg the period of the analyses, a = De/[ep + (1 — ep)K] with De the effective diffusivity, €p the particle void fraction, and K the linear equilibrium constant, f> = 3(1 — e)De, with e the bed void fraction, t is the time, z the distance along the column, u the superficial velocity of the mobile phase, and the parameters 7 ^ , t]^, and A^y are defined by the following equations U;
h,j
=
+ sin
(6.72)
— sin I R
(6.73)
- cos
(6.74)
6.3 The General Rate Model of Chromatography
309
3000
Figure 6.7 Chromatograms (top, breakthrough curves; bottom, elution peaks) calculated for three packing materials differing only by their particle size distrbutions (see Figure 6.8. Curve 1, sample B; curve 2, sample C; curve 3, sample made of identical particles. G. Carta, J. S. Bauer, AIChE ]., 36 (1990) 147 (Fig. 3). Reproduced by permission of the American Institute of Chemical Engineers. ©1990 AIChE. All rights reserved.
Equation 6.71 can be integrated term by term to yield the time average concentration of the sample in the eluent between any two times, t\ and ti, giving the following relationship
c_
4r
Co
k=X
A . \k(t2-h)1 sm I -r-^- sin -^-—V 2r J [ 2r keZ cos
2r
ur
2r
) (6.75)
Figure 6.7 compares the elution band profiles calculated for the packing materials having the size distributions of samples B and C shown in Figure 6.8 and for a hypothetical uniform packing material having identical particles. The broad and skewed distribution (sample C, curve 2) provides a peak height that is 1.6 times higher than the profile computed for homogeneous particles having the same mean radius (0.01 cm). The latter curve is very close to that obtained with the packing material having a symmetrical particle-size distribution (sample B, curve 1). With a very heterogeneous sample (made of 80% of 20 pm particles and 20% of 200 ]im particles), the retention time of the band was approximately 0.8 times that calculated for a packing material made of 20 ^m particles only. The band had a front similar to that of a band eluted on a homogeneous material with 200 }im particles, but a long tail [41].
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310
Sample B
3.3
r
0.2 0.1
JZL
0L
2 4
In,
5 8 10 12 14 16 18
C.4
Figure 6.8 Particle size distributions of the packing materials used to calculate the chromatograms in Figure 6.7. G. Carta, ]. S. Bauer, AIChE /., 36 (1990) 147 (Fig. 1) . Reproduced by permission of the American Institute of Chemical Engineers. ©1990 AIChE. All rights reserved.
Scmple
C.3
0.2
c
|—1
0.1
0
2
5
run n nn n 11
14 17 R, cm *!0"
20
23
26
In conclusion, the particle size distribution may significantly affect the shape of intraparticle diffusion-controlled chromatographic curves, particularly if this distribution is strongly skewed toward the large particle diameters.
6.4 Moment Analysis and Plate Height Equations As mentioned earlier, a closed-form, analytical solution of the inverse Laplace transform has not been derived yet in the case of the general rate model for a pulse injection. However, Kucera [30] and Kubin [31] have shown that the first five moments of the solution can easily be calculated from their solution of the general rate model giving the band profile in the Laplace domain (Eqs. 6.58 to 6.64a).
6.4.1 Moments of a Chromatographic Peak By definition of the moments of a distribution, the nth moment of the band profile at the exit of a chromatographic bed of length z — L is /•OO
Mn = / C(t, z = L)tn dt
(6.76)
JO
The nth absolute or normalized moment is _ Mn F
"
Mo
Jo°° C{t, L) tn dt
(6.77)
f™C(t,L)dt
and the nth central moment is
J?C(t,L)dt
(6.78)
6.4 Moment Analysis and Plate Height Equations
311
The theorem of Van der Laan [42] permits calculation of the moments of the response of the chromatographic column to a pulse injection, from the analytical solution in the Laplace domain
M. = W*Si?%2
(6.79,
Moments are often used in connection with the different formulations and applications of the general rate model because this model can often be solved algebraically in the Laplace domain and, although this solution cannot be inverted into the spatial domain, the moments of this solution can most often be derived as analytical expressions. However, the use of band moments encounters serious problems both on the calculation and the application fronts. The derivation of analytical expressions for the moments of a band in chromatography is tedious. It involves successive differentiations of the Laplace transform solution of the chromatography model used. Several more expedient methods have been proposed to simplify these derivations for axial chromatography [43,44]. A simple and generalized method was described by Lee et al. [45] for the moments in chromatographic elution peaks with any geometric configuration (axial or radial) and any kinetic models. The first moment of the general rate model solution (Eqs. 6.58 to 6.64a) is given by [30,31]: tR = }i1 = r ( ! + fci) + ^ where k\ is given by
(6-8°)
k1=F{ep + {l-ep)Ka)
(6.81)
The second moment is given by o2
= fi2 = Hz-Hi DL
u
\ 2[1 (l ++kl)h? ++ u
( 4
k2 F 1 60D p
(6.82) dp 6kf
F2u(l-ep)2K
Note that the rate coefficient kj used in Eq. 6.82 was defined in Eq. 5.70 and has dimensions LT~^. By contrast, in the lumped kinetic models, the rate coefficient (km in Eq. 6.43 or kt in Eq. 14.3) has dimensions T" 1 . The third, fourth, and fifth moments are given by more complicated expressions and can be found in the literature [30,31]. In practice, only the first and second moments of a band are determined, the first to characterize its retention and calculate the equilibrium constant, the second to characterize and study the band spreading, hence the mass transfer kinetics. Moulijn et al. [46] extended the Kubin-Kucera model by incorporating surface diffusion. They derived the solution of their new model in the Laplace domain and calculated explicit expressions for the first and the second moments of the band. Haynes and Sarma [32] obtained the expressions for the first and second moments for the band solution of a general rate model including macropore
Linear Chromatography
312
^ ^ ~ 11st s t moment moment
4
2nd moment 3rd moment 4th moment
4
r\ /
r\
1 \ \ 3 3
\
11
Y-Axis
/
/
-55
-3
22
A / \
VI
-1
J
f
/
~>»^ 1
\
22
1
3
\
33
5
/
deviation) Time (standard (st ndard deviation)
Figure 6.9 Distributions of the Contributions to the First Four Moments of a Chromatographic Band. Plots of the contributions of the signal to the moments versus the time.
and micropore diffusion, with external film mass transfer resistance, and eddy diffusion. To avoid some of the errors associated with moment analysis, more sophisticated methods such as those using the Fourier transform have been developed [47,48]. In spite of great expectations in the 1970s, the higher moments have never been of significant use in practice because of the large errors made in their derivation from experimental data. The errors made in the numerical integration increase with increasing order of the moment and become unacceptably large for the moments of order higher than two. This lack of accuracy results from the excessively large influence of the noise and the baseline drift (i.e. low frequency noise) on the determination of the integration boundaries. On the theoretical front, the reason is illustrated in Figure 6.9. This figure shows a Gaussian band profile and the distributions of the contributions of the recorded signal to the different moments. The contributions to the high-order moments are important at large distances from the center of the band (at several standard deviations from the center of the Gaussian profile), when the signal itself is very small, hence the signal to noise ratio is low. This explains why large errors are made, because the relative error made on the small values of the signal is large, these errors are amplified in the calculation of the high-order moments and there is an important uncertainty on the estimate of the time when the integration of the moment should be stopped. On the practical front, this phenomenon is illustrated in Figure 6.10. A slightly tailing peak was obtained for the injection of an unretained compound into a chromatograph in which the column had been replaced by a zero-volume connector. A base-line correction has been made and the band profile is well accounted for by a peak shape model obtained as the convolution of Eq. 6.27 and the exponential decay
6.4 Moment Analysis and Plate Height Equations
313
2 0th moment
1st moment
10
t C(t)
C(t)
15
o
11 -
5
0
0 00
0.1
0.2 time (min)
0.3
00
2nd central moment
2 -
0.1
0.2 time (min)
0.3
10 10
104 (t-μ1)3C(t)
10 (t-μ1) C(t)
r.
ft
2
o
5 -
i
3
zL
"o
00
•
0 0
1 A 0.1 0.1
CVJ
0
-2 0.2 0.2 time (min)
0.3 0.3
0
Yf 0.1
3rd central moment
0.2 time (min)
0.3
Figure 6.10 Comparison of the theoretical and experimental distributions alog the time axis of the contributions to the area and the first three moments of a chromatographic band. Plots of the contributions of the signal to the moments versus the time.
function (Eq. 6.137). The solid lines in the four Figures 6.10 show the distributions of the contributions to the peak area (zeroth moment) and to the first three higher moments, as calculated from the best fit of the exponentially convoluted gaussian to the experimental data. The symbols correspond to the integration of the signal implementing the definition of these moments. The comparison between the two sets of data shows an excellent agreement between the line and the symbols for the moments of zeroth and first order (Figures 6.10, top). Already for the second moment, however, the contributions of the high retention time data points do not trend rapidly toward 0 and the calculation of the moment diverge. For the third moment, the result is still worse. This figure illustrates clearly why the use of the band moments to derive fundamental information on the behavior of a chromatographic column, although most attractive on a theoretical basis, has failed completely.
6.4.2 Plate Height Equation in the General Rate Model From the moment analysis of the solution of the general rate model in the Laplace domain one can obtain the expression for the column HETP. By definition of the
Linear Chromatography
314
height equivalent to a theoretical plate, we have (6.83) If the width of the injection pulse is negligible, we obtain from Eqs. 6.81 and 6.82 Udr,
60FDp
6Fkf
kp
\2
u Fkads
(6.84)
with (6.85)
kp = Equation 6.84 is equivalent to 1 _ Nap
1
1
(6.86)
Ndisp
where N^igp is given by Eq. 6.50, Nf and Np by Eqs. 6.68 and 6.69, and Nrea = (FA:a(jsL)/M. Equation 6.86 is used for the calculation of Nap when the profile predicted by a particular model is compared to a Gaussian profile. This equation is general. It includes the numbers of dispersion units, film mass transfer units, pore mass transfer units, and adsorption-desorption units. Equations 6.56 and 6.70 are particular cases of this general equation. Equation 6.84 shows that the column HETP is the sum of the independent contributions of the axial dispersion (molecular diffusion and eddy diffusion), the film mass transfer resistance, the pore diffusion, and slow kinetics of adsorptiondesorption. By comparing Eqs. 6.84 and 6.57a, we obtain: 6kf
(6.87)
60Dp
Equation 6.87 shows that the lumped mass transfer coefficient in the solid film driving force model is related to the film mass transfer coefficient, the intraparticle diffusion, and the adsorption rate constant. This equation is a more complete form of Eq. 6.57b. It reduces to that equation when the adsorption-desorption kinetics is infinitely fast. We can write Eq. 6.84 under a dimensionless form: 1+Jti
vDn
v
1
60FDn
6FSh
F~D
(6.88)
where h = H/dp is the reduced plate height, v = Pep = (udp)/Dm is the reduced velocity, Sh= (kfdp)/Dm is the Sherwood number and D= (kadsdp)/u is the Damkohler number. Equations 6.84 and 6.88 are the general plate height and reduced plate height equations, respectively. All the plate height equations that have been reported
6.4 Moment Analysis and Plate Height Equations
315
and discussed in the literature can be derived from these two equations, which in turn stem from the general rate model of chromatography, the most sophisticated and comprehensive model of chromatography so far. The differences between the various expressions reported in the literature for the dependence of the column HETP on the experimental parameters result from the use of different expressions for the axial dispersion Di and the mass transfer coefficient, kf. For example, Van Deemter et ah [4] have assumed that the axial dispersion coefficient is given by DL = yDm + Audp
(6.89)
The first term of Eq. 6.89 accounts for molecular diffusion and the second term for eddy diffusion. Inserting Eq. 6.89 in Eq. 6.84, we obtain: H = A + - + Cu
(6.90a)
h = a + - + cv
(6.90b)
The first of these two equations is the classical Van Deemter equation and the second its dimensionless form. Giddings [29], Huber [49], and Horvath and Lin [50] have used alternate models to account for the relationships between the rate of variation of the solute concentration in the stationary phase, its mobile phase concentration, and the various parameters characterizing the chromatographic system used. This explains the differences in the plate height equations they derived, as we see in the next section.
6.4.3 Coupling of Molecular and Eddy Diffusion Giddings [29] has argued that the Van Deemter equation (Eq. 6.90a) is too simplistic, because it ignores the coupling that exists between the flow velocity and the radial diffusion in the void space of the packing, around the particles. The engine for this coupling is the eddy diffusion and the turbulence it causes ("rugged flow" [51]). Since both the eddy diffusion and the radial diffusion are simultaneously responsible for the transfer of molecules between different flow paths of unequal velocity, the parameter A in the Van Deemter equation is in effect a function of the velocity. Giddings proposed to replace the term A by a term accounting for this coupling, «/(l + bu ) , so the axial dispersion term becomes D
(6 91a)
^ = TT^i
-
Huber [49] proposed to replace Eq. 6.89 by DL = 7Dm +
r
\v
(6.91b)
and also to assume that the mass transfer coefficient in the mobile phase {i.e., kf in Eq. 6.84) is velocity dependent, by writing that kf is proportional to w05.
316
Linear Chromatography
In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness S. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for 5, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to v a 3 3 Sh = fc/^ = Dv a 3 3 (6.91c) um Horvath and Lin assumed that 5 is equal to the thickness of the Nernst diffusion layer, Dm/kf, with 6= ^
(6.91d)
and using Eq. 6.91c, we obtain '
&
Introducing this value of the stagnant film thickness, they derived a value of the axial dispersion coefficient: DL
=
Dm +
i + nv-o- 33
(6l91f)
Inserting Eqs. 6.91c and 6.91f in Eq. 6.88 gives the Horvath and Lin equation [50]. Arnold et al. [53] claimed that Eq. 6.91c is valid only for large values of the reduced velocity. According to them, the Horvath and Lin equation is valid only for v > 50. When v < 50, and according to Pfeffer and Happel [52], the Sherwood number is constant and so is the film mass transfer coefficient. Furthermore, since kjr is independent of the reduced velocity at low values of the reduced velocity, according to Eq. 6.91d, the stagnant boundary layer thickness, 5, is nearly constant and independent of the velocity. Under these conditions, Eq. 6.91f would reduce to Eq. 6.89. Thus, at low reduced velocities the Van Deemter equation would be more accurate than the Horvath and Lin equation. However, Nelson and Galloway [54] have shown that, although the Sherwood number becomes constant at low velocities for a single particle, the situation is different and more complex for a densely packed bed. In this case, the Sherwood number becomes proportional to the Reynolds number. If this dependence is substituted into the thickness of the Nernst layer (Eq. 6.91d), we obtain Eq. 6.91a, the Giddings plate height equation. Note, however, that the meaning of the numerical constants in Eq. 6.91a is entirely different in the two models.
6.4 Moment Analysis and Plate Height Equations
317
h 5
4 Knox
3
Figure 6.11 Comparison of the main plate height equations. Dimensionless Van Deemter equation (Eq. 6.90b), Knox equation (Eq. 6.92) and dimensionless Golay equation (Eq. 6.105a). Arbitrary but typical values of the coefficients: a = 1.5; b = 1.5; c = 0.08.
Van Deemter
2
1 Golay
0 0
5
10
15
ν
20
At high reduced velocities, Horvath and Lin [50] showed that their equation reduces to the Knox [55] empirical equation: -
™,0.33
(6.92) Ycv v where a, b, and c axe constant parameters. The Van Deemter and the Knox equations are compared in Figure 6.11. Unless experimental results are very accurate and cover a wide range of reduced velocity, it is difficult to decide on an empirical basis which model accounts better for the data. Although the work of Katz et al. [56] concludes unambiguously that the Van Deemter equation is superior, this work has not been accepted as definitive in view of contradictory evidence brought by a large number of other studies. Experimental results from Unger et al. [57] are reproduced in Figure 6.12. They compare the plots of the reduced column plate height, h, versus the reduced mobile phase velocity, v, obtained for spherical and irregular silica particles with average particle sizes ranging between 3 and llfim. Several conclusions can be drawn from these results. First, there is a definite correlation between h and v, independent of the particle size. Other results show this correlation to be independent of the diffusivity of the solute, at least in a narrow range [56,57]. Second, the coefficients of this correlation depend somewhat on the retention factor of the components considered. Third, there is practically no difference between spherical and irregular particles (note that it is extremely difficult to define the diameter of an irregular particle, even though it can be measured based on the permeability of a column or the average volume of the particle). The pore distribution and the pore connectivity inside the particles are certainly more important parameters than the particle shape, but this concept is still poorly understood [58]. Finally, in the range of reduced velocity of the mobile phase within which efficiency measurements can be carried out with reasonable accuracy, the data could be fitted nearly as well by the Knox or the Van Deemter reduced plate height equations, a conclusion reached independently by other workers [56,59]. Experimental results based on chromatographic measurements are inconclusive. An earlier theoretical work by Arnold et al. [53] summarized reasons for H
Linear Chromatography
318 100 SiO 2
h'= 0.3
o
• a
c
= 6.1 £1.5 jJtn = 38i1A pm
s
10
So"
100
10100
SiO 2
angular
-— I
1 k"= 1.8 •
k'=0.3 o
d p ^ = 10.9 £2.5 pm = 5.S i 1.8 jjm
O
= Z8i1.3 urn
•
•
• *<
i.-
1-
I
1
10
•
r-,
i
V
100
Figure 6.12 Comparison between the reduced column plate height vs. reduced mobile phase velocity plots measured for a retained and a nearly unretained solute on a series of spherical (top) and angular (bottom) silica packings. Reproduced with permissionfromK.K. Unger et al, J. Chromatogr., 149 (1978) 1 (Figs. 2 and 3).
which the Van Deemter equation [4] should be preferred at low values of the reduced velocity (below around 50), i.e., in most of the range of practical importance in liquid chromatography, while either the Knox [55], the Giddings [29] or the old version of the Horvath and Lin [50] equations could be chosen at higher values of the reduced velocity. This conclusion is challenged by more recent results obtained by Tallarek et al. (see Figure 6.14). The axial and radial dispersion coefficients in a chromatographic column were measured by Tallarek et al. as a function of the mobile phase velocity, using pulsed field gradient nuclear magnetic resonance (PFGNMR) [60]. This method allows the determination of the change that occurs in the spatial distribution of a population of ^^H atoms during a short period of time. Using methanol as the mobile phase and n-butylbenzene (k'o = 3.3) as the probe, they also measured the HETP of the same column with the conventional method. The two plots are compared
6.4 Moment Analysis and Plate Height Equations
\=2fh
+ 2U(l+
319
b) *
•a & ^jijpcr B'HO$**^
1
GIDDINGS HUBER ——- UOHVATII
10 red. flow velocity v
Figure 6.13 Comparison of the plots of the reduced axial plate height vs. the reduced flow velocity obtained (a) by PFGNMR and (b) using a conventional chromatographic method. Column packed with 50^m particles of porous C18 silica. The lines shown are the best fits of the experimental data (symbols) to the correlations suggested by Giddings (x = 1), Huber (x = 0.5), and Horvath and Lin (x = 0.33). In either case/ the best fit of the data to the Knox equation coincides with that to the Horvath and Lin correlation. Reproduced with permission from U. Tallarek, E. Bayer, G. Guiochon, ]. Am. Chem. Soc, 120 (1998) 1494 (Fig. 6). ©1998 American Chemical Society.
in Figure 6.13. Although it is difficult to compare directly data pertaining to an unretained and a retained compound, the similarity and difference between the two curves are obvious. The PFGNMR curve is much lower than the HPLC one because the effects of any extra-column volume and of any deviation from constancy of the radial distribution of the mobile phase velocity have no effect on the dispersion coefficient measured with this method. However, the two curves are qualitatively similar. These data were fitted to the following equations 27 v
2A 1 + cav~x
Cv
(6.93)
where A and to are numerical coefficients and x is equal to 0.50 (Huber equation [49]), 0.33 (Horvath equation [50]) or 1 (Giddings equation [29]). In both cases, the data seem to fit much better to the coupling term of Giddings (Eq. 6.91a) than to those of Horvath and Huber. If the same data are plotted in natural instead of logarithmic coordinates (see Figure 6.14), an interesting conclusion emerges. The strong downward curvature of the plot at high velocity is obvious, illustrating the major importance of the coupling between eddy and molecular diffusion and confirming the validity of the work of Giddings [29]. This also demonstrates how incorrect can be the extrapolation of HETP data acquired at too low values of the reduced velocity. Finally, the dotted lines in Figure 6.14 show the variation of each of the three contributions to the reduced HETP in Eq. 6.93 with the reduced velocity. The axial diffusion term becomes negligibly small for v > 10. The eddy dispersion term increases linearly
Linear Chromatography
320
O PFGSE • PFGSTE D APGSTE
b)/
40
60
80
a
)
100
red. flow velocity v
Figure 6.14 Plot of the axial reduced plate height vs. the reduced mobile phase velocity, (a) Comparison of the results measured by three PFGNMR methods (PFGSE, PFGSTE and APGSTE, with 5 = 2 ms and A = 22 ms) and best fit of these experimental data (symbols) to the Giddings equation (solid line). The individual contributions to this equation are represented by the dotted lines, (b) Best fit of a subset of these data limited to v < 15 to the van Deemter model and extrapolation (B = 1.35, A = 0.14, C = 0.11, n = 0). Reproduced with permission from U. Tallarek, E. Bayer, G. Guiochon,}. Am. Chem. Soc, 120 (1998) 1494 (Fig. 8). ©1998 American Chemical Society.
with increasing velocity in that range, then tapers off for 10 < v < 40 and becomes practically constant for v > 50. The mass transfer resistance term is smaller than the eddy dispersion term in the whole practical range of operation of HPLC in either analytical or preparative applications. However, it becomes the largest contribution for v > 80. The curves presented in Figure 6.14 should be compared with those shown by Giddings [29] (Figure 2.10-3, p. 55 in Ref. [29]) in his discussion of the coupling term contribution, in which he contrasts the coupled and the classical plate height equations.
6.4.4 Plate Height in Perfusion Chromatography Frey et ah [61] compared the plate height equations under linear conditions for chromatography with traditional and with perfusion columns. In the latter case, convection takes place in large macropores inside the packing particles. For a column packed with a material having a uniform porosity, in the pores of which there is no convection, the reduced plate height is expressed by the following relationship [29-31] = /Zdisp + frext + him +
(6.94)
where /Zdisp i s the combined contribution to the plate height equation of the deviations from plug flow in the column, of axial diffusion and of dispersion in the extra-column volumes of the instrument, hext is the contribution of the external
6.4 Moment Analysis and Plate Height Equations
321
mass transfer. In its simplest form, h^p can be expressed as: A+^
(6.95)
More precise relationships for /z<jisp were discussed earlier, in the previous section (see Eqs. 6.91a, 6.91b, and 6.91f). A and B in the equation above are characteristic of the packing material and Pe — udp/Dm is the particle Peclet number, with u the interstitial velocity. In the chromatographic literature, the particle Peclet number is frequently named the reduced velocity. Typical values of A and B in a well-packed column are 1.5 and 1.6, respectively [56]. For such a column, used at a moderate to high Peclet number, /zaisp is a small contribution to the overall reduced plate height of the column. The contribution of the external mass transfer resistance, hexi, is given by [61, 62]
with Km = e,. + (1 -
£i)Keq
(6.97)
where ee is the external porosity of the bed, e,- is the intraparticle porosity fraction and Keq is the equilibrium constant. For moderate to high values of the particle Peclet number, hext is also a small contribution to the overall reduced plate height. The contribution of a slow adsorption kinetics to the reduced plate height is given by [29]
f^
ft (6.98,
where K is the desorption rate constant [29]. The contribution of intraparticle diffusion to the reduced plate height is [29-31] /lint = 30e A[e + (1 - e )K4 ]2 ; e e eq hi chromatography, intraparticle diffusion gives the largest contribution, h-my to the reduced plate height. As a result, the major goal in the design of a chromatographic experiment or in the production of a packing material is to reduce it. Equation 6.99 does not apply to packing materials that exhibit a bimodal pore size distribution or in the pores of which convection takes place. Perfusion chromatography refers to any chromatographic system in which the intraparticle convective velocity is different from zero [63], a fact that was first recognized by Guttman and Dimarzio [64] and by Kreveld and Van Den Hoed [65] and was examined in detail later [63, 66,67]. It is now well established that the overall mass transfer resistance in chromatographic columns can be substantially reduced when the fluid phase flows through some of the pores of the adsorbent
322
Linear Chromatography
particles [61,63-67]. Intraparticle fluid flow (i.e., convection) can take place in the particles of a chromatographic system when the macropores of the adsorbent particles have a sufficiently large diameter. This could allow fluid flow through these particles [61,63-67]. Liapis and McCoy [63] have assumed that the bimodal pore structure of perfusive adsorbent particles is made of a macroporous region, in which mass transfer takes place through intraparticle convection and pore diffusion, and a microporous region made of spherical microparticles in which mass transfer takes place through pure diffusion. Frey et ah [61] developed a model for the analysis of mass transfer in spherical particles having a bimodal pore distribution and derived the following expression for h^t in perfusion chromatography. )2
D'
where T is the tortuosity of the macropores, A a correction factor accounting for hindered diffusion and steric effects, De the effective diffusivity in these pores, and Dapp the apparent diffusivity given by (6.101) where Pe is the pore Peclet number and / accounts for the diffusional resistance to mass transfer in the porous regions or subsidiary particles. It is given by the following equation [32,61] -l
(6.102)
where e, x, and u are the volume fraction of the macropores in the particles, the tortuosity of and the fluid velocity in the macropores, and e and r are the volume fraction of the mesopores in the subsidiary particles, and the tortuosity of their pores. De and De are the effective diffusivity in the maropores and in the pores of the subsidiary particles and are given by De
=
^ -
D"e = ^f
(6.103)
(6.104)
where A and A are the hinderance parameters in the macro- and micropores, respectively, which are functions of the relative size of the solute molecules and of the pores. Although the other plate height contributions, /z^isp a n d ^int> may also be affected by intraparticle convection, it is assumed that they are not and that they
6.4 Moment Analysis and Plate Height Equations
323
are still accounted for by Eqs. 6.95 and 6.98. To derive the plate height equation detailed above for perfusion chromatography, Frey et al. [61] made two more important assumptions, in order to simplify the derivation 1. The concentration gradients in the macroporous and in the microporous regions of the particles are both linear. 2. Only the component of the intraparticle velocity that is parallel to the column axis is significant. Grimes et al. [68] has argued that these assumptions are unrealistic and lead to an underestimation of the plate height. The concentration gradient in the macroporous and the microporous regions of the particles are not linear and the components of the fluid velocity in the radial (r) and angular (8) directions of the macroporous regions of the particles must be considered [69]. Grimes et al. [68] determined the first and second moments of the response of a chromatographic column to a pulse injection from the exact Laplace transforms of the solutions of the general rate model for a column packed with porous particles having an unimodal pore-size distribution and for a column packed with porous particles having a bimodal pore-size distribution (with intraparticle convection and pore diffusion in the macropores and with pore diffusion in the microporous region, i.e., the model of perfusion chromatography. Their model has the advantages of 1. Accounting for both linear and nonlinear concentration gradients developing in the pores of the particles 2. Considering the intraparticle, the convective and the diffusive mass transport mechanisms in the pores of the particles as separate mass transport mechanisms, each one characterized by its own individual and proper driving force, rather than as a convective flux augmented of a diffusive flux in the pores of the particles 3. Considering the effects of the radial and the angular components of the velocity vector of intraparticle convective flow in the pores of the spherical particles. The band moments are employed to obtain an explicit expression of the HETP for the above cases [68]. The equation derived for the plate height, H, by Grimes et al. [68] is physically correct. It can account for nonlinear concentration gradients in the macroporous and in the microporous regions of the particles as well as for the intraparticle fluid flow along the radial and the angular directions of the particles. Therefore, the expression for the second moment could be used to characterize band width in nonlinear conditions. Details of different possible chromatographic models, of their solutions in the Laplace domain, and of the (very complex) HETP equations derived are given [68-70]. Unfortunately, so many parameters have been introduced in these equations to characterize the properties of the pore structure of particles that the practical usefulness of this work is doubtful.
324
Linear Chromatography
6.4.5 The Golay Plate Height Equation By integration of the mass balance equation for linear chromatography, Golay [71] derived a plate height equation for open tubular columns in gas-liquid chromatography. The model used in this work is a simplified rate model. The simplifications are due to the very simple structure of an open tubular column for gas-liquid chromatography. The column is assumed to be a straight cylindrical tube with a constant circular cross section. The mobile phase is a noncompressible gas (i.e., a liquid). The stationary phase is a liquid film of constant thickness coating the inner wall of the column. The equation applies also if the column is coiled (provided the coil diameter is much larger than the column inner diameter), and a correction can be applied to take the effect of the gas compressibility into account, if needed. The plate height derived by Golay with this model is 2Dm H
+
l + 6k'0 + llk'02udj k'j fc^ D m + 2 4+ ( 1 + ^ )
where k'o is the retention factor, dp is the column diameter, Dm and D; are the molecular diffusivity of the solute in the mobile (i.e., gas) and the stationary (i.e., liquid) phases, respectively, a is the partition coefficient or ratio of the equilibrium concentrations of the solute in the two phases, and F is the ratio of the crosssectional areas of the liquid (stationary) and gas (mobile) phases. The second part of the last term is often replaced by ud^/k^Di. The equation is now classically written
2D m u
1 + 6 ^ + l l f f udj
k'o
2
24fl +k')2 Di
96fl + fc') D
(6 105b)
-
Its dimensionless form is -A-file1
- I - 1 1 A"'2
96(1 + ^)2
k1
/ df\2
24(l + fc'Q)2 \dv)
n
D
(6.105c)
A typical plot of Eq. 6.105c is shown in Figure 6.11 and compared to the classical Van Deemter and Knox equations. As in liquid chromatography df is of the order of a tenth of dp and Dm is not much larger than D;, the coefficient of the v term in Eq. 6.105c is between 0.01 (k'o = 0) and 0.1 (at large values of k'o). Thus, the coefficient selected for the curve in Figure 6.11 corresponds to a rather poor column. The first term of the RHS of Eqs. 6.105a and 6.105b accounts for static, axial, molecular diffusion. In the case of a straight cylindrical tube, there is no tortuosity nor constriction, 7 = 1. There is no eddy diffusion either, nor any A term in Eqs. 6.105a, by contrast to Eq. 6.90a. This is expected in the case of a cylindrical open tube, where the flow is ideally laminar, with no local eddies. The second term in Eq. 6.105a accounts for the dynamic diffusion related to the nonhomogeneity of the gas phase in the radial direction, because of the Poiseuille radial flow profile,
6.4 Moment Analysis and Plate Height Equations
325
and its coupling with the retention. This term is absent in other plate height equations, although it could be incorporated in the C term of Eq. 6.90a. Originally [71], it was anticipated that this term would be small compared to the last term, the resistance to mass transfer in the stationary phase, or the C term of Eq. 6.90a. However, it has been possible to prepare and use open tubular columns for GC with very thin films of liquid phase. With most modern open tubular columns for gas chromatography, the second term is generally more important than the third one [72]. At the difference of the Knox and Van Deemter equations whose parameters are empirical constants, the Golay equation contains only parameters whose values can be measured or estimated independently of any chromatographic result. This reflects the sheer simplicity of a Poiseuille flow profile in a straight, cylindrical, empty tube compared to the complexity of the flow pattern in a packed column. The Golay equation is in excellent agreement with the experimental results obtained by hundreds of investigators [72,73]. From the origin, it was obvious that this equation should apply to all forms of chromatography, whether the mobile phase is a gas, a liquid, or a dense fluid (also referred to as a supercritical fluid). Its validity in liquid chromatography has been demonstrated experimentally [74]. However, open tubular columns have not yet been used for analytical applications and they are of no interest in preparative applications so it would be useless to elaborate further.
6.4.6 Dispersion and Partitioning in Short Coated Tubes The equations of Golay [71] and Aris [21] can be applied to chromatographic columns which are long after the Taylor [75]-Aris [76] definition, i.e., within which the residence time is long compared to the characteristic time for radial transport. This is the situation found in practically all actual chromatographic systems and certainly in all those used in analytical or preparative practice. When the residence time becomes shorter, this approach becomes questionable for several reasons. For example, the asymptotic state may not have been reached yet, or the peaks may be unsymmetrical. These "short-time" situations may be encountered when trying to apply chromatographic concepts to the study of dispersion in connecting tubes, or in some applications, such as hollow-fiber liquid chromatography. Shankar and Lenhoff [77] have derived a solution in the time domain, using series expansion. This solution can be implemented by numerical computation for the determination of concentration profiles inside a tube coated with a retentive layer, when the fluid flow is laminar. This solution is valid for systems that are either short or long after the Taylor-Aris definition. The results of Shankar and Lehnhoff [77] confirm the validity of the procedures currently used for the interpretation of HPLC data, which are based on moment analysis of the experimental data and on the additivity of the first and second moments of the distributions (see next section). These results also show that this approach is not valid for short systems. Multiple peaks or shoulders on a single band may appear, due to the interaction between transport and retention phenomena. Then the method of moment analysis is inadequate to characterize the
326
Linear Chromatography
solutions. The study of short systems should be done either using the complete time domain solution of Shankar and Lehnhoff or a two-phase pseudocontinuum model. Golay and Atwood [78,79] have performed a similar analysis using a numerical solution of the chromatographic problem in cylindrical coordinates and reached substantially the same conclusions.
6.4.7 Evaluation of Transport Parameters from Chromatographic Peaks The determination of transport parameters such as axial dispersion, intraparticle diffusion and the mass transfer coefficients from the profiles of chromatographic peaks through the use of a transfer function, either in the Laplace or in the frequency domains, is a well known approach [80,81]. Most experimental procedures are based on the acquisition of transient responses and the parameter identification of this response and known or assumed relationships between the transport coefficients and the profiles recorded for the pulse, the step, or the frequency responses of the packed bed [80,82,83]. The transport coefficients can be determined either from the response to a single experiment or, more commonly, by performing a series of successive experiments under different conditions. In a linear chromatographic system, the broadening of the response peak results from the combined effects of axial dispersion and the mass transfer resistances. By making measurements over a range of flow rates, it is possible to separate the contributions of dispersion and of the mass transfer effects and to determine the axial dispersion coefficient and the overall mass transfer coefficient. The method is applicable to the measurement of both macropore and micropore diffusion coefficients, although, if both resistances are significant, it may be difficult to determine their relative importance. Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. In principle, Fourier analysis does not suffer from these shortcomings. By calculating the amplitude ratios and the phase lags for a series of frequencies, an unlimited number of equations can be obtained for the calculation of the transport parameters. However, due to the interactions that take place between the transport parameters at low frequencies and to the amplification of the experimental errors that take place at high frequencies, the advantages of the Fourier analysis on the method of moments are quite limited. Boersma-Klein et al. [85] carried
6.4 Moment Analysis and Plate Height Equations
327
out pulse response experiments in packed beds in order to determine the transport parameters, including the axial dispersion coefficient, the internal diffusion coefficients, and the film mass transfer coefficient. The transfer function of the Kubin-Kucera model, including the surface diffusion term [46] was solved in the Fourier domain. Boersma-Klein et ah applied moment analysis, Fourier analysis, and time domain fitting. They showed that, for Gaussian peaks, Fourier analysis has no advantages compared to moment analysis. For asymmetric peaks, however, the combination of Fourier analysis and time domain fitting is the preferred method. By applying Fourier analysis and fitting in the time domain, the internal tortuosity, the axial dispersion coefficient, and the internal diffusion coefficient could be determined from a single experimental run [85]. With the considerable improvement in the numerical techniques that have taken place in the last 25 years and which allows now for very rapid numerical solutions of chromatographic models, the best approach appears to be the determination of initial estimates of the mass transfer parameters by moment analysis, combined with a final optimization of these estimates of the parameters by direct matching of the experimental response curves with the profiles derived from the general rate model in the time domain. Miyabe used systematically the method of moment for the estimation of the surface diffusion coefficient [86], using the following equations (6.106) = —(Sax + Sf + 5d)
(6.107)
with So = Sax
e+(i_e)(ep+jOpK)
= ^-Sl
(6.108) (6.109)
K Sf = (l-e)^(ep+ppK)2
(6.110)
Rl In these equations, UQ is the superficial velocity of the mobile phase, £ the interparticle void fraction, tp and pp the porosity and the density of the packing material, respectively, K the adsorption equilibrium constant, Di the axial dispersion coefficient, Rp the particle radius, kf the external mass transfer coefficient, and De the intraparticle diffusivity. The terms 6ttX, 6f, and 5$ are the contributions to fi2 °f axial dispersion, the external (or fluid-to-particle) mass transfer, and intraparticle diffusion, respectively. In order to obtain the characteristics of the column the values of y.\ and ji2 must be first corrected for the extra-column band broadening contributions that
328
Linear
Chromatography
are usually obtained b y measuring the first a n d the second m o m e n t of the b a n d of a n unretained c o m p o u n d after substituting a zero-volume connector to the colu m n [86]. The column plate number, N, is equal to }i\ /ft 2 a n d its HETP is written Sd]
(6.112)
A plot of H/ (2UQ) versus 1/MQ is a straight line with a slope equal to Di and an ordinate equal to (Sf + S^)/SQ. The coefficient of external mass transfer is estimated using one of the several correlations available for it (see Chapter 5, subsection 5.2.5, correlation of Wilson and Geankoplis [62], Kataoka et al. [87], or the penetration theory [88]). Correcting for the contribution due to the external mass transfer resistance gives the last term in the plate height equation, S^, hence the intraparticle diffusion coefficient, De. This coefficient is the sum of the contributions of pore and surface diffusion. The pore diffusivity can be derived from known correlations (see Chapter 5, subsection 5.2.6, e.g., those of Mackie and Meares [89], Satterfield [90], or Brenner and Gaydos [91]). Therefore, this procedure gives the surface diffusion coefficient. It has been used by Miyabe to derive estimates of the surface diffusion coefficients for alkyl benzenes and alkyl phenols on columns packed with Qg silica [86,92,93] and on monolithic silica columns [93,94]. They were used by Hong et al. to measure the surface diffusion of rubrene on Symmetry Cis [95]. Finally, a word of caution is in order. Few systematic experimental studies have been made in this field so far and the database acquired is still limited. A small number of adsorbents has been investigated. It is difficult to assess the consistency of the published results and their accuracy. A number of the correlations used in the intermediate calculations are of limited accuracy when applied to porous media. The alternate approach of performing parameter identification with raw calculation power cannot give convincing results. The approach holds great promises but independent determinations of the kinetic parameters is dearly needed.
6.5 The Statistic Approach A "microscopic probabilistic" method can be used for the modeling of linear chromatography. In this case, the probability density function at / and t of a single molecule of solute is derived. The "random walk" approach [29] is the simplest method of that type. It has been used to calculate the profile of the chromatographic band in a simple way, and to study the mechanism of band broadening. In 1955, Giddings and Eyring introduced another, more sophisticated probabilistic approach, the stochastic model, for the description of the molecular migration in chromatography [5]. Their molecular dynamic approach is based on statistical ideas and treats the chromatographic process as a Poisson process. These authors considered the random migration of a single solute molecule along a chromatographic column. They derived an expression for the elution profile,
6.5 The Statistic Approach
329
or residence time distribution of a molecule in the column, assuming random adsorption-desorption processes, with a single type of site on the stationary phase and impulse injection. They ignored the dispersion in the mobile phase and assumed that every molecule spends identical to time in the mobile phase. With these assumptions, they showed that the probability density function is given by the equation: P(t - t0) = ^ I i ( X ) exp[-kdt - Mo]
(6.113)
where ka and k^ are the first-order rate constants for the adsorption and the desorption of the component, respectively, Ij is the Bessel function of the first order and first kind, and X is given by: X = y/4kakdt0t
(6.114)
When the rate constants of adsorption and desorption are examined at molecular level, ka and k^, respectively, are the probabilities per unit time that a molecule enters or leaves the stationary phase. Thus, during one adsorption-desorption event the average residence times of a molecule in the stationary and the mobile phases are TS = 1/fc^ and Tm = l/ka, respectively. Since the molecules spend to time in the mobile phase and on average t'R time in the stationary phase, we can express the average number of adsorption-desorption events as:
n = tk = t± Is
(6.115)
Tm
With these variables we can rewrite Eq. 6.113 as
For the calculation of the distribution at large values of X, the first term of the asymptotic expansion of the Bessel function, I\ (X), can be used. The following distribution is obtained: P(t - t0) =
™>e-\y*-v™)
=
"e-W*'«-V*)
(6.117)
McQuarrie [96] has extended the stochastic theory to the case of multiple adsorption sites and to a column with a single type of site, but with various input distribution functions.
6.5.1 The Use of Characteristic Functions The characteristic function of the random variable X is the mathematical expectation E{eiuX}, where to is an auxiliary variable and i is the imaginary unit. For a continuous random variable, the characteristic function, f(co), is calculated by the following equation [97]: /•CO
= /
eiu>xf(x)dx
(6.118)
330
Linear Chromatography
The characteristic function of a random variable and the probability density function of the random variable form a Fourier transform pair. Accordingly, the characteristic function completely describes a random variable. Dondi and Remelli [6] applied the characteristic function method to the stochastic theory of chromatography and analyzed a number of chromatographic models. They obtained basically the same results as the previous authors but extended them to more complex models, including a general model making no assumptions regarding the distributions of the entries in the stationary phase and the residence time on a sorption site. The method also accounts for axial dispersion and the resistances to mass transfer in the mobile phase using a random walk model. These authors have also studied the convergence of the band profile toward the Gaussian shape and the rate of this convergence, using the Levy distance between the actual profile and a Gaussian having the same mean and variance. The Levy distance decreases in proportion to the number, n, of entries of the molecule in the stationary phase [6]. Later these authors derived an HETP equation using Poisson distributions for the dispersion phenomena [98]. The use of the characteristic functions facilitated the extension of the stochastic theory to two-site [99] or to generic multiple-site heterogeneous surfaces [100] that are very complex to handle otherwise [5,101,102]. Thus, the stochastic theory is able to model the band profiles that are due to any unimodal or multimodal distribution of adsorption energies. The stochastic model was further extended to describe the effect of mobile phase dispersion and size exclusion effects as well [7,8,103]. By means of the characteristic function method, closed form expressions of the band profiles are obtained in the Fourier domain, from which the statistical moments of the chromatographic peaks can be calculated directly, even if the transformation into the time domain is possible by numerical means only [104]. When axial dispersion in the mobile phase is neglected, the characteristic function of band profile is obtained as [6,99,100]: = exp (j^—
- n)
(6.119)
Note that the above equation and Eq. 6.116 form a Fourier pair. Equations 6.116 and 6.119 are completely equivalent but the solution via the characteristic function is strikingly simpler than the time-domain solution. The effect of the mobile phase dispersion on the retention time can be handled in the same manner as the effect of the nonconstant number of adsorptiondesorption events on the stationary phase time [8,9,98]. Felinger et al. modeled the mobile phase dispersion by a one-dimensional random walk and by the first passage time distribution arising from the random walk. When combined with the stochastic process of adsorption-desorption, this approach leads to a rather general stochastic representation of the chromatographic process. They obtained the following solution via the characteristic function method [9]: (pR(co) = e x p
(6.120)
6.5 The Statistic Approach
331
Equation 6.120 is the Fourier transform of the band profile. The statistical moments of the band profile are calculated rather simply making use of the moment theorem of the Fourier transform. The first moment of the peak about the origin is lii = to + nxs
(6.121)
The second central moment is !
(6.122)
Since the stochastic models gives the corrected retention time in the form of t'R = nrs and the retention time as fj? = fo + nT$> w e c a n write the second central moment as fi = TT5— +
(6.123)
By means of the moments we can calculate the reciprocal of the column efficiency as: 2
.,
The above equation can be compared with Eq. 6.56, which was obtained with the lumped kinetic model. The two expressions are identical only if n = Nm, i.e. the average number of adsorption-desorption events in the microscopic model and the number of mass transfer units in the macroscopic kinetic model are analogous terms. Felinger et dl. showed that not only the first and the second moments but also the whole band profiles obtained as the solutions of the microscopic model and the macroscopic lumped kinetic model are completely identical [9]. Equation 6.120 gives the characteristic function of the band profile recorded with a destructive detector, such as the flame ionization detector because the mobile phase process is modeled with the first passage time distribution. In destructive detectors, the molecule is destroyed as soon as it enters the detector cell, therefore it is not possible that one molecule is detected twice due to backward diffusion. On the other hand, with UV detection in HPLC, a molecule might diffuse back to the detector cell, just after it has left the cell. This distinction, in theory, gives different band profiles for destructive and nondestructive detectors. In practice, however, the difference between the band profiles calculated by the two approaches is minuscule and experimentally cannot be measured. The band profile in the case of a nondestructive detector is obtained if not the first-passage distribution but the probability distribution of a diffusing molecule is used to describe the mobile phase process. The first moment, the second central moment, and the column efficiency that are obtained with a nondestructive detector are as follows Mjisn + 1
332
Linear Chromatography Ndisp + 2 „
NdiSp + 1
,
N ^ + l n ^+ l j
V
-
>
The above expressions are slightly different from Eqs. 6.121 to 6.124. The value of NdjSp, however, is usually very large, thus practically no difference is seen between Eqs. 6.121 to 6.124 and Eqs. 6.125 to 6.127. 6.5.1.1
Heterogeneous Adsorption-Desorption Kinetics
When the surface of the stationary phase is heterogeneous (see Chapter 3, Section 3.3), the molecules adsorb on the surface with different affinity energies. With the stochastic model, one can take into account rather flexibly the adsorption energy distribution of the various sites. Figure 6.15 shows a chiral separation, where models of increasing complexity were fitted to the tailing peak of the more retained phenylglycine enantiomer measured on a Chirobiotic T column. In order to reach a good fit, the presence of at least three types of sites (one nonselective and two enantioselective sites) had to be assumed [105]. The stochastic model is a kinetic model whose rate and time constants can be related to the affinity energy of adsorption. The Frenkel equation connects the adsorption energy, Ea, and the average stationary phase residence time of one adsorption event Ts [106]: TS = TOeE«/KT
(6.128)
where To is a constant typically around To = 1.6 x 10~ 13 s at room temperature. By means of the Frenkel equation, we can translate an adsorption energy distribution into a stationary-phase residence-time distribution, from which we can easily obtain the statistical moments of the peak. For a generic multiple-site surface, when there are M types of sites, each with a relative abundance of pj, the z'th moment of the distribution of the average residence times is M m
i = LPirij
( 6 - 129 )
i=i
where rSij is the average residence time on site of type ;' during one adsorption event. When the the distribution of average residence times is continuous, we shall have
tn{ = J rlsf(Ts)drs
(6.130)
For the first moment and the second central moment of the peak shape we get, respectively [8]: Fi
=
*o + nm\
_
2DLL{k0 + lf
V-2 -
^3
(6.131a) (6.131b)
333
6.5 The Statistic Approach
3
3.5 lime [min]
Figure 6.15 Separation of 10 ]A of 0.005 mol/L racemate of phenylglycine on a Chirobiotic T, 5 fim, 150 x 4.6 mm column. Mobile phase, water, 35 °C, 1 mL/min; detection, UV, 254 nm. The L-enantiomer elutes before the D-enantiomer. Dots, experimental chromatogram; solid line, chromatogram recalculated on the basis of stochastic theory with one non-selective and two selective sites. The narrow dashed line in the insert shows the best-fit model based on one non-selective and one selective site fitted to the whole peak; the narrow dotted line shows the same model fitted to the front of the peak. ReprintedfromP. Jandera, V. Backovskd, A. Felinger, J. Chromatogr. A, 919 (2001) 67 (Fig. 3) by courtesy ofElsevier.
where the retention factor now is k'o = nm\/toEquation 6.131 allows the calculation of the number of theoretical plates and that of the plate height equation for a generic heterogeneous surface [8]: 1
2OT2
N
nm{
H
=
(6.132a)
k> m2
+ 2-
(6.132b)
6.5.1.2 Stochastic Analysis of Molecular Migration in a HPLC Column The analysis of some simple chromatograms by means of the stochastic approach furnishes interesting and revealing, even sometimes surprising, details regarding the migration of analyte molecules through the column [107]. The number of adsorption-desorption events that a molecule undergoes during its migration along a modern reversed phase chromatographic column is at least several thousands, thus Eq. 6.117 can be approximated with the following Gaussian function P(t - k) =
: exp
-
(t/Ts - nf in
(6.133)
334
Linear Chromatography
Since the model given by Eq. 6.133 concerns only the stationary phase process, the mobile phase dispersion and extra column broadening effects have to be corrected for via the deconvolution of the whole chromatogram with the peak of the unretained marker that contains the necessary information regarding these contributions to the band profiles. The resulting net chromatogram carries the contribution of the stationary phase processes only. The evaluation of the RPLC chromatograms of nonionizable analytes shows that the number of sorption events varies between n = 13000 and 20000 on a 150 x 3.9-mm column and that it is not affected strongly by the retention time of the analytes [107]. Fly-times in the mobile phase between a desorption and the subsequent adsorption vary roughly between rm = 3 to 5 ms. During that fly-time, the mobile phase travels a distance that is 1.5 to 2.3-times the particle diameter. The sojourn time in the stationary phase, however, strongly correlates with the retention of the analytes, so the retention of analytes and the selectivity of their separation are mainly due to the variations of Ts, between 8.4 ms (for k' = 1.75) and 47 ms (for k' = 12.7). The number of dp = 5//m particles that are packed in a 150 x 3.9-mm column is about 1.8 x 1010. The comparison of the number of sorption events and the number of particles in the column shows that one molecule visits only a tiny fraction of the stationary phase: in this example, only one out of every two million particles is visited by any single molecule, although every particle contains approximately 4.2 x 1010 Ci 8 ligands [107].
6.5.2 Transport Equation in Chromatography with a Finite Speed of Signal Propagation It is well known that the diffusion equation (Eq. 6.22) that is used to model transport phenomena in chromatography has an infinite speed of signal propagation. The concentration C{z,t) is positive everywhere for any positive time [108]. As a consequence, the solutions of Eq. 6.22 can be regarded as being accurate only in the neighborhood of the band maximum, at ZM = ut. One way to derive a diffusion equation without the limitations implied by the infinite velocity of propagation of matter is the replacement of Eq. 6.22 by the so-called telegrapher's equation. Masoliver and Weiss [109] have proposed a generalized telegrapher's equation which is applicable to chromatography. Their equation is derived using a generalization of the concept of persistent random walk [110], itself based on the model of random walk with two states. Their equation is written 32C 3C 92C +2A =u 2 2
w
^t w- ^
„ dC
,,.,„, . (6 134a)
-
with
2A = -!- + =?i
(6.134b)
1 (6-134c)
6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography
Figure 6.16 Concentration profiles obtained with a finite rate of signal propagation. Solution of Eq. 6.134a. T+ = 8, T_ = 2, Co = 1, u = 1 for increasing values of time: t = 1; t = 3; f = 5. Reprinted from }. Masoliver and G.H. Weiss, Separat. Sci. Technal., 26 (1991) 279 (Fig. 1) by courtesy of Marcel Dekker Inc.
r\
335
\\\
where T+ and T_ are constants having the dimension of a time. A solution of this equation has been derived, using the Fourier-Laplace transform and the inverse Laplace transform [109]. Unlike the solution of the diffusion equation, which is positive everywhere, the solution of Eq. 6.134a is equal to 0 for z2 > (u i)2. Figure 6.16 shows some concentration profiles calculated as solution of Eq. 6.134a. When time increases, the asymmetry of the concentration profiles and the sharpness of the cutoffs at the endpoints, z = ±ut, increase. For sufficiently long times, however, their solution approaches the shifted Gaussian profile predicted by the solution of Eq. 6.22.
6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography As we discussed in connection with the various models of linear chromatography presented here, the elution profile predicted by any model of chromatography can be represented more than reasonably well by a Gaussian profile even when the column efficiency is only 100 theoretical plates. At 25 theoretical plates, the difference between the profiles predicted by the models and a Gaussian curve is still small. When the efficiency is about 1000 theoretical plates or higher, as typical of modern analytical chromatography, there is no possibility of showing a difference between the actual band profile and a Gaussian curve. Nevertheless, it is not uncommon to observe that an actual band profile is unsymmetrical. A complete theory of linear chromatography should be able to justify this observation, to offer reasons for the apparent contradiction, and possibly to suggest remedies in serious cases. If the mass transfer kinetics becomes extremely slow, the peak exhibits a significant degree of asymmetry and tailing. In Figure 6.17 we show a band profile calculated with the Lapidus and Amundson solution in a case in which the apparent column efficiency is only 5 theoretical plates. We see that slow mass transfer kinetics could be responsible for considerable band broadening and for a tailing peak, but we note that the peak retention time is nearly half the peak width. Chromatograms similar to the one in Figure 6.17 are observed only in unusual cases, such as in affinity chromatography, where the mass transfer or the adsorption-
336
Linear Chromatography
Figure 6.17 Effect of a slow rate of adsorption-desorption kinetics on the shape of the band profile and its asymmetry. Dimensionless plot of C^ versus fd. Chromatogram calculated with the Lapidus and Amundson model, with N ap = 5. Reprinted by-permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 6), with kind permission of Springer Science and Business Media.
desorption kinetics is really very slow. The tailing bands that are observed in most practical cases of analytical chromatography cannot be explained by a slow kinetics alone.
6.6.1 Extra-column Source of Band Tailing One empirical model that is widely used for the evaluation of band asymmetry is the exponentially modified Gaussian (EMG) function [111-113]. It is the convolution of the unit area Gaussian function and of an exponential decay of unit area. By definition, the convolution of two functions
For the purpose of this convolution, the Gaussian profile is written i2
G(r) =
cr\J2K
exp
R
/
L
1-1
\ 2"
(6.136)
The exponential decay is written f or t < 0
0
H{t) = l 1
-eT
t/r
fori>0
(6.137)
6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography
337
Figure 6.18 Comparison of the exponentially modified Gaussian (EMG) and the Gaussian profile. Dimensionless plot of Qj versus t^. Solid line: Gaussian profile with N = 5000. Dotted line: EMG function with N = 5000 and r/tR = 0.02. Reprinted by permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 7), with kind permission of Springer Science and Business Media.
The convolution of these two functions gives
exp [ - - ^
(6.138)
or, after integration, (6.139) Where erfc is the complementary error function. The EMG model seems to be justified only if extra-column sources of band broadening are at work and if they are important. Such sources may include a tailing injection profile. As an example, we show in Figure 6.18 a comparison between a Gaussian band profile calculated using Eq. 6.29 (solid line) and a band calculated with an EMG function (Eq. 6.139), for r/tR = 0.02 (dotted line), both profiles corresponding to a column efficiency of N = 5000 theoretical plates. We see that, as expected, the EMG function can produce significant band asymmetry and tailing, even at high column efficiency. In addition to the EMG function, many other mathematical models have been suggested to account for the profiles of experimental peaks and to determine characteristic shape-parameters, such as a number of theoretical plates, a skew and an excess. These parameters are related to the second, third and fourth moments (Eq. 6.77) of the peak, respectively. For example, the Gram-Charlier series (GC) [96,114,115] and the Edgeworth-Cramer series (EC) [115,116] have been
338
Linear Chromatography
used for peaks that have a low or negligible skew. Olive and Grimalt have used the log-normal function (LNF) [117] for peaks that have low to high values of the skew. They have shown that, at low values of the asymmetry factor, the number of theoretical plates, the skew, and the excess derived from an LNF model are close to those obtained with the GC or EC series. On the other hand, at high values of the asymmetry factor, the parameters derived from the LNF model are close to those derived from the EMG function. Therefore, these authors recommended the use of the LNF model as an alternative to EMG, and a general method of characterization of chromatographic peaks [118]. The problem with GC, EC, LNF, and to a degree with EMG, is the lack of physical background in relating these models and the fundamental equations of chromatography, mainly its mass balance equation. These functions can easily be used as peak-shape models to which experimental data are fitted. Because of the flexibility arising from the number of parameters, excellent results are generally observed, but this agreement affords little additional information, as the parameters obtained have no physical meaning.
6.6.2 Surface Heterogeneity and Complex Isotherm Peak tailing could also be explained by the heterogeneity of the surface of the stationary phase. For the sake of simplicity in this illustration, assume the existence of two different adsorption sites on the surface (e.g., alkyl groups bonded to a silica surface and residual silanols). These sites include the first type of sites, or ordinary sites, covering most of the area of the surface, and the second type of highly active sites occupying only a very small fraction of the surface area. On such a surface model, peak tailing could happen for one of two different reasons. First, the isotherm behavior may be linear for the ordinary sites but nonlinear for the high-energy sites, because their saturation capacity is very small. Second, the kinetics of adsorption-desorption on the high-energy sites may be too slow, while it is fast on the ordinary or low adsorption energy sites. We now discuss these two options. In the first case, since the active sites occupy a very small fraction of the surface and the interaction between active sites and solute molecules is very strong, the active sites will become saturated for a rather low concentration of the solute, a much smaller concentration than would be needed to saturate the ordinary types of sites. In other words, although the isotherm for the main adsorption sites could be considered linear in the range of concentrations achieved in the column, the same is no longer true for the high active site isotherm. It is not linear, and we should use a nonlinear isotherm instead, e.g., a Langmuir isotherm, to account for the band profiles [119]. In Figure 6.19, we compare two band profiles corresponding to the injection of the same amount of solute and calculated as numerical solutions of the equilibriumdispersive model for a nonlinear isotherm, as described in Chapter 10. These profiles were calculated using a one-site (solid line) and a two-site isotherm model (dotted line), respectively. For both sites, we chose Langmuir isotherms. For the single-site model, this isotherm is q = 24C/(1 + 6C). For the two-site model, the
6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography
339
Figure 6.19 Comparison of the band profiles calculated with a one-site and a twosite model, using the Langmuir isotherm model. Dimensionless plot of C^ versus frf. Solid line: one-site Langmuir isotherm model q = 24C/(1 + 6C). Dotted line: twosite bi-Langmuir isotherm: q = 24C/(1 + 6C) + 2.4C/(1 + 600C). In both cases, N = 5000. Reprinted by permission ofKluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 8), with kind permission of Springer Science and Business Media.
isotherm is q = 24C/(1 + 6C) + 2.4C/(1 + 600C). This combination of two Langmuir terms indicates that the fraction of the surface occupied by the active sites is 1000 times smaller than the fraction occupied by the ordinary sites, or practically 0.1% of the total surface area of the adsorbent. On the other hand, the solute has a 100 times stronger affinity for the active-site surface than for the bulk of the surface. As a result, the presence of the active sites increases by only 10% the retention factor at infinite dilution. Although a Langmuir isotherm and not a linear isotherm is used for the single-site surface, the band profile obtained is Gaussian, because the sample amount is low enough, the value of b\C is small, and under these conditions the isotherm has a linear behavior. For the two-site surface, however, since the term b^C is large for the active sites, the isotherm behaves nonlinearly. The result is a band profile that exhibits significant asymmetry and tailing. Obviously, if a linear isotherm had been used for both sites, the band profile would have been Gaussian, regardless of whether a single-site or a two-site model is chosen. Thus, if the sample size were a hundredfold smaller than the size used for calculating the profiles in Figure 6.20, the band profile obtained with the two-site model would be Gaussian, because then both isotherms would behave linearly.
6.6.3 Heterogeneous Kinetics If, because of the small sample size used and the numerical values of the parameters, the isotherm for a two-site surface behaves linearly, a certain degree of peak tailing can still be explained by assuming that the kinetics of adsorption-
Linear Chromatography
340
One-Sra 0 = A
- Two-Sis 1
Figure 6.20 Comparison of the band profiles calculated with one site and two-site linear isotherm models. Dimensionless plot of Q versus td. Solid line: one-site model, fast kinetics of adsorption-desorption (Nap = 2000, D = oo). Dotted line: Two-site model, fast kinetics of adsorption-desorption on the ordinary sites, slow kinetics on the active sites (Di = oo, D2 = 20). Reprinted by permission ofKluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, in NATO ASI Series C, vol 383, 61 (Fig. 9), with kind permission of Springer Science and Business Media.
Q|
= 20
i
•"
/
desorption (A/D) on the ordinary sites is fast, while it is slow on the active sites. Such an assumption is reasonable because we know that when a molecule is adsorbed on an active site, its desorption from this site is slow. There is a relationship between the average residence time on a site and the energy of adsorption. We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm (q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model (q — 24C + 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two types of adsorption sites, Eq. 6.65a is modified to take into account the kinetics of adsorptiondesorption on these two site types. As expected, the profile calculated with the two-site model and slow A/D kinetics for the active sites exhibits a very significant degree of tailing. Situations of this type are expected to arise in reversed phase chromatography using chemically bonded silica, as there might be some unreacted silanol groups or other sites that may behave as active sites toward strongly polar molecules. If the A/D kinetics is very fast on both types of sites, the band profiles derived with either of the two isotherm models would obviously be identical.
6.7 Extension of Linear to Nonlinear Chromatography Models
341
6.7 Extension of Linear to Nonlinear Chromatography Models Most of the models discussed in this chapter have been applied to nonlinear chromatography. In contrast to the analytical applications of chromatography, a large amount of a rather concentrated feed solution is injected in preparative chromatography. Because the concentration of the bands is high during all or most of their migration, equilibrium isotherms do not behave linearly any longer, and this has several important consequences. First and foremost, different concentrations no longer all move with the same velocity (see Chapter 7). Depending on the direction of curvature of the isotherm, the high concentrations move faster or slower than the low ones. This causes a profound deformation of the band profiles during their migration. Second, when they become nonlinear, isotherms also become competitive.They should be written as qi — f{C\, Ci,...). There is competition between the different components of the mixture for access to the stationary phase. As a result, and unlike the case of linear chromatography, the elution profiles of the different components are not independent, but they are affected by the presence of the other components, particularly the profiles of those peaks that are closely eluted. For all these reasons, the mathematical aspects of the theory become much more complex. The mathematics of nonlinear chromatography are so complex that even for a single solute, there is no analytical, closed-form solution available, except with two simplified models, the ideal model and the Thomas model [120]. The ideal model is based upon the assumption of an infinite column efficiency. Its solutions are discussed in detail in Chapters 7 to 9. The Thomas model is based upon the assumptions that there is a slow Langmuir adsorption-desorption kinetics and that there are no other mass transfer resistances, nor any axial dispersion. The system of equations of this model has been solved by Goldstein [121], and this general solution has been simplified for pulse injection by Wade et al. [122]. In all other cases, the problem must be solved numerically. The Thomas model is discussed with other kinetic models in Chapter 14 and 16. In contrast to linear chromatography, which has only single-component problems, the general problem in nonlinear chromatography is the multicomponent problem, or at least the separation of a binary mixture. In this case, we must use one differential mass balance equation for each of the components of the mixture (and each additive in the mobile phase if they are significantly adsorbed). Because of the competitive behavior of the phase equilibrium, these partial differential equations are coupled, which renders the problem most complex. There is no analytical solution in the case of the Thomas model for two components. Even the ideal model cannot be solved completely in closed form for a binary mixture, in the case of the competitive Langmuir isotherm [123], the simplest nonlinear isotherm (Chapter 8). The equilibrium-dispersive model of chromatography has been applied successfully in nonlinear chromatography, but numerical solutions have to be calculated [124,125]. The reaction-dispersive [126] and the transportdispersive [127,128] models of chromatography also have to be solved numeri-
342
REFERENCES
cally for nonlinear isotherms. The general rate model of chromatography and its extension, which includes reaction in the mobile and stationary phases, are solved numerically with nonlinear isotherms [129,130]. As discussed in Chapter 14, the experimental data can be fitted well to each of the different lumped kinetic models of chromatography using a Langmuir isotherm, as in linear chromatography. However, unlike the case of linear chromatography, the apparent number of theoretical plates derived from this fitting is concentration dependent. This means that Eqs. 6.84 or 6.86 are no longer valid in nonlinear chromatography, a result that contributes to making more difficult the analysis of nonlinear chromatography. However, in overloaded elution Eqs. 6.84 and 6.86 are a good approximation as long as the adsorption-desorption or the mass transfer kinetics are not too slow [131].
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Chapter 7 Band Profiles of Single-Components with the Ideal Model Contents 7.1 Retrospective of the Solution of the Ideal Model of Chromatography 7.2 Migration and Evolution of the Band Profile 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5
Continuous Part or Diffuse Boundary of the Profile Characteristic Lines of the Mass Balance Equation Origin of the Concentration Shock Propagation of Concentration Shocks Time Needed for the Formation of a Shock on a Continuous Profile
. . . 349 351 351 354 355 356 359
7.3 Analytical Solutions of the Ideal Model
363
7.3.1 General Closed-Form Solution 7.3.2 Case of the Langmuir Isotherm 7.3.3 Concentration Profile along the Column 7.3.4 Profile of a Vacancy 7.3.5 Case of the Bi-Langmuir Isotherm 7.3.6 Case of the Freundlich Isotherm 7.3.7 Linear Isotherm 7.3.8 Asymptotic Solution 7.4 The Ideal Model in Gas Chromatography 7.4.1 Linear Isotherm and Sorption Effect 7.4.2 Nonlinear Isotherm and Sorption Effect 7.5 Practical Relevance of Results of the Ideal Model 7.5.1 Experimental Investigations of the Helfferich Paradox 7.5.2 Comparison of Experimental and Calculated Elution Band Profiles
364 364 367 368 369 372 375 375 377 378 379 379 380 382
Introduction The solution of the single-component problem does not contribute directly to the understanding of the separation mechanism in nonlinear chromatography. So one might wonder why study the solution for one component of the ideal model, in which we assume that the column efficiency is infinite. This last assumption seems to deprive the exercise of any realism and relevance. On the contrary, we believe that the investigation of this problem is worthwhile for several reasons. First, it is the simplest possible nonlinear chromatographic problem. Its solution gives general background information regarding the more complex multicomponent problems. More specifically, it provides an understanding of the influence of the thermodynamics of adsorption on the shape of the band profiles. Second, although it is already a complex problem, the single-component profile in the ideal 347
348
Band Profiles of Single-Components with the Ideal Model
model is given by a general, closed form solution, valid for all isotherms. Third, this solution allows discrimination between and determination of the equilibrium isotherm models. Finally, comparisons between experimental band profiles and the profiles predicted by the ideal model show only small differences when these profiles are obtained for large sample sizes injected into efficient columns. This observation gives a measure of the relative influences of the thermodynamics and the column efficiency on the band profiles in nonlinear chromatography. For single-component systems, the theoretical solutions obtained are easy to compare to experimental profiles. They differ only by the smoothing effect due to axial dispersion and to the finite kinetics of mass transfers in actual columns. In many cases, because of the qualities of the stationary phases currently available, these effects appear to be secondary compared to the major role of thermodynamics in controlling the band profiles in overloaded elution. Admittedly, the influence of the finite column efficiency on the band profiles prevents a successful quantitative comparison between theoretical and experimental band profiles. However, these profiles are similar enough at high concentrations and the solutions of the ideal model indicate which are the trends to be expected. Furthermore, the theoretical analysis of the single-component problem in the ideal model provides some of the fundamental concepts in nonlinear chromatography, such as the notions of the velocity associated with a concentration, of concentration shocks, and of diffuse boundaries [1,2]. It also provides an understanding of the relationship between the thermodynamics of phase equilibria, the shape of the isotherm (i.e., convex upward, linear, convex downward, or S-shaped) and the band profiles. Finally, it provides an explanation of the relative importance of the influences of the thermodynamics and the kinetics on the band profile. These concepts will provide a most useful framework for understanding the phenomena that occur in preparative chromatography. Assuming that the column efficiency is infinite, the ideal model neglects the influence of all the kinetic contributions (axial dispersion, kinetics of mass transfer in the chromatographic column, and kinetics of adsorption-desorption) on the band profile. This model focuses on the influence of the thermodynamics of phase equilibrium on the band profiles. It studies the relationship between the band profiles of a component and its equilibrium isotherm under given experimental conditions. Although modern chromatographic columns usually have high efficiencies, these efficiencies are still finite. Under the combined influences of axial dispersion and the kinetics of mass transfer and of adsorption-desorption (or more generally of the retention mechanism), the true band profiles are smoother, the true boundaries are less sharp, and the true bands are less tall than predicted by the ideal model. These effects can be accounted for properly by using the equilibriumdispersive model (Chapters 10 to 13) or a kinetic model (Chapters 14 and 16).
7.1 Retrospective of the Solution of the Ideal Model of Chromatography
349
7.1 Retrospective of the Solution of the Ideal Model of Chromatography As we have seen in Chapter 2 (Eq. 2.36), if we assume constant and instantaneous equilibrium between the two phases, and no axial dispersion, we obtain the following equation for the band profile of a single component:
where q is the stationary phase concentration in equilibrium with the mobile phase concentration C, as given by the isotherm (see Chapter 3, Section 3.2), u is the mobile phase linear velocity (u = L/IQ, with L column length and to holdup time), and F is the phase ratio, F = Vs/Vm = (1 — e)/e, where e is the total porosity of the packing material, including both the interparticle and intraparticle void volumes. In order to complete the mathematical formulation of the problem, appropriate initial and boundary conditions (Chapter 2, Section 2.1.4, and Chapter 6, Section 6.2) must be used. Finally, we must define the functional space in which solutions of a partial differential equation are sought. In the case of Eq. 7.1, solutions will be sought in the space of (discontinuous functions) with bounded variations, denoted BV(O) in mathematics [3]. Discontinuities should be allowed, for reasons made clear in the next section. Roughly speaking, the variation of a function in a domain d is the integral of the norm of its gradient (in the sense of distributions) over Q. If the variations are bounded, neither oscillations nor discontinuities can develop too much, which is required in the present case. In fact, the variation should decrease in the course of time. The mathematical formulation of the single-component problem was given first by Wicke [4], who mentions Eq. 7.1 in passing, as a particular case of the equilibrium-dispersive model. Wicke did not pursue the study of the ideal model, apparently considering it as too unrealistic. A year later, Wilson [5] derived the same equation, being the first to formulate the model, to recognize its importance, and to study its properties. We note, however, that both authors neglected in their equation the accumulation term for the mobile phase, dC/dt, probably because, at the time, chromatography was carried out with very high retention factors. If, however, we replace q by C + Fq in the equation derived by Wicke or Wilson, it becomes identical to Eq. 7.1. Thus, in spite of the error made, the conclusions derived in [5] remain qualitatively correct. Using the Langmuir equilibrium isotherm (Chapter 3, Eq. 3.47), Wilson showed that his equation could propagate concentration discontinuities, a conclusion that is also valid for Eq. 7.1, but he did not recognize that the solution also includes a diffuse boundary on the band rear. In 1943, DeVault [6] derived the correct Eq. 7.1, made a more rigorous study of its properties, and demonstrated the equation giving the diffuse boundary. He showed that this diffuse boundary takes place on the rear of the profile when the isotherm is convex upward, and on the front of the profile if the isotherm is convex downward. He suggested that reversing the equation of the diffuse boundary
350
Band Profiles of Single-Components with the Ideal Model
could permit the calculation of the equilibrium isotherm [6]. The methods of elution by characteristic points and frontal analysis by characteristic points (Chapter 3, Sections 3.5.2 and 3.5.3) are derived from this observation [7-10]. Finally, he explained in physical terms the origin of the concentration discontinuity at the band front and its interaction with the diffuse boundary. Because the amount of any compound injected in the column is conserved, the area under the concentration profile must remain constant. Then the location (when a concentration profile along the column is considered) or the elution time (in the case of elution profiles) of the concentration shock is such that the area under the concentration profile is equal to the amount injected [6]. Weiss arrived independently at the same results and described in detail the progressive change of the band profile during its migration for a Langmuir (Eq. 3.47) and for a Freundlich isotherm (Eq. 3.56) [11]. The theory of chromatography expanded rapidly in the late 1940s. The plate theories [12,13], and the solutions of the partial differential equations were developed in parallel [8,14,15]. The case of linear chromatography was essentially solved by the work of Lapidus and Amundson [14] and Van Deemter et ah [15] in the 1950s. Progress in the theory of nonlinear chromatography was hindered, however, by the slow evolution of the theory of hyperbolic systems of partial differential equations. The proper mathematical tools appeared only in the 1950s and were applied to the study of the chromatographic problem only much later [1,16]. Glueckauf [7,8,17,18] was one of the early investigators who made major experimental and theoretical contributions to the solution of all fundamental problems of chromatography. For the single-component problem, he measured singlecomponent isotherms and illustrated experimentally and theoretically the development of the band profile in a pure solvent, when the isotherm is convex upward, convex downward, or sigmoidal. For a convex upward isotherm, the amount of compound at equilibrium in the stationary phase increases less rapidly than the concentration in the mobile phase. Because of this thermodynamic property, the high mobile phase concentrations are less retained and move faster than the low ones.1 Elution profiles have a sharp front and a diffuse rear boundary. The converse is true for a convex downward isotherm, where the elution profiles of the bands have a diffuse front and a sharp rear boundary. For a sigmoidal isotherm, the profile depends on the final maximum concentration of the band with respect to the inflection point [17]. Thus, the single-component problem of the ideal model was practically solved as early as 1949. Unfortunately, the method used by Glueckauf and his choice of symbols made his derivation and his results extremely difficult to understand and use. Many intermediate steps of the derivations are not supplied, and considerable work is required from the reader to reconstruct them. Some fundamental mathematical results were not available at the time, and the concept of shock could not be han1 There is a very important distinction to be made here, the reasons for which will be clarified later, in the discussions of Eqs. 7.3 and 7.7 and Figure 7.4. The movement of a given concentration of a certain species and that of the actual molecules of that species are controlled by different laws. The situation is somewhat analogous to what happens when a stone falls in a pond of stagnant water. The water does not actually move outward, although ripples are formed and propagate. Similarly in chromatography, mobile phase concentrations and molecules associated with a concentration propagate at different velocities. Perturbations and actual samples have different migration rates
7.2 Migration and Evolution of the Band Profile
351
died rigorously. For these reasons, and also because there was a lack of interest in the preparative applications of chromatography but a strong interest in its analytical applications in the 40 years that followed, there were few developments in nonlinear chromatography. Rhee et al. [1,19] gave a rigorous solution using the method of characteristics and the shock theory in the case of a Langmuir isotherm. Finally, Shirazi and Guiochon derived a simple equation for the calculation of the shock location for any isotherm that has no inflection point in the concentration range experienced by the profile [20]. The case of an isotherm of type II (Figure 3.1) with an inflection point has been discussed by Glueckauf [17] and more recently by Hong [21], who gave a graphic solution.
7.2 Migration and Evolution of the Band Profile The migration of an injected band and the progressive change of its profile can be conveniently studied using the theory of characteristics [1]. Equation 7.1 can be rewritten as ^ + ^ ^ ^ - 0
(72)
dt+1
{7 2)
F^dz-°
-
The study of the properties of this equation shows that its solution has a continuous or diffuse boundary, but also that the equation propagates discontinuities (or shocks) and that a stable concentration shock must take place on one side of the profile [22].
7.2.1 Continuous Part or Diffuse Boundary of the Profile Equation 7.2 shows that to each mobile phase concentration, C, is associated a migration velocity, uz, given by uz =
U
—r
(7.3)
The velocity uz depends only on the concentration, C, with which it is associated. Thus, the velocity associated with a given concentration is constant, and each concentration propagates along the column at a constant velocity [1,2]. The point
representing this concentration in a (t, z) space moves along a straight line. These linear trajectories are called characteristic lines or characteristics. Figure 7.1a illustrates in the three-dimensional space z, t, C the propagation of a Gaussian profile injected at t = 0. This figure demonstrates how the shock concept arises naturally. Note that the velocity uz is associated with a concentration, but that molecules do not migrate at this velocity. As explained in Chapter 3 (Figure 3.38), the molecules contained in a volume of solution at concentration C migrate at the velocity Us of the shock, given by the slope of the isotherm chord (see later, Eq. 7.7 and Section 7.2.4). Since a concentration C on a diffuse boundary migrates at the velocity uz, while no matter is transported at that velocity, the concentration should be considered as a signal having its own propagation speed.
Band Profiles of Single-Components with the Ideal Model
352
C 3 1 T—
Characteristics lines
3'
\ ~—~/l /
Injection jnr.fi le L 2
Figure 7.1 Migration and shape evolution of an injection profile as predicted by the ideal model in the case of a convex upward isotherm. Characteristics associated with concentrations and shock formation, velocity associated with a concentration, propagation of a band (i.e., of a concentration signal), and formation of a shock (see text), (a) Injection of a pulse with a Gaussian profile. Reproduced with permission from B. Lin, S. Golshan-Shimzi, Z. Ma and G. Guiochon, Anal. Chem., 60 (1988) 2647 (Fig. 1), ©1988 American Chemical Society, (b) Injection of a continuous linear ramp. Reproduced with permission from T. Ahmad, F. Gritti, B. Lin and G. Guiochon, Anal. Chem., 76 (2004) 977 (Fig. 2). © 2004 American Chemical Society.
The concentration at point 3 is sufficiently high to be in the nonlinear region of a Langmuir isotherm. The horizontal straight lines 1-1', 2-2', 3-3', 4-4', 5-5', and 6-6' are the characteristics of the corresponding concentrations on the profile and represent the propagation or trajectories of these concentrations. The lines a - a ' , b-b', c-c', d-d', and e-e' are the projections of the characteristics in the plane C(t,x)=0. Similarly, Figure 7.1b shows the progressive formation of a shock at the base of a continuous concentration ramp of an additive with a Langmuir isotherm [23]. In this case, however, there is not yet intersection of the projections of the characteristics on the horizontal plane. The two figures 7.1a and b show that at some distance L the projections of these characteristics intersect (although they do not need to intersect all at the same point as shown fortuitously in the figure). When the characteristics get closer to each other, the front becomes self-sharpening. A shock begins to form when the first two characteristics intersect. The formation of a shock or discontinuity is necessary, otherwise two different values of the concentration would exit from the column at the same time, which is physically impossible. The shock layer theory was developed to solve this contradiction. Figure 7.2 illustrates in the three-dimensional space, z, t, C, the propagation of a rectangular injection profile, the formation of a diffuse rear boundary, and the propagation of a frontal shock. The rectangle Olf O 2 ,O 3 , O4 is the injection profile in the column at t = 0, with O\ - O2 its front and O4 - O 3 its rear. The dashed lines u)\ — \\ and o?2 - A2 are two characteristics describing the trajectories of two concentrations associated with the rear, diffuse boundary of the profile. The divergence of these lines shows the dispersion effect. At column length z = A, the band profile is given by Alf A2, A3, A4. At this position, the injection plateau has
7.2 Migration and Evolution of the Band Profile
353
Figure 7.2 Migration and shape evolution of a rectangular injection pulse predicted by the ideal model in the case of a convex upward isotherm. Illustration of the concentration dependence of the propagation velocity of concentrations on a continuous profile and of the shock velocity. Reproduced with permission from B. Lin, S. GolshanShirazi, Z. Ma and G. Guiochon, Anal. Chem., 60 (1988) 2647 (Fig. 1) ©1988 American Chemical Society.
shrunk but not disappeared completely yet. At column length z = B, the injection plateau has shrunk to a single point, Bi, B3, and the band profile has become B\, Bj, B4. From there, the maximum concentration of the band profile decreases during migration. At column length L, the band profile is described by the lines I4, L2, L4. The diffuse boundary of the solution, C(z, t), is given by the intersections of the characteristics issued from all the points of the rear of the injection profile with the plane z = L. The elution profile is obtained as the intersection of the characteristics issued from the injection profile and the plane z = L. The concentration profile along the column at time ta is obtained as the intersection of the characteristics from the injection profile and the plane t = ta. The shock migrates along the line O\,A\, B\,L\. Note that, as will be explained later, from O\ to B\ the shock has a constant height and a constant velocity, and that its trajectory in the plane Ozt is a straight line. Beyond B\, by contrast, the shock height decreases with increasing time and migration distance, its velocity decreases with increasing time, and its trajectory is no longer a straight line. From Eq. 7.3, we also derive that, if there is a concentration C in the continuous elution profile, its elution time must be given by the equation: (7.4)
where L is the column length and tp the width of the injection profile. This is because, with a convex upward isotherm, the front of the injection gives a stable shock while its rear gives a diffuse boundary (see discussion later). Equation 7.4 gives the diffuse part of the elution profile. Equation 7.4 provides a means to the determination of isotherms (Chapter 3, Section 3.5).
354
Band Profiles of Single-Components with the Ideal Model
We have seen that the velocity uz associated with a concentration C in a chromatographic column depends on this concentration. In the case of a convex upward isotherm (e.g., Langmuir isotherm or, more generally, type I isotherms in Figure 3.1), d2q/dC2 is negative and dq/dC decreases constantly with increasing concentration. Therefore, according to Eq. 7.3, the velocity associated with a concentration C increases monotonically with increasing concentration. This progressive increase of uz with increasing concentration results in the higher concentrations moving faster than the lower concentrations. This gives rise to a steep or self-sharpening front, when the concentration increases, and to a diffuse rear boundary, when the concentration decreases. On the band front, the high concentrations move faster than the low ones, and the top of the band tries to pass the lower concentrations, which is physically impossible. Accordingly, the band profile cannot be entirely continuous, and a shock is formed. From a physical viewpoint, the following explanation can be provided. For a convex upward isotherm the fraction of molecules adsorbed at equilibrium decreases with increasing concentration. Therefore, the high concentrations are less retained than the lower ones, and they migrate faster along the column, since, on the average, they spend a greater fraction of their time in the mobile phase. However, high concentrations cannot pass low ones. Thus, the band profile becomes unsymmetrical, with a vertical front and a diffuse rear. For convex downward isotherms, the argument is reversed.
7.2.2 Characteristic Lines of the Mass Balance Equation The characteristic lines of Eq. 7.1 are straight lines that are issued from the points, Ci=o(t), of the injection profiles and have a slope defined by the following equation
"i i ±u ^ dx
(7.5,
The mass balance equation (Eq. 7.1) can be rewritten along the characteristic lines as
§ = ° dx u -r = A(7.6b) dt 1 F | L This equation shows that the concentration remains constant along each characteristic line. In this sense, we can say that concentrations move at a constant velocity and propagate along straight horizontal lines in the space 0, x, t, C. The slopes of these characteristic lines of Eq. 7.1 are uz = w/(l + Fdq/dC) (Eq. 7.6b), the velocity associated with the concentration C. This property has important consequences. A solution of Eq. 7.1 for a given set of initial and boundary conditions is a function C(x, t) that gives the concentration of the solute everywhere in the column, during the whole time of the experiment. The boundary condition is the injection
7.2 Migration and Evolution of the Band Profile
355
profile, C(x = 0,t). Thus, a characteristic line is a straight line in the (x, t) plane that is issued from a point of the boundary condition (i.e., with a certain concentration, C(0, t), and that gives the trajectory of the corresponding concentration on a diffuse (i.e., continuous) front. The mathematical method of characteristics uses this property to derive the solution of the partial differential equation 7.1. It is illustrated in Figure 7.1. It consists in drawing the fan of characteristic lines that are issued from the boundary condition and in determining the times at which the different concentrations pass successive positions in the column. Because the velocities uz are a function of the concentration, the profile of the solution evolves progressively. If this fan is divergent, the progressive derivation of the solution is straightforward. Usually, however, if the injection profile is continuous, there comes a time when the projections of two characteristic lines in the (x, t) plane intersect. This leads to a situation that is physically unacceptable: there would be two different values of the concentration of the compound in the same place in the column, at the same moment. When this happens, the solution consists in a concentration shock in the region where the characteristics intersect (see Figure 7.1, i.e., where the fan of characteristic lines is convergent. The shock begins to appear when and where the first two characteristic lines intersect (see later, Section 7.2.5). Then the tangent to the concentration profile is vertical. Afterward, the shock grows progressively.
7.2.3 Origin of the Concentration Shock As observed by DeVault [6], there can be only a single value of the concentration in any given point of the (t, z) space. In the framework of the ideal model, in which the column efficiency is infinite, this propagation phenomenon results in a concentration discontinuity or shock appearing at the band front. If the isotherm is convex downward, which occurs less frequently, the derivative dzq/dC2 is positive, then the velocity associated with a concentration decreases with increasing concentrations. Therefore, the converse effect occurs: a shock appears on the rear part of the band profile, since the low concentrations now move faster than the high ones but cannot pass them either. For this type of isotherm, the profile obtained is a diffuse front and a rear discontinuity. The mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical background has been reviewed in connection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rouchon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse boundary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can
356
Band Profiles of Single-Components with the Ideal Model
be applied [29]. Rouchon et al. [26] have used a rule formulated by Lax [27], which also gives the correct weak solution of the physical problem stated by Eq. 7.1. It is a property of the ideal model that Eq. 7.1 can propagate discontinuities of the function, C(z,y) [1,2]. This model has the advantage of allowing rigorous definitions and of giving clear explanations of the concept related to the effects of nonlinear equilibrium isotherms. On the other hand, from the physical point of view, the existence of a concentration shock is unrealistic, as are several other consequences of the ideal model. In the case of a shock as defined in mathematics, a macroscopic change of the concentration and of the physical properties depending on the concentration would take place over a distance smaller than a molecular diameter, which is clearly impossible. On a concentration discontinuity, an infinite concentration gradient would take place, resulting in an infinite diffusive mass flux. In practice, as we discuss in Chapter 14, a shock layer, or very sharp front, is formed rather than a true shock [30]. In a shock layer, a dynamic equilibrium takes place between the influence of thermodynamics (which tends to build up a concentration discontinuity) and the influence of both axial dispersion and the mass transfer kinetics (which tends to erode and disperse the shock layer). The shock layer theory is discussed in Chapter 14. The propagation of a shock layer is, mathematically and physically, quite similar to that of a shock wave. It is important to note here that this theory demonstrates that the migration velocity of the shock layers is the same as the velocity predicted by the ideal model for the shock itself [2,30]. On the other hand, the thickness of the shock layer is a function of the kinetics of mass transfer and of the axial dispersion.
7.2.4 Propagation of Concentration Shocks The migration velocity of the stable concentration shock is given by the following equation i +
t
AC
where Aq and AC are the amplitudes of the concentration shocks in the stationary and mobile phases, respectively [1]. Thus, the velocity of the shock, Us, is related to the slope of the chord of the isotherm, in contrast to the velocity associated with a concentration, which is related to the slope of the isotherm at concentration C. These two concepts are illustrated in Figure 7.3a, which shows a typical convex upward isotherm, its initial tangent, the tangent at concentration C = Co, and the corresponding chord. It is interesting to compare the value of Us and uz associated with different values of the concentration. This question has been discussed in Chapter 3, Figure 3.38. In the case of a convex upward isotherm, the shock velocity is higher than the limit velocity of the compound at infinite dilution (i.e., under linear conditions), since Aq/AC < (dq/dC)c=Q. Thus, the band widens.
7.2 Migration and Evolution of the Band Profile Qmg/mL
357 C mg/mL
Tangent dQ/dC
b
202
-201
200
TOO
200
300
C mg/mL
Figure 7.3 Illustration of the relationship between the slopes of the isotherm tangent and chord, the velocity associated with a concentration, and the velocity of a tracer, (a) Equilibrium isotherm, initial tangent, tangent at C = Co, and corresponding chord of the isotherm, (b) Migration of a labeled tracer in a mobile phase containing a concentration Co (lower trace, black line), and concentration signal for the unlabeled component (upper trace, gray line). See also Figure 3.38. On the other hand, in the general case, the shock starts from the baseline and extends up to the top of the band, Qvi, SO the shock velocity is US{CM) =
(7.8)
This velocity is lower than the velocity associated with the concentration CM on the diffuse rear side of the band profile: (7.9)
uz(CM) = c=c M
because q/C\c=cM > dq/dC\c=cM- We can consider the maximum of the band either as the top of the concentration discontinuity on the band front or as the beginning of the diffuse, continuous, rear part of the profile (in the case of a convex upward isotherm). We see that the migration velocity of the band maximum is higher if it is considered as a point on the continuous part of the profile than if it is regarded as the top of a shock. For a convex downward isotherm, the converse is true. Thus, while the band widens and spreads, the shock height is constantly eroded, and the band area remains constant. This is in agreement with the second law of thermodynamics. Equation 7.7 cannot be used directly to calculate the trajectory of a concentration shock in elution chromatography, however, since the height
Band Profiles of Single-Components with the Ideal Model
358 C mg/mL
t=2 min
t=2 min
3t=5min
2
1 Z cm 20
40
Figure 7.4 Illustration of the migration velocities along the column. Same experimental conditions as Figure 7.6c. (a) Profile C(z) along the column at 2 min, for a certain sample size. The velocity of the concentration C is uz(C); that of the slice of height C and width dz is LZS(C). (b) Profiles in the column at times 2 and 5 min. Points A and A' are at the same concentration C on each profile. The migration of a slice dz of constant amount is shown. Points M and M' indicate the peak maximum; points ZQ and z'Q indicate the points at the end of the tail, where C = 0.
of the shock is not constant during the elution of a narrow injection pulse, but decreases with increasing migration distance. Accordingly, the shock velocity decreases during its migration (Eq. 7.7). Note that the velocity of a shock, given by Eq. 7.7, is also the velocity of the molecules in a volume of mobile phase at concentration C. Figure 7.3b illustrates how the injection of a plug of labeled molecules on a concentration plateau permits the determination of both concentration and shock velocities in the same experiment, provided selective detectors are available for the labeled and unlabeled molecules (see Chapter 3, Section 3.5.4). The concentration perturbation moves at the velocity associated with the concentration, uz (positive peak on the gray line), while the labeled molecules (solid line, lower trace) move at the shock velocity, Us- The retention time of the tracer peak is therefore: ts{C)=tp + ^
AC
(7.10)
This equation cannot be used directly to calculate the retention time of a concentration shock of known height, as will be discussed later but is useful to find the retention time of a tracer pulse on a plateau of concentration C. This important point is further illustrated in Figure 7.4, which shows concentration profiles along the column length. Note the reversal in direction of asymmetry compared to the classical elution profiles (e.g., Figures 7.6 and 7.8). This is due to the fact that what elutes first from the column is what has migrated farther along it at a certain time. A concentration C migrates at velocity uz{C) and
7.2 Migration and Evolution of the Band Profile
359
the molecules contained in a slice of width dz and height C move at the velocity US(C) (Figure 7.4a). The velocity associated with a concentration, uz, is larger than the shock velocity, Us, or the average velocity of the molecules in the slice. The distance between the points A and A' (intersections of the horizontal line at concentration C and the two profiles) is equal to the product of «Z(C), which is constant, and the time interval between the two profiles (3 min). The average velocity of the molecules contained in the slice Az shown in Figure 7.4a is the shock velocity, Us, which is not constant because, during its migration along the column, the band spreads, each slice broadens and its velocity decreases. This is illustrated in Figure 7.4b where the slice moves from A to B'. Thus, the position, height and width of the corresponding slices are different in the profiles at 2 and 5 min. Differences are also observed in other points of the profile. The front shock moves faster (M to M') than the concentration at C = 0 (ZQ to ZQ). Therefore, the profile broadens and spreads while its height decreases. The concentration C moves faster, and farther, than the slice that has broadened and slowed down. This illustrates how the velocity associated with a concentration is greater than the velocity of molecules. In the following section (Section 7.3, we will analyze the trajectory of a stable concentration discontinuity injected into the column, i.e., as one of the boundaries of a rectangular injection profile. This derivation could be done directly in the case of a Langmuir isotherm [1,2,31] (see Section 7.3.3). It can also be derived as a particular case of a more general solution. First, however, in a last subsection, we determine the time it takes for a concentration shock to form on a simple continuous profile.
7.2.5 Time Needed for the Formation of a Shock on a Continuous Profile In the previous two sections, we discussed the origin and the stability of a concentration shock under ideal, nonlinear conditions and studied the migration of a rectangular pulse under nonlinear conditions. Depending on the curvature of the isotherm, one of the two vertical boundaries of a rectangular injection profile is a stable discontinuity and propagates as a shock along the column. Its maximum concentration is given by Eq. 7.22 (see later). The other vertical boundary is not a stable discontinuity because the velocity associated with each concentration decreases along this part of the concentration profile (i.e., it decreases with increasing concentration if it is a front boundary, decreases with decreasing concentration if it is a rear boundary). This other vertical boundary of the injection profile becomes a diffuse boundary. Its profile is given by Eq. 7.4. In most cases, however, the injection profile is not rectangular but it is continuous. In this case also, a concentration shock will form on the side of the profile along which the velocity uz increases with increasing concentration. However, there are no reasons for the shock to appear immediately on a continuous injection profile. As explained earlier, on a stable continuous profile, each concentration has an associated velocity that is lower than the velocity associated with the concentration immediately in front of it and higher than the velocity associated with the
360
Band Profiles of Single-Components with the Ideal Model
Figure 7.5 Boundary condition corresponding to the injection of a linear concentration gradient. Reproduced with permission from X. Ahmad, F. Gritti, B. Lin and G. Guiochon, Anal. Chem., 76 (2004) 977 (Fig. 1). © 2004
Co °
American Chemical Society.
concentration just behind it. Since the slope of each characteristic line issued from a point of the injection profile is proportional to the velocity associated with the corresponding concentration, these characteristic lines form a diverging fan and do not ever intersect. By contrast, if a continuous profile is self-sharpening, i.e., if the velocities associated with the different concentrations on this profile are such that the late concentrations move faster than those in front of them, the characteristic lines of this arc of the profile form a converging fan. A concentration shock appears as soon as the first two characteristic lines intersect [1]. Then this shock grows until its height is equal to the maximum height of the profile. Again, this growth is not instantaneous and the time it will take for a stable profile to form depends on the injection profile. 7.2.5.1
Linear Injection Profile
The simplest continuous injection profile is a linear profile, i.e., a linear concentration gradient [23]. So, let us assume as the boundary condition of the problem a trapezoidal injection profile with a linear ramp ending in a plateau of constant concentration (see Figure 7.5). This profile is the classical boundary condition for the strong solvent in linear gradient elution (see Chapter 2, Section 2.1.4). Its equation is 0 < t < th th
c = ^ C = Co
= Caii
(7.11a) (7.11b)
where fy, is the duration of the linear gradient and r\ = t/t^is the reduced time during the injection process. Time t^ is defined as the time when the concentration in the mobile phase at the column inlet becomes constant and equal to Co- The slope of the concentration gradient is C-o/i/,. We will assume a Langmuir equilibrium isotherm. The general equation of the characteristic line that begins in a point of this boundary condition, i.e., from the point x = 0, t = tjtf;, (0 < tj < 1) is t
=
k + ^x
(7.12a) (7 12b)
'
7.2 Migration and Evolution of the Band Profile
361
Each one of these characteristic lines is associated with a concentration (given by rj), the concentration of the point on the injection profile from which it issues. The concentration profile inside the column remains continuous as long as no characteristics intersect. A shock begins to form when two different characteristics intersect for the first time because it is not possible to have two different values of the concentration at the same point in the same time. The condition for the intersection between the characteristic lines corresponding to r\\ and r\i is given by *=
Solving this equation for x gives the position, xCr where the two characteristics intersect [23]. Introducing xc into Eq. 7.12b gives the time coordinate of the intersection point. These two values are:
_ t
-
u \(l
Fa
i
When r\\ tends toward r\i, these two equations tends toward limits that give the coordinates of the intersection point of two characteristic lines that are infinitely close to each other, hence that are tangent to a curve that is defined as the envelope of these characteristic lines. These limits are Xc
= zr«pi_o
tc
Fa
=
(7.15b)
Both xc and tc increase with increasing tj, so the envelope begins at the position corresponding to rj = 0 (see Eq. 7.11a). Hence, the shock begins to form in the column, at time t = IQ/S and at the location x = x0/S given by the following equations:
k
(7 16a)
-
The time £o,s is the relaxation time of the shock formation [23]. It is proportional to tjf, hence inversely proportional to the slope of the concentration gradient. It is also inversely proportional to the product bCo that characterizes the degree of nonlinear behavior of the isotherm at the end of the experiment, when the concentration plateau becomes equal to Co. If fy = 0, we have the boundary condition of a conventional breakthrough curve. The boundary condition has a front shock and this shock is stable, so it is
362
Band Profiles of Single-Components with the Ideal Model
formed instantaneously, at the column inlet. Then, it propagates along the column at the shock velocity [1,2,30]. If fy, ^ 0, the relaxation time is finite, as long as the equilibrium behavior is nonlinear. However, if bCo tends toward 0, which corresponds to the linear behavior of the isotherm, the relaxation time for the formation of the shock increases indefinitely. When the product BCQ is small and uth
th
-
2(£R
- to)
n-\i\
the relaxation time is longer than the retention time of the compound considered under linear conditions and the shock does not have time to form before the continuous profile begins to leave the column. 7.2.5.2 Retention Time of the Solvent Front in Gradient Elution ChromatograLinear gradient elution is a popular analytical method in which the concentration of a strong solvent in the mobile phase is increased linearly with time. This allows an exponential decrease of the retention factors of the sample components. This, in turn, allows the elution of the strongly retained components of a complex sample within reasonable retention times and as relatively narrow and high peaks, thus making easier their detection and more precise their quantitation. The concentration profile of the strong solvent is linear at the column inlet. However, when the injected profile migrates along the column, its shape changes progressively. Its front moves more slowly than its rear (the strong solvent isotherm is convex upward), its initial slope increases progressively until it becomes infinite, and a shock appears on this front. Then, both the shock height and its velocity increase as time passes. Equation 7.15 provides the value of the concentration gradient for which the beginning of the elution of the solvent front coincides with the birth of a shock at the elution of the solvent front. Equation 7.16 shows that the elution front of the solvent will exhibit a shock of infinitely short height if XQ/S — L, or
U
th
1
ZFabL
(718b)
The corresponding retention time, or breakthrough time, of the solvent front is = fR
(7.19)
As expected, this is the retention time under linear conditions of the strong solvent or additive in the pure weak solvent since, as long as no shock is formed, the zero solvent concentration migrates at the constant velocity UQ/(1 + k). When the shock is formed, however, it grows and it moves faster, at a velocity that increases with increasing shock height [2]. If the gradient slope is higher than the value given by Eq. 7.18, we have XQIS < L and the shock forms inside the column, before the strong solvent front can break through. The trajectory of this shock,
7.3 Analytical Solutions of the Ideal Model
363
hence its breakthrough time (xc = L), can be calculated using Eq. 7.14. Howeve^ this derivation is not straightforward and there is no algebraic solution to this problem. The elution of a strong solvent shock of finite amplitude causes the effect known, in thin-layer chromatography, as demixion. The solvent front is eluted before the limit retention time of the strong solvent at infinite dilution and it is extremely sharp. The components eluted before the solvent front are eluted under isocratic conditions in the pure weak solvent. The sudden breakthrough of the strong solvent constitutes an extremely steep gradient of the strong solvent inside the column. It is accompanied with the very rapid elution of numerous components that are poorly or not resolved. Such a situation must be avoided in analytical applications. If the gradient slope is lower than the value given in Eq. 7.18, XQIS > L, the shock has no time to form inside the column. The solvent front elutes at the same time tp_ (Eq. 7.19), regardless of the gradient slope. The front of the strong solvent is not vertical and the gradient analysis is carried out under continuous gradient conditions. However, the concentration gradient is not linear unless the retention of the strong solvent in the pure weak solvent is negligible, which happens some times in RPLC since the retention factors of methanol and acetonitrile in pure water are often small. 7.2.5.3 Other Continuous Boundary Conditions The approach described in the previous subsection remains valid. However, the characteristic lines issued from the boundary condition have now a more complex equation. If the boundary condition is C(x = 0, t) = Cof(t), the equation of the characteristic line corresponding to concentration C will be <7 20)
"
where T = / ~ 1 ( C / C Q ) is the inverse function of the injection profile, /(£). Algebraic calculations can rarely be pursued farther, because of the difficulties encountered in the inversion of the profile equation but numerical solutions are always easy to calculate.
7.3 Analytical Solutions of the Ideal Model The diffuse part of the profile is given by Eq. 7.4. The remaining problem is the determination of the retention time of the concentration shock. A solution of this problem in the case of a Langmuir isotherm is contained in the book by Rhee et ah [1]. It has been reformulated and used for the determination of adsorption isotherms [31]. A simple, more general solution, valid for any isotherm, was published by Golshan-Shirazi and Guiochon [20]. It gives the shock retention time as the solution of an algebraic equation which can be solved in closed form for the Langmuir isotherm and for a few other isotherms. In the case of the bi-Langmuir
364
Band Profiles of Single-Components with the Ideal Model
isotherm, however, the root of a fourth-degree algebraic equation must be calculated numerically.
7.3.1 General Closed-Form Solution We assume that the injection profile, or boundary condition for Eq. 7.1, is a rectangular pulse of width tp and height CQ. The area of the injection pulse is proportional to the sample size. It is given by
where n is the sample size (in number of moles), S the cross-sectional area of the column, e the total packing porosity, and u the mobile phase velocity (L/IQ). The product eSu is equal to the mobile phase flow rate, Fv. The product eS is the part of the column cross section that is available to the stream of mobile phase. If tp is negligibly small, we have a Dirac problem. It tp is very large, we have two successive concentration step injections, or Riemann problems, a positive one followed by a negative one. In practice, tp is finite; in fact, it is often of the same order of magnitude as the natural bandwidth, 4<7t, of the peak observed under linear conditions. The diffuse profile of the band (its rear profile in the case of a convex upward isotherm) is given by Eq. 7.4, derived by DeVault [6]. The maximum concentration of the profile, CM, is obtained by writing that the total peak area (i.e., the area of the elution profile at any length, z) remains constant during its migration and that it is equal to the area of the injected rectangular pulse [20]. The integration of the elution profile along the diffuse boundary leads to the equation: q-CM
dq_ dC
(7.22)
The solution of this algebraic equation gives the maximum concentration of the band. Introducing this concentration into Eq. 7.4 permits the calculation of the retention time of this maximum concentration on the rear diffuse part of the profile, hence of the retention time of the front shock. A closed-form solution of Eq. 7.22 requires knowing the equilibrium isotherm equation, q(C). It can easily be derived for the Langmuir isotherm, in which case we obtain the same result as previously obtained by calculating directly the retention time of the shock [31]. The solutions of Eqs. 7.4 and 7.22 in the case of a few classical isotherms are given in the following. Especially important are the solutions of the ideal model for the Langmuir isotherm, which most pure components seem to follow at least approximately, and for the bi-Langmuir isotherm, which is often found to account reasonably well for isotherms in reversed phase chromatography.
7.3.2 Case of the Langmuir Isotherm In the case of the Langmuir isotherm [q — aC/(l + bC), a, b, numerical coefficients], the equations giving the velocity associated with a concentration (Eq. 7.3)
7.3 Analytical Solutions of the Ideal Model
365
and the shock velocity (Eq. 7.7) become respectively: uz
=
- —
1
~r 1+5C
and, obviously, uz > Us. An example of a Langmuir isotherm is given in Figure 7.6a. The corresponding band profile is given in Figure 7.6b; the numbers associated with the features of this profile are the tag numbers of the corresponding equations. Rearrangement of Eq. 7.4 gives the rear diffuse boundary of the elution profile
( f ) t-tp-to
-1
(724a)
-
(7.24b)
The solution of Eq. 7.22 gives the retention time of the band maximum and of the front shock (Figure 7.6b):
tR = tp + t0 + (tRfl - t0) (l - yjh^j
(7.25a)
tR = tp + t0 + aFt0 (l - ^ZPj =tp + t0 + k'oto (l - yjh^
(7.25b)
where Lf is the loading factor or ratio of the sample size to the column saturation capacity (Eq. 3.51):
with n (mole), the sample size and k'g = aF.2 Note that the band profile ends after a finite period of time, at te = tp + t o ( l + aF) = tp + f o (l + k'o) = tp + tR/0
(7.27)
There is no band spreading in the ideal model, and the band profile ends at te = tR,o + tp- The solution of the ideal model in the case of the Langmuir isotherm (Eq. 7.25) serves as a basis for the RTM method of rapid determination of isotherm parameters (Chapter 3, Section 3.5.5). The maximum concentration of the profile is the concentration obtained on the continuous part of the profile (Eq. 7.24), corresponding to the retention time of the band front (Eq. 7.25). Combination of these two equations gives [31]:
CM = 2
A// ^ - —
(7.28)
Usually, elution profiles are considered. In this case, b, S, L, and kg are constant, and the loading factor is proportional to the sample size, n. However, if the migration of a band along a given column is considered, it should be remembered that its loading factor is not constant, but inversely proportional to the migration distance of the band front.
Band Profiles of Single-Components with the Ideal Model
366 Q mg/mL adsorbent
C mg/mL
12
T
Langmuir
1
2 C mg/mL
C mg/mL
1.6 Dimensionless
2
(t-tp-to)/to3
7.14 C mg/mL
tmin
0.14
a-tp-tc •-4 fo)/to
Figure 7.6 Comparison of the solutions of the ideal model for a Langmuir and a biLangmuir isotherm. Band profiles for different sample sizes (n) in mg on each profile, (a) Plots of a Langmuir and a bi-Langmuir isotherm. The Langmuir isotherm is used to calculate the profiles in (b), the bi-Langmuir isotherm for those in (c). Coefficients of the Langmuir isotherm: a = 13.1, b = 0.803. Coefficients of the bi-Langmuir isotherm: a\ = 9.6, bi = 6.4, «2 = 7.1, bi = 0.11. Inset: enlargement of the low-concentration part of the isotherms, (b) Band profile for the Langmuir isotherm in (a) Each feature is accompanied by the tag number of the corresponding equation in the text. Inset: same for the bi-Langmuir isotherm, (c) Band profiles obtained for the Langmuir isotherm. Main figure, concentration, C vs. (i — to —tp)/t(j. Inset: reduced coordinates bC vs. (t — to — tp)/(tR,Q ~~ £o),same values of the loading factor. Isotherm in (a), (d) Band profiles obtained for the bi-Langmuir isotherm in (a). Inset: enlargement of the profiles for low sample sizes n, sample size in mg. In contrast to linear chromatography, the band height does not increase linearly with increasing sample size under nonlinear conditions. It increases in proportion to the square root of the loading factor at low values, but tends toward infinity when the loading factor tends toward unity. If we consider Eq. 7.24, we see that the band profiles will be identical for all components with a Langmuir equilibrium isotherm if we use a reduced plot where F — bC is plotted versus k'/k'o = (t - to)/(tRiQ - t0) (see Figure 7.6c, inset) [31]. Therefore,
7.3 Analytical Solutions of the Ideal Model
367
in dimensionless units, the band profiles of any compounds with a Langmuir isotherm are a function only of the sample size, provided that the time origin of each profile is at the end of the sample injection, since the rear diffuse boundary originates from the collapse of the rear discontinuity of the rectangular injection pulse. This assumes that either we may neglect fp, or we take as time origin the end of the injection of the wide rectangular pulse, not its beginning as has been done above. Then all the band profiles will have the same rear profile, given by either one of the following two equations: *
=
or
(7.29b) (7.29c)
while the retention time of the front shock depends only on the sample size, or in reduced terms, on the loading factor Lf (Eq. 7.25). In summary, the Langmuir and bi-Langmuir isotherms are shown in Figure 7.6a. The inset of this figure illustrates the expansion of the isotherms around the origin and the difference between them. Figure 7.6b summarizes the equations needed to calculate the band profiles. Figure 7.6c describes the profiles calculated as solutions of the ideal model in the case of a Langmuir isotherm, for increasing sample sizes [31]. The system of reduced coordinates just described is used for the presentation of the profiles in the inset. The principles of velocity associated with a concentration and of shock velocity are illustrated in Figure 7.6c. As the sample size increases, the front shock elutes earlier. The rear diffuse profiles are overlaid because the velocity associated with a concentration depends only on the isotherm and on this concentration, not on the sample size.
7.3.3 Concentration Profile along the Column Equations 7.24 to 7.29 are valid only when applied at the outlet of the column (z = L). They give the elution profile or history of the concentrations at the column exit. It may be necessary to use the concentration profile along the column at a certain time, t. This profile is obtained by replacing L with z in Eq. 7.4
1+
bCf
(7.30)
Equation 7.30 can be solved for the migration distance, z(C), of concentration C at time t *
^
-
(7.31)
368
Band Profiles of Single-Components with the Ideal Model
Equation 7.30 can also be solved for C k'Qz/u t - t p - z/u
1
(7.32)
Similarly, the time at which the shock passes at the point of abscissa z is given by[l] (7.33)
u
u
The maximum concentration of the band is obtained by combining Eqs. 7.31 and 7.32
°M=b
(7.34)
Equation 7.31 remains valid when C = 0. So, the migration distance of the zero concentration is uff
_ i \
zo = „ , J
(7.35)
This point marks the end of the profile opposite to the shock.
7.3.4 Profile of a Vacancy In most cases, chromatography is performed with a simple initial condition, C(t = 0,z) = q(t = 0,z) = 0. The column is empty of solute and the stationary and mobile phases are under equilibrium. There are some cases, however, in which pulses of solute are injected on top of a concentration plateau (see Chapter 3, Section 3.5.4). The behavior of positive concentration pulses injected under such conditions is similar to that of the same pulses injected in a column empty of solute and they exhibit similar profiles. Even under nonlinear conditions (high plateau concentration), a pulse that is sufficiently small can exhibit a quasi-linear behavior and give a Gaussian elution profile. Its retention time is linearly related to the slope of the isotherm at the plateau concentration. Measuring this slope is the purpose of the pulse method of measurement of isotherm data. Large pulses may also be injected and they will give overloaded elution profiles similar to those obtained with a column empty of solute. On a concentration plateau, however, another kind of perturbations can be made. Negative concentration pulses or vacancies can be injected. Because the velocity associated with a concentration varies in opposite directions, depending on whether the concentration increases or decreases, the shape of a vacancy profile will be the converse of that of a positive concentration pulse. If the isotherm
7.3 Analytical Solutions of the Ideal Model
369
Figure 7.7 Chromatograms calculated with the ideal model for (1) a large pulse on a plateau of the same compound and (2) for a vacancy of the same amount on the same plateau. Plateau concentration, lmM; concentration of the positive pulse, 2mM; concentration of the vacancy pulse, 0. Column length, 10 cm.Reproduced with permission from G. Zhong, T. Fornstedt, G. Guiochon, }. Chromatogr. A, 734 (1996) 63. (Fig. 5).
e.5 7 Time (min)
is convex upward, conventional band profiles exhibit a front shock. Accordingly, the profiles of the vacancies will exhibit a rear shock. This effect has been discussed [32] and experimental proofs reported [33]. Figure 7.7 shows the profiles calculated for a large positive and a large negative pulse of a compound on a plateau. Similar experimental proofs have not yet been reported in the case of a convex upward isotherm. The phenomenon is also discussed later in Chapter 13, in the case of actual columns in which case concentration shocks are replaced with shock layers (which are discussed in Chapter 14).
7.3.5 Case of the Bi-Langmuir Isotherm The use of this isotherm model is justified by the existence of a bimodal energy distribution on the adsorbent surface [34-36]. It accounts well for many adsorption systems in reversed phase liquid chromatography and for isotherm data obtained with numerous chiral selective stationary phases [36]. The profiles obtained with this isotherm exhibit a long tail at the end of the large concentration band (see Figure 7.6b,d). The bi-Langmuir isotherm is written
a2C
+ a2)C
a2h)C2
(7.36)
with numerical values such that b2 » b\ (much larger adsorption energy on the active site, 2) and a2fb2 -C fli/&i (much larger saturation capacity for site 1). Figure 7.6a shows the bi-Langmuir isotherm which exhibits the minimum sum of squares distance to the Langmuir isotherm in the concentration range 0-1 mg/mL.
370
Band Profiles of Single-Components with the Ideal Model
Figure 7.6b (inset) shows the equations used to describe the band profile. Equation 7.4 for the rear diffuse boundary becomes
+ hC)2 (737)
The limit retention time at zero concentration becomes tRfl = tp + t0[l + F(ai + a2)]
(7.38)
The maximum concentration of the band is obtained from Eq. 7.22, rewritten as
where Lft\ is the loading factor calculated for the sample size and the column saturation capacity for the first site. Cmax is the positive root of Eq. 7.39. However, this equation cannot be solved in closed form for Cmax, as it is a fourth-degree algebraic equation. The solution of Eq. 7.39 can be calculated numerically only. Alternatively, the retention time f j$ (hence the maximum concentration of the profile) can be obtained by numerical integration of the equation Cdt = Cotp = ^-
(7.40)
For this numerical calculation, the integral can be approximated by a sum n
y cAt (c,-_i) — t(cA] I
(7-41)
j=i
with Cj — c,-_i + 4c, Co = 0, t(co) = £R,O + ^ and ^( c 0 solution of Eq. 7.4. The floating boundary is determined by comparing each successive result with the profile area, Cotp = n/Fv. Then Cmax = nAc, and t& = t(cn). Figure 7.6d shows the band profiles for samples of increasing sizes corresponding to the bi-Langmuir isotherm in Figure 7.6a. The bi-Langmuir isotherm has been chosen so that it is very close to the Langmuir isotherm used to calculate the profiles in Figure 7.6c. As seen in Figure 7.6a, the two isotherms are very close for mobile phase concentrations up to about 1 mg/mL. Comparison of the band profiles in Figure 7.6c and 7.6d shows important differences for small and medium size samples. The inset of Figure 7.6d shows an expansion around the tail of the bi-Langmuir profile. The profiles corresponding to the bi-Langmuir isotherm are more strongly arched and have a marked tail. In the concentration range 0-0.8 mg/mL, where the two isotherms are very similar, the shock fronts at the same sample size are eluted at the same times. At n = 0.3 mg, the retention factor of the shock, (£R — to — tp)/to, is 2.5 minutes for both the Langmuir and bi-Langmuir isotherms. For large sample sizes, the front of the bi-Langmuir profile elutes later than that of the Langmuir band (the retention time is 1.5 minute for n = 1.7 mg with the Langmuir isotherm, and n — 3.8 mg with the bi-Langmuir isotherm). The differences observed between the profiles reflect the differences between the
7.3 Analytical Solutions of the Ideal Model
371
Q mg/mL adsorbent
60
C mg/mL Q mg/mL adsorb&nt
30
(t-tp-to)/to C mg/mL
80
200
1
(t-tp-to)/to
2
Figure 7.8 Solutions of the ideal model in the case of a bi-Langmuir isotherm, (a) BiLangmuir isotherms with variable «2 and \>i coefficients and constant qs. Inset: lowconcentration part of the isotherm (with liquid phase concentration scale at the top of the inset), (b) Band profiles corresponding to the isotherms in (a). Only the rear dispersive boundaries of the profiles are shown. The position of the front shock depends on the sample size. Inset: end of the profiles, on a different scale, (c) Bi-Langmuir isotherms with constant a-i and variable b^ coefficients. Inset: low- concentration part of the isotherms, (d) Band profiles corresponding to the isotherms in (c). Inset: enlargement of the end of the profiles.
slopes of the tangents and chords of the two isotherms at a given value of the concentration C. This result shows the importance of accurate measurements of the adsorption isotherms for the correct prediction of band profiles. The effect of the isotherm parameters on the band profiles is instructive. Figures 7.8a and 7.8b illustrate the influence on the isotherm (Figure 7.8a) and on the diffuse rear of the band profiles (Figure 7.8b) of the term ai of the bi-Langmuir isotherm at constant column saturation capacity (i.e., constant a\/b\ + «2/^2)- This term corresponds to the contribution of "active" sites. The increased tailing is illustrated in the inset. Since i# Q is equal to F(fli + aj), the retention time increases
372
Band Profiles of Single-Components with the Ideal Model
linearly with increasing «2- A slight change in the curvature of the diffuse rear is seen in the main figure. Figures 7.8c and 7.8d illustrate the influence on the isotherm (Figure 7.8c) and the band profiles (Figure 7.8d) of the value of the saturation capacity as affected by the b% term of the bi-Langmuir isotherm. Since all the isotherms have the same initial slope, all the profiles end at the same time; tRto is independent of b\ or hi (Figure 7.8d). Strong changes in the curvature of the rear part of the profile are observed with changes in the column saturation capacity. The inset shows that there are no effects on the limit retention time, £#,0 in this case. As shown in Figures 7.6c and 7.8, the band profile obtained with a bi-Langmuir isotherm may exhibit a very long tail, depending on the value of the Langmuir isotherm coefficients for the second type of sites. This phenomenon may explain some instances of band tailing reported in analytical chromatography (Chapter 6, Section 6.6).
7.3.6 Case of the Freundlich Isotherm With this isotherm (see Chapter 3, Section 3.2.2.4), the relationship between q and Cis q = aCl/i3
(7.42)
with p > 1. For a very small sample size, the retention is nearly infinite because the ratio q/C tends toward infinity when C tends toward 0. However, when the sample size is finite, the band front is eluted after a finite time. Equation 7.4 gives the profile of the rear, diffuse part of the profile: aF In contrast to the band profiles that correspond to the Langmuir and bi-Langmuir isotherms, the profiles corresponding to a Freundlich isotherm do not end at a finite time, but tend asymptotically toward a concentration 0. The maximum concentration of the band is given by Eq. 7.22: CM
=
(7.44)
_Fvt0Fa(p -
where n is the sample size (number of moles) and Fv the mobile phase flow rate. Hence, the retention time of the band front is Fa( p
np \FvtQFa(p-l)
(7.45)
The profile obtained exhibits asymptotic tailing, as expected (see Figure 7.9d). This band profile is not unusual, although it may be difficult to distinguish in practice from the bi-Langmuir profile obtained in the case of an adsorbent having some very energetic, "active," adsorption sites. Isotherms of this type are often observed in the case of proteins separated by reversed phase or by hydrophobic interaction chromatography (Chapter 3). It is often difficult to decide whether
373
7.3 Analytical Solutions of the Ideal Model C mg/mL
Q mg/mL adsorbent
AS
4.4Dimensionless
200
30
C mg/mL
80
C mg/mL
60
0.5
0.6 \o.2
n: 18.4
\p.03
0
40
2
0.5
(t-tp-to)/fo C mg/mL
1.2 Bilangmuir
0
6°
1
2.5 :i
(t-tp-to)/to
Figure 7.9 Solutions of the ideal model for a Freundlich, a Langmuir, and a bi-Langmuir isotherms. Comparison with the profiles obtained with a Freundlich, a Langmuir, and a bi-Langmuir isotherm, n, sample size (mg) at the shock of each profile, (a) Freundlich isotherm used to calculate the band profiles in (d). Comparison with a Langmuir and a bi-Langmuir isotherm. Coefficients of the Freundlich isotherm: a = 13, p = 1.5; of the Langmuir isotherm: a = 13.1, b = 0.058; of the bi-Langmuir isotherm: a\ = 10.4, b\ = 0.192, «2 = 2.7, £>2 = 0.007. Inset: Low-concentration part of the isotherms. Note that the Freundlich isotherm is tangent to the ordinate axis, (b) Band profiles obtained with the Langmuir isotherm in (a). Inset: dimensionless profiles, (c) Band profiles obtained with the bi-Langmuir isotherm in (a). Inset: Profiles of small sample-size bands, (d) Band profiles obtained with the Freundlich isotherm in (a). Note that the bands tail indefinitely. Inset: band profiles for very low sample sizes. the band tailing is due to an isotherm of the Freundlich type or to sluggish mass transfer kinetics. There is, however, a marked difference between the band profiles calculated for isotherms which look very similar at first glance. Figure 7.9 compares the band profiles for a Freundlich, a bi-Langmuir, and a Langmuir isotherm (Figure 7.9a) over a large concentration range. The isotherm parameters were chosen for the three isotherms to be as close as possible. The inset
Band Profiles of Single-Components with the Ideal Model
374 Cmg/mL o
b
a on I/p
53 P=O.3
n
n=325.
q=aC+aC
=37^1
p=0.6
n=174/
n=l 1
10
n=7f/ n=2T/
/
I 0
n. 20
40
c}
n=L3/| 20
40(t-tp-to)/to
Figure 7.10 Band profiles solution of the ideal model in the case of a convex-downward isotherm. Isotherm: q = a(C + C1/P); coefficients, a = 13 and left: p = 0.3; right: p = 0.6. Sample size, n, in mg. of Figure 7.9a shows an expansion around the origin and illustrates the characteristic deviation of the Freundlich isotherm, which becomes tangent to the vertical axis. The Freundlich and bi-Langmuir isotherms can be made to follow each other closely at concentrations between 2 and 50 mg/mL. The deviations are larger with the Langmuir isotherm. Nevertheless, the band profiles in Figures 7.9b and 7.9c, calculated from the Langmuir and the bi-Langmuir isotherms, are markedly different from those in Figure 7.9d corresponding to the Freundlich isotherm. As shown in Figure 7.9b, these profiles differ still more importantly from those calculated with the Langmuir isotherm in Figure 7.9a. For example, to obtain the elution of the front shock at IR = 3 min, the components having Langmuir, biLangmuir, and Freundlich isotherms would require 1.2,0.5, and 2.2 mg of sample, respectively. Similarly, a shock retention time of 2 min is observed for sample sizes of 11,5, and 9 mg, respectively. Furthermore, whereas the profiles corresponding to the Langmuir and bi-Langmuir isotherms end at a finite time (with an infinite efficiency column), they tail indefinitely with a Freundlich isotherm. Lastly, comparison of Figures 7.6a and 7.9a shows a large difference in the column saturation capacity. In Figure 7.6c the shock retention time is 2 min for a sample size of 0.8 mg, but in Figure 7.9b this requires 11 mg. These results illustrate the importance of an accurate description of the equilibrium isotherm when one wants to predict band profiles in chromatography. They also illustrate
the importance of a large saturation capacity to allow for large loadings, hence for high production rates and low separation costs in preparative chromatography. Finally, to illustrate one case of band profiles calculated with a convex downward isotherm, we have selected a simple, but completely empirical, isotherm model, q = a(C + C1^), with p < 1. The result is shown in Figures 7.10a and 7.10b, with p = 0.3 and 0.6, respectively. The band front is diffuse and a rear shock appears. The direction of the convexity of the band profile itself depends on the
7.3 Analytical Solutions of the Ideal Model
375
value of p compared to 0.5 (but in both cases the isotherm remains convex downward). This result illustrates the importance of the initial slope and curvature of the isotherm on the band profiles.
7.3.7 Linear Isotherm Under conventional chromatographic conditions, i.e., in elution or frontal analysis, the solution of the ideal, linear model is trivial: the boundary condition is merely transported along the column at a velocity given by Eq. 7.3 with dq/dC = a, a constant equal to the initial slope of the isotherm. The situation is different, however, in simulated moving bed (SMB) chromatography or with other processes using periodic column switching. An interesting effect arises in SMB: even under linear isotherm condition, the front of the raffinate zone and the rear of the extract zones are stable concentration shocks. This explains the effectiveness of the process. The solution of the ideal, linear model with the SMB boundary and initial conditions [37] provides a useful and most informative illustration of the behavior of this process (see Chapter 17).
7.3.8 Asymptotic Solution A general asymptotic solution of Eq. 7.1, valid for any isotherm, has been derived [20]. This solution gives the profile toward which the band of a single-component that migrates along a column of infinite length tends, and for which H = 0. This limit cannot be a Gaussian profile, within the framework of the ideal model, since there are no sources of band spreading. However smooth the injection profile can be, a concentration discontinuity will take place eventually, as long as the equilibrium isotherm is not linear. For the asymptotic solution [20], the concentration is different from 0 between the following two times, corresponding to C = 0 at one end of the peak and the elution of the concentration shock at the other te
= zg'{0)
(7.46a)
tR
= zg'(Q)±^\2Apzg"(0)\
(7.46b)
where Ap is the area corresponding to the injection (i.e., Ctfp for a rectangular pulse) and the function g(C) is related to the equilibrium isotherm, q(C), by g(C) = ±(C + Fq)
(7.47)
The concentration is 0 at t = te, on a diffuse profile. A concentration discontinuity takes place at t = t^. The sign in Eq. 7.46 is negative for a convex-upward isotherm (e.g., Langmuir), positive for a convex-downward isotherm. Depending on the sign of the isotherm curvature at the origin, te is longer (convex-upward isotherm) or shorter (convex-downward isotherm) than t%. Between these two times, te and £#, the asymptotic solution is (7 48)
-
Band Profiles of Single-Components with the Ideal Model
376
C Convex-downward
CM M Convex-upward
T/
Figure 7.11 Asymptotic solution of the ideal model.
tR
te
t
This solution is a right triangle, lying on one side of the right angle (Figure 7.11). The vertical side of this triangle appears first in the case of a convex-upward isotherm, last with a convex-downward isotherm. The direction, height, and width of this right triangle depend only on the value at the concentration origin of the first two derivatives of the isotherm, i.e., on its initial slope and curvature. For example, the profile obtained is the same for the Langmuir isotherm and for the parabolic isotherm having the same initial slope and curvature [38]. The maximum concentration of the band is CM
(7.49)
=
Actually, the injection has a finite width, tp. In the case of a convex upward isotherm, the shock originates from the front discontinuity of the injection. For a convex downward isotherm, it originates from the rear discontinuity of the injection. These two discontinuities enter the column a time tp apart. Accordingly, the retention times of the front shock and of the last point of the asymptotic profile (C = 0) in the case of a convex upward isotherm are given by tR,s
=
tp+zg'(0)-^\2Apzg"(0)\
tR,o = tp+zg'(0)
(7.50a)
(7.50b)
In the case of a convex upward isotherm, these times are given by tR,s = tp + zg'(0)
(7.50c)
tR,o =
(7.50d)
h+zg'(0)
This situation is illustrated in Figure 7.12 which should be compared to Figure 7.11 for which the width of the injection pulse was neglected.
7.4 The Ideal Model in Gas Chromatography
377
C 1
Figure 7.12 Asymptotic solution of the ideal model with a finite injection width, tp.
2
0
a
b
c
d
time
7A The Ideal Model in Gas Chromatography The understanding of the process of band migration and the calculation of the band profiles in gas chromatography (GC) is more difficult than in HPLC for three reasons. First, because the gas phase compressibility is considerable, it cannot be neglected as it is in HPLC. The mobile phase density varies along the column, in inverse proportion of the pressure (Boyle-Marotte law applies well to all conventional carrier gases in the pressure range used). Thus, a mobile phase mass balance is always needed. Furthermore, the equilibrium isotherm varies along the column, so even the behavior of a plateau under nonlinear conditions is related to an arc of the isotherm, not to a single point, making frontal analysis far more difficult to apply. Finally, the partial molar volume of a solute is several orders of magnitude smaller when it is adsorbed on the stationary phase than when it is in the mobile phase. Accordingly, the flow velocity of the mobile phase is higher upstream than downstream a high concentration zone. The combination of this change in the pressure balance within the column and of this variation of the flow rate that compensates it lead to an effect called the sorption effect [8,39—41] that does not exist in HPLC. At about the same time as Rhee et ol. [19,30] were studying nonlinear liquid chromatography, Jacob et al. [22,42,43] used the same method of characteristics and discontinuity to describe the migration of bands of single compounds in GC and to calculate their profiles, using the ideal model and considering successively linear and nonlinear isotherms. To simplify the problem, besides the conventional assumptions of the ideal model (instantaneous phase, infinite mass transfer kinetics, no axial dispersion, infinite column efficiency), they posited that the pressure was constant throughout the column, which reduces the three effects listed above (the mobile phase density decreases along the column, the local isotherm varies along the column, the partial molar volumes of the compounds in the two phases differ by orders of magnitude) to the last one and leaves the sorption effect to be tackled [22,42,43].
378
Band Profiles of Single-Components with the Ideal Model
These authors showed that the elution profiles are the combined result of the propagation of a concentration discontinuity and of the conventional continuous migration of a concentration profile. They showed what was established earlier in this chapter for the simpler case of HPLC that, if the profile is continuous, a velocity is associated with each concentration. This is so even in the linear case because of the sorption effect. This may also be true because of the nonlinear behavior of the isotherm, although, in some rare cases, both sorption and isotherm effects can cancel each other (see later). When the velocity of a concentration increases with increasing concentration on the peak front, this front eventually becomes discontinuous while the rear of the band is always a continuous diffuse boundary. The opposite effects take place when the velocity of a concentration decreases with increasing concentration. Considering both types of contributions to elution profiles (sorption and isotherm effects) is necessary to understanding the elution of a zone in GC, whether the isotherm is linear or nonlinear [22,42]. The properties of chromatographic systems and the evolution of elution band profiles were discussed within the frame of this model. No analytical solutions were obtained, although the band profiles were correctly described. The discussion was mostly qualitative rather than quantitative in nature.
7.4.1 Linear Isotherm and Sorption Effect If the isotherm is supposed to be linear, the equilibrium isotherm does not intervene in the band profile and the global effect is derived from the flow properties. The characteristic method applies. It shows that, in linear gas-chromatography, although the isotherm is linear, the sorption effect causes the velocity associated with a given concentration to decrease with increasing concentration. [22]. A slice of mobile phase having a given mole fraction, X, moves with the velocity ^ 1 + ^ (751) K (l+fc'(lX))2 ' According to this equation, the larger the mole fraction, the larger the characteristic velocity, V. Then, the highest concentration of a symmetrical injection profile will travel faster than the zero concentration at its base, which travels at the velocity of analytical peaks, the velocity associated with C = 0. The peak becomes increasingly unsymmetrical as it travels down the column, with a steeper and steeper leading edge and a more and more strongly tailing end. If the column is long enough, a discontinuity will build up and the front edge becomes vertical. The front shock of a rectangular injection is stable, its rear shock is not and becomes a continuous profile (the situation is similar to the one illustrated in Fig 7.2, although the origin of the dependence of the velocity on the concentration is quite different). The sorption effect, not a nonlinear isotherm, explains the asymmetry of the elution profiles that was observed for high concentration bands of light hydrocarbons on high efficiency open tubular columns [22]. y=
7.5 Practical Relevance of Results of the Ideal Model
379
7.4.2 Nonlinear Isotherm and Sorption Effect At finite concentrations, the assumption of a linear isotherm and a constant k' is often not valid. Assuming that the number of moles of solutes in the liquid (stationary) phase is n and the mass of stationary phase is m, the liquid-vapor equilibrium of the solute can be written [42,43] nL = mg(pX)
(7.52)
where p is the local pressure of the mobile (gas) phase, X is the mole fraction of the solute, and g{pX) is the equilibrium isotherm. The equilibrium constant between the two phases, K(X), can be defined as [42,43] ng
vg p
p vg
where Vg is the free gas volume in the column and k' is the derivative of K(X), so k' = dK/dX. Equation 7.51 remains valid. So, the progressive variation of the profile of a high concentration band can be studied using the derivative of V by respect to X, which is: dV
VdX
2k'-k"(l-X)
l + fc'(l-X)
where k" = dk'/dX = d2K/dX2. Depending on the sign of N = 2k'{X) k"(X)(l — X), V may increase or decrease with increasing X. So, three different cases are possible, depending on whether N is positive, zero, or negative [44]. 1. If N > 0, V increases with increasing X. Then, the peak shape is similar to the shape observed under conditions of pure sorption effect. In this case, the sorption effect is enhanced by the isotherm effect. This is always the case for convex upward isotherms since their second derivative is always negative. 2. If N = 0, V does not depends on X. The sorption effect is exactly compensated by the isotherm effect. This implies that the finite concentrations have no significant influence on the shape of the elution band, and that the peak shape is similar to the one observed under analytical conditions. 3. If N < 0, V decreases with increasing X. The high concentrations move more slowly than the low ones. This takes place when the isotherm curvature is positive and sufficiently large. Then, the band has a rear discontinuity and a smooth continuous diffuse boundary at its front. Convex upward isotherms have a positive curvature, hence a positive value of k". In this case the isotherm effect acts in the direction opposite to that of the sorption effect and reduces it or completely reverses it, if it is large enough.
7.5 Practical Relevance of Results of the Ideal Model Because the ideal model is based upon the assumption that the column efficiency is infinite, we expect serious discrepancies to arise between the band profiles it
380
Band Profiles of Single-Components with the Ideal Model
predicts and those which are recorded with real columns. Although actual chromatograms always exhibit a marked blur of the concentration shock predicted by the ideal model, the shock being replaced by a shock layer of finite thickness (as is explained in Chapter 14, Section 14.1.4), the calculated chromatograms look very much like the real ones, as much as a good caricature looks like the original while it highlights its specific features. This point is discussed in more detail in Chapter 10, which deals with the solution of the equilibrium-dispersive model for a single component. However, to demonstrate the practical relevance of the ideal model, in this section we compare solutions of this model with experimental data. This comparison will illustrate the similarities between the solutions of the ideal model and actual band profiles at a high degree of column overload.
7.5.1 Experimental Investigations of the Helfferich Paradox Recently, Samuelsson et al. [45] made a careful experimental investigation of the concentration signals obtained at the exit of a column upon the injection of different perturbations on a concentration plateau. The Helfferich Paradox states that, if, after a column has been equilibrated with a constant finite boundary condition (i.e., with a stream of mobile phase with Qn/ef = Co = constant ^ 0), two different signals are observed upon injection of a tracer of the same compound. The tracer must be identifiable, either because its molecules are isotopically labeled or because one enantiomer is used for the plateau concentration and the other for the tracer, in an achiral chromatographic system. Two chromatographic systems were used in this study. The first one was unlabeled and tritium-labeled isosorbide mononitrate (ISMN) eluted by acetonitrile on a column packed with a polymeric resin, using a radiodetector and a diode-array UV detector. This column does not separate ISMN from tritium labeled ISMN (T-ISMN). Experiments were made by injecting pulses of T-ISMN on various concentration plateaus of ISMN and measuring the tracer retention time or, conversely, by injecting pulses of ISMN on plateaus of T-ISMN and measuring the retention times of the perturbations. The second system consists in the two enantiomers of methyl mandelate retained but not resolved on a Kromasil-Cis bonded silica column, with methanol/water as the mobile phase. Pulses of one enantiomer were injected on plateaus of the other. Fractions were collected and analyzed on a chiral column. The isotherm parameters of the first system were determined from the simultaneous processing of the retention times of the perturbations and the sample peaks and those of the second system from staircase FA data. Figure 7.13 shows examples of the signals recorded with the first system. The top two chromatograms show the elution of a pulse of ISMN on a plateau of TISMN, under linear conditions. The solid line (UV signal) gives the total concentration of labeled and unlabeled ISMN measured by a nonselective detector. It is the perturbation response measured under conventional conditions. The dotted line is the perturbation of the actual component on the plateau, the T-labeled ISMN, recorded with the radiodetector. The second chromatogram shows the results of this experiment calculated with the equilibrium-dispersive model. The agreement is excellent. The bottom two chromatograms show the results obtained
7.5 Practical Relevance of Results of the Ideal Model
381
10 11 Time [min.]
10 11 Time [min.] >40 6 20
b
I
is
en
7
8 9 Time [min.]
7
8 9 Time [min.]
12
70 I60 d °2 50* 10
1
1&().5
'
ci
0
04
10
Figure 7.13 Plateau perturbation and tracer peak, aj Chromatogram of ISMN on a plateau of T-ISMN under linear conditions. Solid line: UV signal; dotted line radiodetector signal, a// Calculated chromatogram under the same conditions as above, b; Chromatogram recorded as in a/ above but under nonlinear conditions, bjj Converse chromatogram of b/, perturbation of a plateau of ISMN by a pulse of T-ISMN. Reproduced with permission from ]. Samuelsson et al., Anal Chem., 76 (2004) 953 (Fig. 4). ©2004 American Chemical Society.
under nonlinear conditions. The third one corresponds to an injection of ISMN on a plateau of T-ISMN. The solid line shows the total perturbation as observed under conventional conditions while the dotted line shows the actual perturbation of the plateau of T-ISMN. The dotted line in the last chromatogram shows the elution of the T-ISMN tracer peak on a plateau concentration of ISMN. The plateau concentrations being the same in the last two chromatograms, the perturbations
Band Profiles of Single-Components with the Ideal Model
382
40 30 20
\
10 Q
10
\
i
\
12 Time [min.]
14
Figure 7.14 Retention times of the plateau and tracer peaks. Experimental chromatograms showing the perturbation peak and the tracer peak recorded over a series of concentration plateaus. The lines are plots versus the plateau concentration of the retention times of the plateau perturbations (fat solid line) and of the tracer peaks (dashed line). The fat solid line is also the elution profile of a large rectangular pulse (with an initial condition QK;ef = 0) given by Eq. 7.4. That of the dashed line is Eq. 7.10. Reproduced with permission from J. Samuelsson et at, Aval. Chem., 76 (2004) 953 (Fig. 6). ©2004 American Chemical Society.
of T-ISMN on a plateau of ISMN and the converse perturbation have the same profiles. It is easy to see in Figure 7.13 how the combined result under conventional conditions, when recorded with a nonselective detector, is the solid line in the third chromatogram. This result is confirmed by the results obtained with the racemic mixture of methyl mandelate. Finally, Figure 7.14 shows the dependence of the retention time of the perturbation peak on the plateau concentration (Eq. 7.4) and that of the retention time of the tracer pulse (Eq. 7.10). It is important to note that the perturbation and the tracer peaks coincide under linear conditions but that they are resolved under nonlinear conditions.
7.5.2 Comparison of Experimental and Calculated Elution Band Profiles The experimental band profiles in Figures 7.15a and 7.15b are overlaid with the profiles calculated as solutions of the ideal model, using the isotherms determined previously by frontal analysis on the same column [46]. Note that using isotherms determined by the ECP method would lead us into a circular argument. In the first case (Figure 7.15a), the profiles were obtained for phenol on a column packed with Cis bonded silica (reversed phase LC), eluted with a 20:80 mefhanol-water mixture, with loading factors between 2 and 11%. In the second case (Figure 7.15b), the profiles were obtained for benzyl alcohol on silica (nor-
383
7.5 Practical Relevance of Results of the Ideal Model
BENZYL ALCOHOL THF/n-heptane (15/85) on SiOj
PHENOL Methanol/water (20/80) on CIS Silica
1 2 3 4 5
1-41 2.82 4.23 5.64 7.05
mg mg mg mg mg
2.5 1G.0 10-5 1L0 11.5 t2.0 12-3 I3.Q .1X5 W.Q W.3 15.0
7.0 7.5
8.0
6.5 9.0
9.5 10.0 10.5 110 11.5 12.0 12.5 13.0
Figure 7.15 Comparison between the solution of the ideal model and experimental band profiles. Phenol on Cis silica, sample loading factors (%): 1, 2.1; 2, 4.3; 3, 6.4; 4, 8.5; and 5, 10.7. Benzyl alcohol on silica, sample loading factors (%): 1, 0.47; 2, 0.95; 3,1.9; 4,3.8; 5,4.6; and 6, 5.7. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 462 (Figs. 1 and 2), ©1989 American Chemical Society.
mal phase LC), eluted with a 15:85 THF/n-heptane mixture, with loading factors between 0.5 and 6%. When the experimental profiles (symbols) are compared to those calculated (solid lines), they are found to be very similar, differing mainly by the very steep front of the experimental profiles as opposed to the vertical front of the ideal model, and by the diffuse tail. We see in these two figures that the front of the peak straightens up, that its retention time increases, and that the difference between the experimental and the profiles calculated with the ideal model decreases with increasing value of the loading factor. The diffused tail finds its origin in the finite column efficiency, due to the axial dispersion and the finite rate of the mass transfer kinetics. At low concentrations the rear of the band profile tends to become identical to the profile observed with small samples. It returns only asymptotically toward the base line. Elution profiles of several continuous linear boundary conditions of A-tertbutyl phenol on a Kromasil-Cis column having more than 20 000 theoretical plates were recorded under various experimental conditions. Examples of these experimental results are shown in Figure 7.16 where they are compared to profiles calculated from the ideal model [23]. This demonstrates how, under nonlinear conditions, a linear concentration ramp injected into a column transforms into a curved
REFERENCES
384
50-
O
0
500
1000
1500
2000
2500
t(s)
Figure 7.16 Comparison between the calculated (dotted lines) and the experimental (solid lines) elution profiles of 4-tert-butyl phenol. Mobile phase, 65:35 (v/v) methanol/water; linear velocity UQ = 0.175; phase ratio, F = 0.749; column length, L = 25 cm; Langmuir isotherm constants, a = 6.84, b = 0.0437 1/g. Concentration of 4-tert-butyl phenol, 75 g/1. The durations of the injection ramp, th, from left to right are 450, 650,1000,1400, and 1900 seconds. Reproduced with permission from T. Ahmad, F. Gritti, B. Lin and G. Guiochon, Anal. Chan., 76 (2004) 977 (Fig. 5c). © 2004 American Chemical Society.
concentration profile, with formation of a shock and illustrates the growth of this shock and the decrease of its elution time with increasing slope of the injected ramp. There is an excellent agreement between the experimental results, the retention time of the solute front when a shock is not yet formed at elution, and the breakthrough profiles calculated numerically. These results justify the attention we have devoted to the ideal model. This model can provide a rather accurate picture of the band profile we may expect to observe at high concentrations.
References [1] H.-K. Rhee, R. Aris, N. R. Amundson, First-Order Partial Differential Equations - II. Theory and Application of Hyperbolic Systems of Quasilinear Equations, PrenticeHall, Englewood Cliffs, NJ, 1989. [2] B. C. Lin, S. Golshan-Shirazi, Z. Ma, G. Guiochon, Anal. Chem. 60 (1988) 2647. [3] A. I. Vol'pert, S. I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Martinus Nijhhof, Dordrecht, The Netherlands, 1985. [4] E. Wicke, Kolloid Z. 86 (1939) 295. [5] J. N. Wilson, J. Am. Chem. Soc. 62 (1940) 1583. [6] D. DeVault, J. Am. Chem. Soc. 65 (1943) 532. [7] E. Glueckauf, Nature 156 (1945) 205. [8] E. Glueckauf, Proc. Roy. Soc. A186 (1946) 35. [9] H. C. Thomas, J. Chem. Phys. 20 (1952) 157.
REFERENCES [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]
385
E. Cremer, G. H. Huber, Angew. Chem. 73 (1961) 461. J. Weiss, J. Chem. Soc. (1943) 297. A. J. P. Martin, R. L. M. Synge, Biochem. J. 35 (1941) 1359. L. C. Craig, J. Biol. Chem. 155 (1944) 519. L. Lapidus, N. R. Amundson, J. Phys. Chem. 56 (1952) 984. J. J. van Deemter, F. J. Zuiderweg, A. Klinkenberg, Chem. Eng. Sci. 5 (1956) 271. E. Glueckauf, J. Chem. Soc. (1947) 1302. E. Glueckauf, Disc. Faraday Soc. 7 (1949) 12. G. Guiochon, L. Jacob, Chromatographic Review 14 (1971) 77. H. K. Rhee, R. Aris, N. R. Amundson, Philos. Trans. Roy. Soc. London A267 (1970) 419. S. Golshan-Shirazi, G. Guiochon, J. Phys. Chem. 94 (1990) 495. J. Hong, Bioprocess Technol. 9 (1990) 495. L. Jacob, P. Valentin, G. Guiochon, J. Chim. Phys. (Paris) 66 (1969) 1097. T. Ahmad, F. Gritti, B. Lin, G. Guiochon, Anal. Chem. 76 (2004) 977. R. Courant, K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, NY, 1948. P. Lax, Commun. Pure Applied Math. 7 (1954) 159. P. Rouchon, M. Schonauer, P. Valentin, G. Guiochon, Separat. Sci. Technol. 22 (1987) 1793. P. Lax, Commun. Pure Applied Math. 10 (1957) 537. P. Lax, in: E. H. Zarantello (Ed.), Contributions to Non-Linear Functional Analysis, University of Wisconsin, Madison, WI, 1971, p. 603. O. A. Oleinik, Amer. Math. Soc. Transl. Ser. 2 33 (1963) 285. H.-K. Rhee, R. Aris, N. R. Amundson, Phil. Trans. Roy. Soc. London A269 (1971) 187. S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 60 (1988) 2364. G. Zhong, T. Fornstedt, G. Guiochon, J. Chromatogr. A 734 (1996) 63. P. Sajonz, T. Yun, G. Zhong, T. Fornstedt, G. Guiochon, J. Chromatogr. A 734 (1996) 75. D. Graham, J. Phys. Chem. 57 (1953) 665. R. J. Laub, ACS Symp. Ser. 297 (1986) 1. S. Jacobson, S. Golshan-Shirazi, G. Guiochon, J. Amer. Chem. Soc. 112 (1990) 6492. G. Zhong, G. Guiochon, Chem. Eng. Sci. 51 (1996) 4307. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 506 (1990) 495. C. Bosanquet, D. G. Morgan, in: D. H. Desty (Ed.), Vapour Phase Chromatography, Butterworths, London, UK, 1957. F. Helfferich, Ind. Eng. Chem. Fundam. 6 (1967) 362. G. Klein, D. Tondeur, T. Vermeulen, Ind. Eng. Chem. Fundam. 6 (1967) 339. L. Jacob, G. Guiochon, Bull. Soc. Chim. France 3 (1970) 1224. L. Jacob, G. Guiochon, Chromatogr. Rev. 14 (1971) 77. L. Jacob, G. Guiochon, J. Chim. Phys. (Paris) 67 (1970) 185. J. Samuelsson, P. Forssen, M. Stefansson, T. Fornstedt, Anal. Chem. 76 (2004) 953. S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 61 (1989) 462.
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Chapter 8 Band Profiles of Two Components with the Ideal Model Contents 8.1
General Principle of the Solution 8.1.1 Statement of the Problem and Its Constraints 8.1.2 Properties of the System of Mass Balance Equations 8.2 Elution of a Wide Band With Competitive Langmuir Isotherms 8.2.1 Position of the Two Concentration Shocks 8.2.2 Rear Diffuse Profiles of the Two Components 8.2.3 The Intermediate Plateau on the Rear Diffuse Profile of the Second Component . 8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms 8.3.1 Retention Time of the Second Concentration Shock 8.3.2 Maximum Concentration of the Two Components in the Mixed Zone 8.3.3 Elution Profile of the First Component between the Two Shocks 8.3.4 Retention Time of the First Shock 8.4 Method of Calculation of the Ideal Model Solution in a Specific Case . . . . 8.4.1 Case 1: Wide Injection, L < z,, or L p > L^2 8.4.2 Case 2: Injection Plateau Eroded, Pure First Component Plateau Present, zj < L < z} or I* > Lfr2 > L]f2 8.4.3 Case 3: Narrow Injection and Mixed Zone, z; < L < ZK or Lj 2 > L^2 > Lj2 8.4.4 Case 4: Touching Bands, Second Component Plateau Present, ZK < L < zi or Lf,2 > L/,2 > Lji2 8.4.5 Case 5: Resolved Bands, zL < L or L^2 > L/,2 8.4.6 Influence of the Width of the Injection Pulse 8.5 Dimensionless Plot of a Two-component Band System 8.6 The Displacement Effect 8.6.1 Origin of the Displacement Effect 8.6.2 Intensity of the Displacement Effect 8.7 The Tag-Along Effect 8.7.1 Origin of the Tag-Along Effect 8.7.2 Intensity of the Tag-Along Effect 8.8 The Ideal Model in Gas Chromatography 8.9 Practical Relevance of the Ideal Model References
390 390 391 395 396 398 400 401 403 404 405 406 407 408 409 410 411 413 413 414 416 418 418 419 419 420 421 423 436
Introduction. Retrospective Wilson [1] and DeVault [2] both discussed the theoretical statement of the twocomponent problem and its solution in the case of the elution of a pulse of a bi387
388
Band Profiles of Two Components with the Ideal Model
nary mixture. DeVault showed that with Langmuir competitive isotherms, the two band fronts are self-sharpening, as in the single-component case [2]. He further showed that before complete resolution is achieved between the two bands, three zones can be distinguished. In the front zone the first component is pure, and its concentration in this zone is higher than in the feed solution. In the rear zone, the second component is pure, and its concentration tends to be smaller than in the middle zone, where both components are present [2]. These are the first theoretical suggestions of the displacement and the tag-along effects and of their linkage to the competition of the two components for interaction with the stationary phase. A further step was made by Offord and Weiss [3], who assumed, without demonstration, that the velocities associated with the concentrations Q and C2 found simultaneously on a diffuse boundary (in the mixed zone) are equal. This assumption, which provides the key to the calculation of the diffuse boundaries of the two components in the mixed zone, was analyzed much later by Helfferich and Klein [4] when they developed the concept of coherence. Glueckauf was the first to investigate in detail the mathematical properties of the ideal model and to give a comprehensive analysis of the chromatographic separation process in the elution mode for two solutes following the competitive Langmuir adsorption model [5,6]. Glueckauf [5] calculated the individual band profiles of the two components during their migration and progressive separation. He showed the existence of two discontinuities, one in front of each component band, and the existence of a concentration plateau on the rear diffuse boundary of the second band. Coates and Glueckauf determined experimentally the individual band profiles of the two components by collecting fractions with an "automatic device" [7]. Except for the severe tailing of the second component band (which could possibly have been accounted for by a bi-Langmuir, or a LangmuirFreundlich isotherm or might have been due to slow mass transfer kinetics), good agreement was obtained between experimental and theoretical profiles. Glueckauf extended these theoretical results to two-component separations with a competitive Freundlich1 isotherm [8]. Unfortunately, the mathematical derivation reported was complex, the solution described was not entirely rigorous, and it was obfuscated by a system of symbols which has not been used by other authors. The work of Glueckauf remained almost ignored by chromatographers, in part because of its complexity, in part because chromatographers, who are mainly analysts, have been concerned exclusively with the linear regime of chromatography during the past 40 years. It was better known by chemical engineers, undaunted by the mathematics, but the result was hardly better. Chemical engineers were slow to recognize that chromatography is not merely a scaled-up analytical method. It is a sophisticated, specialized adsorption process, with unparalleled separation power for closely related nonvolatile compounds. The problems that plagued Glueckauf's solution, especially its complexity, were related to the poor state of development of the theory of partial differen1 For obvious practical reasons (see Chapter 3, Section 3.2.2.4 and Chapter 4, Section 4.1.9), the chromatographer will avoid at all costs using a stationary phase with a Freundlich isotherm. Thus, we do not think that the discussion of the solution for a Freundlich isotherm is relevant.
Band Profiles of Two Components with the Ideal Model
389
tial equations at the time. Progress in this theory in the 1950s and early 1960s allowed a reformulation and a clarification of the solution on a more general, more complete, and more rigorous basis. This was permitted by the development of the theory of characteristics and of the shock theory (see Chapter 7, Section 7.2.2). Using these tools, Rhee, Aris, and Amundson discussed in 1970 the chromatographic cycle and the separation of components in the column from a pulse injection as well as the progressive development of a separation in a time-distance space [9]. The primary purpose of their paper was to explain the mathematics of one-dimensional, isothermal, ideal equilibrium models of chromatography and the theory of simple waves [10]. The basics of shock theory, the concept of weak solutions, and the selection of the weak solution of the physical problem have been discussed in Chapter 7, Section 7.2.3. Their approach is particularly well suited to the study of liquid-solid chromatography. Also in 1970, Helfferich and Klein [4] developed a theory of multicomponent chromatography based on the ideal model of chromatography, the concept of coherence, and the use of the competitive Langmuir isotherm model [4]. They used the fo-transform to calculate the composition trajectories in distance-time diagrams. Although this approach could be extended to the study of overloaded elution chromatograms [11,12], this theory has been applied primarily to the calculation of displacement chromatograms (see Chapter 9). It is better designed for ion-exchange chromatography, although its extension to adsorption chromatography is straightforward. This approach is discussed mainly in the next chapter on displacement chromatography. The theory of simple waves applies to large-volume injections, i.e., to the profiles obtained upon injection of rectangular profiles which are so wide that the injection plateau has not been entirely eroded when the band elutes. Then, simplifications of the solution occur because there is a constant state, the concentration plateau. This solution is not valid in overloaded elution chromatography when the injection volume is sufficiently small that the injection plateau has eroded and disappeared by the time the band elutes from the column. It is important to discuss this solution, however, because it takes a finite time for the profile of even a narrow rectangular injection to decay, and the band profile during that period is given by the simple wave solution. Also, this solution is the basis for a method of determination of competitive equilibrium isotherms (Chapter 4, Section 4.2.4). Finally, Kvaalen et al. have shown that the system of equations of the ideal model for a multicomponent system (see later, Eqs. 8.1a and 8.1b) is strictly hyperbolic [13]. As a consequence, the solution includes two individual band profiles which are both eluted in a finite time, beyond the column dead time, to = L/u. The finite time that is required for complete elution of the sample in the ideal model is a consequence of the assumption that there is no axial dispersion. It contrasts with the infinitely long time required for complete elution in the linear model. This difference illustrates the disparity between the hyperbolic properties of the system of equations of the ideal model of chromatography and the parabolic properties of the diffusion equation. In summary, there is an analytical solution of the two-component ideal model in one single, very specific case: that of a rectangular pulse of a mixture of two
390
Band Profiles of Two Components with the Ideal Model
components which follow the competitive Langmuir isotherm [5,6]. Otherwise, numerical solutions are required for the determination of the individual band profiles of the mixture components (Chapter 11). Unfortunately, none of these earlier publications, and especially not the landmark papers of Glueckauf [5], Rhee et al. [9], Helfferich and Klein [4], contain the details of the solution that are needed in order to calculate the exact individual band profiles predicted by the ideal model knowing the isotherm coefficients and the relevant characteristics of the column and the mixture. These equations were derived later [12,14] and will be discussed in this chapter. We present here the principles of that solution and discuss its properties and their consequences. For a detailed discussion of the derivation of the equations, the reader is referred to [14]. Use of the equations collected in Tables 8.1 to 8.5 (see Appendix, at the end of this chapter) permits the determination of the individual band profiles of the components of a binary mixture knowing the parameters of the system. Only simple algebraic calculations are required.
8.1 General Principle of the Solution There is no general solution for the individual band profiles of the components of a binary mixture that is comparable to the solution of the single-component band profile problem. The solution that we present here is valid only when the adsorption behavior of the two components follows the Langmuir competitive model. In this section, we state the problem in the general case. In the next two sections, we derive successively the solutions in the cases of a wide and a narrow rectangular injection pulse, respectively. As the injection of a finite amount of the binary sample cannot be made instantaneously, the band profile in the column is the profile obtained in the case of a wide rectangular injection until the top concentration plateau formed during the injection is eroded. Then the solution must take into account the progressive decrease in the maximum band concentration during its migration. In turn, the main consequence of the band decay is a decrease in the propagation velocity of the two concentration shocks.
8.1.1 Statement of the Problem and Its Constraints In the ideal model, we assume that the column efficiency is infinite, hence the rate of the mass transfer kinetics is infinite and the axial dispersion coefficient in the mass balance equation (Eq. 2.2) is zero. The differential mass balances for the two components are written:
where Q and qi are the concentrations of component i in the mobile and the stationary phases, respectively. Since the column efficiency is infinite and the mobile
8.1 General Principle of the Solution
391
and stationary phases are in constant equilibrium, the concentration of each component in the stationary phase, qi, is related to the concentrations of both components in the mobile phase by the isotherm equation (Chapter 3, Section 3.2): ^
(8.2)
Equation 8.2 states that the two components compete for interaction with the stationary phase, following the Langmuir competitive model. Since the stationary phase concentration of each component at equilibrium is a function of both concentrations in the mobile phase, the two partial differential Eqs. 8.1a and 8.1b are coupled. This coupling increases considerably the complexity of the mathematical problem, compared to the single-component case. The initial condition for the solution of the system of Eqs. 8.1a and 8.1b corresponds to a column filled with pure mobile phase: t < 0,
Ci(f,z) = C 2 (f,z)=0
(8.3)
The boundary condition corresponds to the injection of a rectangular pulse of finite width, tp, and height, Cj, C^- Because the column efficiency is infinite in the ideal model, we can write the boundary condition: 0 < t < tv,
d(f,0)=C°,
C2(t,0) = c£
(8.4)
and tp < t,
Ci(f,0) = C2(f,0) = 0
(8.5)
More generally, the injection will be characterized by the concentration profile, Ci(t,0) =
8.1.2 Properties of the System of Mass Balance Equations The system of Eqs. 8.1a and 8.1b is the classical system of reducible, quasilinear,2 first-order partial differential equations of the ideal model of chromatography [1, 2,4-6,9-17]. The properties of these equations have been studied in detail [4,9,10, 18-24]. We discuss here those properties that are important for the understanding of the solutions of the ideal model in the case of elution or displacement of a binary mixture. They are the existence of characteristic lines, called characteristics, the coherence condition, and the properties of the hodograph transform. Because the feed injected is a mixture, a binary mixture in the case discussed here, there will be a region in the column where the two components are simultaneously present. This region propagates along the column while it progressively separates into two bands of pure compounds. In the mixed zone, the concentrations, C\ and Ci, of the two compounds that coexist at a certain time, in a certain point, are not randomly associated. They must satisfy a constraint, their velocities must be equal. So, they will propagate together, as long as they exist. However, 2 This means that Eqs. 8.1a and 8.1b are linear with respect to the derivatives but the coefficients of these derivatives depend on the variables.
392
Band Profiles of Two Components with the Ideal Model
as already explained in Chapter 7 (Section 7.2.4), matter, i.e., the molecules of the solutes in a slice at concentrations C\ and C2, does not propagate at the velocity of the concentrations C\ and C2 but more slowly (in the case of a convex upward isotherm). In the process, the corresponding slice becomes thicker (Chapter 7, Figure 7.4) and more dilute. This permits the separation. Because dilution proceeds ceaselessly in chromatography, there comes a time when the concentrations Q and C2 disappear. It is easily shown [14] that the velocities associated with a given concentration of each component are given by:
uz2
U
=
-FT-
(8.6b)
where the differentials D ^ / D Q are the two directional derivatives: Dqi DQ
_ "
DC2
dqx dC2 dqx + dC~1 dC~1dC2 dC2
dC2 dCi
K
'
Both velocities uZi\ and uZ/2 are functions of both concentrations, C\ and C2. The most important property of the set of Eqs. 8.1a and 8.1b is the equality of these two directional derivatives. As a consequence, the characteristic velocities uZ/\ and uZ/2 associated with the set of concentrations (Q, C2) are equal: «z,i = «z,2
(8.8)
This equality defines a relationship between the concentrations C\ and C2 that defines which are the couples of concentrations that may coexist during the separation. Equation 8.8 states the coherence condition of Helfferich and Klein [4], which had been postulated previously by Offord and Weiss [3] and by Glueckauf [5,6,8]. As shown by Helfferich and Klein, coherence is a state of stability that develops from arbitrary noncoherent conditions, so the distance-time regions where noncoherence prevails are finite [4]. Using the condition in Eq. 8.8, characteristic lines can be constructed to describe the migration of the two component bands and their progressive separation [10,18,25]. Inserting Eqs. 8.6a and 8.6b into Eq. 8.8 and using the equality of the two directional derivatives (Eqs. 8.7a,b), we can write: dq2
dq2 dC2
(8 9)
'
This equation was derived first by Offord and Weiss [3] and later by Glueckauf [5], using different approaches. In the case of competitive Langmuir isotherms, Eq. 8.9 becomes: - (a - 1 + afciQ - b2C2)r - b2Cx = 0
(8.10a)
8.1 General Principle of the Solution
393
where r = dC\/'dC2. We will use Eq. 8.10a in the next two subsections to discuss the characteristic pathways and the individual elution band profiles. Equation 9.13 is the generalization of Eq. 8.10a in the case of an n-component system. Equation 8.10a has two roots of opposite signs. Let r\ be the positive one and r2 the negative one. These two roots define two families of characteristic lines. One line of each family goes through any point of the (Q, C2) plane which is external to the parabola [cc — 1 + cOo\C\ — b2C2]2 + ^.ab\b2C\C2 = 0. Equation 8.10a, which is a linear relationship between C\ and C2 can be rewritten for future use as Q = rC2 - (* ~ ^
(8.10b)
abr + b
Another method of representation of the solutions of the system of Eqs. 8.1a and 8.1b is the use of the graphs obtained with the hodogmph transform [10,19-22]. Fundamentally, the hodograph transform gives the integral curves of the two characteristic fields, C(A, t), or the eigenvectors of the matrix associated with the system of Eqs. 8.1a and 8.1b. For practical purposes, the hodograph transform exchanges the roles of the functions, C\, C%, and of the coordinates, t, z. The two concentrations C\ and C2 are defined at any time, at any point of the column, except at the shock locations. They are solutions of the system of Eqs. 8.1a and 8.1b and give the concentration profiles along the column at a certain time, or, conversely, the elution profiles, i.e., concentration profiles in time, at the column exit, or point z = L. The hodograph transform of these profiles are the column location, x{Ci,C2), and the time, t(Ci,C2), where the two concentrations C\ and C2 are found simultaneously. The hodograph transform can be performed whenever the Jacobian:
f|f = r'
(an,
has a finite value, different from zero. In the general case of a binary mixture, the Jacobian matrix (Eq. 8.11) is of rank 2. However, the hodograph transform of the solution of Eqs. 8.1a and 8.1b becomes singular if this matrix does not have its full rank [10,19-22]. This is always the case when the solution has a constant state (e.g., C\ and C2 are constant in a certain (z, i) domain) and, for example, when the injection is a rectangular pulse wide enough to prevent the complete erosion of the concentration plateau during its elution; see Figure 8.1. In such a case, the characteristics of the (t, z) space are mapped into a single point, F, of coordinates C®, C2 and two lines intersecting in that point and intersecting the axes of coordinates in A and B. This linear relationship between the concentrations C\ and C2 that may be found on either the front or rear profiles demonstrates that only certain couples C\, C2 can travel together. These solutions are called simple wave solutions. The equations of the two lines of the hodograph transform are related to the competitive isotherms of the two components. The hodograph transform of the profiles in Figure 8.1 is shown in Figure 8.2. In the case of a competitive Langmuir isotherm, these lines are straight lines [10,21,26] and the determination of their slopes and intercepts permits the calculation of the four isotherm parameters for the components of a binary mixture [26]
Band Profiles of Two Components with the Ideal Model
394 C mg/mL
4 Δt At,1
8.4
C2o
C11o°/C °"l/2 C /C2o2=1/2
8.23b 8.13a 8.13a
8.25 8.25
N.
C1A
Δt2
8.13b 8.13b
C2B
L=10 Cm cm
8.33 Δt1 At,
8.4 8.4
C1o
\
V" i 8.26a
8.23b
Mixed Zone
First First Zone
\ ^
'
8.28 8.28
Third Zone
\
0 8
tR,1
tR,2
tF,1
tE
tI
tR,2+tp
8.17
8.19
8.23a
8.27
8.29
7.27
14 t
min
Figure 8.1 Elution profile of a wide rectangular injection pulse of a 1:2 binary mixture. Individual elution profiles of the two components. Band profiles: thick solid line, first component; thin solid line, second component. Bands of a 1:2 mixture, calculated under the same conditions as the figures below. Conditions for the calculations: Phase ratio: F = 0.25; column length 10 cm, diameter 0.46 cm; Flow rate: Fv = 1 mL/min; Relative retention: a = 1.2; retention factor: kol= 6.0; other isotherm coefficients: &i = 6; bi = 7.2; sample size: 66.7 ^mol; injection volume: 2.5 mL.
(see Chapter 4, Section 4.2.4). The two slopes are the roots Y\ and r2 of Eq. 8.10b. As long as the plateau at the top of the injection pulse is not completely eroded, for example, in the case of the injection of a wide rectangular pulse, the boundary conditions for Eq. 8.10b are C\ = Cj and C2 = C\, and Eq. 8.10b becomes cthC^r2 - (« - l + a&jC? - b2Cl)r - b2C\ = 0
(8.12)
Equation 8.12 has two roots, T\ (positive) and r2 (negative), as seen in Figure 8.2. The two intercepts are:
a-1
(8.13b) b2 We have shown (see [27]) that the root r\ of Eq. 8.12 satisfies the inequalities: C° b C° 4- ^ ^ < ri < ; i q 0*
(8.14)
In most practical cases, b\C\ » (oc — 1)/'oc, so, as a first approximation, we can assume that r\ ~ C®/C%, and Cf is quite small. When C° is small, however, this approximation may cease to be valid.
8.2 Elution of a Wide Band With Competitive Langmuir Isotherms
395
C1 mg/mL
3.5
A
Figure 8.2 Hodograph transform of the elution profile of a wide rectangular injection pulse of a 1:2 binary mixture. F
O
C1
0 0
B
o
C
2
mg/mL
C2
4
Note that Eq. 8.13a is made of three equations and Eq. 8.13b of two. The first equation in each case is derived from the equation of the characteristic lines (Eq. 8.10b). The second equation in each case is obtained by observing that the product of the roots of Eq. 8.12 is - ( ^ ^ / ( f t ^ C ! ? ) , hence r2 = -(fc 2 Cj)/(a&iC^i). The concept of the hodograph transform is important because its use can be extended to the representation of the solutions of the equilibrium-dispersive model (Chapter 11), a model that accounts accurately for a wide majority of the experimental results. The hodograph transform of the solution of the system of Eqs. 8.1a and 8.1b will help us to understand this solution in the case of a wide rectangular injection pulse [14,26]. The hodograph transform is also useful for the derivation of the solution in the case of a narrow rectangular injection pulse, a case in which points A and F no longer exist [10-13,15]. The composition of the rear diffuse boundaries of the two component bands is still transformed into part of the segment FB and the segment BO, while the concentrations Cf of the first component on the rear of the first shock and C^-C^ of the two components on the rear of the second shock are the coordinates of the intersection points of the C\ axis and the segment FB with a straight line parallel to segment AF [14].
8.2 Elution of a Wide Band With Competitive Langmuir Isotherms The solution of this problem is based on the simple wave theory summarized in the previous section and on the existence of a constant state that corresponds to the elution of the remaining part of the injection plateau of a wide band [12,14, 15,26]. In this case, we have achieved only partial separation. We will consider successively the elution profiles of the first and second components (see Figure 8.1). We assume a rectangular injection of concentrations C° and C°, and width
396
Band Profiles of Two Components with the Ideal Model
tp, large enough that the elution profiles of the two components have a plateau at the injection concentrations Cj and Cj, respectively. We assume also that the two components follow the competitive Langmuir isotherm model [15,26] (Eq. 8.2 and Chapter 4, Section 4.1.2). The hodograph transform gives us the following information in the case of competitive Langmuir isotherms [14,15,26]. As explained in the previous section, the transform of the individual elution band profiles corresponding to a rectangular injection of a binary mixture is a plot of C\ versus Ci, which is made of two straight lines intersecting at the point C\, C°; see Figure 8.2. These lines are the characteristic lines through the constant state (C°, C°). They intersect the axis of coordinates in points A and B. During the elution of the two components, the composition of the eluent is represented by a point in the Q , Ci plane which moves from O (0, 0) to A (Cf, 0), F (C°, C°), B (0, Cf), and O (Figure 8.2). However, the point representing the eluent composition does not move at a constant velocity on this trajectory. Other considerations, which follow, are needed to complete this description.
8.2.1 Position of the Two Concentration Shocks The velocities associated with the concentrations Q and C2 are given by Eqs. 8.6a, 8.6b, and 8.8. The velocities of the concentration shocks are given by [10]:
Since the competitive Langmuir isotherms are convex upward, the velocities uZ/; associated with a concentration of either component increase with increasing concentration of each of the components. Thus, the front shock of the rectangular injection profile is stable for both component profiles while the rear shock of this injection is not stable and collapses. The rear parts of the two band profiles will be diffuse boundaries. Thus, the two segments OA and AF of the hodograph plot, Figure 8.2, correspond to shocks and the representative point jumps from O to A and later from A to F. The intermediate points on the segments OA and AF do not correspond to any actual eluent composition, in the framework of the ideal model. In practice, however, if the column efficiency is high enough, the eluent composition will follow paths that are very close to these lines; see Chapter 11. Since the front shock is stable and is a shock of the pure first component concentration, its velocity is given by Eq. 7.7 [14,15]. At the band front, the concentration jumps from 0 to Cf (Eq. 8.13a), hence:
Since the concentration Cf is constant as long as the injection plateau has not been eroded, the retention time of the front of the first component (a shock with
8.2 Elution of a Wide Band With Competitive Langmuir Isotherms
397
an amplitude equal to the segment OA in the hodograph plot, Figure 8.2) is Z
(8.17)
1+
where z is the migration distance. We note that Cf (Eq. 8.13a) is higher than Cj (r2 is negative, by definition), so the first component band front is eluted faster when component 2 is present in the sample than when it is not, and that the retention time of the front of the first band decreases with increasing concentration C2, at constant values of the sample volume and of Cj. This is the origin of the displacement effect [14,15,28] (see Section 8.6). When the second component front appears, the concentration of the second component jumps from 0 to C2, while at the same time the concentration of the first component falls from C^ to C°. This is a simultaneous concentration discontinuity. It corresponds to the segment AF in the hodograph plot (Figure 8.2). Afterward, the concentrations of both components in the eluent remain constant and equal to their concentrations in the feed until the end of the injection plateau and the beginning of the diffuse rear profiles. This second front shock moves at the velocity associated with the concentration discontinuity AC2 — C2 in the presence of a concentration C° of the first component (see Figure 8.1, conditions at the rear of the second shock). From Eq. 8.15 and since Aq2 is equal to qz{C\, C2), this velocity is Us 2 = ^-r- = s 2 ' l+F ^ 1+
%
Fa
(8.18) *
With a wide rectangular injection, the shock velocity USi2 is also constant and the retention time of the second shock is [14,15]
tR 2
' ~I
Fa2
(8.19)
A concentration plateau is eluted, at the same composition as the feed. After the end of this plateau, the concentration of the first component decreases progressively from Cj to 0 while the concentration of the second component decreases from C° to Cf (Eq. 8.13b), the point representative of the eluent composition following the segment FB in the hodograph plot, Figure 8.2. The rear shock of the rectangular injection is not stable, because the corresponding characteristics fan out, as they did in the case of a single-component band, Figure 7.1. We now calculate the time when the injection decay begins and the equation of the rear part of the two band profiles [14]. The last point on the concentration plateau for the first component is also the point at concentration C° on a continuous profile, where it moves at the velocity M2/I, as given by Eq. 8.6a. Using the competitive Langmuir isotherm equations, the directional derivatives (Eq. 8.7a) become
bi(C2-LL,.LJJ DQ
(1 + b i d + hC2)2
(1 + hCx + b2C2)2
(s20)
Band Profiles of Two Components with the Ideal Model
398
since the ratio dC2/dC\ is the reverse of the slope r\ of the straight line FB (Figure 8.2). Replacing C° - C\lrx by Cf (Eq. 8.13b) gives (8.21)
1+ The velocity of the beginning of the continuous rear part of the profile of the two components is therefore (see Eq. 8.8): u u (8.22) < l = < 2 = «z,l(C?, C2°) = Fgl(l+&2Cf)
1+
1+
where 7 = a / ( l + \>2C\) = {cOo\rx + b2)l(b\r\ + b2). Its elution time is3 [14]: z
(8.23a)
1+
Thus, the width of the residual injection plateau at the column outlet (z = L and t0 = L/u) is Fa2t0
= h,i - tR/2 =
-1
(8.23b)
The condition for the validity of the equations derived in this subsection is that ^R,2 (Eq. 8.19) needs to be shorter than tFi\, so the front shock of the second component elutes before the injection plateau ends. Otherwise, the plateau vanishes, the front shocks are eroded, and their velocities are no longer constant. Then, the problem becomes the elution of a narrow rectangular injection pulse; see next section.
8.2.2 Rear Diffuse Profiles of the Two Components The retention time of a concentration C2 of the second component on the continuous profile is derived from its associated velocity (Eq. 8.21):
1+ -
&1C1 + &2C2)2
1+
(8.24)
Fa2
Elimination of Q between Eqs. 8.10b and 8.24 gives the concentration C2 at the column exit [14,15] C2 =
7
t-tp-to
-1
(8.25)
3 There is no tp term in Eqs. 8.17 and 8.19 because they correspond to the elution of the front shock of the injection, whereas we are now dealing with the elution of the rear boundary, which corresponds to the rear shock of the injection, a shock that entered the column a time tp later.
8.2 Elution of a Wide Band With Competitive Langmuir Isotherms
399
with z = L and to = L/u. Elimination of Ci between Eqs. 8.10b and 8.24 gives: Ci =
1
(8.26a)
Hence the retention time of a concentration C\ on the first component profile in the mixed zone is 7
(8.26b)
Equations 8.25 and 8.26a give the rear continuous profiles of the two components behind the feed concentration plateau in the mixed zone. They are valid as long as concentration C\ is different from 0, which eventually happens at time t-g. This time is given by writing that C\ is zero in Eq. 8.26a [14]:
l(t°K1 - t0) =
(8.27)
Combination of Eqs. 8.25 and 8.27 shows that when the first component concentration becomes zero, at t = tg, the second component concentration becomes equal to Cf. Then, past time tg, the remaining fraction of the second component band in the column is eluted as a band of pure second component (see previous section). This band profile of pure second component has two regions. First, in the region between times t% and t\, we have a plateau at constant concentration Cf, which corresponds to the point B in the hodograph plot, Figure 8.2. Second, in the region after time t\, the remaining fraction of the second component is eluted as a diffuse rear boundary. This corresponds to the segment BO in Figure 8.2. The rear part of the second component elution profile is given by the same equation as we have used previously for the rear, diffuse part of the elution profile of a pure compound, Eq. 7.4, or for a Langmuir isotherm, Eq. 7.12b, which we can rewrite as [14] l
R,2
t-tp-to
-1
(8.28)
Equation 8.28 gives the rear part of the profile. The time when the concentration C\ is eluted on this diffuse profile is (see Eq. 8.22 for the definition of 7): t0 1 +
(8.29)
This is also the time when the elution of the concentration plateau, C|, ends.
400
Band Profiles of Two Components with the Ideal Model
8.2.3 The Intermediate Plateau on the Rear Diffuse Profile of the Second Component One last problem remains: the connection between the two parts of the diffuse rear profile of the second component, which are given by two different equations, Eq. 8.25 for the mixed zone and Eq. 8.28 for the last zone, for the pure second component [14,15]. Comparison of these two equations shows that Eqs. 8.25 and
8.28 do not give the same time for the elution of the concentration Cf. The velocity associated with a concentration C2 on the continuous part of a mixed zone profile (simultaneous elution of the two components) is given by Eq. 8.21 (uz>\ = uZ£, Eq. 8.8):
At the end of the first component profile, when the concentration of the first component becomes zero and that of the second component is Cf, the limit of uZi2 is
On the other hand, the velocity associated with a concentration C® of a pure compound is given by Eq. 7.7: 1
(8 32)
\ (l+b2Cf )
"
Obviously, the term «i(l + foCf) *s different from a%. The difference between the two retention times, t£, when the concentration C | appears on the continuous rear profile of the second component at the end of the mixed zone, and t\, when it disappears on the continuous rear profile of the pure second component band, is (Eqs. 8.28 and 8.29):
7(71)., F- 7 ( 7 - 1 ) , ta2 2 *o — 2 i'RA2 ~~ HU (0.33) From its definition (Eq. 8.22), 7 can be transformed using Eqs. 8.13a and 8.13b into 7
l + bCf
hr + b
+
bn + h
Thus, 7 is always larger than unity and At2 is always positive. During the time between tg and f 7, the concentration of the second component cannot decrease: it is "too early" for the elution of a continuous rear profile of the band of the first component. A concentration discontinuity is impossible, as a rear discontinuity would be unstable. The concentration cannot increase either, which, in isocratic,
8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms
401
isothermal elution would be against the second law of thermodynamics. Thus, the concentration of the second component must remain constant during that time and a concentration plateau at C2 = Cf is formed [14]. This is the origin of the tag-along effect [28] (see Section 8.7). Figure 8.1 shows the elution profile of a wide injection band. The numbers of the corresponding equations are given on the different segments of the profile. These equations are listed in Table 8.1, as a summary. Finally, we can now answer the question, how is a wide injection pulse defined? It is an injection whose width is such that there is an elution plateau at the feed concentration. Thus, its width, tp, must be such that tR/2 < tF/1
(8.35)
where tR2 and tpi are given by Eqs. 8.19 and 8.23a, respectively. The condition is equivalent to: tp
>
b2C\
fl2-
b2C°
(8.36)
In summary, the wide rectangular profile is characterized by two concentration shocks, at times tRi\ and tRi2, for the first and the second component, respectively, by a residual of the injection plateau, and by a concentration plateau at Cf having a length At2. These characteristics define the three zones of the elution chromatogram of a binary mixture (Figure 8.1): the pure first component zone, the mixed zone, and the pure second component zone. Analytical solutions are provided to calculate the individual band profiles for a binary mixture, Table 8.1. We now study the elution profile of a narrow injection pulse, when Eq. 8.36 is no longer verified.
8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms The difference between the elution of a narrow band and the elution of a wide band, discussed in the previous section, is that in the former case the injection plateau has eroded during elution and, therefore, tR/2 (Eq. 8.19) is larger than t^t\ (Eq. 8.23a), making Eq. 8.36 no longer valid at z = L. After the injection plateau has disappeared, the heights of the two concentration shocks decrease continuously. Accordingly, the velocities of these shocks decrease and the solution of the problem becomes more complex. We will show that the first shock continues to move faster than the second shock and that the two bands eventually separate.
Band Profiles of Two Components with the Ideal Model
402 C mg/mL 7
o
o
C°11/C /C°2 =1/2 = l/2 C \
C2
L-10 cm L=10
8.41a
M
A \
8.13
C2 B
C1
8.46
8.25 8.25 Δt2 8.33
\
\
8.44
A'
\
\
First Zone Fir.t zone
Mixed Zone
1 |
8.41b
C1
M
^
8.26a
^
0 6
tR,1 8.47
tR,2 8.39
\
|(
V \
8.28 8.28
Third Zone Third Zone
!
\
tE 8.27
tI 8.29
7.27 121
12
toR,2+p
Figure 8.3 Schematic of the solution of the ideal model for the elution of a narrow rectangular pulse of a binary mixture. Individual elution profiles of the two components. Same conditions as for Figure 8.1, except injection volume = 200 }iL. The numbers on the different parts of the profiles, under the time of specific events and by important concentrations are the numbers of the corresponding equations.
The presentation in this section summarizes the results of publications that detail the derivation of the analytical solution of the elution problem in the case of a narrow rectangular pulse injection of a binary mixture following the Langmuir competitive equilibrium isotherm, using two different methods [12,14,15]. The first method is based on the theory of characteristics and on the theory of shock wave propagation. The second method is based on the concept of coherence and the use of the /i-transform described by Helfferich and Klein [4]. The primary assumption made in this derivation, as in the study of the propagation of a wide rectangular injection pulse, is that the concentrations of the two components in the mixed zone at a given time and location have the same velocity (Eq. 8.8). The solution of the problem is long and complex. However, its derivation follows the same line as the one presented in the previous section, the simpler case of a wide rectangular injection pulse. The essential features of the solution are similar in both cases. They originate from the competition that takes place between the two components when they interact with the stationary phase. The major features, such as the displacement and the tag-along effects, are common to both solutions. Accordingly, it does not seem necessary to reproduce here the details of the derivations, which can be found elsewhere [12,14,15]. Figure 8.3 illustrates a schematic of the solution of the ideal model in the case of a narrow rectangular pulse of a binary mixture with a competitive Langmuir
8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms
403
isotherm. The solution includes three zones: a zone of pure first component (between ts.ti and £#,2), a mixed zone (between ^ 2 and t^), and a zone of pure second component (between fg and fiR2 + *?)• The main features of this solution are two concentration shocks, one in the front of the first band (at £R,I)/ the other one between the first and the mixed bands (at £2^2); two diffuse boundaries for each component, one in the mixed band and one in the band where each component is pure; and a single concentration plateau at C | of length At2 [14]. Note that this solution for a narrow injection band assumes that the injection plateau has eroded away before the column exit. The equations giving the location and characteristics of these features are summarized in Tables 8.2 to 8.5. The choice of equations depends on the degree of resolution achieved between the two bands, i.e., complete resolution (Table 8.5), touching bands (Tables 8.3 and 8.4), or overlapping bands (Tables 8.1 and 8.2). Since the equations giving the rear diffuse profiles of the two components in the mixed zone and of the second component in the third zone where it is pure are the same for a narrow or for a wide injection band, it seems logical to begin here the description of the chromatogram. Equations 8.25 to 8.29 and 8.33 apply to both cases. By contrast, the retention times of the two concentration shocks, t^i and t^2r and the elution profile of the pure first component between the two shocks are different and must be calculated separately.
8.3.1 Retention Time of the Second Concentration Shock Behind the second shock, the concentrations of the two components, C\ and C2 are still given by the same equations as derived for the decay of the feed concentration plateau in the previous subsection, Eqs. 8.25 and 8.26a. These equations give the rear continuous profiles, which decrease from Cf1 and C^ to 0 and Cf, respectively. The smaller the sample size, the longer the retention time of the shock, the lower the values of C^ and C^, and the narrower the mixed zone. If we assume that r\ is equivalent to C\/C\ (see discussion of Eq. 8.14), another important general result is obtained. When the sample size decreases, the time when the mixed zone ends, t-£, as well as the time, t\, when the plateau of the second component ends, does not change if one neglects the trivial influence of their linear dependence on tp. Like t-£, the length {At2) and the height (Cf) of the plateau at the rear of the second component profile do not depend on the sample size, as long as the retention time of the second shock is shorter than £g. As we have shown, the heights of the two concentration discontinuities decrease constantly during their migration, as soon as the plateau at the feed concentration has vanished. The velocity associated with the maximum concentration of the band on the continuous (rear) side is always larger than the shock velocity. This is why the shock erodes continuously. The trajectory of the second shock can be obtained and its retention time calculated. It is longer than the retention time of the same shock in the case of a wide rectangular injection because the shock is less high, hence its velocity is lower (Eq. 8.15). The slope of the trajectory of the second shock, dz/dt, is given by Eq. 8.18,
404
Band Profiles of Two Components with the Ideal Model
provided that Cj and C2 are replaced by C\ and C2, respectively: dz rlt
u 1 -i
(8.37)
Fa2 l+fciCi+fc 2 C2
The retention time of a concentration C2 of the second component on the diffuse profile is given by Eq. 8.24. Combination of Eqs. 8.37 and 8.24 permits the derivation of the retention time of the second shock [14], which is given by (8.38)
t = tpF + u At the column outlet, z = L,IQ = L/u, and we obtain tR,i = tp + to[l + ?Ffl2(l - y ^ ) 2 ]
(8.39)
where L, is an apparent loading factor given by Lf = h (7 is given by Eq. 8.34) and Lfi2 is the loading factor calculated as usual, for the single second component (seeEq. 3.44, Lf/2 = {bxC^tp)/(t°R2~^o) = (^2^2)/(eSLfc'o^), with ??2 the amount of second component in the feed and k'0,2 the retention factor of the second component under linear conditions). We have observed that in most cases r\ is very close to the relative composition, C\/C\ (see Eq. 8.14). Thus, the apparent loading factor, which determines the retention time of the second shock, is proportional to the sample size at constant composition. L'j is not equal to the sum Lffi + Lf/2, however. If we make the approximation r\ = C®/C2, we obtain from Eq. 8.40, L, = L^2 + ^01/(^02^/4) = ^/,2 + ^f\l>K- At constant sample volume and concentration of one component, L, increases with increasing concentration of the other component.
8.3.2 Maximum Concentration of the Two Components in the Mixed Zone By reason of continuity, the concentrations of the two components in the back of the second shock are given by Eqs. 8.25 and 8.26a, for the retention time of the shock. The corresponding values are [14]
1L±
(8.41a)
8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms
Cff =
-,
^-L
405
(8.41b)
We can now give a description of what happens when the sample size is decreased by reduction of the injection band width, tp. When tp becomes smaller than the threshold given in Eq. 8.36, the feed concentration plateau has vanished before elution, the second component concentration behind the second shock is lower than C°, and its velocity decreases. Thus, the top concentration plateau in the first component profile, which is caused by the displacement effect, begins to erode. This erosion is rapid (it is complete when Zj in Eq. 66 of Ref. [14,15] is equal to L). When this plateau has also disappeared, the two concentration discontinuities begin to slow down progressively, the distance between them increases regularly, and, eventually, the second shock will be eluted at a time for which C^, as given by Eq. 8.41a, is equal to 0. This time is such that
The corresponding value of t% is given by combining Eqs. 8.39 and 8.42. We have the touching band case. In this case, C^1 is equal to cf, and the second component plateau is intact. For still lower values of tp, the width of the plateau at C | decreases. Band interaction ceases to have an influence on the profile of the second component band when the second shock is eluted at time tj (Eq. 8.29). This situation corresponds to the point K and column length z = z%, as discussed later.
8.3.3 Elution Profile of the First Component between the Two Shocks In front of the second shock, at t-R^, only the pure first component is moving. It is pushed by the second shock, whose image in the hodograph plot (Figure 8.2) is no longer in F but is at the point of coordinates given by Eqs. 8.41a and 8.41b. The migration of the first component profile in front of the second shock is described by the solution of Eqs. 8.10a or 8.10b, but not by the solution of Eq. 8.12. We must replace C\ and C£ in Eq. 8.12 by the values Cf and Cf (Eqs. 8.41a and 8.41b). Now the concentration of the first component on the front side of the second shock is given by Eq. 8.13a where C° is replaced by C^ [14]:
This equation gives the ratio of the concentrations of the first component on both sides of the shock. Since the point C^1, C^1 remains on the same continuous line of the hodograph plot, r\ in Eqs. 8.41a and 8.43 is still the solution of Eq. 8.12 with Cf. Combination of Eqs. 8.41a and 8.43 gives [14]: -i 1 — oc + a* /L'f
Cf = 7 -—
y=-
(8.44)
406
Band Profiles of Two Components with the Ideal Model
Between the two shocks, a zone of pure first component elutes. Now, the stationary phase concentration c\\ of the first component in this zone is a function of its single mobile phase concentration, C\. Thus, the velocity associated with Q is given by Eq. 7.3. The rather complex derivation [14] gives the profile of the first component as the time when the concentration C\ is eluted [14]:
»•
1-
ffit'-1'
(S.45,
At the column outlet (z/w = L/u = to), we have
«,, - «„)
(MO
There is no closed-form solution of this equation giving C\ as a function of time.
8.3.4 Retention Time of the First Shock Since Eq. 8.46 cannot be solved to give C\ explicitly as a function of i, it is not possible to derive a closed-form equation for the retention time or the concentration of the first shock [14]. However, it is possible to calculate f^j and C^ from a mass balance around the first component. This involves integration of the continuous rear profile of the first component, i.e., of Jt R'2 C\dt where C\ is related to t through Eq. 8.26a (a finite integral with known boundaries), and of ft R/1 C\dt where C\ is related to t through Eq. 8.45 (a finite integral with a known and a floating boundary). The floating boundary, t^\, is derived as the time for which the sum of these two integrals corresponds to the amount of first component contained in the sample [14]. The concentration boundary limits are Cf and C^ for Eq. 8.46, with C^ the unknown maximum concentration of the first component at t = tg j . Hence 1
Cxdt |
(8.47)
where n\ is the amount of the first component in the injected feed and Fv is the mobile phase flow rate. In practice, the second integral is replaced by a sum, HLi Q'1+2Q-1'>1 (ti-i - k), with C u = Q _ u + AC,, C0/1 = Cf (Eq. 8.44), t0 = tR/2, and n = (C^* — Cf-) I AC\, where n is the unknown, and AC\ is kept small for the sake of accuracy. For each value, Qt\ of the concentration, £; is derived from Eq. 8.46. It is also convenient to replace the first integral in Eq. 8.47 by a similar sum, although it can be calculated analytically. Increments are added to the second sum until the correct profile area is reached, which gives n, hence tn = £R j . Solutions of the ideal model can be calculated easily under the various possible conditions corresponding to the progressive separation and dilution of a pulse of a binary mixture during its migration along a column. Solutions giving the elution chromatograms at the end of columns of increasing lengths or the changes in elution profile of a rectangular pulse of decreasing bandwidth at the exit of a
8.4 Method of Calculation of the Ideal Model Solution in a Specific Case
407
given column are shown in the next few figures. The features of these solutions as described will become apparent in the discussion presented in the next section.
8.4 Method of Calculation of the Ideal Model Solution in a Specific Case The ideal model supplies simply and rapidly a sketch of the chromatogram obtained under conditions of strong column overload. It is quite useful to consider its solution as a first step toward the understanding of the problems that occur during the development of a separation. In practice, there are two simple steps to follow: first to determine, among five possible types of chromatograms, which one corresponds to the separation considered, then to apply the corresponding set of equations. Considering that the injection profile is a rectangular pulse, the ideal model can supply five different types of chromatogram • Case 1. The injection plateau has not yet eroded. This is the case of the wide injection (Figure 8.1, Eqs. in Table 8.1). • Case 2. The injection plateau has eroded, but the plateau of the pure first component band, at concentration Cf, has not yet disappeared (Eqs. in Table 8.2). • Case 3. The injection plateau and the plateau of the pure first component have been eroded, there is a mixed zone. The resolution between bands is incomplete. This is the case of a narrow injection pulse (Figure 8.3, Eqs. in Table 8.3). • Case 4. The two bands are separated, there is no mixed zone left, but there is still a plateau on the top of the second band profile, residual of the interaction between the two bands. This is the beginning of the "touching bands" situation (Eqs. in Table 8.4). • Case 5. The two bands are completely separated and the profile of the second component band no longer has a plateau (Eqs. in Table 8.5). In order to decide which type of chromatogram to consider and what set of equations to use, it is sufficient to calculate the four characteristic column lengths, Zj, Zj, ZK, and zi, or the four corresponding characteristic loading factors, Li 2, hi 2 , L^2, and Lh 2 , and to compare them to the length of the column considered or to the loading factor used for the different cases, respectively. The characteristic lengths and loading factors of the four intermediate points, I, J, K, and L are as follows. Point I, between cases 1 and 2: "ts
L/i
—
' ""•"
T
n
11 + ^ 1
(8.48)
(8-49)
Band Profiles of Two Components with the Ideal Model
408 C mg/mL Cmg/mL
70
a
T
b
70 C1o/C2o = 3/1
C1o/C2o = 1/3
L=0.81 cm
L=0.87 cm
35
35
0
0 0.3
0.6
0.9
1.2
0.3
0.6
0.9
t min
1.2
Figure 8.4 Case 1. Elution profile of a rectangular injection band at the end of a column of length shorter than zj (see Eq. 8.48). Same conditions as for Figure 8.1. The injection plateau is conserved, (a) L = 0.9 z7 = 0.87 cm; 1:3 mixture, (b) L = 0.9 zj = 0.81 cm; 3:1 mixture.
Point /, between cases 2 and 3: z
!
(8.50)
=
(8.51) 1-
Point K, between cases 3 and 4: (8.52) tt-1
(8.53)
Point K, between cases 4 and 5: tst-1
(8.54)
^iri
(8.55)
We discuss briefly these five types of chromatograms and the derivation of the four characteristic lengths.
8.4.1 Case 1: Wide Injection, L < Zj, or Lf/2 > Li2 In this case (Table 8.1, Figures 8.1, 8.4, and 8.10a, c, and d), the injection plateau subsists at Cj, C^ while a plateau of pure component 1 at a concentration Cj4
8.4 Method of Calculation of the Ideal Model Solution in a Specific Case C mg/mL
a
70
409 b
70
L=0.97 cm L=0.9 cm
35
35
0
0 0.3
0.6
0.9
1.2
0.3
0.6
0.9
t min
1.2
Figure 8.5 Situation intermediate between Cases 1 and 2. Elution profile of a rectangular injection band at the end of a column of length equal to zj (Eq. 8.48). The injection plateau has just disappeared. Same conditions as in Figure 8.4, except: ( a) L = Z\ = 0.97 cm; 1:3 mixture, (b) L = Z| = 0.90 cm; 3:1 mixture.
higher than Cj is eluted between the two shocks and before the mixed zone, and a plateau of pure second component is eluted just after the mixed zone and before the diffuse rear profile of the second component. The formation of this chromatogram has been discussed in the first part of this chapter. The corresponding equations are listed in Table 8.1 and also indicated in Figure 8.1. Other examples are provided in Figures 8.4 and 8.10. The point at concentration C° on the diffuse profile of the second component (C% > C > Cf, Eq. 8.25) moves faster than the front shock of the second component, so the width of the plateau between t^ and tpfi decreases. The plateau disappears when i ^ = £f,i, where t^ and i^i are given by Eqs. 8.19 and 8.23a, respectively. Hence (see Eq. 8.36 with to = L/u), the corresponding column length is given by (8.56) When L > z\, the injection plateau has disappeared when the band is eluted from the column. We can distinguish several intermediate cases before the separation becomes complete, eventually.
8.4.2 Case 2: Injection Plateau Eroded, Pure First Component Plateau Present, zj < L < Zj or hi 2 > Lf/2 > Ll2 When the injection plateau has eroded, the maximum concentration of the second component begins to decrease below Cj, and the second component shock slows down. The pure first component plateau concentration at C^ is no longer stable, because the rear concentration shock of the first component also has to slow down. A rear diffuse profile of the first component appears between the two shocks (see
410
Band Profiles of Two Components with the Ideal Model
Eqs. in Table 8.2). The concentration of the first component on this diffuse profile is given by Eq. 8.45. Introducing C\ = Cj4 in this equation gives the elution time of the rear of the plateau at Cj = Cj*. If the top of the plateau at C = C^ is not completely eroded, the time at which the first point of the rear diffuse profile of the first component passes at length z is obtained by replacing C by Cf in Eq. 8.45 z t
t
+
F fll z
tpb2C°2(<x-l)
+
The front shock of the first component profile, at t^t\, moves at the constant velocity of a shock of pure first component at concentration Cf. Its trajectory is given by [14] Fai
^
(8.58)
1 + hCf) and the intersection takes place at the distance 1-
b2C°2(cc-l)
(8.59)
The equations given in Table 8.2 permit the calculation of the profiles of this type. Examples of such profiles are given in Figures 8.5 (where L = zj) and 8.10b. Beyond the distance Zj, the plateau of the first component, Cf, has disappeared, and the chromatogram is typical of overloaded elution with a narrow injection pulse.
8.4.3 Case 3: Narrow Injection and Mixed Zone, Zj < L < ZK or L
/,2 ^ Lf,2 > tf/2
This case has been discussed in detail in the third section of this chapter. The relevant equations are listed in Table 8.3 and Figures 8.6 and 8.9e to 8.9h. Figure 8.3 indicates which equation applies to each feature of the chromatogram. Another example of such profiles is provided in Figures 8.9e to 8.9h. The degree of separation between the two bands increases with increasing migration distance along the column. We need to indicate here what happens when total resolution of the two bands takes place. Total resolution between the two bands takes place first at point K, where the concentration of the first component becomes 0 at the time of passage of the front shock of the second component, and £#,2 = t% [14]. The concentration of the second component on the rear of this shock is Cf • As long as there is band interference, Eq. 8.26a is valid and gives the profile of the first component in the mixed zone. The end of this elution profile at t-£ is given by Eq. 8.27, rewritten for a column of length z by replacing to by z/u. Equation 8.38, which gives the trajectory of the second shock, is also valid. Inserting t% in Eq. 8.38 and solving
8.4 Method of Calculation of the Ideal Model Solution in a Specific Case
411
aa
C mg/mL mg/mL 9-
L=10cm L=10 cm
6-
b 9-
V
L=10cm L=10 cm
6-
V \
3-
\
3-
\
0
0 6
9
12 12
6 6
99
ttmin min
12 12
Figure 8.6 Case 3. Elution profile of a rectangular injection band at the end of a column of length intermediate between Zj and ZR (Eqs. 8.50 and 8.52). The two bands overlap. Same conditions as in Figure 8.4, except: (a) L = 10.0 cm; 1:3 mixture, (b) L = 10.0 cm; 3:1 mixture.
for z gives the column length, takes place
at which complete separation of the bands just
(8.60) Beyond ZK, the two bands are resolved. However, the second component band still has a plateau at C^. This plateau will erode progressively (see next section).
8.4.4 Case 4: Touching Bands, Second Component Plateau Present, zK Lft2 >LLf2 j i*-
J
'
j
i*-
Once the two bands are completely separated, as in Figures 8.7, 8.8, and 8.9a to 8.9d, the diffuse profile of the rear boundary of the first component is given entirely by Eq. 8.46. The mixed zone has disappeared, Cf has become equal to 0 or vanished, so Eq. 8.26a is not applicable. The equations giving the features of the profiles are found in Table 8.4. An example of the profiles obtained in this case is found in Figure 8.7, where L = z^For the second component, the rear boundary is given by Eq. 8.28. The second plateau at Cf subsists, but its width, At2, is no longer given by Eq. 8.33, since the point E has vanished, but is given by At% — t\ — t^- This width is given by Eq. 27 of reference [29] = h - tRi2 = (*o
- f0)
cc-1
(8.61)
Thus, the retention time of the second band front is now given by tRi2 =
ti-M2
(8.62)
Band Profiles of Two Components with the Ideal Model
412 C mg/mL C
a a m L=17.6 C cm
66 8.41b 8.41b
bb
66
L=16 . 7 5 cm L=16.75 cm •
.62 8.62 \
\ \
3-
\
\
0 10 10
t(;R,1
[\ \
o t%. t +tt pP R,22+
t t,-t .2 E=t RR,2
3 3
•
\
0 20 20
10 10
Y 15 15
\ t t
min
20 20
Figure 8.7 Situation intermediate between Cases 3 and 4. profile of a rectangular injection band at the end of a column of length equal to z^ (Eq. 8.52). The two bands are just resolved, but a plateau still exists at the top of the second component band. Same conditions as in Figure 8.4, except: (a) L = 17.6 cm; 1:3 mixture, (b) L = 16.75 cm; 3:1 mixture. The numbers on the different parts of the profiles are the numbers of the corresponding equations.
The first component band ends at the retention time given by Eq. 8.46 for C\ = 0. Thus (8.63) The diffuse rear boundary of the pure first component band is given by Eq. 8.46. Equation 8.26a no longer applies since the mixed zone has vanished. The retention time of the first component cannot be derived in closed form but can be obtained by numerical integration of the first component band profile ni
(8.64)
where C\ is given by Eq. 8.46. The numerical calculation is done by replacing the integral by a sum, Ef=i(Q,l + Q_i,i)/[2(t(-_i - *,-)]/ with C a = Q_u + AClr Q,l = 0, to = *£/ n = C^ax/ACi, and by deriving £,- from Eq. 8.46 for each value of Q. The increments are summed up until Eq. 8.64 is satisfied, and t^\ = tn. The concentration plateau of the second component erodes and disappears after the band has migrated over the distance Z£ (8.65)
8.4 Method of Calculation of the Ideal Model Solution in a Specific Case C mg/mL
a
b
L=22.97 cm L=22.97 cm
4
413
L=62.2 L=62.2 cm
4
8.64
2
2 7.25a
0
0 17
tE
tR,2
26
48
56
64
t
min
72
Figure 8.8 Situation intermediate between Cases 4 and 5. Elution profile of a rectangular injection band at the end of a column of length equal to zi (Eq. 8.54). The plateau at the top of the second component band has just disappeared. Same conditions as in Figure 8.4, except: (a) L = 22.97 cm; 1:3 mixture, (b) L = 62.2 cm; 3:1 mixture. The numbers on different parts of the profiles are the numbers of the corresponding equations.
8.4.5 Case 5: Resolved Bands, zi < L or Lh2 > Lj 2 Beyond L = zi, the two bands are resolved. The second component band has the same profile as a band of the same amount of the pure second component injected alone, with the same tp. Its retention time is given by Eq. 7.25a and its rear profile by Eqs. 7.24 or 8.28. The band of the first component, however, remains forever affected by its interaction with the band of the second component, as indicated by Eqs. 8.46 and 8.64. The equations of the two band profiles are listed in Table 8.5. Figure 8.8 illustrates the profiles obtained in this case. One will note, comparing Figures 8.4 to 8.7, the progressive widening of the bands, and the dilution of the feed in the mobile phase. This effect is more important for the second component than for the first one because the latter has been somewhat concentrated by the displacement effect.
8.4.6 Influence of the Width of the Injection Pulse Figures 8.4 to 8.8 illustrate the changes in the individual band profiles of the components of a binary mixture during their migration and separation on columns of increasing lengths. Obviously, the effect is controlled by the loading factor, which decreases in proportion to the inverse of the column length. The same effects can be obtained by reducing the size of the feed sample injected on a given column. This is illustrated in Figures 8.9 and 8.10, which show the individual band profiles of samples of increasing sizes, for 1:3 and 3:1 mixtures. The sample sizes are increased by varying the pulse width at constant feed concentration, hence the sample volume injected on a 10-cm-long column. For narrow sample pulses, i.e., for low column loadings, well-resolved bands are observed (tp = 0.089 min; Figures 8.9a and 8.9b, case 5). For tp = 0.1 min, the two bands are just resolved, corresponding to "touching band" separation, Figures 8.9c and 8.9d (case 4). For tp = 0.5 and 0.75 min, the two bands interfere strongly, but a certain degree of
414
Band Profiles of Two Components with the Ideal Model
separation is achieved (Figures 8.9e to 8.9h, case 3). Finally, for a wide injection (tp = 2.3 min, Figure 8.10), a pure first component plateau and a feed plateau are achieved for the first component in the case of a 1:3 mixture (strong displacement effect, case 1), while the feed plateau has just eroded away in the case of the 3:1 mixture (case 2). The situation described in Figures 8.9e to 8.9h is the most common case in preparative chromatography. A relatively narrow injection pulse results in "overlapping bands" for the mixture components. Increasing the injection pulse width from 0.5 (Figures 8.9e and 8.9f) to 0.75 min (Figures 8.9g and 8.9h) causes the maximum concentration of both bands to increase and the retention times of the front shock of both component bands to decrease. At the same time, the rear boundary of the second component profile remains the same. The end time of the chromatogram, at C2 = 0, increases only as tp. Thus, the cycle time, denned as the time from injection to t°R 2 , is easily calculated. These phenomena have important practical consequences for the optimization of the experimental conditions in preparative chromatography. For example, for a 1:3 mixture, if one is interested in collecting the first component at 99% purity, the amount of this first component eluted between the two shocks divided by the cycle time gives the production rate. The amount of first component eluted between the two shocks divided by the amount injected gives the recovery yield. Figures 8.9e to 8.9h illustrate how the recovery yield of the first component decreases while its production rate increases when the sample size is increased. For the 3:1 mixture, by contrast, the retention time of the second component front decreases rapidly with increasing sample size when the bands interfere strongly. This explains how some late eluting impurities may surface underneath a first eluted main component, far ahead of their expected retention time, when a preparative column is too strongly overloaded. For a 3:1 mixture, the production rate of the first component will depend much on the sample size, the recovery yield, and the required purity. For the second component, loading beyond the "touching band" situation does not increase the amount of purified compound produced. The recovery yield decreases, all the excess amount of second component injected being eluted in the mixed zone. This topic is addressed in detail in Chapter 16 on optimization. Comparison of Figures 8.9a to h and Figures 8.10a to d shows that increasing the sample volume beyond a critical value results in a plateau at the injection composition. Further increase in sample volume merely increases the length of this plateau, without affecting the production.
8.5 Dimensionless Plot of a Two-component Band System In Chapter 7 (Section 7.3.2), we have shown that the solution of the ideal model in the case of a single component following the Langmuir isotherm model can be plotted in a dimensionless graph such that the band profile depends on the single value of the loading factor, Ly (Eq.7.26) [30]. In this plot, the reduced concentration
8.5 Dimensionless Plot of a Two-component Band System C mg/mL
415 b
a
8
8 C1o/C2o=3/1
C1o/C2o=1/3 tp= 0.089
tp= 0.089
4
4
0
0 c
9
d 9
tp=0.11
tp=0.12
6
6
3
3
0
0 e
18
f 18
tp=0.5
12
tp=0.5
12
6
6
0
0 g
24
h 24
tp=0.75
tp=0.75
16
16
8
8
0
0 4
8
12
4
8
t min
12
Figure 8.9 Elution profile of rectangular bands of various widths at the end of a 10-cm-long column. Left (a, c, e, and g): 1:3 mixture. Right (b, d, f, and h): 3:1 mixture, (a and b) Injection band width, tp = 0.089 min; case 5. (c and d) Injection band width, tv = 0.11 min (1:3) or 0.12 min (3:1); case 4. (e and f) Injection band width, t p = 0.50 min; case 3. (g and h) Injection band width, tp = 0.75 min; case 3.
is bC, the reduced time is (t — £O)/(£R,O ~~ to)/ where b is the second coefficient of the Langmuir isotherm, to is the column holdup time and tR$ the retention time at infinite dilution (t^o = Fato). Similarly, we can define a dimensionless graph for the individual elution profiles of the components of a binary mixture [31]. There are severe restrictions, however, that limit the usefulness of this plot. The loading factors of both components and their relative retention must remain the same. The column saturation capacities may change, and the feed composition must be adjusted to keep the two loading factors constant. Under such constraints, we obtain the plots illustrated in Figures 8.11a to 8.11c, corresponding to mixtures of different compositions and different relative column saturation capacity. The dimensionless plot is in Figure 8.11d. It shows the overlay of the chromatograms in Figures 8.11a to 8.11c. From the restrictive conditions imposed, we see that the dimensionless plot is not very
Band Profiles of Two Components with the Ideal Model
416 C mg/mL
b
a
70
70 C1o /C2o =1/3
C1o /C2o =3/1
tp=2.3
35
35
0
0 d
c 70
70
tp=4
35
35
0
0 3
7
11
15
3
7
11
15
t, min
19
Figure 8.10 Elution profile of rectangular bands of various widths at the end of a 10-cmlong column. Left (a and c): 1:3 mixture. Right (b and d): 3:1 mixture, (a and b) Injection band width, tp = 2.3 min; case 1 (1:3 mixture) or (3:1 mixture), (c and d) Injection band width, tp = 4 min (case 1). general.
8.6 The Displacement Effect The interaction between the bands of different components of the feed eluted successively is due to the competition between these components for interaction with the stationary phase. Although the results of these band interactions are often important and can be spectacular under certain circumstances, their origin can be traced to the fact that the differences between the equilibrium constants of the two components are rather small. For example, if we have a 1:1 binary mixture of two components with a competitive Langmuir isotherm and a value of the relative retention & = ail'a\ = 1.2, which is not unusual in practice, the proportion of the second component in the stationary phase at equilibrium is only 20% larger than the proportion of the first component. Under such conditions, the molecules of the component present in large excess in the feed may crowd those of the other component out of the stationary phase, whether the major component is the more or the less retained one. Thus, the interactions that take place between these two components result in an important displacement effect of the first component, at the front of the second component band, and/or in a significant tag-along effect exhibited by the second component band. In this section, we discuss the former, the displacement effect. We note for further reference that one of the properties of the competitive
8.6 The Displacement Effect
417 a
0.2 -r
b 0.2 -•
Q Q,,,=5, Q.,^10 s,1=5, Q s,2=10
Qs,1=10, Qs,2=5 Feed: 2/3
Feed: Feed: 1/6
0.1 0.1 0.1
0
•
0 10
20
30
10
20
t, min
0.2
c T
0.6
t, min
30 d
T
Lf,1=5% Qs,1=Qs,2=7.5 Lf,2/Lf,1=3
0.4 --
Feed: 1/3
0.1 0.2 --
0
0 10
20
30 t, min
0.3
0.7
(t-to)/(tro,2-to)
1.1 1.1
Figure 8.11 Dimensionless plot of the individual elution band profiles for a binary mixture. Plots of h\, C\ and &2> Gi versus (t — £O)/(£R,O,2 ~~ *o) for both components. Constant loading factor LfA = 5%. L = 25 cm; N = 5000 plates; F = 0.25; u = 0.125 cm/s; k'o = 6.0; a = 1.20. Langmuir isotherms with different column saturation capacities for both components, as indicated in the figures, (a, b, c) Chromatograms obtained under three different sets of experimental conditions, (d) Overlaid chromatograms in Figure 8.10a to c, plotted in dimensionless coordinates &,, Q versus (t — £o)/(*R2 ~~ *o)- Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 62 (1990) 217 (Figs. 1 and 2), ©1990 American Chemical Society.
Langmuir isotherm is that the selectivity of the chromatographic system for the two components:
s
«2
K0,1
(8.66)
is constant, independent of the composition of the mixture (Chapter 4). Thus, the competitive Langmuir isotherm cannot account for a change in the selectivity with increasing concentration. This is one of the limitations of the Langmuir isotherm model.
418
Band Profiles of Two Components with the Ideal Model
8.6.1 Origin of the Displacement Effect Physically, the displacement effect is easy to understand. The equations of the competitive Langmuir isotherm4 show that the amount of one component adsorbed at equilibrium with a given mobile phase concentration decreases with increasing concentration of the other one [27]. When the frontal concentration discontinuity of the second component arrives in a point of the column, the equilibrium concentration of the first compound decreases instantaneously and a large fraction of the molecules of the first component which are in the stationary phase are expelled or displaced. The stationary phase concentration of the first component drops, while its mobile phase concentration surges. In other words, the second component being more strongly adsorbed than the first one, it expels from the stationary phase and desorbs a fraction of the molecules of the first component. For example, in case 4 all the amount of the first component injected is displaced in front of the second component band. This explains why in Figure 8.9c the sample size is nearly 25% larger than in Figure 8.9a, while both chromatograms exhibit "touching bands." This effect is fundamental in displacement chromatography (see Chapter 9) and explains the displacement of the band of one component by another band and the eventual formation of an isotachic train.
8.6.2 Intensity of the Displacement Effect According to Eq. 8.43
£)
(8.67)
the concentration of the first component is higher in the first zone of the chromatogram (on the front side of the shock between the first and second zones) than in the second zone (on the rear side of that shock). As noted before, this is the displacement effect. We can now discuss the factors that influence the intensity of this effect. In elution chromatography, the intensity of the displacement effect is conveniently measured by the ratio [27]
ct ^t
h
(868)
The ratio ocb\ Ibi is equal to the ratio of the column saturation capacity for the two components, qs,2/({s,l- As we know (see Chapter 4), the Langmuir model is truly applicable only if this ratio is equal to unity. As we have noted, the root r\ of Eq. 8.4 is practically equal to the ratio of the concentrations of the two components in the feed, C^/C®. As a consequence, we have Dh
1
+
4
See Eq. 8.2: q{ =
1+t
^
"cfU c , and Chapter 4, Section 4.1.2.
(8.69)
8.7 The Tag-Along Effect
419
where the loading factor is the ratio of the amount of a component injected to its column saturation capacity (see Eqs. 3.44 and 7.26). The intensity of the displacement effect increases linearly with increasing ratio of the loading factors of the two components [27]. If the column saturation capacity is the same (or nearly the same) for the two components, the intensity of the displacement effect is controlled by the relative composition of the feed. The higher the concentration of the second component, the stronger the effect. If the column saturation capacities for the two components are different, the Langmuir isotherm model becomes inconsistent with thermodynamics [32]. Experience shows that in such a case the Langmuir model rarely accounts well for experimental data and a more sophisticated isotherm must be considered [32]. Unfortunately, there is no analytical solution of the ideal model for any isotherm other than the Langmuir isotherm.
8.7 The Tag-Along Effect This effect results in a long plateau on the rear part of the elution profile of the second component when the column is strongly overloaded and the loading factor of the first component is much larger than the loading factor of the second one. Like the displacement effect, the tag-along effect is a consequence of the competition between the molecules of the two components for interaction with the stationary phase. At constant concentration, the second component is less retained in the presence of a finite concentration of the first one than when it is alone.
8.7.1 Origin of the Tag-Along Effect When the concentration of the first component is very high, its molecules tend to crowd the molecules of the second component out of the stationary phase, even though this second component is more strongly adsorbed [27]. For example, with the Langmuir model the amount of second component adsorbed is given by qz — («2C2)/(1 + b\C\ + bzCz). If C\ is large enough, qz for a constant value of Cz may become much lower than ( ^ Q ) / (1 + ^2^2), the concentration of the second component adsorbed at equilibrium when pure. As a consequence, the velocity associated with a constant concentration C2 of the second component increases with increasing concentration C\ of the first component. This is obvious in Eq. 8.22, which gives the velocity in the mixed zone: .n — ..n — ..11 ..1 U
Z,1 -
M
Z,2
^
-
7(l+b1C1+b2C2)
2
At the end of the mixed zone, the concentration Q tends toward 0 and the concentration of the second component toward Cf, given by Eq. 8.13b. The limit of the velocity associated with the concentration C2 is «&(0, Cf) =
i +
U Fai
(8.71)
420
Band Profiles of Two Components with the Ideal Model
This velocity is always higher than the velocity associated with the same concentration C | on a continuous rear profile of the pure component 2, i.e., in the third zone of the chromatogram. This latter velocity is given by Eq. 7.7 or 8.32: uin{C)
=— ^ - =
%
(8-72)
Accordingly a concentration plateau appears on the rear of the second component profile, at the concentration Cf given by Eq. 8.13b [5,14,15,27]:
d =
*"*
= C° - ^
(8.73)
We have shown that the length of this plateau, in time units, is given by Eq. 8.33 [27]:
Ah = Ftol \a21 - J = Faz^J^to
(8.74)
If we assume again that r\ is approximately equal to C^ZC®, we can simplify this equation and obtain ,8.75)
In summary, the shock velocity of a concentration Cf of the second component in the presence of the first component is always larger than the velocity associated with the same concentration C | of the pure second component on a disperse boundary. As the sample size increases, the front shock of the second band approaches the rear of the first band. For still larger sample sizes, interference between the two bands begins, leading first to touching bands, then to the formation of a mixed zone. The front shock of the second component moves faster, and thus farther into the rear of the first band. Physically, although the second component is more strongly retained than the first one, the first component molecules can expel a significant fraction of the molecules of the second component if they are in large excess. These expelled molecules of the second component migrate with the first component band, forming a wider, more dilute, mixed zone (Figures 8.9f and 8.9h). Note that, although a large excess of first component can desorb an important fraction of the second component, total displacement is impossible. There is always some second component behind the first one.
8.7.2 Intensity of the Tag-Along Effect It is convenient to choose the length of the plateau on the rear of the second component profile as a measure of the intensity of the tag-along effect. We shall define
8.8 The Ideal Model in Gas Chromatography
421
this intensity as [27] L//2
L
/-2
(fc 0,2 - fc o,i)
As with the displacement effect, the intensity of the tag-along effect depends essentially on the ratio of the two loading factors. When the loading factor for the first component is much smaller than that of the second component (j^-
f
i T ^ o ir
w
(8-77)
;
L/,i C\qs,2 % Thus, for a given value of the relative composition of the feed, the intensity of the tag-along effect, like the intensity of the displacement effect, depends strongly on the ratio of the two column saturation capacities. Note, however, that the solution of the ideal model for a binary mixture that is discussed in this chapter assumes that the Langmuir competitive model is valid. But the Langmuir competitive model is truly valid only if c\Si\ = cjS/2.
8.8 The Ideal Model in Gas Chromatography Jacob et al. used the method of characteristics to discuss the general properties of the system of mass balance equations in multicomponent preparative gas chromatography (GC) [34-36], assuming either a linear or a nonlinear isotherm. The GC problem is more complicated than the HPLC one because the gas mobile phase is much more compressible than a solution and the mobile phase velocity is very different inside and outside a high concentration band because the partial molar volumes of compounds are much larger in the gas mobile phase than in the condensed stationary phase (the sorption effect). They showed that the method of characteristics applies to multicomponent systems as well as to single component
422
Band Profiles of Two Components with the Ideal Model
ones (see Chapter 7) [34,35]. Due to this complexity of GC and to the difficulties in handling both the sorption and the nonlinear competitive isotherm effects, however, they discussed only the evolution of high concentration GC band profiles for two solutes, assuming a linear isotherm, therefore considering only the sorption effect. Assuming a linear isotherm for both compounds implies that each solute interacts only with the stationary phase, independently of any other feed component, which is admittedly an unrealistic assumption in preparative chromatography. The elution and separation of a binary mixture was illustrated in the case of a rectangular injection as the boundary condition (see Figure 8.12) [34]. After a short migration distance, the band consists in three successive, partly separated zones, one for each component and a mixed zone. A band of pure component A (the less retained one) is eluted between two concentration shocks. The concentration of A in this first band is higher than it was in the feed. The second shock separates the band of pure component A from the mixed zone that contains A and B. This mixed band is made up of a first part having the same composition as the feed, followed by a diffuse boundary for the two compounds. Upstream of the end of the continuous profile of A, there is a small band of pure component B which begins with a plateau and ends as a diffuse boundary. During further migration, the band separates and the plateau of B disappears. As in the case of a single compound, only one of the two discontinuities of the rectangular injection profile is stable. In the multicomponent case it generates two concentration shocks, one at the front of the zone of pure component A while the other separates this pure component from the mixture of the two components. The other discontinuity of the injection (the rear one in the case of sorption effect with linear isotherm) collapses into a continuous profile. Thus, the sorption effect in GC has consequences most similar to those of a convex upward isotherm in HPLC.
pa
PI
N\ it t
Figure 8.12 Elution of a binary mixture with sorption effect (GC) and linear isotherms. Three successive chromatograms at the beginning of the separation. Reproduced with, permission from L. Jacob and G. Guiochon, Chromatogr. Rev., 14 (1971) 77(Fig. 4).
8.9 Practical Relevance of the Ideal Model
423
8.9 Practical Relevance of the Ideal Model As in the single-component case, we may expect significant differences between the band profiles predicted by the ideal model and the profiles recorded with actual samples. Depending on the case, the differences can be from negligible to considerable. Basically, the higher the column efficiency, the closer the calculated and the experimental band profiles. Similarly, the higher the loading factor, the smaller the differences between theoretical and experimental profiles. Detailed comparisons between the individual band profiles calculated with the ideal model and with the equilibrium-dispersive model are presented in Chapter 11. It is shown there that good agreement is observed when the apparent loading factor, m = N[k /(1 + k )}2Lj exceeds 50 for the two components. The theory of the hodograph transform and the relationship derived between the equations of the two lines given by this transform in the case of a binary mixture and those of the competitive equilibrium isotherms were briefly presented in Section 8.1.2. The theory is easily extended to multicomponent mixtures, although in this case we must represent the hodograph transform in an n-dimensional coordinate system, Q , C2, • • •, Cn, or in its planar projections. If the solution presents a constant state (Figure 8.1), it is a simple wave solution, and there is a relationship between the concentrations of the different components in the eluent at the column exit (Figure 8.2). This result is valid for any convex-upward isotherm. In the particular case in which the competitive Langmuir isotherm applies, these relationships are linear. The hodograph transform is valid only within the framework of the ideal model. It has been shown, however, that the hodograph plots derived from actual chromatograms are very similar to those predicted by the ideal model [18]. If the column efficiency exceeds 100 to 200 theoretical plates, there is no significant difference between the hodograph plot obtained with the ideal model and the plot derived from the profiles calculated with the equilibrium-dispersive model, except very near the axes of coordinates (Figure 8.13). Figure 8.14a compares the
Figure 8.13 Influence of column efficiency on the hodograph plot for a wide rectangular band. H = 0.02 cm; column length (L, cm) and number of theoretical plates (N): 1, 1.0 (50); 2, 2.0 (100); 3,4.0 (200); 4, 8.0 (400). Reproduced with permission from Z. Ma and G. Guiochon,}. Chromatogr., 603 (1992) 13 (Fig. 2).
424
Band Profiles of Two Components with the Ideal Model
TIME (MIN)
Cp (me/ml)
Figure 8.14 Comparison between experimental band profiles obtained for a wide pulse injection and the corresponding hodograph plot with those calculated with the ideal model, (a) Experimental elution profiles for a wide rectangular injection of 2-phenylethanol (squares) and 3-phenylpropanol (circles). Profiles calculated assuming competitive Langmuir isotherms with numerical parameters derived from the single component isotherms measured by frontal analysis (solid lines). Sample relative composition: 1:1. (b) Hodograph transform of the data in Figure 8.14a. A.M. Katti, Z. Ma and G. Guiochon, AIChE }., 36 (1990) 1722 (Fig. 7). Reproduced by permission of The American Institute of Chemical Engineers. ©1990 AIChE. All rights reserved.
elution profiles of a wide band injection predicted by the ideal model and by the equilibrium-dispersive model (Chapter 11) with the experimental band profiles obtained in the case of a mixture whose adsorption behavior deviates somewhat from the Langmuir model [26,37,38]. Figure 8.14b compares the hodograph plots determined experimentally, from the individual band profiles obtained by fraction collection and analysis, and those calculated from the single-component isotherms, assuming a competitive Langmuir isotherm model. Although there is excellent agreement between the prediction of the simple wave solution and the hodograph transform of the experimental data when the isotherm model is known, the significant differences between the two hodograph plots in Figure 8.14a show that the Langmuir isotherm model is not entirely supported by the experimental results in this case [19,20]. Finally, Figure 8.15 shows the results of a theoretical study regarding the individual elution profiles of the components of a ternary mixture [39]. Figure 8.15a shows the profiles obtained by simulating the analysis of collected fractions; Figure 8.15b shows the trajectories of the discontinuities issued from the injection profile, the regions of the (t, z) plane where the plateaus at the injection concentra-
8.9 Practical Relevance of the Ideal Model ±*-
425
;i
C
«
M*d
o»jMrn»-lal
I
*
CI
.
C2
a
a
r. M
ot
•? .
t fl
lfl
IT
14
It
IB
c
\
1r
i
Figure 8.15 Elution of a wide injection band of a ternary mixture, (a) Experimental chromatogram and profile predicted by the model, (b) Trajectories of the concentration shocks and simple wave regions, (c) Hodograph transform in the C\,Ci plane, (d) Hodograph transform in the Ci, C3 plane, (e) Hodograph transform in the C\,C^ plane. Reproduced with -permission from R. ZenhUusern, Ph.D. Thesis, Eidgenosische Technische Hochschule, Zurich, Switzerland, 1993.
tion are stable, and those where simple wave solutions take place. Figures 8.15c to 8.15e show the projections of the hodograph transform in the three coordinate planes, (Q, C2), (C2, C3), and (Q, C3), respectively. These results demonstrate that the ideal model supplies us with useful tools to account for experimental data in complex cases and that, unless the experimental isotherms deviate strongly from the Langmuir model, these results are particularly simple to use.
Band Profiles of Two Components with the Ideal Model
426
Appendix - The Five Different Cases of Separations of Binary Mixtures This section summarizes the analytical solutions of the ideal model with a Langmuir competitive isotherm in the different cases identified5. Table 8.1 Case 1; Elution of a Wide Rectangular Injection Pulse L <
[l + ^ ] (Eq. 8.49)
z,(Eq.8.48)arL / < 2 >
The injection plateau is eroded only in part. Auxiliary Parameters 1. Positive Root of Eq. 8.12:
:° - b2c\
oc-l
cc -
2. Parameter 7, Eq. 8.34: b2
7 =
= 1
(a b2
b2
First Zone, Pure Component 1 1. Elution Time, Eq. 8.17: tR,i =
1+
Fax
bxCf
2. Concentration, Eq. 8.13a: nfa
>
fa
Second Zone, Mixed Band 3. Retention Time of Second Component Shock, Eq. 8.19 : tR,2 = to 5
1+
Fa2
° + b2C°
Definitions: F, phase ratio (F = (1 — e)/e); to, holdup time (tg = L/u; L, column length; u, mobile phase velocity); tp, duration of the rectangular pulse; a\, ai, b\, &2, coefficients of the Langmuir isotherm; and a = aila\, separation factor.
Appendix
427
4. Composition of the Concentration Plateau. It is the same as the feed composition: Ci = C° and C2 = C°. 5. End time of the Concentration Plateau, Eq. 8.23a: Fa?
1+
tF,i =
6. Width of the Residual Concentration Plateau, Eq. 8.23b: Fa2t0
-1
7. Elution Profile of the First Component, Eq. 8.26a: 1
Ci =
17
-1
b2/(ocr1)
8. Elution Profile of the Second Component, Eq. 8.25: C2 =
1
7^^
b2
9. End Time of the Second Zone, Eq. 8.27:
10. Concentration of the Second Component at the End of the Second Zone, Eq. 8.13b:
a 1
—
b2
—
00
\^n
*
C?
—
t\
Third Zone, Pure Component 2 11. End Time of the Concentration Plateau and Beginning of the Rear Continuous Profile, Eq. 8.29: t0
Fa2
12. Duration of the Concentration Plateau at Cf, Eq. 8.33:
At2 = t!-tE = Ft0^ [« 2 2 _ fll] =
(tm2
- t0 )
428
Band Profiles of Two Components with the Ideal Model
13. Elution Profile of the Pure Second Component, Eq. 8.28:
t-tp-to
-1
14. End of the Elution Profile:
Table 8.2 Case 2; Elution of a Rectangular Injection Pulse6 <
2J
L
<
[l + fy] >
zf (Eqs. 8.48 and 8.50) or
L /(2 >
-8-49 The injection plateau is completely eroded but a plateau subsists at the top of the first component band. Auxiliary Parameters 1. Loading Factor of the Second Component Eq. 7.26: n2b2
eSLk'Q Q2
2. Apparent Loading Factor, Eq. 8.40:
l
R,2
First Zone, Pure Component 1 1. Elution Time, Eq. 8.17:
1+
Ffli
2. Concentration of the First Plateau, Eq. 8.13a:
6
Same definitions for the parameters F, to, tp, a-y, «2, b\, &2/ «, 7, and ri as in Table 8.1a.
Appendix
429
3. End Time of the Plateau at Q = Cf [14], Eq. 8.50:
z
/
1-
=
4. Concentration Profile of the First Component behind the Plateau End, Eq. 8.46: to
5. Concentration of the First Component at the Front of the Second Shock, Eq. 8.44: _ 1 {!-*)/*
Second Zone, Mixed Band 6. Retention Time of Second Component Shock, Eq. 8.39: tR,2 = tp + t0[l + F a 2 7 ( l -
y/u}
7. Elution Profile of the First Component, Eq. 8.26a: Ci =
1
7 ^i
8. Elution Profile of the Second Component, Eq. 8.25: 1 t-tp-to 9. End Time of the Second Zone, Eq. 8.27: tE = tp +10 + ^{t°Rrl - t0) = 10. Concentration of the Second Component at the End of the Second Zone, Eq. 8.13b: «-l _r0 C° CB_ -b2 r\
Band Profiles of Two Components with the Ideal Model
430
11. Maximum Concentration of the Second Component (at the Top of the Second Shock), Eq. 8.41b b2
'/
12. Concentration of the First Component at the Rear of the Second Shock, Eq. 8.41a: 1
b2
_ R. f
Third Zone, Pure Component 2
13. Duration of the Concentration Plateau at Cf, Eq. 8.33:
At2 = t l - t E = Fto2 fal - Bl ] =
- to)
14. Beginning of the Rear Continuous Profile of Pure Second Component, Eq. 8.29: ti — tp + to
.21
1+
15. Elution Profile of the Pure Second Component, Eq. 8.28: 1
° - to
°-R,2
t-tp-to
-1
16. End of the Elution Profile:
Table 8.3 Case 3; Elution of a Narrow Rectangular Injection Pulse7 zj
< L < zK (Eqs. 8.50 and 8.52) or § ^
\ J.
L
fl > [ ^ ]
2
«(*-!+*! C^«)2
5 ^ (Eqs. 8.51 and 8.53).
The separation of the two bands is incomplete. 7
Same definitions for the parameters F, to, f p , «i, «2/ bi, &2/«, 7, and r\ as in Table 8.1a.
Appendix
431
Auxiliary Parameters 1. Loading Factor of the Second Component, Eq. 7.26 nzb2 Lf 2
'
=
0,2
l
° - t0
R,2
2. Apparent Loading Factor, Eq. 8.40:.
_
L
f2
First Zone, Pure Component 1 1. Elution Time, Eq. 8.47. The retention time of the first shock cannot be calculated in closed form. It is derived as the lower boundary of the finite integral of the two profiles of the first component (in the first and second zones), such that this integral corresponds to the mass of first component injected. 7 tR,2
-1
Ut-tp-t0
dt+
I fRA cxdt\
(see text for the calculation of a numerical solution) 2. Concentration Profile of the First Component, Eq. 8.46
=h 3. Concentration of the First Component at the Front of the Second Shock, Eq. 8.44: 1-JL
/
Second Zone, Mixed Band 4. Retention Time of Second Component Shock, Eq. 8.39:
+ Fa2j(l - yjv 5. Elution Profile of the First Component, Eq. 8.26a: Ci =
1
432
Band Profiles of Two Components with the Ideal Model
6. Elution Profile of the Second Component, Eq. 8.25: 1
C2 =
b2
t-tp-t0
7. End Time of the Second Zone, Eq. 8.27:
tE = h + k + ^(t°RA - t0) = 8. Concentration of the Second Component at the End of the Second Zone, Eq. 8.13b: /v-1 C° B
2
a.
_
i
_
Q
u1
2 ab\r\ + b2 r\ 9. Maximum Concentration of the Second Component (at the Top of the Second Shock), Eq. 8.41b:
Cf = 10. Concentration of the First Component at the Rear of the Second Shock, Eq. 8.41a:
f Third Zone, Pure Component 2 11. Duration of the Concentration Plateau at C|, Eq. 8.33: At
t
At2 = tl-tE
7 \n 7 n 1 Vn ' cc I -' d- «ij*J= Fa ~ Ft - \a
Vt
t
=
0
2
2
(a) 2
t0 =
7(7 ~
12. Beginning of the Rear Continuous Profile of Pure Second Component, Eq. 8.29: 10
1+
Fa2
13. Elution Profile of the Pure Second Component, Eq. 8.28: 1
14. End of the Elution Profile:
t°R,2-t0 -1 t-tp-t0
Appendix
433
Table 8.4 Case 4; Elution of a Narrow Rectangular Injection Pulse8 zK
< L < z L (Eqs. 8.52 and 8.54) or [^1]
2
- <^
> Lff2 >
.
-
1
1
(Eqs. 8.53 and 8.55) The separation exceeds touching band resolution but a plateau subsists on the top of the second component band. Auxiliary Parameters 1. First Loading Factor of the Second Component, Eq. 7.15: n2b2
biC^tp
2. Second Loading Factor, Eq. 8.40:
First Band, Pure Component 1 1. Retention Time, Eq. 8.64 The retention time of the first shock cannot be calculated in closed form. It is derived as the lower boundary of the finite integral of the profile of the first component, which is now pure, such that this integral corresponds to the mass of first component injected. «i = rqtp o, = || /ftE (see text for the calculation of a numerical solution) 2. Concentration Profile of the First Component, Eq. 8.46: t = tp + t0 + (fRil - t0) 3. End of the Elution Profile of the First Component, Eq. 8.63:
8
Same definitions for the parameters F, tg, tv, a\, ai, b\, \>i, a, "f, and r\ as in Table 8.1a.
Band Profiles of Two Components with the Ideal Model
434
Second Band, Pure Component 2 4. Retention Time of the Second Component, Eq. 8.62: 2
1 + CC^P
oc-1
5. Concentration of the Plateau, Eq. 8.13b n 2
-i
abiri + h2
c° 2
r\
6. Duration of the Concentration Plateau at Cf [29], Eq. 8.61:
At2 = ti-
tR/1 = {t%2 -
k)
d-1
cc-1
7. Beginning of the Rear Continuous Profile of Pure Second Component, Eq. 8.29: = tp +10 1 + 8. Elution Profile of the Diffuse Rear Profile, Eq. 8.28 A,-tQ t-tp-t0 9. End of the Elution Profile:
Appendix
435
Table 8.5 Case 5; Elution of a Narrow Rectangular Injection Pulse9 -I 2
zL < L (Eq. 8.54) or Lf/2 <
(Eq. 8.55).
The complete separation of the two bands has been achieved. Auxiliary Parameters 1. First Loading Factor of the Second Component, Eq. 7.26 n2b2 _ eSLk'0,2
f2
'
4,2 -
First Band, Pure Component 1 1. Retention Time, Eq. 8.66. The retention time of the first shock cannot be calculated in closed form. It is derived as the lower boundary of the finite integral of the profile of the first component, which is now pure, such that this integral corresponds to the mass of first component injected. tE
—| /
c
d\
J tR
(see text for the calculation of a numerical solution) 2. Rear Concentration Profile of the First Component, Eq. 8.46 - t0) 3. End of the Elution Profile of the First Component, Eq. 8.63
Second Band, Pure Component 2 4. Retention Time, Eq. 7.25a
5. Elution Profile of the Diffuse Rear Profile, Eq. 7.24b 1
fR2 -10 t-tp-t0
6. End of the Elution Profile:
9
Same definitions for the parameters F, to, tv, a\, a-i, b\, \>i, a, 7, and t\ as in Table 8.1a.
436
REFERENCES
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
J. N. Wilson, J. Am. Chem. Soc. 62 (1940) 1583. D. DeVault, J. Am. Chem. Soc. 65 (1943) 532. A. C. Offord, J. Weiss, Nature 155 (1945) 725. R Helfferich, G. Klein, Multicomponent Chromatography, M. Dekker, New York, NY, 1970. E. Glueckauf, Proc. Roy. Soc. A186 (1946) 35. E. Glueckauf, Disc. Faraday Soc. 7 (1949) 12. J. I. Coates, E. Glueckauf, J. Am. Chem. Soc. 69 (1947) 1309. E. Glueckauf, J. Chem. Soc. (1947) 1302. H.-K. Rhee, R. Aris, N. R. Amundson, Trans. Roy. Soc. (London) A267 (1970) 419. H.-K. Rhee, R. Aris, N. R. Amundson, First-Order Partial Differential Equations - II. Theory and Application of Hyperbolic Systems of Quasilinear Equations, PrenticeHall, Englewood Cliffs, NJ, 1989. R. W. Geldart, Y. Qiming, P. C. Wankat, L. N.-H. Wang, Separat. Sci. Technol. 21 (1986) 873. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 484 (1989) 125. E. Kvaalen, L. Neel, D. Tondeur, Chem. Eng. Sci. 40 (1985) 1191. S. Golshan-Shirazi, G. Guiochon, J. Phys. Chem. 93 (1989) 4143. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 461 (1989) 19. B. C. Lin, S. Golshan-Shirazi, Z. Ma, G. Guiochon, Anal. Chem. 60 (1988) 2647. J. Weiss, J. Chem. Soc. (1943) 297. R. Courant, K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, NY, 1948. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, Ronald Press, New York, NY, 1953. A. Jeffrey, T. Tanuiti, Non-Linear Wave Propagation, Academic Press, New York, NY, 1964. J. H. Seinfeld, L. Lapidus, Numerical Solutions of Ordinary Differential Equations, Academic Press, New York, NY, 1964. G. B. Whitman, Linear and Non-Linear Waves, Wiley, New York, NY, 1971. A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman, London, UK, 1976. H.-K. Rhee, NATO Adv. Study Inst. Ser. E (1981) 183. H.-K. Rhee, N. R. Amundson, Chem. Eng. J. 1 (1970) 241. Z. Ma, A. M. Katti, B. Lin, G. Guiochon, J. Phys. Chem. 94 (1990) 6911. S. Golshan-Shirazi, G. Guiochon, Chromatographia 30 (1990) 613. G. Guiochon, C. Guillemin, Quantitative Gas Chromatography, Elsevier, Amsterdam, 1988. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 506 (1990) 495. S. Golshan-Shirazi, S. Ghodbane, G. Guiochon, Anal. Chem. 60 (1988) 2630. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 517 (1990) 229. M. D. LeVan, T. Vermeulen, J. Phys. Chem. 85 (1981) 3247. M. Z. El Fallah, S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 511 (1990) 1. L. Jacob, G. Guiochon, Chromatogr. Rev. 14 (1971) 77. L. Jacob, G. Guiochon, J. Chim. Phys. (Paris) 67 (1970) 185. L. Jacob, G. Guiochon, J. Chim. Phys. (Paris) 67 (1970) 291. A. Katti, Z. Ma, G. Guiochon, AIChE J. 36 (1990) 1722. A. Katti, M. Czok, G. Guiochon, J. Chromatogr. 556 (1991) 205. R. Zenhausern, Ph.D. thesis, Zurich, Switzerland (1993).
Chapter 9 Band Profiles in Displacement Chromatography with the Ideal Model Contents 9.1
Steady State in the Displacement Mode. The Isotachic Train 9.1.1 9.1.2 9.1.3 9.1.4 9.1.5
The Operating Line Influence of the Displacer Concentration The Watershed Point Case of a Trace Component Analysis of the Isotachic Pattern When There is Selectivity Reversal at High Concentrations
9.2 The Theory of Characteristics 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.2.6
439 442 444 445 446 446
450
Determination of the Characteristic Parameters Application to Displacement Chromatography Wave Interactions Critical Value of the Displacer Concentration Plateau Concentrations and Bandwidth Critical Column Length for Isotachic Train Formation
9.3 Coherence Theory 9.4 Practical Relevance of the Results of the Ideal Model References
450 453 454 457 457 461
461 467 468
Introduction In displacement chromatography, a rectangular band of sample is introduced into the column which has first been filled with a low- or moderate-strength eluent, or carrier. This mobile phase is weaker than the one that would normally be used to elute the sample, hence the retention volumes of the mixture components under these conditions are usually larger than those that would be observed under typical elution conditions, but this is not necessary. By contrast with elution, upon completion of the sample injection, the stream of mobile phase pumped into the column is replaced by a stream of a displacer solution. The displacer is a compound that is more strongly adsorbed than any component of the mixture. The displacer concentration must be high enough for its breakthrough curve to interfere with the sample bands. Depending on the experimental conditions, the individual band profiles vary in degree of complexity. If the column is long enough, however, a steady state is reached and a succession of quasi-rectangular bands of 437
438
Band Profiles in Displacement Chromatography with the Ideal Model
the pure sample components is eluted, pushed by the displacer front. Whereas elution chromatography is a well-known and relatively simple chromatographic method, displacement chromatography, although more than 60 years old, remains much less familiar to separation scientists. This is due in part to their long-term association with linear chromatography and analytical-scale separations by elution, in part to the higher degree of complexity of displacement separations. The immense growth of linear elution chromatography as a method for analytical separations and quantitative analysis has overshadowed the use of displacement chromatography, which has remained essentially ignored by the scientific community until recently, in spite of several important applications. The difference between elution and displacement chromatography was recognized by Tswett [1] at the turn of the century. However, Tswett [2], and after him Kuhn and Lederer [3], used the elution mode exclusively. Displacement chromatography was developed by Tiselius [4], at a time when on-line detection was not available and when the sensitivity of the analytical methods used offline was many orders of magnitude less than it is today. Tiselius clearly defined and discussed the various modes of chromatography [5]—elution, frontal, and displacement. Glueckauf [6,7] gave the first theoretical analysis of the method using the ideal model, a model that has remained the preferred tool of the users of displacement chromatography. He also showed the effect of the mobile phase flow velocity on the development of the isotachic train [8]. Beyond this pioneering work, two contributions were critical, those of Helfferich and Klein [9] and of RheeetoZ. [10]. The "theory of interference" [9] was originally developed for stoichiometric ion exchange systems. In this approach, the set of concentrations Q of the components of a system (mixture components and retained components of the mobile phase) is replaced by a new set of variables, hi, using the ^-transform. However, adsorption processes can be viewed as equivalent to ion exchange processes by introducing a fictitious component to account for the exchanged ion [9]. Thus, an ncomponent adsorption system becomes equivalent to an (n+l)-component ion exchange system. Therefore, the same approach may be applied to ion exchange and adsorption, provided that we may assume the validity of both the ideal model and the Langmuir competitive isotherm model. The method of Helfferich and Klein has been widely used to account for experimental results obtained in displacement chromatography. For example, it was used for the calculation of the separation of a 15-component rare earth mixture [11,12]. Later, concentration profiles in the column effluent calculated using the /t-transform method were compared with experimental profiles for a four-component ion exchange sorption system [13] and for the separation of two phenols in reversed phase chromatography [14]. Very good agreement was reported (see Chapter 12). Like Helfferich and Klein [9], Rhee et al. [10] studied the separation of multicomponent mixtures by displacement chromatography using the restrictive assumption of the validity of the Langmuir isotherm model and the ideal model. They used a different approach, based on the method of characteristics, and studied the interactions between concentration shocks and centered simple waves [15]. This approach is more directly suited to adsorption chromatography than the
9.1 Steady State in the Displacement Mode. The Isotachic Train
439
method of Helfferich and Klein, but the results of both methods are equivalent [16]. More recently, Basmadjian and Coroyannakis have modeled displacement separations, neglecting accumulation in the mobile phase and band dispersion [17]. They reexamined some of the pioneering work of Glueckauf [8] using geometrical concepts to predict adsorption behavior, for example, by illustrating the significance of the watershed point. They investigated the displacement of a less adsorbed species with a more strongly adsorbed species and vice versa. Other valuable contributions have been made by Geldart et at. [18], Frenz [19], and Jen and Pinto [20,21] who attempted to introduce the column efficiency. Other important theoretical work on displacement chromatography has been carried out using other models (equilibrium-dispersive or kinetics) of chromatography and will be discussed in Chapters 12 and 16. A comprehensive review of the history and development of displacement chromatography and its applications has been published by Frenz and Horvath [22]. On the experimental front, systematic investigations by Horvath and his coworkers [22-26] have reintroduced this technique and demonstrated its great potential advantages for the purification of gram-size or sub-gram-size samples of chemicals or biochemicals using conventional analytical HPLC instruments. The experimental results verify for many different systems the effect of various operating parameters as determined by theoretical calculations. Another important aspect of this work is the identification of displacers and operating conditions leading to an isotachic train. These authors capitalized on various technological advances, such as high-efficiency columns and fast high-performance liquid chromatography. Some of these results are accounted for with the ideal model. Others are not, and will be discussed further in Chapter 12 as part of topics concerning the equilibrium-dispersive model.
9.1
Steady State in the Displacement Mode. The Isotachic Train
In displacement chromatography, the feed is injected in a column saturated by a weak solvent such that the feed components are retained (Figure 9.1, steps 1 and 2). Immediately, or very soon after the feed introduction is completed, a stream of displacer solution is pumped through the column (Figure 9.1, step 3). The displacer is a compound that is more strongly retained than any component of the feed. The feed components are then expelled from the stationary phase by the displacer and form a series of bands eluting ahead of the displacer breakthrough front. When the experimental conditions are properly chosen, steady state is achieved before the sample leaves the column. A series of successive separated bands is formed and moves along the column at the velocity of the displacer front. This series of bands is called the isotachic train. The formation of the isotachic train requires a certain time and is a complex phenomenon which we describe in this chapter [9,10]. Following completion of the displacement separation, the column must be regenerated (Figure 9.1, step 4) and reequilibrated with pure carrier (Figure 9.1,
440
Band Profiles in Displacement Chromatography with the Ideal Model
Feed Column Inlet
Carrier
Regenerant Displacer
Carrier
Step #
Column Outlet Carrier
Product Carrier
Regenerant Displacer
Figure 9.1 Schematic of displacement separations. step 5). In the following example we consider a system of n components dissolved in a weak solvent. These components are ranked from 1 to n by increasing order of retention, i.e., by increasing order of the initial slope of their adsorption isotherm between the weak solvent and the adsorbent. The first n — 1 components are the sample components. The last one is the displacer. Thus, the subscript n stands for the displacer. Several experimental setups have been used to implement displacement chromatography in the laboratory, using conventional HPLC hardware. Figures 9.2a and 9.2b show two examples. In Figure 9.2a, a single 10-port valve is used. The column is equilibrated with the carrier coming from pump A (position A). Then, the feed is injected or pumped into the sample loop. Before switching to position B, pump B is used to purge the line with displacer. After the feed injection is complete, the valve is switched to position B, resulting in the displacer driving the feed load to the column. In order to control the displacement analysis, fractions are collected at the column exit, and/or the column effluent flows through
9.1 Steady State in the Displacement Mode. The Isotachic Train
441
(a) Displacer Pump B
Position A Position B
Outlet B
Outlet A
(b) Displaces Outlet PumpB
..Feed
. \ \j
On-line HPLC ; On-line Analyzer]
Detector Fraction Collector
Figure 9.2 Schematic of experimental setups for displacement chromatography. (a) Schematic employing a single 10-port valve, (b) Schematic employing a 6-port and a switching valve.
442
Band Profiles in Displacement Chromatography with the Ideal Model
the sample loop of an on-line chromatograph equipped to carry out fast analyses. In Figure 9.2b, a single 6-port valve is used in conjunction with a switching valve. In both cases, the use of two pumps allows smooth continuous switching. When large feed volumes are used, or whenever automation is required, a feed pump may be employed. When a single pump is available, the setup can be modified. In this case, the carrier flow is stopped, and the inlet of the pump is switched to the displacer reservoir. In most cases, the column eluent is analyzed at high speed to determine the degree of separation. This is particularly important for the development of preparative methods using displacement chromatography.
9.1.1 The Operating Line The one requirement for performing displacement chromatography is that the competitive isotherms of the mixture components and the displacer should be convex upward and should not intersect each other (Figure 9.3a). When the equilibrium isotherms are convex, their fronts are self-sharpening (see Chapter 7 for an explanation of the relationship between the sign of the isotherm curvature and the shape of the band front). In the ideal model, since the column efficiency is infinite, each front becomes a concentration shock. We now assume that an isotachic train will form and that an asymptotic solution exists. This will be demonstrated later (Section 9.2). In an isotachic train, all the fronts move at the same velocity. A shock separates two successive bands of pure components, and between the two shocks that limit a band, the concentration of the pure component is constant. Thus, the height, or concentration, of each component zone must be such that its front shock velocity travels at the same velocity as the displacer front. Therefore, the plateau concentrations and the migration velocity can be controlled by choosing the displacer concentration. The velocity of the displacer front depends on the flow rate, on the displacer concentration, and on the displacer isotherm. The isotachic migration of all component fronts requires that we have: US/1 = Us,2 = • • • = Us4 = ... = Us,n
(9.1)
The velocity of the displacer front is (see Eq. 7.5)
If we assume a Langmuir isotherm for the displacer, the retention time of its front is
The velocity of each band front is given by a similar equation, replacing qn/Cn in the denominator by ifr/Q for each component i. In almost all cases, the initial displacer concentration in the mobile phase, as well as the initial concentration of all the mixture components, is 0. Combination of Eqs 9.1 and 9.2 gives Q
C2
Q
Cn
9.1 Steady State in the Displacement Mode. The Isotachic Train Packing Q mg/g Packing
a
443 b
C mg/mL
Operating Line
3 Displacer
Cn
1
Displacer
2
C3 •
C2
C1 1 1
C1
C2
C3
Cn
C mg/mL
2
3
Volume of Eluent or Time
Figure 9.3 Illustration of the operating line and determination of the plateau concentration of the individual bands, (a) Single-component isotherms with Cn > C\/CIit (Eq. 9.7). (b) Profiles of the zones in an ideal isotachic train. The term ift/Q in these equations is the slope of the chord of the isotherm of component i (Figure 9.3a). Equation 9.4 shows that in order to form an isotachic train, in which all the bands move at the same velocity, the concentration of each band is given by the intersection of the corresponding isotherm and the chord of the displacer isotherm. This chord is called the operating line (Figure 9.3a). The isotachic train (Figure 9.3b) forms as a series of bands whose heights, Cf, are given by the intersection of the single-component isotherms and the operating line. By changing the slope of the operating line, i.e., by changing the displacer concentration, it is possible to change the heights (or concentration) of all the bands (Figures 9.4a and 9.4b). In the case of a Langmuir isotherm for the displacer and the feed components, this height is obtained by equalling Eqs. 9.2 written for the displacer and for component i, and solving for the band plateau concentration, Cf bnCn
(9.5)
It is not possible to adjust the band heights for each component independently. For a given concentration of the displacer, the band height or plateau concentration of each sample component at steady state is constant. Therefore, in order to conserve the area of each band, which must remain equal to the area of the injected profile, the width of each band is proportional to the amount of the corresponding compound injected. As the amount of any feed component injected in the column increases, the width of its band in the isotachic train increases (Figures 9.4a and 9.4b). If the amounts of several components in the sample increase, the widths of their bands increase at constant height (Figures 9.4b and 9.4c). Although the previous discussion supplies a correct description of the isotachic train in the ideal model and of its steady propagation, it does not provide any clue regarding the individual band profiles during the progressive formation
Band Profiles in Displacement Chromatography with the Ideal Model
444 C mg/mL C
a
b Cn=125 mg/mL
Cn= 100 mg/mL Cn=100 100 100
100 100
•
•
100
50 50
100
•
50 75 50 50
•
75 50 0 50 0 • 0 20.5
23.5
26.5
0 • 0 20.5
23.5
26.5
c
Cn=125 Cn=12 5 mg/mL
100
Figure 9.4 Influence of the displacer concentration on the zone profiles in the isotachic train at different sample sizes, (a) Isotachic train obtained by displacement of a ternary mixture. Cn = 100 mg/mL. (b) Displacement of the same sample by a more concentrated displacer. (c) Displacement of a larger size sample. Cn = 125 mg/mL.
100
100 50
75
0 20.5
23.5 23.5
26.5
26.5
t, min
of the isotachic train. This information can be obtained by a more general theory which shows how the individual component profiles are progressively transformed into an isotachic band train (Section 9.2). In displacement chromatography, it often occurs that the concentration of some of the feed components in the isotachic train is higher than their concentration in the feed. This is in contrast to overloaded elution chromatography, where dilution of the feed always occurs.
9.1.2 Influence of the Displacer Concentration We have shown in Figure 9.4 that when the displacer concentration increases, the height of the zones increases and their retention time and their width decrease (Figures 9.4a and 9.4b). When the displacer concentration decreases, the concentration plateaus decrease until the displacer chord or operating line intersects the single component isotherm (Figure 9.3a). When the operating line becomes tangent to the initial slope of an isotherm, the band of the corresponding component is eluted under conditions of overloaded elution. This situation takes place when the displacer solution is too dilute or the component too weakly retained (i.e., Figure 9.5a). As shown by Rhee et al. [10], there is a critical displacer concentration below which displacement of a compound band cannot take place. If the
9J Steady State in the Displacement Mode. The Isotachic Train
445
isotherms of the compounds studied are Langmuirian, the condition for possible displacement of feed component i by displacer n can be written
The critical displacement concentration is
and a successful displacement requires Cn > On crlt . For a multicomponent mixture, successful displacement is impossible for those components, /, for which Cn < C!n c r i r The bands of these components are elution bands with a front shock and a rear diffuse boundary (see Chapters 7 and 8). The critical displacement concentration for a multicomponent mixture depends only on the isotherms of the displacer and the lesser retained components of the mixture. It does not depend on the feed composition or on the nature of the intermediate components, provided they are all less retained than the displacer, their isotherms are convex upward, and these isotherms do not intersect. The height of the concentration plateau of component i, Cf, is given by Eq. 9.5, which can be rewritten as
Cf = i f— - 1 j + ^ y Cn
(9.8)
This equation shows that Cf is independent of the concentration and the relative composition of the feed. The plateau concentration depends only on the displacer concentration and on the parameters of the isotherms of the corresponding component and the displacer.
9.1.3 The Watershed Point The watershed point of component i is the displacer concentration (Cn — ID*) corresponding to the intersection of the displacer isotherm and the initial tangent to the isotherm of that component. This concept was first identified by Glueckauf [6,7]. This critical concentration is given by Eq. 9.7. No displacement of component i is possible if the displacer concentration is below its watershed point (Figure 9.5a). Under conditions where Cn = w*, the tail of the first component will end at the front of the second component band. As the displacer concentration falls below the watershed point, the rear boundary of the first component separates completely from the isotachic train. In this case, which is not exceptional, one or a few early eluting components appear as independent elution bands before the isotachic train. This may be impossible to avoid—for example, if the solubility of the displacer in the carrier is insufficient. Note that the relationship between the isotherms of the feed components and the operating line implies that the successive bands of the isotachic train should have increasing heights (in terms of concentration; because the detector response may vary widely from component to component, the actual band heights may
Band Profiles in Displacement Chromatography with the Ideal Model
Q mg/g Packing
a
Operating Line
b
C mg/mL
Cn Displacer
446
3
Displacer
2
1
C3 3
C2 2 1
C2
C3
Cn
Volume of Eluent or Time
Figure 9.5 Displacement at the watershed point of the first component, (a) Single- component isotherms with Cn = Ci/Crjt. (b) Profiles of the zones in the ideal isotachic train.
vary). Compounds that are well resolved {i.e., have large a. values) will have significantly different heights. Compounds that have separation factors close to 1 will have bands of nearly the same height since their isotherms are close.
9.1.4 Case of a Trace Component The previous derivation shows that the band height is independent of the concentration of a component in the feed but depends only on its isotherm and on the concentration at the intersection of this isotherm with the operating line. The derivation of Eq. 9.8 makes no exception for trace components. The width of the bands, on the other hand, is proportional to the feed concentration and becomes extremely small for a trace component. Thus, the ideal model predicts a considerable enrichment of trace components in displacement chromatography. Figure 9.6 illustrates how the band of a trace component is squeezed between the bands of two major components, with a plateau concentration intermediate between those of the two wide bands [27]. Although, in practice, axial dispersion and the mass transfer resistances combine to spread the trace component band and to prevent it from reaching a maximum concentration that should be of the same order of magnitude as that of the main component bands, this phenomenon provides an original procedure for trace enrichment. It will be discussed later, in Chapter 12.
9.1.5 Analysis of the Isotachic Pattern When There is Selectivity Reversal at High Concentrations As discussed earlier in this chapter, one of the characteristic features of the competitive Langmuir isotherm is that the selectivity between any pair of components is constant, independent of their concentrations. As a consequence, the requirements for their successful separation by displacement chromatography would merely be that the selected displacer is retained more strongly than either feed
9.1 Steady State in the Displacement Mode. The Isotachic Train
447
20001
£1600-]
I
Component 1
Displacer 1000-
500-
130
Component 8 ^~^~ (Trace Component)
Component 3 '
iff
134 136 TIME (MIN)
138
Figure 9.6 Displacement chromatogram of a three-component system with one trace component. Reproduced with permission from T. W. Chen, N. G. Pinto and L. van Brocklin, J. Chromatogr., 484 (1989) 167 (Fig. 6).
components and that its concentration exceeds a certain minimum value (see Eq. 9.7). Then, with a sufficiently long column, a displacement train would eventually be formed, with the bands of the two components separated and exiting the column in the order of increasing initial slope of their respective single component isotherm. Some experimental results suggest that displacement becomes unsuccessful and does not separate bands of pure compounds if the single component isotherms of two feed components cross each other [28,29]. In an adsorption process applied to compounds of similar polarity, the larger molecules tend to bind more strongly to the sorbent than the smaller ones. Therefore, the single-component isotherms of the larger molecules tend to have the larger initial slope. However, the larger molecules tend also to occupy more space on the surface of the sorbent. Therefore, they are expected to have a lower saturation capacity and, as a result, we can anticipate that the single-component isotherms of compounds with large and small molecules will often cross each other. In many instances, the single-component isotherm data of the individual components of a mixture fit well to the single-component Langmuir model. However, because these compounds have different saturation capacities, the multicomponent or competitive Langmuir isotherms of these compounds are not thermodynamically consistent (see [30,31]). Even when the experimental data are forced to fit to this model and the residuals are small, the resulting competitive isotherms do not account for the selectivity reversal that takes place at some intermediate concentration. As discussed in Chapter 4, the IAS theory and the LeVanVermeulen isotherm are able to explain the crossing of the isotherms, the selectiv-
448
Band Profiles in Displacement Chromatography with the Ideal Model
ity reversal, and the consequences of this phenomenon for the phase equilibrium of the two components. Antia and Horvath [32] have provided an analysis of the isotachic patterns in displacement chromatography that take into account the effects of a selectivity reversal. They used for this purpose the multicomponent isotherm generated by the IAS theory. In order to examine the possibility of performing separations even when the single-component isotherms of two components intersect and cross over, they analyzed the stability of the displacement train and the criteria for displacement and for the stability of the resulting boundaries that are established in an isotachic train. Their analysis is based on the separation of a mixture of two components, A and B, whose single-component adsorption equilibrium isotherms are represented by the relationship qA = qA(CA,Cs) and q% = <7B(QA/CB)- The mixture is separated with a displacer D, of concentration CQ and isotherm qu — qoiCo), having an affinity for the sorbent that is greater than that of either component of the feed mixture considered. According to Antia and Horvath [32], a necessary condition for the establishment of fully resolved bands in this process is expressed as: lim
U{CA,^B.)
lim ^CA'CB-)
<
qo
(9 9 a )
>
S?L
(9.9b)
Therefore the development of concentration bands in the column downstream of the displacer front is governed by the binary adsorption equilibrium of A and B, not just their single component isotherm as predicted by Glueckauf using Langmuir isotherm. In these inequalities, C^ and Cg are the concentrations that the components A and B, respectively, would attain in an isotachic displacement train in which these two species would be fully resolved into two pure component bands. The values of C*A and C | depend only on the two pure component adsorption equilibrium isotherms and on the displacer concentration. They are calculated from the Glueckauf relationship
,q)
=
qo_
{9W)
C
Using the competitive adsorption isotherms generated with the IAS theory and Eq. 9.10, Antia and Horvath showed that the stability conditions 9.9a and 9.9b led to the identification of three different operating regions in displacement chromatography in the case when the single-component isotherms of two solutes intersect. The isotherms of A, B, and the displacer are shown in Figure 9.7a and the corresponding hodograph plot of C# versus CA is shown in Figure 9.7b. The operating lines are drawn on the isotherm plot (Figure 9.7a) from the origin to the point on the displacer isotherm that corresponds to the selected displacer concentration. The operating lines are transformed into lines in the hodograph plot, lines that are called tie lines and are shown as dashed lines in Figure 9.7b. Each tie
9.1 Steady State in the Displacement Mode. The Isotachic Train
449
c [mM| 10
20
Figure 9.7 Operating lines when the isotherms of two feed components intersect, (a) Isotherms of A, B, and the displacer (solid lines) and two possible operating lines (dashed lines), (b) Corresponding hodograph plots showing tie lines (dashed lines) and the unit selectivity line. The operating and tie lines demarcate the space into three operating regions (see text). Reproduced with permission from F. Antia, Cs. Horvdth, J. Chromatogr., 446 (1991) 119 (Fig. 3).
10
20
c A [mM] line is obtained from an operating line; it connects the concentrations CA and C% determined by the intersection of the operating lines with the respective parent isotherms. The unit selectivity line, CB + CA = Cx is also drawn in the hodograph plot. The two tie lines and the corresponding operating lines shown in Figure 9.7 are for the cases in which CA = Cx and C% = Cx. The stability conditions indicate that a separated isotachic sequence AB is stable if the corresponding concentrations CA and Cg both lie below the line of unit selectivity, i.e. below the tie line passing through CA = Cx. On the other hand when both C*A and Cg lie above the unit selectivity line, i.e., when the tie line lies above the line passing through Cg = Cx, a stable isotachic pattern with the opposite elution order (BA) is expected. The stability conditions also require that when CA and Cg lie on opposite sides of the unit selectivity line, i.e., when the operating lines and the corresponding tie lines lie between those shown in Figures 9.7a and 9.7b, a completely separated pattern cannot be obtained. The operating region between these lines is called the separation gap. The operating lines to the left of the separation gap yield displacement trains with the original selectivity and those on the right of it yield displacement trains with the reversed selectivity, hi the separation gap, no complete separation is possible. Carta et at. [33] observed that there is a selectivity reversal in the case of the
450
Band Profiles in Displacement Chromatography with the Ideal Model
separation of a mixture of the two amino acids a-aminobutyric acid (ABA) and isoleucine (He). Displacement chromatography of mixtures of these two components on a cation-exchange resin resulted in the complete separation of these two compounds at low concentrations. At higher concentrations, in contrast, the nonideal equilibrium effects resulted in an incomplete separation. This behavior is explained by considering the nonideality of the equilibrium uptake of these amino acids by the resin. A semi-empirical model, taking into account the heterogeneity of the ion-exchange resin, was used to fit the isotherm data. The equilibrium uptake model used in the analysis of the experimental results for the ABA/Ile separation provided an excellent representation of the single component uptake behavior. The same model can be used to predict multicomponent equilibrium. Such a prediction, although only approximate, appeared to be consistent with the anomalous development of the component bands in displacement chromatography. This model predicts a selectivity that varies with concentration and that reverses under certain conditions.
9.2 The Theory of Characteristics Rhee et al. developed a theory of displacement chromatography based on the mathematical theory of systems of quasi-linear partial differential equations and on the use of the characteristic method to solve these equations [10]. The h- transform is basically an equivalent theory, developed from a different point of view and more by definitions [9]. It is derived for the stoichiometric exchange of adsorbable species (e.g., ion exchange), but as we have discussed, it can be applied as well to multicomponent systems with competitive Langmuir isotherms by introducing a fictitious species. Since the theory of Rhee et al. [10] is based on the use of the characteristics and the shock theories, its results are comprehensive; e.g., the characteristics of the components that are missing locally are supplied directly by this theory, while in the /i-transform they are obtained as trivial roots, given by rules and definitions. As expected, the two methods give basically the same results [16]. Both theories are valid only in the case of competitive Langmuir isotherms. This restriction should be kept in mind, because deviations from Langmuir behavior are the most probable explanation for the not so inconsequential disagreements observed sometimes between theoretical predictions and experimental observations reported in the literature [14,34].
9.2.1 Determination of the Characteristic Parameters The mass balance equations of the ideal model for an n-component mixture have been written (see Eqs. 7.1): ^ dz
^ dt
^ = 0 dt
(9.11)
with qi=fi(C1,C2,---,Cn)
(9.12)
9.2 The Theory of Characteristics
451
Equation 9.12 is the isotherm equation. Equations 9.11 can be rewritten [10] in a dimensionless form by using the reduced variables x = z/L and r = ut/L, where L is the column length: dx
3T
3T
Assume that the isotherm of each component, i, is given by the Langmuir competitive isotherm model (Eq. 4.5):
and the mixture components are numbered in increasing order of their affinity for the stationary phases, so their rank is also their elution order. The solution in the (x, T) plane consists of n + 1 constant states connected by n shocks or simple waves. Across each wave, there exists a one-parameter representation of the state. Thus, there are n parameters to characterize the n waves and the n — 1 intermediate constant states. With Langmuir competitive isotherms, the n parameters which we call the n characteristic parameters {<X>\,OJ%, • • • ,con) are determined by solving the following nth-order algebraic equation V -*M-
=1
(9.15)
This equation is the generalized coherence equation. Equation 8.9a is the particular case of Eq. 9.15 in the case of a binary mixture. The definition of the characteristic parameters is analogous to the H-function from which the h roots used in the /i-transform [9] are obtained:
t ,,hf:\ , = i
(9-i6)
Thus, the characteristic parameters, OJI, are analogous to the h roots, h{. Neither Eq. 9.15 nor Eq. 9.16 can be solved explicitly for n > 2, but their roots can be calculated by simple numerical techniques. Equations 9.15 and 9.16 have n real, positive roots co^ and h^, respectively. The roots of Eq. 9.15 can be arranged in the following order: 0 < QJ-[ < d\ < CU2 < Cl2 < • • • < OJj < Clj < COj+l < fly+i < • • • < C0n < ttn (9.17)
Equation 9.15 represents the transformation of the concentration space (Q or q{) into the <x> space (a?,). The solution of Eq. 9.13 in the (x, T) plane consists of n waves. The advantage of the transformations just discussed is that the value of only one of the characteristic parameters, o?j (or hi), changes when we move across each one of these waves, the others parameters remaining unchanged. By contrast, the value of several concentrations change across a wave, and even the number of concentrations that change across a wave varies during the propagation of the band train.
452
Band Profiles in Displacement Chromatography with the Ideal Model
If we identify a characteristic parameter, cok and the corresponding kth wave, and if the value of this parameter is equal to cok behind the wave and to ofy. ahead of the wave, then, if cork < a>[, the fc-wave is a shock wave [10]. This shock wave propagates along a straight line in the (x, T) plane, and the line slope is given by .=l a} where the superscripts I and r stand for the left- and right-hand side of the shocks, respectively. By contrast, if a>k > cu£, the fc-wave is a centered simple wave in the region where to changes continuously [10]. A Jc-simple wave is represented by a family of straight characteristics (Q), in the (x, r) plane, and their slopes are given by
The simple wave region is bound by the kih and the (k — l)th constant states, on its left- and right-hand sides respectively, so, Eq. 9.19 can be rewritten as Jl k-X of.
n <J.
The only variable in Eq. 9.20 is cok, which varies in the following range: cu[ < cok < aip.. Thus, one can generate in this range a family of straight lines, each of which carries a fixed value of co^ and thus a fixed set of solute concentrations. Finally, when cUj. = a;£, the Icth wave does not exist and we have Q ^ = Q ^ - I During the development of a displacement chromatogram from the injection of the feed to the formation of an isotachic train, several successive stages are reached. At the beginning, all feed components are simultaneously present in a narrow spatial region. As some components move faster than the others, regions will appear where one or more component is absent from the mobile phase. As the separation progresses, these regions will become wider and more and more numerous. Usually, the mobile phase ahead of the band train is free of all components and the displacer solution does not contain any mixture component. If some components are missing locally (and eventually, as all the components will separate, they will be missing from all other components' bands), the corresponding roots cannot be obtained directly from Eq. 9.15. However, this equation implies that for each missing component, /, there exists a characteristic parameter, whose value is equal to «y. Since the mobile phase is pure, it contains no components nor displacer, and we have tok = ak
f o r k = 1 , 2 , ••• , n
(9.21)
If a particular species, /, is absent from the solution, Eq. 9.15 shows that one of the characteristic parameters should be equal to aj, and it follows from Eq. 9.17 that we have either COj =
fly
(9.22)
9.2 The Theory of Characteristics
453
or w;-+i = Oj
(9.23)
If the most strongly adsorbed species in the mixture is absent from the solution, it is obvious from Eq. 9.15 that con = an
(9.24)
Equations 9.21,9.22,9.23, and 9.24 are equivalent to the set of rules given by Helfferich and Klein [9] for the determination of the trivial roots of h.
9.2.2 Application to Displacement Chromatography In the initial state of displacement chromatography, before injection, no component is present, so the set of n characteristic parameters for the mobile phase is given by Eq. 9.21 I[u)\,cvl2,---,u>ln]
= I[a1,a2,---,an\
(9.25)
where the superscript i represents the initial conditions (pure mobile phase, without components and displacer). The feed is injected as a rectangular pulse of width tp. When the injection ends, all the components are present in the column, except the most strongly adsorbed one, the displacer. Thus, the characteristic parameter of the displacer is given by Eq. 9.24, while the characteristic parameters of the n — \ feed components are obtained by solving Eq. 9.15. Their values are denoted oo^w^,- • • ,wn_v given by
The set of n characteristic parameters for the feed is
F[u/1,cof2,---,u>fn_1,u/n=an]
(9.26)
where the superscript / represents the sample injection (all components present in the mobile phase, except the displacer), and, thus, o;{ < a\, OJ2 < a2, • • •, ^n-i < an-i- Kw
e
compare Eqs. 9.25 and 9.26, we see that for any component, k,
except the displacer, we have coj, < wlk, which is the stability condition for a shock wave. This shows that n — \ shock waves will develop upon injection of the feed and that they propagate from the origin. Obviously, no shock is observed for the displacer when the feed is injected, since col = Wn. After the feed injection is completed, a continuous stream of displacer solution is pumped into the column until the displacer breaks through. Since the n — 1 feed components are absent from the displacer solution, the n — 1 values of their characteristic parameters, according to Eq. 9.24, are equal to «i,a2l • • •, an-\. The last root of Eq. 9.15 can be obtained by solving it, which gives
The value of to* is a function of the displacer concentration and can be adjusted between 0 and an. Thus, the order of the whole set of a^ is not fixed. It depends
454
Band Profiles in Displacement Chromatography with the Ideal Model
on the concentration of the displacer. If the displacer concentration is such that «y_i < w* < a.j, to* becomes the ;th root of Eq. 9.15, upon injection of the displacer solution, we have D[tof, to{, • • •, cof_lr tof, tof+1, • • •, Jn\ =
(9.28)
D[tof = alt co{ = a2, • • -,a^_x = fly-i, (vj = to*, wj+1 = dj+1, • • •, c4i = ««]
where the superscript d represents the displacer stream (no components but the displacer in solution). Comparing Eqs. 9.26 and 9.28, we see that, upon displacer introduction, j — 1 simple waves (corresponding to the values of to on the left side of to^ in Eq. 9.28) and n — j shock waves (corresponding to the values of to on the right side of a)1- in Eq. 9.28) will develop. Starting from the point (0, rp = (tpu)/L), the /th wave remains a shock if to* < to- or becomes a simple wave if to* > an. Thus, there are two sets of shocks in displacement chromatography. The first one originates as a result of the feed injection. The second set results from the displacer injection. These two sets will be involved in a series of interactions which have been analyzed in detail [10].
9.2.3 Wave Interactions Since the transformation of {Q} into {to^} defined by Eq. 9.15 is independent of both x — z/L and T — ut/L, the solution scheme is applicable to any point in the {x, T) plane if an initial discontinuity is imposed at that point. If discontinuities are imposed at two or more points along the x or the T axis, the solution may be derived by constructing separately the wave solutions for each of these points. However, after a finite period of time, any two wave solutions centered in two different, but adjacent, points will meet each other. An overlap region appears, where the solution is influenced by both sets of data. Such a phenomenon is called wave interaction. From what was explained in the previous section, this interaction takes place during the formation of the isotachic train of any mixture. 9.2.3.1 Patterns of Interaction The following rules result from a detailed investigation of wave interactions [10, 15]. • The interaction between two shocks is instantaneous, while interactions involving simple waves are progressive. • When two shocks of the same kind interact, they become superimposed. • A shock absorbs a simple wave of the same kind and its strength decreases. • When two waves of different kinds interact, they move across each other.
9.2 The Theory of Characteristics
455
Figure 9.8 The distance-time diagram illustrating the progressive formation of an isotachic train for a binary mixture. H.-K. Rhee and N. R. Amundson, AICHE J, 28 (1982) 423 (Fig. 2). Reproduced by permission of the American Institute of Chemical Engineers. ©1982 AIChE. All rights reserved.
9.2.3.2 Applications As an example, Figure 9.8 illustrates the development of a displacement train for a binary mixture. It provides a distance-time diagram indicating the boundaries of the different domains defined in the previous theoretical discussions. The column is equilibrated with the carrier mobile phase (I), and the injection of a sample (F) of a binary mixture (components A\ and A2) is performed during a finite period of time (TO = utp/L), after which the displacer (P3) stream is pumped into the column. There are two concentration shocks: one at the origin (0, 0), the stable front shock of the first component, the other at the end of the injection (0, To), the front shock of the displacer. Thus, two wave solutions appear in the (x, r) plane. According to Eq. 9.25, for an initially empty bed, the characteristic parameters are l{a\,ai, a$). State F represents the injection of A\ and Ai into the column; according to Eq. 9.26, the characteristic parameters are F(ct^,0*2,83), where o>{ and 0*2 are the roots of the quadratic Eq. 9.15, with 0 < o>{ < a\ < cv2 ^ a2- The replacement of the carrier by the feed changes two of the characteristic parameters, hence two shocks develop from the origin. As they proceed in the direction of the x-axis, 102 increases from wl^ to a.2 across the 2-shock, so there is no second solute on the downstream side of this shock. Similarly, to\ increases from o>{ to a.\ across the 1- shock, and there is no solute A\ on the downstream side of the first shock. The new intermediate constant state is represented by U(ur[, «2/fl3)- Since the values of to are known for states I and F, it is easy to derive the slopes of the characteristic lines for these two shocks using Eq. 9.18. These slopes are given by =
1 + Fco{
(9.29a) (9.29b)
When the displacer, A3, is introduced into the column, the two mixture components are no longer present in the mobile phase. Their characteristic parameters
456
Band Profiles in Displacement Chromatography with the Ideal Model
are equal to a.\ and a2, respectively. The third root, co*, given by Eq. 9.15, depends on the displacer concentration, C|. It is given by a3-co*
= 1
(9.30a)
or (9.30b) So, upon switching the mobile phase stream from the feed solution to the displacer solution, the values of all three characteristic parameters are changed. Accordingly, three boundaries arise from the point (0, To). Depending on the displacer concentration, C*, the value of co* is between 0 and 03, so the ordering of the parameters {tOk} also depends on the displacer concentration. For example, if co* < a\ < o2, state P3 is represented by P3 [a.\, a2, co*]. The comparison of the characteristic parameters of states F and P3 shows that the second and third boundaries are shock waves while the first boundary could be either a shock (if co* < w|) or a simple wave (if co* > coy). The values of co for the intermediate constant states P2 and V are P2\co*,a\,a.3] and V[to*,002,113], respectively. Thus, state P2 contains only the pure component A2 and corresponds to the band of pure component A2. Across the 3-shock, CO3 increases from a2 to «3, so the displacer is absent from the band of pure second component. The slopes of the characteristic lines can be derived using Eqs. 9.18 and 9.19: s3
=
s2
=
1 + Fco* / * 1+ - ? —
(9.31a) (9.31b)
fl2
If co* < cJ-y, the first wave (1) is a shock and the slope of its trajectory is given by Eq. 9.18 si = 1 + F ^ i ^ (9.32) a2 If co* > coly, the wave (1) is a simple wave and with the slope of its trajectory is given by Eq. 9.19 ai = 1 + F^A
(9.33)
a2 for co* > co\ > co^. State Pi is represented by Pi[a?*,«2,83], hence a band of pure component A\ is formed. Once the values of CO{ for the different states involved have been identified and calculated, all the wave interactions can be analyzed in detail [15]. The same approach can be applied to multicomponent samples [10]. As mentioned earlier, the ^-transform and the o;-transform give the same results. Distance-time diagrams, such as the one just discussed and shown in Figure 9.8, can be constructed using the ^-transform as well [9,11,12,14].
9.2 The Theory of Characteristics
457
9.2.4 Critical Value of the Displacer Concentration As the displacer concentration increases, co* decreases (Eq. 9.27), and it moves to the left-hand side in Eq. 9.28. The number of shock waves increases. If we are interested in obtaining an isotachic train after the development is over, we want to obtain n rear shocks. This is possible only if the displacer concentration is large enough, and co* becomes the first in the ordered set of characteristic parameters in Eq. 9.28. This requires that a;* < a.\. Introducing this condition in Eq. 9.27 gives c
« > v (v On V f l l
As we have already shown, using a simple approach based on the assumption that a steady state can be reached (Eqs. 9.6 and 9.7), the critical displacer concentration for successful displacement of a multicomponent mixture depends only on the adsorption isotherm of the displacer (aM and bn) and on the initial slope of the isotherm of the less retained solute at infinite dilution («i). On the other hand, if the displacer concentration is so small that co* is the last element in the ordered set of characteristic parameters of the displacement (Eq. 9.28), there are no shocks, and no isotachic train is possible. This happens when co* > an-\. Combining this condition and Eq. 9.27 gives (9.35) ht-1
We can generalize Eqs. 9.34 and 9.35. If the displacer concentration is such that fly_i < co* < fly, we have \
1
/„
\
(9.36) In this case, we observe the successful formation of an isotachic train including n — j bands, preceded by a series of / — 1 bands eluted separately or interfering, but not included in an isotachic train. The /th wave is a shock if co* < co- and a simple wave if a;* > ur-.
9.2.5 Plateau Concentrations and Bandwidth In the final stage of the development of an isotachic train, we obtain n plateaus and as many shock waves: one plateau and one shock wave for each pure band /. Since only component / is present in the /th band, and all the other components are missing from that band: Pl,f CO;
* =
CO
coFl>i = ak
for
k^j
(9.37a)
where co^ '^ is the value of co for the component k in the band j when the isotachic train is formed and the band has a concentration plateau. Since at this final stage
458
Band Profiles in Displacement Chromatography with the Ideal Model
of the development w* is the smallest root of Eq. 9.15, while the other (n — 1) roots are given for every pure component by a^ (k ^ j), we have co*
=
^ - ^ = ••• =
a
J—p = . - • =
a
"
(9.38)
or
Cf = f ( ^ - 1 ) + ^ Q
(9.39)
' fcy \an J anbj where C? is the plateau concentration of component j in the isotachic train. Equation 9.35 (which is identical to Eqs. 9.5 and 9.8) shows that the concentration plateau of each solute in the isotachic train does not depend on the feed concentration and its composition, but depends only on the displacer concentration and on the parameters of the isotherms of the displacer and the solute itself. The concentration Cj can also be obtained by solving the equation:
Using Eq. 9.35, and applying the conservation of mass of each component during the experiment, we can derive the time width of the band (9.41) Since Sy = ATJ/AXJ is the slope of the characteristic line, we obtain the bandwidth along the column: (9-42) The slope of the characteristic lines can be calculated using Eq. 9.18: s = si = s2 = • • • = s» = 1 + Fa;* = 1 +
Fa< ;
(9.43)
Since the paths of the n shocks are now known, we can locate each of them at the final stage of the isotachic train without the need for a detailed analysis of the interactions between the bands. The shocks are located using Eqs. 9.41 to 9.43. In order to calculate the minimum bed length required for the formation of an isotachic train, however, we must analyze all the interactions involved. In addition, if the concentrations of the solutes are chosen so that to* is larger than a)f for one or more components, the corresponding waves of these solutes are simple waves, and we must also analyze the successive interactions between these simple waves and each of the shocks in order to obtain the solution [10]. An excellent
459
9.2 The Theory of Characteristics
ite by clevulopm
Figure 9.9 Application of displacement chromatography to the separation of 15 rare earth cations of euxenite on a strong acid cation-exchange resin. Feed amount: 20,000 mequiv. Column cross-sectional area: 100 cm2; total porosity: 0.4. Displacer concentration: Q : 0.1 mequiv/cm3. Flow rate: 2.0 mL/s. Component, mole fraction, separation factor «: [La, 0.006,1], [Ce, 0.015,4.7], [Pr, 0.002,10.7], [Nd, 0.008,21.9], [Sm, 0.01,67.6], [Eu, 0.001,93.3], [Ga, 0.035,95.5], [Tb, 0.016,457], [Y, 0.622,692], [Dy, 0.095,1072], [Ho, 0.035,3890], [Er, 0.09, 6761], [Tm, 0.015, 22380], [Yb, 0.042, 46770], [Lu, 0.008, 85110]. Reproduced with permission from F. Helfferich and D. B. James, J. Chromatogr. 46 (1970) 1 (Fig. 8).
example of a distance-time diagram published by James and Helfferich [11] is reproduced in Figure 9.9. It illustrates the separation of a mixture of 15 rare earth cations by ion-exchange displacement chromatography. We show in Figures 9.10 the separation of a ternary mixture (solutes Alr A2 and A3), displaced by a solution of A4, as predicted by the ideal model and calculated by Rhee et al. [10]. In this case, the displacer concentration is higher than the critical value, co* < a\, and the isotachic train finally formed includes the bands of all three components. For clarity, the authors have shown two separate chromatograms for each time at which they calculated the individual band pro-
460
p- L JOJ
Band Profiles in Displacement Chromatography with the Ideal Model o.e
, T
0.! -
n
=
r 1 -1 LJ
15 1 1i
Figure 9.10 Progressive formation of the isotachic train in displacement chromatography. Profiles along the column. Solute Al (dash-dot), solute A2 (dash-dot-dot), solute A3 (solid), solute A4 (dash). H.-K. Rhee and N.R. Amundson, AICHE J, 28 (1982) 423 (Fig. 10). Reproduced by permission of the American Institute of Chemical Engineers. ©1982 AIChE. All rights reserved.
files. The concentration profiles of the first and third components are given in the bottom chromatogram, the profiles of the second component and of the displacer in the top one. Note that these chromatograms are concentration profiles along the column length, x. The first component to elute is the one that has moved farther to the right. All the components are introduced at the same concentration, 0.05 M, except the displacer. However, the bands in the isotachic train are more
9.3 Coherence Theory
461
concentrated than the feed. The feed mixture is introduced between t = TQ = 0 and To = (utp)/L = 10 (L, column length). So, in Figure 9.10a (T = 13), the injection is finished and the displacer has been introduced for only a short time, at a concentration C4 = 0.20 M. The bands of components 1 and 2 already exhibit a plateau at an enriched concentration due to frontal enrichment (Chapter 8, Section 8.2), and a narrow band of pure third component is already formed. This band widens progressively in Figures 9.10b to 9.10f, where it becomes rectangular. Similarly, the bands of the second and third component are separated, grow, and become narrower. At T = 70, the isocratic train forms and propagates unchanged at a constant velocity.
9.2.6 Critical Column Length for Isotachic Train Formation Even in an ideal column, the reorganization of the distribution of the component concentrations between the injection and the formation of the isotachic train requires a certain time, i.e., it cannot be achieved in less than a minimum migration distance. This distance can be derived from the distance-time diagram.
9.3 Coherence Theory The coherence theory of chromatography [9] is based on the use of the concept of coherence to explain the band profiles observed in ideal chromatography. A chromatographic column subject to a disturbance will, after a period, settle into a "resolved " state, which consists of a series of composition waves, each of them being subject to the coherence condition uc,i = uCij
(9.44)
This condition is valid for all i and /', i.e., for all the compounds involved in the system studied, feed and mobile phase components alike. This theory was originally developed for the cases in which the retention mechanism is accounted for by a stoichiometric reaction such as ion-exchange. However, Helfferich [9] has shown that it can also be applied in the case of adsorption, if assuming a Langmuir competitive isotherm (see Eq. 9.45), by introducing a dummy species, p, that is used to convert the non-stoichiometric adsorption of an n-component system into the stoichiometric reaction of an n + 1-component system. The Langmuir competitive isotherm is given by Hi = ,
t
X \ n
for
' = ! ' 2 ' • • • >n
( 9 - 45 )
In this equation, the solutes are listed in order of their decreasing affinity for the stationary phase. The less retained solute is component n and the more retained is component 1. If all the components follow thermodynamically consistent Langmuir competitive isotherm behavior (in practice, an unusual case, unfortunately),
462
Band Profiles in Displacement Chromatography with the Ideal Model
it can be shown that the coherence condition defines a grid of coherent composition paths to which the system is restricted once the coherence condition is satisfied. Thus, given the feed history, this grid can be used to find the composition route for the column and, therefore to predict the column effluent history. Using the nonlinear /i-transform, Helfferich and Klein [9] defined an orthogonalized composition space which they called the /z-composition space. The transformation from the C composition space to the h space involves the calculation of the roots of the following equation
£ ^ - 1
=0
(9-46)
The application of the ft-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called /i-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that fll > «2 > «3 > • • • > « « .
The first application of the coherence theory to displacement was made by Helfferich and James [12] who calculated the separation by displacement on an ion exchange column of a mixture of 15 rare earths cations (see Figure 9.9). The achievement of this separation is facilitated by the stoichiometric exchange mechanism that is unique to ion-exchange chromatography. The predictions of the /z-transform theory were also compared with the results of experimental measurements made on an ion-exchange system with a four-component feed by Clifford [13]. A procedure for simulating stoichiometric behavior in a system following Langmuir adsorption isotherm behavior was suggested by Helfferich and Klein [9]. It consists in introducing a fictitious component (an + 1-th component) and an adjustable parameter R with any value in the range 0 < R < an. The results of the calculations are not affected by the choice of R. The value of R selected in the work of Frenz and Horvath [14] was equal to the slope of the operating line, with R — (\\IC\, a value that is always positive. The condition R < an yields
which is the condition stated by Glueckauf [6] for the successful achievement of displacement.
9.3 Coherence Theory
463
The ^-transform method was also applied to the calculation of the breakthrough curve for the separation of a binary mixture on a charcoal bed by Tien et al. [36]. By using the ^-transform, the course of the separation can be calculated by transforming the concentration variable via the ^-transform into a coordinate system in which algebraic equations describe the process for any number of components. Displacement development involves two successive step changes in the mobile phase concentration at the column inlet. The first step takes place upon the introduction of the feed. A second concentration step takes place when the feed injection ends and, almost simultaneously, the displacer solution enters into the column. Both step changes give rise to self-sharpening boundaries between which lie the bands of the feed components. As long as the displacer has a higher affinity for the stationary phase than any of the feed components, the rear boundaries of the feed slug remain sharp because the displacer suppresses the adsorption of the component immediately ahead of itself. The calculation of the boundary velocities in multicomponent chromatography can be greatly simplified under certain conditions by an appropriate transform of the variable. The /z-transform was introduced for this very purpose by Helfferich [35]. It is equivalent to the transformation to the "characteristic parameters" that was suggested by Rhee et al. [10] at the same time. The power of the /z-transform is based on the fact that only one of the new dependent variables, h\, hi, • • •, h{, • • •, hn for a n-component mixture changes from one side of a boundary to the other side, in contrast with the possible changes of several of the n concentration variables. As a result the expression giving the velocity of a boundary depends on only one variable when it is written in terms of the variable h\, whereas an expression for this velocity written in terms of the concentrations will depend in general on all the n concentrations, Q. If the h variable that changes across a boundary is h{ and its value ahead (i.e., downstream) of the boundary is /i,/fl and its values behind (i.e., upstream) of the boundary is h^y then the boundary is self-sharpening if h{A < h{j, and the adjusted velocity of this sharp boundary is given by [9]: vs>i = ^-^—=hiAhi)hPi
(9.48)
where P, is given by =
; - i ' ' / - » + * i'
(949)
TTn+1
where q/C = qn/Cn is the slope of the operating line, vs is the adjusted velocity of displacer front and UQ is the mobile phase velocity. If hiiA > hif,, the boundary is diffuse and there is a continuous change in hi from hiA to h{^. The velocity of any value of h{ in this range is given by vhi = hfPi
(9.50)
Thus, the calculation of the boundary velocities is greatly simplified by the use of Eqs. 9.48 and 9.49 instead of the concentration velocities. This is due to the
Band Profiles in Displacement Chromatography with the Ideal Model
464
Table 9.1 Concentrations and /z-function roots in displacement development
Q =0
Q> 0
Carrier Feed Displacer " Trivial roots;
I,--- ,n 1
0C\\,-
• • a
,0i\n
l,l #1,2/ ' " ' / *l,n
2 , - - ,n b
z
hf
i
V-- ,n 1
?*2,F,' ••,K,F L
Roots of Eq. 9.48
replacement of the variables Q with the h{ which represent the natural coordinate of the system [9]. If the components of the system are ranked from 1 to n, component n being the least retained and 1 being the displacer (the most strongly retained compound), the variables hj are obtained as the roots of the h-function defined as: (9.51)
= 1
Equation 9.51 is a polynomial that cannot be solved explicitly, but its roots can be evaluated easily by a numerical technique such as Newton's method. The characteristic parameters, co, employed by Rhee and Amundson [10] which were discussed earlier in this chapter are analogous to the reciprocal of the h{ values. In displacement chromatography, three different solutions, the mobile phase, the feed, and the displacer solution enter successively into the column. For the mobile phase, Q = C% = • • • ,= Cn = 0. For the feed, Q = 0 (displacer) and C2 > 0,C3 > 0,- • -,Cn-l > 0,Cn > 0. For the displacer solution, Q > 0, C2 = C3 • • • = Q_i = Cn = 0. Table 9.1 summarizes the composition of the three solutions used in displacement development and the /z-function roots. When one or more components are not present in mixture (which is true for all three solutions considered here), Eq. 9.51 alone is not sufficient to determine all the h roots. For example, the feed does not contain the displacer, so Cj = 0 and the first term of Eq. 9.51 is 0. Therefore, the resulting polynomial equation has only n — 1 terms and n — 1 roots which we call ^2,F^3,F/ • • • / hn,F- ^ order to complete the set of hi, Helfferich and Klein [9] showed that there is another, trivial root which is given by hi = cc^i = «i/fli = 1. More generally, when component i is absent from the solution (Q = 0), the corresponding trivial root cannot be derived from Eq. 9.51 but is obtained from: ft?
=
fll fll
or
fl,-
(9.52a) (9.52b)
9.3 Coherence Theory
465
if Q = 0 and i > 1, h =
=l
(9.52c)
and ftj = flj/fli when i = 1 (Q = 0). (Note that, in the carrier and the feed, the trivial root corresponding to Q = 0 is hj while, in the displacer solution, the trivial root corresponding to Q = 0 is fy_i.) Equation 9.51, along with Eqs. 9.52a to 9.52c for the evaluation of the trivial roots, allow the calculation of the h{ values for all of three solutions involved in displacement. These values are given in Table 1. For example, the feed solution does not contain any displacer, so C\ =0 and the first term of Eq. 9.51 is equal to 0. Therefore, the polynomial has only n — 1 roots, called h^h^. the trivial root is determined from Eq. 9.52c and is equal to h\ = 1. The mobile phase contains no solutes and no displacer, so none of its roots can be determined from Eq. 9.51. Instead all the roots are trivial roots and are % = 1, hi = oc\r2 = &\l tti, • • • ,hn = a i,n = a\/an. The displacer solution lacks components 1 through n — 1, so only one of the roots can be determined by Eq. 9.51 and it is: «i, n+ i
(9.53)
and the second through the n-th roots are given by h\ — m-y^, hi — OL\$, /13 = In accordance with the above rule, for an n — 1 feed solution and a displacer, the switch from pure mobile phase to feed changes n — 1 of the hi roots (% remaining constant) and so gives rise to n — 1 boundaries that travel along the column. All these boundaries are self-sharpening and their velocities are ordered as v&2 < vhj, < • • • < vhn. The switch from the feed solution to the displacer solution changes all the n values of the h{, giving rise to n boundaries. The boundary associated with hn is the transitional boundary, representing the change of hn from h^ to R\in+\. This boundary may be self-sharpening or diffuse, depending on the relative magnitude of h^ and a^ n + i. In the final pattern, the feed components are separated into single-component zones. For the zone containing the pure component j , hj = «i,y+i and the other hj roots are given by the trivial root ct\ti for the components that are not present within the zone. For example, in the zone containing only the component 2 the roots are h\ = CL\^ = 1, /12 = a i,3/ ^3 — aiA'aru^ ^« = ai,n+iThe velocity of the boundary between the zones of the displacer and of component 2 (the velocity of the displacement front) is given by Eqs. 9.48 and 9.49. vsi = hliahlibPx = 1 x ct^Py
(9.54) x a
x a
with Pi = (a1/3 x «1(4 x • • • x ai,n+i)/(l x #1,2 i,3 i,4 x • • • x #i /rt +i). Therefore, vSii = 1 and, according to the coordinate frame used here, the velocity of the displacer front is equal to 1. Similarly, the isotachic velocity of all the boundaries can be calculated and it is shown that they are also all equal to 1 (that is equal to the velocity of the displacer front). This velocity is the same as the one that can be determined from the slope of the operating line, in accordance with Tiselius [5].
Band Profiles in Displacement Chromatography with the Ideal Model
466
20
Column Length, cm 30 40 SO
60
Figure 9.11 Displacement development graph for the separation of a binary mixture. The crosshatched area represents mixed regions in the transient part of the development, the hatched areas the pure component regions in the pattern. /. Frenz, C. Horvdth, AIChE ]., 31 (1985) 400 (Fig. 3). Reproduced by permission of the American Institute of Chemical Engineers. ©1985 AIChE. All rights reserved. 5 Ojmi
EO 25 Bed Volume
Since the migration velocities of the concentration boundaries are easily calculated with the ft-transform variable, a development graph such as the one in Figure 9.11 can be constructed easily once the isotherms of the feed components and the displacer, the column hold-up volume, the feed volume, its composition, and the displacer concentration are known [14]. The starting points of the boundary trajectories are the beginning or the end of the feed introduction step. The other points of interest in the diagram are those at which the boundaries originating from the beginning of the feed introduction intersect those arising from the end of the feed introduction. At these points of intersections, the boundary velocity changes, as predicted from the knowledge of the ft-function roots that change when two boundaries intersect. When the two boundaries are associated with a change in the same root, they combine into a single boundary after this intersection and the new velocity is given by Eq. 9.48 or 9.50, that is its new value of the boundary velocity is intermediate between the two boundary velocities before the intersection. If the two boundaries do not represent changes in the same ^-function root, then they both continue beyond the intersection point, with velocities that changed due to the change in the value of the P; term in Eqs 9.48 and 9.50. The displacer boundary that is associated with a change in the value of h\ is the only one that does not intersect with any other boundary and, so, its velocity does not change as it moves along the column. After constructing the paths of the various boundaries, the only remaining information that we need in order to construct the displacement chromatogram is the set of concentrations of the components in each part of the development graph. Since the values of the hj are known in each part of the graph, this is accomplished by reversing the transformation of Eq. 9.51. The reverse transform to calculate the concentrations from the known values of hi is given by: C
i =
(9.55) -1
9.4 Practical Relevance of the Results of the Ideal Model
0
K>
TO
467
W
Mobile Phase Concentration, ma/mi
Volume of Effluent, ml
3 1 3 : •> e o Volume of Effluent, ml
"o t i
6 8 O 12
Volume ol Effluent, ml
Figure 9.12 Comparison of experimental band profiles and the profiles calculated using the ideal model and single-component isotherms. (Far left): isotherms (q vs. C, both in mg/mL) of 1, resorcinol; 2, catechol; and 3, phenol. Stationary phase: Cig silica with (top, A) 7.16% carbon load and (bottom, B) 7.97% carbon load. Experimental data points and curves fitted to the Langmuir equation. (Center left): effluent concentration profiles (C, mg/mL, vs. V, mL) for a 15-cm-long column. Calculated (top) and experimental (bottom) profiles. Carrier: water. Displacer 50 kg/m 3 phenol solution in water. Flow rate: 3.33 mm 3 /s, fraction volume 1000 mm3,25DC, Feed: 40 x 10~6 kg each of resorcinol and catechol in 0.4 cm3 water. Stationary phase B. (Center right): same as Center left, except column length 40 cm. (Far right): same as Center left, except column length 25 cm. /. Frenz and Cs. Horvdth, AIChE }., 31 (1985) 400 (Figs. 5 to 8). Reproduced by permission of the American Institute of Chemical Engineers. ©1985 AIChE. All rights reserved.
9.4 Practical Relevance of the Results of the Ideal Model Figures 9.12 compare some experimental band profiles obtained by displacing a resorcinol-catechol mixture by a phenol-water solution. The isotherms of the three components are well accounted for by the Langmuir model (Figure 9.12a). The experimental band profiles obtained by analysis of collected fractions are in excellent agreement with the band profiles calculated using the ideal model. With a 15-cm-long column, the first component (resorcinol) is eluted as a narrow spike followed by a diffuse boundary (Figure 9.12b), while the mixed band containing both catechol and resorcinol is wide. With a 25-cm-long column, the diffuse boundary is confined to the first part of the resorcinol band, and the mixed band is narrower (Figure 9.12c). Finally, with a 40-cm-long column, the isotachic train is fully established, and the mixed zone, due to the finite rate of mass transfer, is very narrow (Figure 9.12d). The main reason for this sharp boundary, not found in many reports on displacement chromatography, is the extremely high concentrations of the bands. They are so high that they correspond to the region of the isotherms close to its horizontal asymptote. For example, in Figures 9.12a, the saturation capacities derived from the values of the coefficients of the Langmuir isotherm are 32 and 26
468
REFERENCES
mg/mL for catechol and phenol, respectively in A; and 61, 46 and 36 mg/mL for resorcinol, catechol and phenol, respectively in B. These values are less than twice as large as the adsorbate concentrations at the largest mobile phase concentration used in the measurements. The nonlinear effects are extremely intense under such conditions, which explains the exceptional sharpness of the band boundaries. Furthermore, the separation factors between catechol and phenol are 1.76 for stationary phase A and 1.74 for stationary phase B. As explained in Chapter 16 (Eq. 16.15.), the larger the separation factor, the narrower the boundaries between bands. Figure 9.12 illustrates how narrow band boundaries become when the effects of a large separation factor and large plateau concentrations are combined. The most conspicuous difference between the experimental and calculated band profiles is the lower maximum concentration peak (center figures) and the lower plateau concentration (far right figures) measured for resorcinol and catechol than calculated. Note, however, that the isotherm was determined in a concentration range which does not extend to the displacer concentration used. Hence, there is a possible error in the exact position of the operating line. In this concentration range, deviations of the adsorption behavior from the Langmuir model is probable, as this model assumes the solution to be ideal. More experimental results are presented and discussed in Chapter 12, together with the results of the equilibrium-dispersive model. Numerous examples of isotachic trains and of band profiles in incompletely developed trains are given there.
References [1] M. S. Tswett, Ber. Deut. Botan. Ges. 24 (1906) 316. [2] M. S. Tswett, Khromofilly V Rastitel'nom Zhivotnom Mire [Chromophylls in the Plant and Animal World], Izd. Karbasnikov, Warzaw, Poland, 1910, partly reprinted in 1946 by the publishing house of the Soviet Academy of Science, A. A. Rikhter and T. A. Krasnosel'skaya, Eds. [3] R. Kuhn, E. Lederer, Naturwissenschaften 19 (1931) 306. [4] A. Tiselius, S. Claeson, Arkiv Kemi Mineral Geol. 16A (1943) 18. [5] A. Tiselius, Disc. Faraday Soc. 7 (1949) 7. [6] E. Glueckauf, Proc. Roy. Soc. A186 (1946) 35. [7] E. Glueckauf, Disc. Faraday Soc. 7 (1949) 12. [8] E. Glueckauf, J. Chem. Soc. (1947) 1302. [9] F. Helfferich, G. Klein, Multicomponent Chromatography, M. Dekker, New York, NY, 1970. [10] H.-K. Rhee, R. Aris, N. R. Amundson, AIChE J. 28 (1982) 423. [11] D. B. James, F. Helfferich, in: Proc. Seventh Rare Earth Conf., Vol. 1, San Diego, CA, 1968, p. 397. [12] F. Helfferich, D. B. James, J. Chromatogr. 46 (1970) 1. [13] D. Clifford, Ind. Eng. Chem. (Fundam.) 141 (1982) 21. [14] J. Frenz, Cs. Horvath, AIChE J. 31 (1985) 400. [15] H. K. Rhee, R. Aris, N. R. Amundson, Philos. Trans. Roy. Soc. London A267 (1970) 419. [16] F. G. Helfferich, Chem. Eng. Sci. 46 (1991) 3320. [17] D. Basmadjian, P. Coroyannakis, Chem. Eng. Sci. 42 (1987) 1723.
REFERENCES
469
[18] R. W. Geldart, Y. Qiming, P. C. Wankat, L. N.-H. Wang, Separat. Sci. Technol. 21 (1986) 873. [19] J. Frenz, Ph.D. thesis, Yale University (1983). [20] S. C. D. Jen, N. G. Pinto, J. Chromatogr. 590 (1992) 3. [21] S. C. D. Jen, N. G. Pinto, Reactive Polymers 19 (1993) 145. [22] J. Frenz, Cs. Horvath, in: Cs. Horvath (Ed.), High-Performance Liquid Chromatography — Advances and Perspectives, Vol. 5, Academic Press, New York, NY, 1988, pp. 211-314. [23] Cs. Horvath, A. Nahum, J. H. Frenz, J. Chromatogr. 218 (1981) 365. [24] Cs. Horvath, J. Frenz, Z. El Rassi, J. Chromatogr. 255 (1983) 273. [25] Cs. Horva'th, in: F. Bruner (Ed.), The Science of Chromatography, Elsevier, Amsterdam, 1985, p. 179. [26] Y.-F. Maa, Cs. Horvath, J. Chromatogr. 445 (1988) 71. [27] T.-W. Chen, N. G. Pinto, L. V. Brocklin, J. Chromatogr. 484 (1989) 167. [28] G. Subramanian, S. Cramer, J. Chromatogr. 484 (1989) 225. [29] Gy. Vigh, G. Quintero, Gy. Farkas, J. Chromatogr. 506 (1990) 481. [30] C. Kemball, E. K. Rideal, E. A. Guggenheim, Trans. Faraday Soc. 44 (1948) 948. [31] M. D. LeVan, T. Vermeulen, J. Phys. Chem. 85 (1981) 3247. [32] F. D. Antia, Cs. Horvath, J. Chromatogr. 556 (1991) 119. [33] G. Carta, A. Dinerman, AIChE J. 40 (1994) 1618. [34] J. C. Bellot, J. S. Condoret, J. Chromatogr. 635 (1993) 1. [35] F. Helfferich, Ind. Eng. Chem. Fundam. 6 (1967) 362. [36] C. Tien, J. S. C. Hsieh, R. M. Turian, AIChE J. 22 (1976) 498.
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Chapter 10 Single-Component Profiles with the Equilibrium Dispersive Model Contents 10.1 Fundamental Basis of the Equilibrium Dispersive Model 473 10.2 Approximate Analytical Solutions 476 10.2.1 The Houghton Solution 477 10.2.2 The Haarhoff-Van der Linde Solution 478 10.2.3 Range of Validity of the Houghton and Haarhoff-Van der Linde Equations . . . 480 10.2.4 Influence of the Sample Size on the Bandwidth 482 10.2.5 Other Plate-Height Equations for Non-linear Chromatography 486 10.2.6 Comparison of the Experimental Band Profiles and the Prediction of These Equations 489 10.2.7 Relaxation Model 490 10.3 Numerical Solutions of the Equilibrium-Dispersive Model 492 10.3.1 Principle of the Finite Difference Methods 494 10.3.2 Estimation of the Numerical Errors Made during the Calculation 495 10.3.3 First Method: Direct Calculation of Numerical Solutions of the Mass Balance Equation 496 10.3.4 Second Method: Replacement of Axial Dispersion by Numerical Dispersion . . . 497 10.3.5 Application of the Second Method 500 10.3.6 Finite Element Method 505 10.4 Results Obtained with the Equilibrium Dispersive Model 509 10.4.1 Comparison of Solutions of the Ideal and the Equilibrium-Dispersive Models . . 509 10.4.2 Comparison of the Results of Different Calculation Methods 513 10.4.3 Results of Computer Experiments 515 10.4.4 Comparison with Experimental Results 518 References 527
Introduction We have shown in Chapters 7, 8, and 9 that the ideal model gives a good first approximation of the band profiles observed under conditions of strongly nonlinear behavior of the isotherm. This approximation is better at high column efficiency. Even then, however, the profiles predicted by the ideal model are angular and have no round corners like those observed experimentally. Indeed, experimental profiles do not show concentration discontinuities but shock layers of various thicknesses (see Chapter 14), while the other features of the solutions are smoothed out, eroded by the influence of axial dispersion and of the resistances to mass transfer in the column. A more accurate model is needed to account for the actual band profiles and to predict them when needed. 471
472
Single-Component Profiles with the Equilibrium Dispersive Model
Preparative chromatography plays a major role as a purification process in the pharmaceutical and fine chemical industries. It is important to calculate and evaluate the performance of a separation unit for the production of a given purified component from a certain feedstock. It is also important to optimize the design and operating conditions, often for minimum cost and sometimes for maximum production rate. This requires the availability of a model of chromatography that gives accurate band profiles, so that both the production rate of the needed component at the stated degree of purity and its recovery yield achieved can be calculated. The computation power of personal computers makes it easy to carry out systematic computer-assisted optimization of any separation process or separation strategy (e.g., recycling techniques, displacement chromatography, simulated moving bed, rotary annular column, use of different operating modes) that can be designed for use in preparative chromatography. This approach permits large savings on the time and cost of process development. In order to meet the requirements just described, we need a model which includes the finite column efficiency, i.e., the contributions to the band profile of the axial dispersion (molecular and eddy diffusions) and of the kinetics of mass transfer through the chromatographic column. Otherwise, it would be difficult to estimate accurately the band profile changes that take place upon variation of the parameters which affect the column efficiency, such as flow rate, packing-particle size, column length, packing procedure, and packing life-time. We have discussed in Chapters 3 and 4 the importance of adsorption isotherms and in Chapter 5 the fundamental basis of diffusion, axial dispersion, and mass transfer kinetics and the nature of their contributions to the shape of band profiles. In Chapter 6, we have shown how the general rate model of chromatography permits the inclusion of axial dispersion and of all possible contributions to mass transfer resistances and how this model can be solved in linear chromatography for the general case. The problem of accounting for the combined effects of axial dispersion and a finite rate of mass transfer kinetics in nonlinear chromatography is much more complex for a combination of reasons. First, the behavior of the different components of a mixture cannot be considered separately. Some approaches that are possible in the single-component case, and lead to approximate analytical solutions, become untractable for binary mixtures. Second, band profiles result from the convoluted interaction between phase equilibria thermodynamics, which tends to sharpen some boundaries, turning them into discontinuities, while dispersing other boundaries, and the kinetics of phase equilibria, which tends to disperse all boundaries. The classical shift-invariant convolution permits a simple calculation of the combined effects of multiple sources of band broadening when the column efficiency is not very low. This approach gives correct results in linear chromatography but is incorrect in nonlinear chromatography [1]. The simplest model that takes axial dispersion and mass transfer kinetics into account is the equilibriumdispersive model. This model permits, with a good approximation, the accurate prediction of the importance of the self-sharpening and dispersive phenomena due to thermodynamics and kinetics of phase equilibria. This, in turn, results in correct prediction of the band profiles and the achievement of often excellent
10.1 Fundamental Basis of the Equilibrium Dispersive Model
473
agreement with experimental data. This model, however, is based on an equation which is exact only in linear chromatography but becomes approximate in nonlinear chromatography (see Chapter 14, Section 14.1.7). The equilibrium-dispersive model had been discussed and studied in the literature long before the formulation of the ideal model. Bohart and Adams [2] derived the equation of the model as early as 1920, but it does not seem that they attempted any calculations based on this model. Wicke [3,4] derived the mass balance equation of the model in 1939 and discussed its application to gas chromatography on activated charcoal. In this chapter, we describe the equilibriumdispersive model, its historical development, the inherent assumptions, the input parameters required, the methods used for the calculation of solutions, and their characteristic features. In addition, some approximate analytical solutions of the equilibrium-dispersive model are presented.
10.1 Fundamental Basis of the Equilibrium Dispersive Model The model of Lapidus and Amundson [5] is the focal point of study of linear and nonlinear chromatography. Since in chromatography we have two independent variables, z and t, and two dependent variables, the concentrations of the solute in the mobile and the stationary phases, C and Cs, respectively, two equations are required for the model to permit the calculation of C(x, t) and Cs(x, t). The model of Lapidus and Amundson considers a set of two partial differential equations for a single component. The first equation is the mass balance equation (Eq. 2.2) (10.1) t/t
Ut
Ui
O^-
where w is the mobile phase linear velocity, F is the phase ratio (F — vs /vm — (1 — e) Ie, where e is the total porosity of the column packing), and Di is the coefficient of axial dispersion. The axial dispersion term is the sum of the contributions of axial molecular diffusion and eddy diffusion. The boundary and initial conditions of Eq. 10.1 have been discussed in Chapter 2 (Section 2.1.4). The second equation of the model relates the two concentrations in Eq. 10.1. Lapidus and Amundson [5] chose a linear kinetic model dt
= kaC-kdQ
(10.2)
where ka and k^ are the rate constants of adsorption and desorption, respectively. Another kinetic model is often used in chromatography, the solid film linear driving force kinetic equation (see Chapter 14):
^f=km(q-Cs)
(10.3)
where km is the lumped mass transfer coefficient and q is the concentration of the compound studied in the stationary phase at equilibrium with the mobile phase
474
Single-Component Profiles with the Equilibrium Dispersive Model
at concentration C in the mobile phase. The equilibrium isotherm is given by q = /(C). As shown in Chapter 6 (Section 6.2.2), in linear chromatography Eq. 10.3 is a particular case of Eq. 10.2 [6]. This is no longer true in nonlinear chromatography. The solution of the model of Lapidus and Amundson is complex, even in linear chromatography (Chapter 6, Section 6.2). However, Van Deemter et al. [7] have shown that, in linear chromatography, the solution of this model is equivalent to a Gaussian profile provided that the mass transfer kinetics is not very slow. We have shown (Chapter 6) that this is true as long as the column efficiency exceeds a few dozen theoretical plates, a rather modest threshold given the current level of column performance. The variance of the Gaussian profile equivalent to the solution of the Lapidus-Amundson model is the sum of the contributions of the axial dispersion and the mass transfer resistances. Therefore, in linear chromatography, the column height equivalent to a theoretical plate (HETP) is given by [7]
do.4) It is convenient to separate these contributions to the column HETP and to distinguish the number of mixing stages, Nnisp, the number of mass transfer stages, Nm, and the apparent number of theoretical plates, N ap = L/H. These various contributions are described by the following equations
(m5)
=P~Y= j k Nm
=
^
(10.6)
Nap
=
^
(10.7) 1
•
~
(10.8)
According to Glueckauf [8,9], in linear chromatography the lumped mass transfer coefficient km in Eq. 10.3 is related to the film mass transfer coefficient, kp and the pore diffusion coefficient, Dp, by F dp dl — + 60D 77&r k'okm " 6k f p
(10-9)
where dp is the particle diameter. Equations 10.4 and 10.9 are very important. Their combination demonstrates that, in linear chromatography, the effects of the axial dispersion and the various contributions to mass transfer resistances are additive for both the general rate model and the two linear driving force models (See Chapter 2, Section 2.2.3 and Chapter 6, Sections 6.2 and 6.3). This result constitutes the fundamental basis of the equilibrium-dispersive model of chromatography. In the equilibrium-dispersive model, it is assumed that: • the mobile and the stationary phases are always in equilibrium and
10.1 Fundamental Basis of the Equilibrium Dispersive Model
475
• the contributions of all the nonequilibrium effects can be lumped into an apparent axial dispersion coefficient. Accordingly, the equilibrium-dispersive model of chromatography for a single component is represented by one single partial differential equation, the mass balance equation dC
rdq
dC
„ 32C
.
and two algebraic equations: the isotherm equation, q = f(C), and the equation relating the apparent dispersion term to the apparent column efficiency:
In linear chromatography, the height equivalent to a theoretical plate, H, and the apparent plate number, Nap, are related to the axial dispersion and to the coefficients of resistance to mass transfer through Eqs. 10.4 to 10.9. As discussed already in Chapter 2 (Section 2.2.6), Giddings [10] has developed a nonequilibrium theory of chromatography and showed that the influence of the kinetics of mass transfers can be treated as a contribution to axial dispersion. As illustrated in Chapter 6, this approximation is excellent in linear chromatography, as long as the column efficiency exceeds 20 to 30 theoretical plates. Rhee and Amundson [11] have studied the effects of axial dispersion and mass transfer resistance in nonlinear chromatography, using the model of Eqs. 10.1 and 10.3. hi the case of the breakthrough curve (frontal analysis for chromatographers, Riemann problem for mathematicians), and assuming the existence of shock layers (i.e., of a stable front or constant pattern, in which the concentration varies rapidly, see Chapter 14), they have shown that Eqs. 10.1 and 10.3 can be combined and reduced to a single equation. The contributions of axial dispersion and mass transfer resistances to the thickness of the shock layer are additive, although there exists a coupling, second-order term which turns out to be negligible in most practical cases of interest in chromatography. However, instead of Eq. 10.4, the following additivity rule must be used
"=
+2
where K is the slope of the isotherm chord, K = (FAq) /AC. For a Langmuir isotherm and for a step injection from 0 to C, the parameter K is given by K — ko/(l + bC). Equation 10.12 is of primary importance. As shown in Chapter 14, it proves that in nonlinear chromatography, as well as in linear chromatography, the contributions of axial dispersion and of the mass transfer coefficient are additive, at least in frontal analysis, assuming the solid film linear driving force model and that the flow velocity dependence of the shock layer thickness in frontal analysis is related to the HETP curve in linear chromatography [12]. However, since K is concentration dependent, the apparent axial dispersion coefficient in nonlinear
476
Single-Component Profiles with the Equilibrium Dispersive Model
chromatography, by contrast with linear chromatography, is concentration dependent. In linear chromatography K = k0; therefore, Eq. 10.12 reduces to Eq. 10.4. Equation 10.12 is valid only in frontal analysis and will be discussed further in Chapter 14. In overloaded elution, neither Eq. 10.4 nor Eq. 10.12 is valid. The former statement is true because the isotherm is no longer linear, the latter because, at the difference of what happens in frontal analysis, the band height in overloaded elution, and hence the amplitude of the shock layer and its velocity vary during the migration of the band. In overloaded elution K in Eq. 10.12 should be replaced by the slope of the isotherm K = Fdq/dC. In the case of the Langmuir isotherm, K would be given by K = fco/(l + bC)2. Accordingly, the situation in overloaded elution becomes more complex than in linear chromatography or in nonlinear frontal analysis, since C and hence K vary all along the elution band profile. In the equilibrium-dispersive model of chromatography, however, we assume that Eq. 10.4 remains valid. Thus, we use Eq. 10.10 as the mass balance equation of the component, and we assume that the apparent dispersion coefficient Da in Eq. 10.10 is given by Eq. 10.11. We further assume that the HETP is independent of the solute concentration and that it remains the same in overloaded elution as the one measured in linear chromatography. As shown by the previous discussion this assumption is an approximation. However, as we have shown recently [6], Eq. 10.4 is an excellent approximation as long as the column efficiency is greater than a few hundred theoretical plates. Thus, the equilibrium-dispersive model should and does account well for band profiles under almost all the experimental conditions used in preparative chromatography. In the cases in which the model breaks down because the mass transfer kinetics is too slow, and the column efficiency is too low, a kinetic model or, better, the general rate model (Chapter 14) should be used.
10.2 Approximate Analytical Solutions There are no closed-form analytical solutions of the equilibrium-dispersive model as there are of the ideal model. However, some approximate solutions of this model, valid only at low concentrations, were proposed by Houghton [13], Haarhoff and Van der Linde [14], and Golshan-Shirazi and Guiochon [15]. These solutions are interesting only if we want to study the phenomena appearing at the onset of column overloading. Then, we may assume that the sample size is small, (i.e., the loading factor Lf1 is small), so the maximum concentration is low and we can replace the equilibrium isotherm by its two-term expansion [13-15] around the origin. In the case of the Langmuir isotherm, for example, this expansion is q = aC{l - bC)
(10.13)
Such an expansion will be referred to as the parabolic isotherm. The simplifying assumption made here is of a physical nature. It restricts the range of validity of 'See Chapter 3, Eq. 3.51 and Chapter 7, Eq. 7.26.
10.2 Approximate Analytical Solutions
A77
the model to low concentrations where there is no significant difference between the actual and the parabolic isotherms. With the parabolic isotherm, we can rewrite Eq. 10.10 as: 3C
AuC +
dC
g
Da
d2C
^ ( l +AC) dg2
{
'
)
with k' -2b—^-r 1 + /co
A
=
I
= LtR'°~t
(10.15) (10.16)
Equation 10.14 is exact for a parabolic isotherm. However, it cannot be solved in closed form without some further simplifications. These simplifications will be of a mathematical nature, rendering the equation approximate. Several approaches are possible at this stage [15]. The physical (parabolic isotherm) and the mathematical (see below) simplifications combine to give an approximate solution. It is important to understand the difference between the two types of simplifications and their different consequences.
10.2.1 The Houghton Solution Originally, Houghton [13] derived his equation with the assumption that the mass transfer kinetics is infinitely fast but that axial dispersion cannot be neglected. In view of the previous discussion (Section 10.1), we can extend the validity of the Houghton approach to the case of a finite rate of mass transfer, by lumping axial dispersion and mass transfer contributions into an apparent dispersion coefficient. Since Eq. 10.14 cannot be solved in closed form, Houghton [13] proposed to neglect AC in the denominator of the last two terms of Eq. 10.14. This simplification assumes that the concentration of the component, C, is low and that the sample size is such that ACmax < 0.05, where C max is the maximum concentration of the band [15]. It should be underlined at this stage that this assumption is incorrect to the first order and that the modified equation obtained no longer conserves mass with the boundary conditions of chromatography [15-18]. Hence, the area of the Houghton profile is not proportional to the sample amount injected or independent of the curvature of the isotherm (coefficient b) [19]. There is an apparent mass loss for convex-upward isotherms (for which A is negative) and an apparent mass gain for concave-upward isotherms (for which A is positive) [15-17,19]. Ignoring the term AC in the denominator of the second and third terms of Eq. 10.14 gives dC
A solution of Eq. 10.17 can be derived using the Cole-Hopf transform. This solution gives the elution profile of a finite width pulse at the end of an infinitely long
Single-Component Profiles with the Equilibrium Dispersive Model
478
column (asymptotic solution. Chapter 7, Section 7.3.8). Jaulmes et al. [20] have simplified the Houghton equation by considering a Dirac injection pulse (impulse or infinitely narrow input pulse). The solution is given by the following set of equations (10.18)
X =
where X is a dimensionless concentration, m the dimensionless sample size, and T the dimensionless time, all given by the equations X
=
T
=
/ uH l + k[ V2Dfl(l4 ~ko) k'o
t
k'0L
to
(1+) m =
Lu " y /c 2Da i +0
]\
(10.19) (10.20)
(10.21)
where L r is the loading factor, or ratio of the sample size to the column saturation capacity as defined for the Langmuir isotherm (see Eq. 3.51), which has the parabolic isotherm for two-term expansion (i.e., has the same tangent and curvature at the origin). An example of a profile calculated with the Houghton equation is given in Figure 10.1. Houghton has shown that the limit of Eqs. 10.18 and 10.19 to 10.21 when the apparent dispersion coefficient, Da, tends toward 0 is Eq. 7.4, the solution of the ideal model for the diffuse rear profile [13]. It is significant, however, that the limit solution is different from the rigorous solution of the ideal model for a parabolic isotherm [15]. This shows that the Houghton equation is not self-consistent. This flaw comes from the simplification made to replace Eq. 10.14 by Eq. 10.17.
10.2.2 The Haarhoff-Van der Linde Solution Haarhoff and Van der Linde [14] have studied the same problem, the determination of the band profile in the case of a moderately overloaded column, a case in which the thermodynamic effect of a nonlinear isotherm perturbs only mildly the band profile. They have used the same approach as Houghton, down to Eq. 10.17. However, since the component concentration is significantly different from 0 only around the band maximum, during a period of time which is only a few times the standard deviation of the Gaussian profile obtained under linear conditions, they have suggested that the effect of the apparent dispersion term on the band profile can be calculated at the limit retention time, f^o- They replaced Dat in Eq. 10.20 by DH£jy). It turns out that this procedure corrects the Houghton equation for its lack of mass conservation. In the Haarhoff-Van der Linde solution, mass is conserved. The solution derived by Haarhoff and Van der Linde [14] gives a profile whose equation is very similar to the Houghton solution. It can be written in the same
10.2 Approximate Analytical Solutions
479
Figure 10.1 Comparison of band profiles derived from the Houghton and the Haarhoff-Van der Linde equations. Parabolic isotherm: q = 20C(l ± 5C). L = 25 cm; F = 0.25; f0 = 200 s; N = 12,500 theoretical plates. Loading factor: 1%. 1, convex-upward isotherm; 2, convexdownward isotherm. The Houghton profiles are identified by squares. The masses lost by the Houghton profiles are 5.3% and -4.5%, respectively. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, J, Chromatogr., 506 (1989) 495 (Fig. 2).
1050 1HW 1150 1200 USI 1300 1350 MOO
Time (*)
dimensionless form as Eq. 10.18, but with slightly different definitions for the dimensionless parameters. Now, X, the dimensionless concentration, T, the dimensionless time, and m, the dimensionless sample size are defined as follows: (10.22)
tR
1 + VQ tRr0
T = m =
'°~- tfo
N
(10.23) (10.24)
where N is the column efficiency, the number of theoretical plates achieved under linear conditions, i.e., determined at low sample size. If we differentiate Eq. 10.18 to obtain the coordinates of the maximum of the band profile (dX/dr = 0), we obtain (10.25) Combining Eqs. 10.22,10.23, and 10.25 gives (10.26) where XM, TM, and CM are the reduced peak maximum concentration, the reduced time, and the true concentration of the band maximum, respectively. A
480
Single-Component Profiles with the Equilibrium Dispersive Model
band profile calculated using this equation is shown in Figure 10.1 and compared to the Houghton solution for the same experimental conditions. The profiles are close, but there is a significant deviation and a nontrivial mass loss in the case of the Houghton profile (see next section). Jaulmes et al. [20,21] have derived from the Houghton solution an approximate equation relating the retention time of the band maximum and its concentration. In liquid chromatography and with the symbols used here this equation becomes / k> tM = tRjo 1 - 2b-^-rCM
•, k' \ - ——o-r
(10.27)
In their equation, these authors neglected to compared to tj^ or fR Q- The last term on the RHS of Eq. 10.27 is negligible, and Eqs. 10.26 and 10.27 are equivalent at high values of N (see also Section 10.2.6). Surprisingly, although the simplifying assumption made by Haarhoff and Van der Linde in the derivation of their equation gives an approximate solution of Eq. 10.17, while Houghton has found an exact solution of Eq. 10.17, the approximate solution of Eq. 10.17 derived by Haarhoff and Van der Linde [14] is a much better solution of the mass balance equation (Eq. 10.14) because their solution conserves mass. Golshan-Shirazi and Guiochon [15] have shown that the Haarhoff and Van der Linde profile equation can also be derived by replacing the term AC in Eq. 10.14 by the value derived from the analytical solution of the ideal model (Eq. 7.4): 1 tRJO + t 2b tRjo-to
p
- t ^ 1 tRfi-t ~2btRfl-t0
=k'0-k>
2bk'o
K
" '
Hence:
l + AC=±±%r = -±-
(10.29)
This new simplification is correct to the first order and conserves mass. This observation explains why the Haarhoff-Van der Linde equation conserves mass while the Houghton equation does not, even to the first order. In the next section we discuss the range of validity of these solutions.
10.2.3 Range of Validity of the Houghton and Haarhoff-Van der Linde Equations This problem has been studied in detail by Golshan-Shirazi and Guiochon [15] who have compared systematically, under conditions of increasing sample size, the results given by the ideal model, the numerical solution of the equilibriumdispersive model and the two approximate analytical solutions of this latter model. The approximate character of the Houghton equation comes first from the simplification made in solving the equation. As a consequence, as noted earlier, the equation gives a profile whose area is not proportional to the sample size. For a Langmuir isotherm, the "apparent response factor" or ratio of the profile area at
10.2 Approximate Analytical Solutions
97S1O0O 1023 1050 1075 HOD 1135 1150 HIS 1300 B29 1250 UTS
Time (»)
481
350
T»me(
Figure 10.2 Comparison between the band profiles derived from the Houghton equation, the Haarhoff and Van der Linde equation and the numerical solution of the equilibriumdispersive model using the Rouchon procedure, (a) Influence of the isotherm model used. 1.2, Numerical solutions of the equilibrium-dispersive model with L <• = 1%, and a parabolic (1, q = 20C(l - 5C)) or a Langmuir (2, q = 20C/(l + 5C)) isotherm. The Haarhoff-Van der Linde profile is identified by squares. Column length: 25 cm; phase ratio: 0.25; to = 200 s (u = 0.125 cm/s); column efficiency: 2500 theoretical plates, (b) Same as (a), except Lf = 5%. 1.3, Profiles calculated with the equilibrium-dispersive model and the parabolic isotherm (1) or the Langmuir isotherm (3). 2, Houghton solution. The Haarhoff and Van der Linde solution is identified by squares. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon,}. Chromatogr., 506 (1989) 495 (Figs. 17 and 18).
time t to the injected profile area (or area at time t = 0) decreases with increasing sample size. The apparent mass loss is already 2.5% for a loading factor of only 0.2%. In addition, the Houghton equation assumes a parabolic isotherm. The approximate character of the Haarhoff-Van der Linde equation comes only from their assumption of a parabolic isotherm. This assumption is bound to fail when the sample size increases because the band profiles are very sensitive to minor changes in the isotherm. Therefore, these equations cannot be expected to hold at high values of the loading factor. Note, in addition, that the concept of loading factor is not consistent with a parabolic isotherm, because the latter has no saturation capacity. Golshan-Shirazi and Guiochon [15] have shown that, because the parabolic isotherm is acceptable only at low sample sizes, the approximate analytical solution (Eqs. 10.18 and 10.22) is valid only for loading factors below about 0.2%. The
482
Single-Component Profiles with the Equilibrium Dispersive Model
numerical solutions of the equilibrium-dispersive model (see next two sections) for a parabolic isotherm and the Haarhoff-Van der Linde [14] profiles are nearly coincidental up to the highest concentrations for which calculations have been carried out [15], confirming the validity of the Haarhoff-Van der Linde equation as a solution of the mass balance equation for a parabolic isotherm. The difference between the profile calculated with the equilibrium-dispersive model for a Langmuir isotherm and the Haarhoff-Van der Linde profiles is due to the fact that the parabolic isotherm used for the latter approximates the Langmuir isotherm only in a limited concentration range (Figure 10.2a). This explains the limited range of validity of the Haarhoff and Van der Linde profile, most solutes having a Langmuir, a Langmuirian, or a more complex isotherm behavior. For example, in the ideal model, the baseline width of the profile obtained with a Langmuir isotherm (Wi) is WL = (2yjLf - Lf)(tRi0
- t0)
(10.30)
while the baseline width of the band obtained with a parabolic isotherm is Wpar = 2^Tf(tRfi
- t0)
(10.31)
The ratio between these two bandwidths is 1 — . Lf/2. It becomes significantly different from unity when the loading factor exceeds 1%. Because of the simplifications made in the calculations, the Houghton profile [13] is not an exact solution for either the parabolic or the Langmuir isotherms. It turns out to be intermediate between the two profiles [15]. It has been shown (Figure 10.2b) that the HaarhoffVan der Linde solution [14] gives a band profile which is essentially correct when bCMax < 0.05 and which is reasonably close to the correct one whenfeCMax< 0.1 (where b is the second coefficient of the Langmuir isotherm, Eq. 3.47) [15]. The Houghton profile is less satisfactory, however, because it does not conserve mass and gives only an approximate band shape.
10.2.4 Influence of the Sample Size on the Bandwidth Analysts studying the behavior of band profiles at increasingly large sample sizes have often mistaken the band broadening due to the nonlinear behavior of the isotherm, which is of thermodynamic origin, for a loss of column efficiency. The column efficiency results essentially from the effects of axial dispersion and the kinetics of mass transfers and is kinetic in nature. Based on the implicit and erroneous concept of a strong dependence of the rate constant of mass transfers on the solute concentration, this quest [22-26] was doomed to fail. Thus, various authors have studied the dependence on the sample size of the apparent column HETP in the nonlinear case [22-25,27-33]. Some have presented correlations between the apparent column efficiency and the sample size. A typical example is given in Figure 10.3 [24]. This work assumes that the kinetic and thermodynamic contributions to the band variance are independent and additive.
10.2 Approximate Analytical Solutions
483
Figure 10.3 Correlation between the column efficiency and the amount of sample injected [26]. Plot of N/NQ versus wx^wSr where NQ is the column efficiency under linear conditions (very small sample size), W
X,N = N0[k'0/(l
+ k'Q)]2(wx/ws),
with wx,
sample size and ws, column saturation capacity. Hence, wxfws is the loading factor, Lj, and WX/N = m (Eq. 10.33). Column: 5x0.46 cm, packed with ODS Zorbax. Sample, benzyl alcohol. Sample size: 30 mg. Symbols: experimental data. Reproduced with permission from J.E. Eble, R.L. Grob, RE. Anile and L.R. Snyder, } . Chromatogr., 384 (1987) 45 (Fig. 7).
Poppe and Kraak [29] have defined the apparent plate number by extending the definition given in linear chromatography H=
Ntap
L /W 16 UR
(10.32)
where t% and W are the retention time of the band maximum and the baseline bandwidth, respectively. Now both W and t% are functions of the sample size. These authors [29] have shown that this apparent plate number Nap is a function of the effective loading factor, m (Eq. 10.24) and is given by
Nf(m)=Nf(N,
(10.33)
This equation was also used by Knox and Pyper [30] and by Eble et al. [24]. It derives directly from the Haarhoff-Van der Linde equation [14]]. Unfortunately, there is no similar equation for the apparent retention factor. Eble et al. [24] suggested an equation similar to Eq. 10.33: k' = k'o g(Vm)
(10.34)
but this relation is an empirical oversimplification and is not equivalent to Eq. 10.26. Equation 10.34 is valid only in the same range as the Haarhoff-Van der Linde equation, for very low values of the loading factor. The solution cannot be obtained by applying dispersion directly to the ideal profile. As observed by Poppe and Kraak [29], diffusion results in the widening of the profile. To keep its area constant, the band has to be made shorter and there is no simple way to calculate directly the new height. We can, however, neglect the consequence of the band broadening and change in the peak height in the concentration range around the center of the band, at midheight. Thus, a more subtle approach was suggested by Knox and Pyper [30].
484
Single-Component Profiles with the Equilibrium Dispersive Model
These authors separated the apparent plate height into two independent contributions, one of kinetic origin, the other of thermodynamic origin. Furthermore, they assumed the kinetic contribution to be nearly constant in the concentration range where chromatographic experiments are carried out, and they calculated the thermodynamic contribution from an approximate solution of the ideal model (Chapter 7). Finally, they used the rule of the additivity of the variances of independent contributions to calculate the apparent plate height [30]. Thus, they assume that the actual band profile results from the convolution of the thermodynamic band broadening, as derived from the ideal model, by the broadening due to the finite column efficiency, and that the band variance is given by the sum of two independent contributions [30].
where Ctot is the total standard deviation of the band derived from its width at a certain fractional height, usually the baseline width or, better, the width at half height, c^ is the bandwidth contribution due to thermodynamics, i.e., to a nonlinear isotherm, and a^m *s the bandwidth contribution due to the mass transfer resistances and to axial dispersion. It is derived from the bandwidth of a profile obtained under linear conditions, i.e., with a sample of very small size, and measured at the same fractional height as Ptot- This model is a sort of clumsy version of the equilibrium-dispersive model in which the solution of the ideal model is convoluted with axial dispersion. However, the convolution is simplistic and incorrect, so the model does not even conserve mass. Equation 10.35 assumes the additivity of the variances of independent contributions, which is only approximate in nonlinear chromatography [1]. Since the column HETP is proportional to the band variance, we have Hap = Hkin + Hth
(10-36)
where H ap , H ^ , and H^ stand for the apparent HETP, as measured from the recorded band profile, and the contributions to the HETP of the mass transfer kinetics and the nonlinear behavior of the isotherm, respectively. We can now relate the apparent column efficiency under overloaded conditions to the loading factor:
^ k .
(10.37)
Equation 10.37 can be rewritten, for the sake of convenience, as N _Nap N0
_
1
where N is the column efficiency measured with a finite size sample and No the efficiency measured with at infinite dilution (i.e., under linear conditions). Dose and Guiochon [1] have demonstrated, however, that the rule of variance additivity does not apply in nonlinear chromatography, as the convolution of the
10.2 Approximate Analytical Solutions
485
thermodynamic band profile by an apparent axial dispersion is a shift-variant convolution [1]. Furthermore, we showed that the result of the convolution is different depending on the relative band height at which the bandwidth is measured to calculate the apparent plate number [27,31]. Nevertheless, at constant relative peak height, an excellent agreement between experimental and calculated results was obtained [31]. Recently, Lucy and Carr [32] have showed that the method can be used in a wide range of experimental conditions to characterize band broadening in nonlinear chromatography. Still, band broadening at high concentrations is basically thermodynamic in nature, since it is due to the curvature of the equilibrium isotherm, while the column efficiency is related to the mass transfer kinetics. As a matter of principle, it is not satisfactory to use a kinetic concept to account for a thermodynamic effect. However, the concept is widespread and seems useful if properly understood, so we discuss it briefly. The apparent plate number can be calculated from the experimental profiles [27]. However, this number depends on the fractional height at which the bandwidth is measured. The value of Nth is calculated from the profiles predicted, under the same experimental conditions, by the ideal model. Finally, N ^ is derived from the band profiles recorded in linear chromatography, e.g., with a very small sample size, using the relationships valid for Gaussian profiles. From Eqs. 7.24 and 7.26, we can derive the band width at half height, u>i/2, and the retention time of the band profile, 11, obtained with an infinitely efficient column. In the case of a Langmuir isotherm, we obtain [31] = 5.54 ( —!— ) = 5.54 w1/2
(10.39)
Equation 10.39 assumes that the sample is injected as a rectangular pulse, without any dispersion in the sampling system. If the latter contributes significantly to the bandwidth, a contribution accounting for this spreading and similar to instrument contributions must be added [27]. This problem has been discussed by Dose and Guiochon [1]. They have shown that this approach is correct for the detector contribution because the dispersion term is given by a shift-invariant convolution but that it is not correct for the injector contribution because the dispersion in the injector changes the boundary condition of the mass balance equation and the corresponding dispersion term is given by a shift-variant convolution. When the loading factor is small, Eq. 10.39 can be simplified by expanding its right-hand side in a power series of Jhf [31]: 5.54
! + \25-T^y)
\Lf\
do.40)
Knox and Pyper [30] and Eble et al. [24] have derived an equation similar to Eq. 10.40, which is slightly different because they define the plate number from the bandwidth at the baseline while we have defined it from the width at halfheight. Because the baseline is often noisy and because the rear of the band profile
486
Single-Component Profiles with the Equilibrium Dispersive Model
is diffuse, the bandwidth at the baseline is poorly defined and its measurement is neither precise nor accurate. Admittedly, when the column is overloaded, the detector response is often nonlinear and the determination of the half-height position may not be very accurate either, unless careful detector calibration has been carried out. With this alternate definition using the baseline width [24,30], Eq. 10.40 becomes [31]:
The two values of the plate height obtained (Eqs. 10.40 and 10.41) are different because the overloaded band profile is not Gaussian (see solution of the ideal model). If we assume that the term within brackets in the RHS of Eq. 10.41 is equivalent to 1, we can write
Equation 10.42 is the equation given by Eble et al. [24]. We see that this equation is valid only at low values of the loading factor. The only correct equation at moderate loadings would be obtained by combining Eqs. 10.38 and 10.39 for a plate height defined from the band width at half-height (or Eqs. 10.38 and 10.41 for a plate height defined from the baseline bandwidth). Equation 10.42 is an expansion of this correct equation for low values of L t.
10.2.5 Other Plate-Height Equations for Non-linear Chromatography Lee [33] derived another plate-height equation that should be valid for nonlinear chromatography. He used the same assumption as made previously by Knox and Pyper [30] and by Shirazi and Guiochon [27]. He assumed that the column HETP is the sum of two independent contributions which are due to the nonlinear behavior of the isotherm and to the mass transfer kinetics, respectively: H = Hih + H^
(10.43)
and he used the following definition of the plate height from the moment equations [32]: H = 2L^%
(10.44)
where }i\ and fi2 are the first moment and the second centered moment of the peak, respectively. Using the analytical solution of the ideal model in the case of a Langmuir isotherm, he derived an equation for H^ as a function of a parameter a which is equal to . Lf. Replacing a with . Lt, his equation becomes
JJ\ ko
W-f)
(10-45)
487
10.2 Approximate Analytical Solutions
Table 10.1 Expressions for the Thermodynamic Contribution to the Plate Height
Source 5.54
[31]
(2 - V M 4 ^r (4jTf-Lf)2 1 -l
16
[31]
1<
. /-,
%I
Vl + fc^(M/
[24]
1+ n
-2
,(V L /- L /) 2
[33]
[30]
-2
4 n 4-J-M2
2xp(Lf)
/
3L
8 (! + *{,)
[34]
L column length; L<- loading factor; (p{Lt), ip(Lc), see text, Eqs. 10.46; .K^ equilibrium constant; Co sample concentration; k0 retention factor at infinite dilution; c empirical constant. Reprinted with permission from W. C. Lee, J. Chromatogr., 606 (1992) 153 (Table 2).
where
cp(Lf) =
llLf-T5Lf
(10.46) (10.47)
Table 10.1 summarizes the equations giving the thermodynamic contributions to the plate height [33]. The H4/, contribution obtained from the relationship derived by Lee [33] and given in Table 10.1 produces a result that is intermediate between the two equations derived by Shirazi and Guiochon. [27,31], i.e., Eqs. 10.39 and 10.41.
488
Single-Component Profiles with the Equilibrium Dispersive Model
Lee also extended the non-equilibrium theory developed originally by Giddings [10] to obtain H kin/ the plate height contribution due to the mass transfer resistances and to axial dispersion, the non-equilibrium contribution. He started from the kinetic equation of the lumped rate constant kinetic model: ^=K(is-q)C-k*dq
(10.48)
By following the same approach as the one developed by Giddings in linear chromatography, he derived an expression for the plate height contribution that results from the non-equilibrium under nonlinear conditions (with a Langmuir isotherm): 2»Q
k'Q[l + 9(Lf)] (m49)
In this equation, 6{Lj) is given by 6{Lf) = — p ^
(10.50)
For linear chromatography, Lj tends toward zero and:
or:
Now, adding the two contributions, Htj, and H]^, we obtain an expression for the plate height in nonlinear chromatography: k
>
+9
^
no 53) 2
( m 5 3 )
where k^ is an effective rate constant that lumps the effects of a slow kinetics of adsorption-desorption and all other contributions of the mass transfer resistances to band broadening, and which is assumed to be independent of the sample concentration. Defining the plate numbers corresponding to H, H^, and H kin as
T^
H = £
Nth
N
and
Ho = A
(10.54)
NQ
Introducing these equations into Eq. 10.43, the following expression can be obtained: — - — + ——
(10 55)
10.2 Approximate Analytical Solutions
489
Figure 10.4 Comparison of experimental band profiles with the best profiles obtained by fitting experimental data to the Houghton equation. Experimental conditions: Erbasil C18 with methanol-water (17:83) at FH = 3 mL/min and 20°C. Solute: benzyl alcohol. Reproduced with permission from A. Jaulmes, C. Vidal-Madjar, H. Colin and G. Guiochon, J. Phys. Chem., 90 (1986) 207 (Fig. 1). ©1986, American Chemical Society.
Equation 10.55 is slightly different from the one given in the literature, which is 1 N
1
1
(10.56)
In nonlinear chromatography, most of band broadening is due to the thermodynamic contribution. As soon as Lf is significant, the kinetic contribution becomes small compared to the thermodynamic contribution and the difference between these two equations has little practical consequence. The contribution H^^/HQ is significant only when mass transfer kinetics is slow and the contribution of the mass transfer resistances to the band profile is greater than the contribution originating from the nonlinear behavior of the isotherm.
10.2.6 Comparison of the Experimental Band Profiles and the Prediction of These Equations Jaulmes et al. [21] have found good agreement at low values of the loading factor between the band profiles predicted by the Houghton equation and the experimental profiles obtained with benzyl alcohol in reversed phase liquid chromatography (Figure 10.4). In principle, the second coefficient, b, of a Langmuir isotherm could be derived from a plot of the maximum band concentration versus its retention time. Equation 10.25 relates these parameters through the dimensionless quantities X^ and XM defined in Eqs. 10.19 and 10.20 (Figure 10.5). However, the values of the parameters derived from these profiles by curve fitting are not constant when the sample size increases but drift slowly, showing that the model is not entirely satisfactory [21]. This observation is explained by the nonproportionality to the sample size of the area of the profile given by the Houghton equation [15]. Better results would be obtained with the solution of Haarhoff and van der Linde. Golshan-Shirazi and Guiochon [27] have compared calculated and experimental results regarding the dependence of the column efficiency on the sample size. They have measured the efficiencies of several columns for different samples at increasing sample sizes and calculated the efficiencies predicted by Eq. 10.39. They showed that there is an excellent agreement, at all sample sizes, between the plots
490
Single-Component Profiles with the Equilibrium Dispersive Model
Figure 10.5 Plots of the maximum concentration of a peak versus its retention time under experimental conditions of overloading onset. Experimental data (symbols) and plot of Eq. 10.27. Same experimental conditions as for Figure 10.4. Reproduced with permission from A. Jaulmes, C. Vidal-Madjar, H. Colin and G. Guiochon, J. Phys. Chem., 90 (1986) 207 (Fig. 2). ©1986, American Chemical Society.
Figure 10.6 Dependence of the apparent column efficiency on the sample size. Experimental results (symbols) and results predicted by Eq. 10.39 for several compounds (see also Figures 7.8a and 7.8b). Reproduced from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 462 (Fig. 5). ©1989, American Chemical Society.
of the measured column efficiencies versus sample sizes and the similar curve predicted by Eq. 10.39. An example of these results is shown in Figure 10.6.
10.2.7 Relaxation Model In Chapters 3 and 4, we discussed the numerical analysis procedure suggested by James et al. [35] and applied by Felinger et al. [36] to calculate solutions of the inverse problem of ideal chromatography and, more specifically, to derive the best possible estimates of the numerical coefficients of an isotherm model together with a figure of merit for any isotherm model selected. The main drawback of this approach is that it is based on the use of the equilibrium-dispersive model since
10.2 Approximate Analytical Solutions
491
the band profiles calculated and compared to the experimental ones are calculated with this model (note that in their own work, James et ol. used the ideal model, introducing a model error into their solutions). More recently, James et ol. [37] discussed another numerical analysis method permitting the calculation of solutions of the same problem but using a kinetic model and providing estimates of both the best coefficients of an isotherm model and of the rate coefficient of mass transfer. These authors studied the singlecomponent case, ignored axial dispersion but took into account the finite rate of the mass transfer kinetics between the mobile and the stationary phase in a chromatographic column [37]. The resulting hyperbolic system contains a nonlinear relaxation term that is treated with a Chapman-Enskog type expansion. The numerical solution for both models (the relaxation and the parabolic approximation models) are derived and compared for various initial and boundary values. Ignoring axial dispersion, the mass balance can be written as: ^(Cm + C's) + ^(uCm + vC's) = 0
(10.57)
In conventional chromatography, the stationary phase does not move, so v = 0. In SMB (see Chapter 17), v is different from 0. Note that in the previous equation, the phase ratio was incorporated into the stationary phase concentration by writing that Cs = FCS, with F the phase ratio. If the mass transfer kinetics is instantaneous, Cs is equal to the equilibrium concentration in the stationary phase and the above equation reduces to the mass balance of the ideal model. -(Cm +qs{Cm)) + ^(uCm+vq's{Cm))
=0
(10.58)
with^(C m ) = Fqs{Cm). If the rate of the mass transfer kinetics is finite, then it can be modeled using one the equations in the following set |
^ C;
+|(r4)
=
=
\{ds-qs(Cm))
(C;^(CM))
(10.59a)
(10.59b)
The right-hand side of this equation is a pulling-back force that is proportional to the deviation from equilibrium. The relative importance of the contributions of the nonlinear thermodynamics of phase equilibrium and of the finite mass transfer kinetics to the band profile is evidenced and quantified by the numerical value of e. When e = 0, Cs = qs{Cm), and Eqs. 10.59a and 10.59b reduce to Eq. 10.58 by summing them up, which gives the equation of ideal chromatography. This asymptotic result can be justified in a few cases. On the other hand, the slower the mass transfer kinetics, the larger will be the value of e. The set of Eqs. 10.59a and 10.59b is called the relaxation model. A third model should be introduced at this stage, which is first-order correction in e to the ideal
492
Single-Component Profiles with the Equilibrium Dispersive Model ].:
•CllH fMib
i
IP (KJ
II
u
aos
(p i
aji
a;
o.j5
u->
Figure 10.7 Comparison of the results of numerical calculations of the relaxation and of the parabolic models with experimental results. Solid lines: experimental results by SeidelMorgenstern [41]. Dot-dashed line: profiles calculated with the relaxation model. Dotted lines: profiles calculated with the parabolic approximation. The figure on each subfigure indicates the duration of the rectangular injection. ReproducedfromF. James, M. Postel, M. Sepulveda, Physica D, 138 (2000) 316.(Fig. 8).
model. This correction can be obtained by a Chapman-Enskog expansion [38-40]. This approximation is valid only for small values of e. It is called the parabolic approximation model [37]. James et al. [37] discussed the numerical solutions of two above mentioned models in detail and compared the numerical solution of two methods for the same value of e for various initial and boundary values. The differences between the solution of the relaxation method and the solution obtained using the parabolic approximation increases with increasing the value of e, of the injection volume and of the concentration of the solute injected into the system. They also showed that the parabolic approximation is valid in a relatively wide range of injected amounts and of relaxation parameters. The parabolic approximation has the big advantage of requiring a much shorter computation time than the calculation of a solution of the complete relaxation model. Figure 10.7 compares the results of numerical calculations of the relaxation and of the parabolic models with experimental results [37]. The calculation parameters correspond to the case of the separation of the Troger's base enantiomers on microcrystalline cellulose triacetate, a chiral phase, using ethanol as the mobile phase [41]. This method needs to be extended to the binary case, in order to be useful for practical applications and to allow the determination of the parameters of competitive isotherms and of mass transfer kinetics in the investigation of the separation of multicomponent mixtures.
10.3 Numerical Solutions of the Equilibrium-Dispersive Model As we have shown clearly in the previous section, there are no closed-form analytical solutions of the equilibrium-dispersive model. There exists only an approximate solution in the case when the loading factor is low, and bCmax is smaller than 0.05 to 0.1 [15], with b, the second coefficient of the Langmuir isotherm and
10.3 Numerical Solutions of the Equilibrium—Dispersive Model
493
the maximum concentration of the elution band. In all other cases, which include most of the cases of practical interest in preparative chromatography, a numerical solution is the only choice. Considerable work has been devoted to the study of numerical solutions of these models, and it is not possible to give here a comprehensive review of the area. The lack of computer availability, the limited development of the theory of partial differential equations, and difficulties in handling the problem of numerical dispersion did not permit the systematic use of this approach for a better understanding of the band profile properties before the 1980s. Guiochon et ah [42] performed some band profile calculations in gas chromatography. They used a linear isotherm but took into account the sorption effect that results from the large difference between the partial molar volumes of the feed components in the stationary {i.e. condensed) phase and the mobile (gaseous) phases [43]. This effect is negligible in liquid chromatography (see Chapter 2). The profiles obtained are similar to those recorded in liquid chromatography under conditions corresponding to a Langmuir isotherm, because of the nature of the sorption effect. The excessively long run times required at the time (early 1970s) prevented a systematic investigation of the influence of the different experimental parameters on the band profiles. Seshadri and Deming [44] have used the Craig model to calculate band profiles in chromatographic systems. However, they selected an unrealistic isotherm,
dz
dt
U dz2
494
Single-Component Profiles with the Equilibrium Dispersive Model
By replacing (C + Fq)/u by G(C), for the sake of simplifying the writing, we have dC
3G(C)
_Dad2C
te+^r-irw
(m61)
A variety of methods can be used to derive numerical solutions of Eq. 10.61. These methods include mainly finite-difference methods and methods of orthogonal collocation on finite elements. We discuss briefly these methods, the properties of the solutions obtained, and some of the problems of numerical analysis encountered in the development and use of algorithms for the computation of solutions of Eq. 10.61 [49,50].
10.3.1
Principle of the Finite Difference Methods
The principle of the finite difference methods consists of replacing the continuous plane (z, t) by a grid obtained by dividing the space and time into a number of small, equal segments (of size h for space and T for time) and replacing each partial differential term in Eq. 10.61 by a finite difference term. Thus, the first order terms
|
and
§
(10.62,
can be replaced by one of the following finite differences:
d • a forward finite difference:
d+1 - d
- dn
"+1
d—d
n
• a backward finite difference: ——
d
- , respectively2 1
and —
d—d
~ 1 / and —
-d
• ox a central finite difference: " + 1
d
—, respectively
d
"~ 1 and — — - — - — , respectively
Similarly, the second-order term:
0
(10.63) d
— id + d
can be replaced by a central finite difference: n+1 -£ ^—^. There are many ways to combine the various finite differences that may be used for each of the terms of the mass balance equation, and there are as many ways to approximate a partial differential equation by a finite-difference scheme. The choice is limited in practice, however, for two reasons. First, we need the numerical calculation to be stable, and there is a condition to satisfy to achieve numerical stability. Second, we need to control the numerical errors that are made during the calculations. Replacing a partial difference term with any of the possible finite difference terms gives a truncation error. These truncation errors accumulate during the calculation of a numerical solution. The error contribution 2
The subscript indices refer to the space coordinate, the superscripts to the time coordinate.
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
495
depends on the way we replace the partial differential term by a finite difference. Thus, we need to evaluate these errors and consider which combination gives the numerical solution closest to the true solution of the system. We can choose between two different approaches for the calculation of numerical solutions of Eq. 10.61. First, we can solve directly Eq. 10.61 and adjust the values of the numerical integration increments to minimize the error made. Second, we can ignore the second-order term in the RHS of Eq. 10.61. In this case, we will choose the space and time increments in such a way that the numerical or truncation error becomes as close as possible (only in linear chromatography can they be equal) to this neglected second-order term and replaces it. We discuss in the next few sections the evaluation of the numerical errors and these two approaches to the calculation of numerical solutions.
10.3.2 Estimation of the Numerical Errors Made during the Calculation Assume that we have a continuous function, G(x), and that we use a Taylor series expansion to calculate its first and second differentials, using a finite difference scheme involving the values of G(x) at x/_i, x,, and x !+ i- We let G,- = G(x;), and write the Taylor expansions for G,-+i and G,_j, +i
3G 3x~
! +
Ax-
4
3G
Ax
ax 4
4(
Ax Ar
3x4
dx
Ax2
2
+
x=xt
d3G 3x3 X=Xi
4
X=Xi
dG
d2G
Ax3 VJ '
(10.64)
d2G dx2
Ax2 2!
93G
Ax3
~3T
4
(10.65)
IT
with Ax = X( — X;_i = Xj+i — X{. Rearranging these equations and dividing them by Ax, we obtain the expressions of a forward finite difference:
Ax
dG dx
Ax ^ 2 dx2
(10.66)
and a backward finite difference: Gi - Gt-_! _ dG " 3x" Ax
Ax 9^G
T 'dx2
(10.67) X=Xf
Thus, when we replace a partial differential term by a forward or a backward finite difference, we make an error which is of the same order as O(Ax), and the coefficient of this error contribution is the second-order partial differential, d2G/dx2.
496
Single-Component Profiles with the Equilibrium Dispersive Model
If we subtract Eq. 10.65 from Eq. 10.64, and divide the result by Ax, we obtain the expression of the centralfinitedifference for the first-order differential: 2Ax
3! dx3 , _
dx v__. Ar-j
Thus, when we replace a partial differential term by a central finite difference, we make an error that is of the order of O(Ax2), and the coefficient of this error contribution is the third-order partial differential, d3G/dx3. Finally, if we add up Eqs. 10.64 and 10.65, rearrange this sum, and divide it by Ax2, we obtain the expression for the central finite difference for the second-order differential: 2
Ax
dx
2
2Axz 4!
+ •••
(10.69)
X=X,
Thus, when we replace a second-order partial differential term by a central finite difference, we make an error of the order of O(Ax2), and the coefficient of this error contribution is the fourth-order partial differential, 3 4 G/3x 4 . In conclusion, when we replace the first- and second-order partial differential terms in a partial differential equation by central finite difference terms, we make errors that are of the order of O(Ax2). For most practical purposes, this second-order error is negligible. By contrast, when we replace the first-order partial differential terms with a forward or a backward finite difference term, we make errors that are of the order of 0{Ax). This first-order error contribution is never negligible. At this stage, two approaches are possible. The first one calculates solutions of the mass balance equation (Eq. 10.60) and uses finite-difference schemes that give a numerical error of the second order. The second approach calculates solutions of the mass balance equation of the ideal model (Eq. 10.72) and uses finite-differences schemes that give an error of the first order. The parameters of the numerical integration are then selected in such a way that the numerical error introduced by the calculation is equivalent to the dispersion term, so the approximate numerical solution of the approximate equation and the exact solution of the correct equation are equal to the first order.
10.3.3 First Method: Direct Calculation of Numerical Solutions of the Mass Balance Equation In this first method, we can write a large number of possible combinations of terms. The number of useful combinations is much lower because of our requirements that (1) the solution should be stable and (2) the error term should be of the second order, O(h2 + T 2 ) , SO that it is small enough and can be neglected. With a calculation scheme that gives a first-order error term, a second-order partial differential term equivalent to a numerical dispersion term would appear. The contribution of this term to the band profile could be difficult to control or cancel. As an example, if we select a central finite difference for the first term in the LHS of Eq. 10.61, a backward finite difference term for its second LHS term, and a
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
497
central finite difference term for its RHS, we obtain the following finite difference equation:
H
ri+1 ri n H
—H
— oH — H
St+1 2/z St-1 , ^n r— <^n _ uj-h Si+l h2zl-n
Si-1
Q Q 7' QX
The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of O(h2 + T). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, i.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment T are selected. This, in turn, would make the computation time very long, hi order to overcome this type of problem, Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows:
2h
+
T
~ ^ M
+
2 ;
h2
with Fq = FaC = k'0C, and G(C) = \{C + Fq) = Cw(l + k'o) = ^
(see Eqs. 10.60
and 10.61). This Lax-Wendroff scheme is widely used in linear chromatography. It gives the exact solution of Eq. 10.61. Unfortunately, this is no longer true in nonlinear chromatography because, then, the numerical error generated in Eq. 10.70 is not equal to (TMZ/O)/2- Nevertheless, Lin et al. [52] used Eq. 10.71 for the calculation of band profiles in nonlinear chromatography. These authors reported serious difficulties in finding conditions giving numerical stability, a result to be contrasted with the resounding success of the second method (next section).
10.3.4 Second Method: Replacement of Axial Dispersion by Numerical Dispersion hi this second method, we look for the numerical solution of the ideal model by ignoring the RHS of Eq. 10.61. In the process we introduce a truncation error which we adjust in such a way that this numerical error accounts as closely as possible for the RHS of Eq. 10.61 which is initially ignored. If we ignore the RHS term of Eq. 10.61, it becomes
In principle, we can use any combination of forward, backward and center finite differences to replace the first two differential terms of Eq. 10.72. However, our choice is limited for two reasons. First, we need to achieve numerical stability of the solution (see Section 10.3.5.1 below). For example, the stability analysis shows that the finite difference scheme which replaces the first term of Eq. 10.72 by a
498
Single-Component Profiles with the Equilibrium Dispersive Model
forward finite difference term and the second term by a center finite difference term: H _ ri W+l r ; - l n+1 n + Gn ~rGn = 0 (10.73) h is always unstable. This calculation scheme should not be considered any further. Second, we need to control the order of the numerical error made, and thus to consider the truncation error.
10.3.4.1 The Lax-Wendroff Two-Step Scheme This calculation scheme proceeds as follows. Knowing the value of C at the grid point (n, j), we calculate the value of C at the grid point n + 1 in two steps. First, we determine C at the point whose coordinates are (n + 1 / 2 , j + 1/2), i.e., halfway between two points of the grid, by _
Cn +Cn
h
_
2 C n + Cn
2 TI
-
2
\r(rj+U
^n ' ~
h 2T
then we calculate C at the mesh points (j, n + 1) from L
n+1 ~Ln-~
[HLn+l/2) ~
(10.76)
^HJ-I^JJ
With the Lax-Wendroff two-step scheme, the difference term found in the RHS of Eq. 10.76 is a central difference term. The computational error made in this calculation scheme is of the order of O(T 2 + h2). Thus, if we are interested in solving the ideal model (Eq. 10.72), the Lax-Wendroff two-step scheme [51] appears to be one of the best choices [53]. Unfortunately, in the nonlinear case, this Lax-Wendroff scheme has a strong tendency to oscillate near shocks and is not much used for this reason, as discussed in the previous section. In linear chromatography, we have: G(C) — (1 + ko)/udC/dt — l/uzfidC/dt, and in this case, the solution of Eq. 10.72 using the Lax-Wendroff two-step scheme can be written as C
n+1 ~ Cn-1 +J_Cn
- Q, = TZ^o Cn+1 ~ 2 C » W
^2
( m 7 7 )
This equation is equivalent to Eq. 10.71 with Da = 0, which is expected because we are now writing the Lax-Wendroff equation for solving Eq. 10.72, which is equivalent to Eq. 10.61 with Da = 0. However, in this chapter we are interested in solving the equilibrium-dispersive model (Eq. 10.61), not the ideal model (Eq. 10.72). So, neither the Lax-Wendroff nor any similar scheme which gives a high-order truncation error is suitable for our purpose. The error made is too small to account for the dispersive effects in an actual column. We need calculation schemes that give a first-order error, O ( T + h), or at least O(T 2 + h) or O ( T + h2); in other words, the truncation error should be equivalent to a second-order partial difference term. With such a scheme, we can adjust
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
499
the coefficient of the error term to replace the axial dispersion term that we have neglected in Eq. 10.72. The values of the time and space increments can be used for that purpose, by choosing them such that the first-order error term becomes equivalent to the second-order term of Eq. 10.61, which we originally ignored. We discuss now some of the schemes that are used for solving Eq. 10.72 because they give a first-order error term. 10.3.4.2 Scheme 1: The Forward-Backward Differences hi this calculation scheme, the first term of Eq. 10.72 is replaced by a forward finite difference while the second term is replaced by a backward finite difference. We obtain the equation n+
\
h
"+
Gn Gn
~
=0
r
(10.78)
Equation 10.78 can be solved easily for Cn+1: (io.79) Equation 10.79 permits the calculation of the concentration at the new space position, n + 1 , knowing the concentration at the previous space position, n (Godunov scheme). This method calculates band profiles along the column at successive time intervals. The elution profile is the history of concentrations at z = L. The forward-backward difference scheme was first used to calculate solutions of the equilibrium-dispersive model of chromatography by Rouchon et al. [46,47]. Since then, it has been widely used. It is particularly attractive because of its fast execution by modern computers [50]. Czok [50] and Felinger [54] have shown how the CPU time required can be further shortened by eliminating the needless computation of concentrations below a certain threshold. The dramatic increase over the last fifteen years of the speed of the computers available for the numerical calculations of band profiles has considerably reduced this advantage of the ForwardBackward scheme over the other possible ones. 10.3.4.3 Scheme 2: The Backward-Forward Differences In this calculation scheme, the first term of Eq. 10.72 is replaced by a backward finite difference, while the second term is replaced by a forward finite difference. We obtain the following finite difference equation: n
L h
«-i
G +
"
T
~
G
*=0
(10.80)
which can be solved for G!n : (10.81)
500
Single-Component Profiles with the Equilibrium Dispersive Model
This last equation permits the calculation of the value of G = (C + Fq)/u at any time, j + 1, knowing its value at the previous time, j . However, the concentration C of the solute in the mobile phase at this time must be calculated from the value of G = (C + Fq)/u, a calculation which can be carried out only numerically, using an appropriate iteration method. For this reason, the backward-forward calculation scheme requires a much longer run time than the forward-backward scheme [50]. The backward-forward finite difference scheme is identical to the Craig model if we choose the time and space increments such that | = u. The Craig model has been used by many authors, including Eble et al. [45], Czok and Guiochon [49,50], and El Fallah and Guiochon [55]. This model affords a good numerical solution of the gradient elution problem, which is very difficult to solve numerically with the forward-backward finite difference scheme [55,56]. 10.3.4.4 Scheme 3: The Forward-Backwardn+i Differences In this scheme, the first term of Eq. 10.72 is replaced by a forward finite difference, while the second term is replaced by a backward finite difference, but this backward finite difference is calculated at the space position n + 1. We obtain the following equation: "+1 = 0
(10.82)
^;\
(io.83)
from which we derive:
+^Y T
Jn+1
=4
+
T
Using this last calculation method, we can calculate the value of C + ^G = C + ^ ( C + Fq) at any space position, n + 1, and for any time, /, knowing its values at the same space position, at the previous time, and at the same time, at the previous space position. However, the concentration, C, of the solute at this space position must be calculated from the value of C + ^ ( C + Fq), which has to be done numerically, using an appropriate iteration method. This implicit calculation scheme was used by Lin et al. [26].
10.3.5 Application of the Second Method To be acceptable, a calculation procedure must be stable, and the amount of truncation error introduced must be the one required. In this subsection, we evaluate the stability condition of each of the finite difference calculation procedures described above, and we determine the amount of truncation error done with each of these schemes. However, because of the great difficulty of these theoretical studies when handling the nonlinear case, we limit this discussion of the stability condition and the truncation error made with the three schemes we have described to the case of a linear isotherm.
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
501
10.3.5.1 Numerical Stability in Linear Chromatography If the isotherm is linear, G is equal to
?
C/UZIQ
and Eq. 10.72 becomes
£ uZ/0 dt
dz
where uZto is the solute velocity in linear chromatography, given by
(10-85)
We use here the Neumann stability analysis [57], which is the most widely used procedure for the determination of the stability of a calculation scheme using a finite difference approximation. In this stability analysis, an initial error is introduced as a finite Fourier series and one studies the growth or decay of this error during the calculation. The Neumann method applies only to initial value problems with a periodical initial condition; it neglects the influence of the boundary condition, and it is applied only to linear finite difference approximations with constant coefficients, i.e., to linear equations. This method gives only a necessary condition for the stability of a numerical procedure. It turns out, however, that this condition is sufficient in many cases. If the Neumann stability analysis is applied to the three calculation schemes that we have discussed in the previous section, the following stability conditions are obtained in the assumption of a linear isotherm: • For the forward-backward finite difference scheme a
= IM > i
(10.86)
• For the backward-forward finite difference scheme 0 < a= ^ ^ < 1
(10.87)
• For the forward-backwardfj+i difference scheme a= ^ ^ > 0
(10.88)
Conditions 10.86 to 10.88 are frequently called the Courant-Friedrichs- Lewy (CFL) convergence conditions [58] and a is called the Courant number. 10.3.5.2 Error Analysis in Linear Chromatography The error analysis is conveniently done by using Taylor series expansions. For the concentrations at the four points of the grid which are the neighbors of the point
502
Single-Component Profiles with the Equilibrium Dispersive Model
in, j), we have C'
-
cj
-
r
ar dC
2 2
hh2d?2Cr
VI' (10.89)
dz
2 (10.90) n
8C
(10.91) (10.92)
By combining Eqs. 10.89 to 10.92 and Eqs. 10.78,10.80, and 10.82, and assuming a linear isotherm, one obtains: For the forward-backward finite difference scheme (Eq. 10.78)
dz
uZ/0 dt
2[U
}
dz2
(10.93)
For the backward-forward finite difference scheme (Eq. 10.80) dC
d2C
1 dC _ h +
[
H~ 2
'Ite2
(10.94)
For the forward-backward n+ i difference scheme (Eq. 10.82)
BC d
J_dC_h j'dt ~ 2
(10.95)
In these equations, uZ/o is the migration velocity of the solute at infinite dilution given by Eq. 10.85, a is the Courant number, and the order of the error made in the three models is the first order, O(h + r). A comparison of Eqs. 10.93 to 10.95 with Eq. 10.61 shows that, although we performed the calculation of a numerical solution of Eq. 10.72, the three models give an exact solution of Eq. 10.61 in linear chromatography, provided that: With the forward-backward finite difference scheme, the space increment, h, and the time increment, T, are chosen such that \{a — 1) — Da/u. Since Da — Hu/1, where H is the column HETP, we obtain for the first scheme the following condition: h{a-l)=h
(10.96)
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
503
With the backward—forward finite difference scheme, the space increment, h, and the time increment, r, are chosen such that | (1 — «) = Da /u or h(l -a)=h(l-
^
)
= H
(10.97)
With the forward-backwardn+\ difference scheme, the space increment, h, and the time increment, T, are chosen such that | ( l + a) = Da/u or h(l + a)=h(l
+ ^r^)
= H
(10.98)
Czok and Guiochon [49] called these three schemes Methods I, II, and III, respectively. They discussed their physical sense and their consequences for the calculation of numerical solutions of the single and multicomponent problems. 10.3.5.3 Extension to Nonlinear Chromatography In the previous two subsections, we have analyzed the numerical stability of the solution and discussed the importance of the truncation errors made. Both discussions have been related to the linear case. Their results are not valid in nonlinear chromatography. On the contrary, their extension may raise some serious questions, and they should be considered only as first-order approximations. The situation regarding the condition for numerical stability is very clear, as it is immediately obvious from the results whether the numerical procedure used is stable or not. During the past 16 years, members of our research group have carried out thousands of band profile calculations under nonlinear chromatography conditions, using computer programs based on the forward-backward and backward-forward algorithms and using the conditions given by Eqs. 10.86 and 10.87, which were derived assuming a linear isotherm for the three methods. A variety of isotherm models have been used. In all cases in which these numerical calculations were performed, except with the anti-Langmuir isotherm q = aC/(l + bC) with b > 0, a stable solution was obtained whenever the proper stability condition was satisfied. On the other hand, as these stability conditions are not very stringent, there is no reason to try to adopt combinations of increments which would not meet the conditions set out in Eqs. 10.86 to 10.88. Equating the numerical dispersion with the axial dispersion raises a much more serious issue. Equations 10.96 to 10.98 have been derived and are valid only for linear chromatography. Applying the relevant condition in linear chromatography renders the numerical dispersion of the corresponding calculation scheme equal to the required axial dispersion. Therefore, starting from Eq. 10.72, we obtain an exact solution of Eq. 10.61. This is no longer true in nonlinear chromatography, because Eqs. 10.96 to 10.98 do not hold when the isotherm is not linear. As a first approximation, we may assume that in nonlinear chromatography we should replace in Eqs. 10.96 to 10.98 the (constant) solute velocity, uZfi, by the local velocity of the solute or velocity associated with the concentration C, uz (Eq. 7.3). If we make this substitution and remember that, as seen in Chapter 7, the value of Mz for a convex-upward isotherm (e.g., the Langmuirian isotherm and the
504
Single-Component Profiles with the Equilibrium Dispersive Model
isotherms most often found in HPLC) increases with increasing concentration we derive the following: With the forward-backward finite difference scheme: h(a -l)=h
( ^
-1) > H
(10.99)
With the backward-forward finite difference scheme: h(l -a)=h(l-
^ )
(10.100)
With the forward-backwardn+x difference scheme: >H
(10.101)
These results show that, when they are used for the computation of band profiles in nonlinear chromatography, the first and last calculation schemes overestimate the axial dispersion term, while the second scheme underestimates it. We have used and are using extensively the schemes described in the previous subsections and Eqs. 10.96 to 10.98 for the calculation of band profiles in nonlinear chromatography, although we know that they are approximate. As an example, Figure 2.7 illustrates the evolution of the band profile of a pure compound injected as a narrow pulse. The calculations were made using the forward-backward scheme. All the intermediate profiles obtained were stored as independent files and are displayed in the figure. Because of size limitations, a low column efficiency of 300 theoretical plates was used. Sections of the surface by vertical planes parallel to the time axis give the elution profiles passing at the corresponding location in the column, including the elution profile shown as the last frame. Sections parallel to the abscissa axis give concentration profiles along the column at the corresponding time. The bottom figure shows sections of the surface by horizontal planes, giving contours of constant concentration. The rapid dilution of the band in the early part of its migration, when the velocity of its front and rear parts are most different, is obvious. The rear diffuse boundary of the profile gives straight lines, in apparent agreement with Eq. 7.4, in spite of the low column efficiency. The error introduced by dispersion in this case is discussed in Chapter 3, Section 3.5.3 dealing with the inverse use of Eq. 7.10 to determine the equilibrium isotherm (ECP method), and Figure 3.33. The propagation rate of the front shock layer decreases rapidly as the band dilutes and its height decreases. Its trajectory in the x, t space is strongly curved. We use frequently the forward-backward scheme with a = 2 and h = H, which satisfies Eq. 10.96 [48-50]. If calculations are made with this scheme and a = 1 + e, with e very small, of the order of 1 x 10~6, the numerical solution obtained is practically identical to the solution of the ideal model (H = 0), as expected. Czok et al. used the backward-forward scheme with different values of a [49,50]. Lin
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
505
et al. used the forward-backwardn+i scheme with a = 2 and h = H/3 [26]. Although, as just explained, all these calculation schemes give approximate solutions in nonlinear chromatography, comparisons between the solutions obtained with the different schemes for the same set of parameters have shown that there is little difference between the profiles calculated with the forward-backward and the backward-forward schemes [49,50]. Since one scheme overestimates and the other underestimates the axial dispersion, this agreement demonstrates that the calculation errors made are inconsequential as long as the column efficiency exceeds a thousand theoretical plates. Column efficiency is usually high in modern liquid chromatography, and the influence on the band profile of a finite column efficiency is small compared to the influence of a nonlinear isotherm, especially at high column loading, which is the most important case in preparative chromatography. The numerical solutions of the equilibrium-dispersive model calculated by this procedure are in good agreement with experimental results (next section), despite the use of two approximations, first that the truncation error is the same in linear and nonlinear chromatography, second that the column HETP is independent of the solute concentration. The reason for this agreement can be explained as follows. At low concentrations and especially on the diffuse boundary (usually the rear, tailing part of the profile) the solute concentration is low and ^nonlinear ~ ^linear' while the velocity associated with a concentration closely approximates MZ o- Thus, the calculation gives very accurate results at low concentrations. At high concentrations, both the deviation of the isotherm from linear behavior and the error made in approximating the axial dispersion increase. However, in this part of the profile, the nonlinear, thermodynamic effect has the dominant role. Accordingly, the error made may be more important but it has a small effect on this part of the profiles. In the case of a single component, the forward-backward scheme is most often the best one, because no iteration is necessary and the computation time is much less than with the other two methods.
10.3.6 Finite Element Method As we have shown above (Eqs. 10.99 to 10.101), the finite difference methods underestimate or overestimate the extent to which axial dispersion affects the profiles of single-component bands. There are cases in which we need a more exact profile that the one calculated with the finite difference schemes. This is of special importance in the case of multicomponent bands (Chapter 11). As we show in the next chapter, the use of finite difference methods causes more important errors in the calculation of the individual band profiles of the components of mixtures than in the calculation of the profiles of single component bands. A more accurate method may be necessary to account for the elution profiles of multicomponent mixtures, especially when the column efficiency is modest [59]. The finite element methods provide this advantage, but at the cost of longer computing times. This may seem paradoxical, because these methods have been developed to improve the performance of numerical calculations of solutions of partial differential equations. Finite element methods are used systematically by
506
Single-Component Profiles with the Equilibrium Dispersive Model
chemical engineers for the study of adsorption in fixed beds [60-62] and by mechanical engineers for the study of flow dynamics [63]. They are considered especially advantageous for the calculations of transitory phenomena which take place in a three-dimensional space, because computation times are much shorter than required for the use of finite difference methods. However, this speed advantage is lost in the case of chromatography, because we consider only one spacial dimension, assuming that the column is homogeneous radially. The advantage of faster computations would be found if we consider the column as a two-dimensional space (i.e., a cylinder with axial symmetry) or a three-dimensional space. This could be quite appropriate for large scale preparative columns, but it would also require that we determine the distribution of the density of the stationary phase in the column, density that controls the local permeability, the phase ratio, and the local equilibrium isotherms. There are no convenient methods to acquire this 3D map of the column properties. 10.3.6.1 Principle of the Finite Element Methods The formulation of the finite element method allows the generation of optimally ordered, accurate finite difference relations [62]. This is achieved through the use of a local approximation and a grid refinement. Whereas finite difference methods use a step function approximation of the solution, the finite element methods use an interpolation on each element. A number of different finite element methods have been designed. Chromatographers [61,62,64] prefer the method of orthogonal collocation on fixed elements [63] to the method of moving element for two reasons. First, in the one-dimensional chromatographic problem the moving boundary method, which is in principle faster than the fixed element scheme, loses this advantage because tracking the moving boundary inside the column requires the solution to be known. Thus, an adaptive procedure [62] is required, costing additional computation time. Second, we need to calculate the elution profiles obtained with high-efficiency columns which, in HPLC, have typically between about 3000 and 15,000 theoretical plates. Such efficiencies correspond to values of the apparent axial dispersion coefficient between 1 x 10~4 and 6xlO~ 4 cm 2 /s and to values of the lumped rate constant, kj, between 10 and 50 s^1. When high-efficiency chromatographic columns are overloaded, very steep concentration gradients are observed [48]. The fixed element method requires less computation time than the moving boundary method to update the elements and for the global interpolation required whenever the recollocation is carried out. A further advantage of this method is that the use of a global spline is not required, thus permitting the achievement of more accurate calculations. The finite element solution is constructed by dividing the space coordinate into NE successive intervals, with NE + 1 nodes having for coordinates S;, such that 0 = Si < S2 < ... < St < ... < SNE < S N E + I = L
(10.102)
where L is the column length. This division discretizes the space domain. Orthogonal collocation is then applied to each element and the solution of each of
10.3 Numerical Solutions of the Equilibrium-Dispersive Model
507
these elements is approximated by interpolation. In the fixed element method, the width of each element is Az = L/NE = S,-+i — S,-. The space variable on each element is z-S{ (10.103) se[o,i] Az On each element, i, Np interior collocation points are chosen as the roots of an Npth degree orthogonal polynomial [65]. The first- and second-order differentials with respect to space are then approximated using two matrices, A and B, obtained by solving the Gaussian-Jacobi quadratures [63,65]. For boundary problems, the endpoints can be included in the calculation of the spatial derivatives. Thus, at thefcthcollocation point on the ith element, we have s =
Np+2
3C
EA
jtkCKi
3s"
32C 3s 2
j = 1,2,...,Np + 2
(10.104)
/ = 1,2,...,Np
(10.105)
Np+2
E
c=cj{
Jfc=l
where Qy is the concentration at the fcth collocation point on the ith element. A finite element assembly algorithm is then implemented, imposing the continuity conditions on the concentrations and flux at the boundaries between elements. Thus, at the node points of the /th element Eq. 10.1 becomes, in the case of a single-component problem, (10.106a)
"37
„ Np+2
Np+2
3C
L
(10.106b)
dt
c=cj; j
=
1,2, . . . , N P
(10.106c)
i
=
2,3, ...,NE
(10.106d)
k=X
At the origin (the inlet node, NE = 1, z = 0), we have, assuming Danckwerts boundary conditions (Chapter 6, Section 6.2.1.2),
—
3z
Q=C..
=—(C--
= 1 f= l
'
(10.107)
At the exit (end of the last node, N,- = NE, Z — L), we have 3C
"37
= 0 ; = Np+2,
i = NE
(10.108)
c=cn
If we choose a kinetic model (e.g., with Eqs. 10.1 and 10.2 or 10.3), we have dC ~dt c=c«
DL Np+2
dt
-^2 E z
k=x
u •«•-—
Np+2
E
^ ^ fe+1
(10.109)
508
Single-Component Profiles with the Equilibrium Dispersive Model
dt
= K\f{Cji) - qji]
(10.110)
If we chose the simpler equilibrium dispersive model (e.g., Eqs. 10.10 and 10.11, and the isotherm, q = fj), we have at all points (replacing in Eq. 10.106 Di, coefficient of axial dispersion, by Da, coefficient of apparent dispersion): 3C dt
Np+2
E
dt
C—C-
NP+2 k=\
Equations 10.106 to 10.111 constitute a set of algebraic equations and first order ordinary differential equations. The two algebraic Eqs. 10.104 and 10.105 are solved using the classical procedure described by Villadsen [65]. The set of ordinary differential equations is easy to solve with the fourth-order Runge-Kutta method. 10.3.6.2 Accuracy of the Numerical Solutions It is more difficult to estimate the errors made with a finite element method than with a finite difference method. An estimate of the error made with the finite element in the calculation of the solution of the mass balance equation (Eq. 10.1) using a finite element method is given by [65] E(AC, AC) < MAz2(n+1-m^\\C\\2n+1
(10.112)
where M is a constant, n is the degree of the approximation polynomial used on each element (in most cases, as in the method described above, n — 3), 2m is the order of the partial differential equation integrated (here a second-order equation), AC is the difference between the calculated and the true value of the elution profile, and E(AC, AC) and \\C\ \n+\ are two functions given by _
lz
(10.113)
and
l|c|U+i =
i:
dz
(10.114)
E(u, v) is the inner product of u and v. In conventional OCFE calculation methods, the exponent of Az in Eq. 10.112 is 6, hence the degree of convergence between the calculated and the true profiles is of the sixth degree with respect to the space increment. One expects the value of | |C| I4 to be rather small in the type of problems dealt with here. The fourth-order Runge-Kutta method used in the OCFE algorithm discussed here introduces an error of the fifth order. Accordingly, we may anticipate that the numerical solutions of the system of partial differential equations of chromatography calculated by an OCFE method will be more accurate than those obtained with a finite difference method [48] or even with the controlled diffusion method [49,50].
10.4 Results Obtained with the Equilibrium Dispersive Model
509
Unfortunately, Eq. 10.112 gives only an overall estimate of the error made. This estimate is averaged out over the entire profile. A more detailed investigation of the nature of the computation errors and of the influence of the parameters which may control these errors would be useful. The problem is of much greater importance in the case of multicomponent chromatograms and will be discussed in more detail in the next chapter. Of special importance in the case of single-component bands, however, is a determination of the plate height dependence on the value of the column capacity factor resulting from the OCFE integration and a comparison of this relationship with those given by Eqs. 10.96 to 10.98. Ma and Guiochon [64] have shown that if the OCFE method is applied to a multicomponent mixture in linear chromatography, the solution obtained contains a Gaussian peak for each component, all having the same efficiency. With a finite difference method the peak efficiency would be given by the appropriate equation, 10.96 to 10.98.
10.4 Results Obtained with the Equilibrium Dispersive Model The equilibrium-dispersive model has great practical importance because it is the simplest model that permits accurate modeling of preparative separations. The columns currently used for preparative applications have high efficiency. They are packed with small particles that have an easily accessible pore network, with a high degree of connectivity. Thus, mass transfers are rapid, and the band profiles are essentially controlled by the thermodynamics of the phase equilibrium used for the separation. The contribution of the finite column efficiency remains important, especially for the proper estimate of the width of the mixed zones and the recovery yield, but this contribution is usually small, hence can be estimated as a correction. The model has been used abundantly in the recent past. Comparisons between the results of different methods of calculation, "computer experiments," in which the influence of the main separation parameters is investigated, and comparisons between experimental and calculated band profiles, leading to the validation of the model, will be discussed.
10.4.1 Comparison of Solutions of the Ideal and the EquilibriumDispersive Models The ideal model (Chapter 7) assumes an infinite column efficiency. This makes the band profiles that it predicts unrealistically sharp, especially at low concentrations. This sharpness is explained by the fact that the ideal model propagates concentration discontinuities or shocks. For a linear isotherm, the elution profile would be identical to the input profile, clearly an unacceptable conclusion. The effects of a nonideal column are significant in three parts of the band profile. The shock is replaced by a steep boundary, the shock layer, whose thickness is related to the coefficients of the column HETP (axial dispersion and mass transfer resistance; see Chapter 14). The top of the band profile is round, instead of being
Single-Component Profiles with the Equilibrium Dispersive Model
510 C mg/mL
2
Lf= Lf» 1%, 1%, 0.71 0 . 7 1 mg
a
b
4
Lf= Lf» 2.6%, 2.6%, 1.9 1.9 mg
Ideal
Ideal
N=10000, m=50
N=3500, m=50
N= 5000, m=26
N=1500, m=21 N= 500, m=7
N= 1500, m=8
1
2
0
0 3
3.5
2.6
4
3.4
4.2
c
9 Lf=11%, 8 mg
Ideal N=1500, m=88
6
N= 850, m=50
Figure 10.8 Comparison between the solutions of the ideal model and the band profiles calculated with the equilibriumdispersive model for various values of the loading factor and the column efficiency. k0 = 2.7, b = 0.803, Langmuir isotherm. See definition of m in Eq. 10.115.
N= 500, m=30
3
0 2
3
(t-tp)/to
4
angular. The diffuse boundary is asymptotic to the horizontal axis, instead of ending at the point t = tgfi + tp, the retention time at infinite dilution. This last effect opposes the hyperbolic character of the ideal model to the parabolic character of diffusion. It is observed, however, that for columns of high efficiencies, the difference between the profiles calculated with the equilibrium-dispersive model and derived from the ideal model becomes rather inconsequential at moderate or high sample sizes, depending on the column efficiency (see Figure 10.8). At low values of the loading factor, around 1% or lower, the discrepancy between the profiles calculated with the ideal and the equilibrium-dispersive model is important as long as the column efficiency is not very high (Figure 10.8a) [66]. At high values of the loading factor (e.g., 10% or higher; Figure 10.8c), on the contrary, the calculated and experimental profiles are very much alike, even at low values of the column efficiency. The larger the deviation of the isotherm from linear behavior, the more the thermodynamics of phase equilibrium controls the band profile (Figures 10.8a to 10.10b). This deviation is measured by the loading factor. 10.4.1.1 The Shirazi number Golshan-Shirazi and Guiochon [27,66] have shown that the degree of agreement between the ideal and the equilibrium-dispersive models depends on the value
10.4 Results Obtained with the Equilibrium Dispersive Model
511
of the effective loading factor, m, a dimensionless number also called the Shirazi number (10.115) The agreement between the profiles calculated with the two models is excellent when m is larger than about 35. It increases with increasing value of this number (Figures 10.9 and 10.10). This degree of agreement increases with increasing column efficiency and increasing loading factor. The influence of the retention factor is comparatively negligible, unless this factor is small. In the practically useful range of k0, between 1 and 10, [fco/(l + ko)]z varies between 0.25 and 0.83, barely a factor 3. Thus, values of m larger than 35 correspond approximately to values of the product NL t larger than 50, i.e., to an efficiency larger than 5000 theoretical plates for a loading factor of 1%, which is still only a moderate degree of overloading. Equation 10.115 has a considerable fundamental and practical importance. It combines parameters of fundamentally different origins, the plate number at infinite dilution, N, which characterizes the intensity of axial dispersion taking place in the column and two parameters of thermodynamic origin, the retention factor at infinite dilution, kQ, related to the initial slope of the isotherm, and the loading factor, proportional to the sample size and related to the saturation capacity of the isotherm. Accordingly, Eq. 10.115 indicates the extent to which the selfsharpening effect on the band profile due to the nonlinear thermodynamics is balanced by the dispersive effect of axial and eddy diffusion and of the mass transfer resistances. 10.4.1.2 Applications of the Shirazi number In Figure 10.8a, the value of m is 50 when the column efficiency is 10,000 theoretical plates and 8 when it is only 1500 plates. The agreement between profiles is good in the former case, poor in the latter. Similarly, in Figure 10.8b, the loading factor has been multiplied by 2.6, so m has become 21 for N = 1500 plates, and the agreement with the ideal model has improved. It is good for N = 3500 plates m = 50, but again poor for N =500 plates m =7. Finally, in Figure 10.8c, the agreement is already fair with N = 500 plates (m = 30); it becomes good with N = 850 plates (m = 50) and excellent with N = 1500 plates. When both the column efficiency and the loading factor are high, there are practically no differences between the solutions of the ideal model and the equilibrium-dispersive model. Thus, the relevance of the ideal model is certain in practical applications of preparative chromatography, where high values of the loading factor are achieved and the column efficiency is often quite significant. The ideal model permits a rapid and simple estimate of the band profiles which can be obtained under given experimental conditions. The influence of the sample size is illustrated in Figures 10.9 and 10.10. For a constant efficiency of 2000 plates (Figure 10.9a), the condition NLf = 50 corresponds to Lf = 2.5%. Under such conditions, compared to the band profile calculated with the equilibrium-dispersive model, the retention time of the band front
Single-Component Profiles with the Equilibrium Dispersive Model
512
bC
C mg/mL
a
Lf=20%
b
m=5
0.8 12 Ideal Ideal N=2000
10%
10
8 0.4 5%
25
4
50
2.5% 1%
105 0.5%
210
0 3
5
t, min
0 0.4
7
0.8
(t-to)/(tro-to)
1.2
Figure 10.9 Comparison between the band profiles predicted by the ideal model and the numerical solution of the equilibrium-dispersive model for a Langmuir isotherm. Constant column efficiency, 2000 theoretical plates, (a) Classical C vs. t profile. Sample size given as loading factor, (b) Reduced profiles, plots of bC vs. (t — to)/(tR
C c moles/L mole
0.06 0.06
bC
a
Lf=5%
0.3 0 - 3 -r
b
M-1000 N=1000 m=35
Ideal Non-Ideal N=2000 Lf=2.5%
0.03
0.2
Lf=1%
N=5000
0.1
Lf=0.5%
N=10,000
0 13
18
t, min
23
0 0.6
0.9
1.2 (t-to)/(tro-to)
Figure 10.10 Comparison between the band profiles predicted by the ideal and the equilibrium- dispersive models for a Langmuir isotherm. Constant apparent loading factor, m = [ko/(l + ko)]2NLj- = 35. (a) Classical C vs. t profile. Sample size given as loading factor; column efficiency, see Figure 10.10b. (b) Reduced profiles, plots of bC vs. (t — to)/ (tRtf — to). Efficiency in number of plates; loading factor; see Figure 10.10a. predicted by the ideal model is shorter by 0.5%, the band height is taller by less than 10% and the band does not tail strongly. The ideal model gives a closer approximation to band profiles than the approximate Houghton or Haarhoff-Van der Linde solutions for situations when Ly > 1% and N > 5000.
10.4 Results Obtained with the Equilibrium Dispersive Model
513
Figure 10.11 Overloaded chromatograms of a pure compound (k'o = 5) calculated according to the three different schemes. Langmuir isotherm, Lf = 10%, L = 25 cm. (a) N = 300 theoretical plates, m = 15. Integration increments: Forward-backward scheme, h = 500 }im, T = 4.0 s; backward-forward scheme, h = 600 \m\, r = 0.4 s; forward-backwardn+i, h = 250 }im, T = 1.0 s. Retention times: ideal model, 335 s; forward-backward scheme, 372 s; backward-forward scheme, 352 s; forward-backwardn+i, 360 s. (b) Same as (a), but N = 5000 theoretical plates, m = 250. Integration increments: forward-backward scheme, h = 30 jwm, T = 0.24 s; backward-forward scheme, h = 36 jim, T = 0.024 s; forward-backwardn+j, h = 15 }im, T 0.060 s. Reproduced with permission from M. Czok and G. Guiochon, Anal. Chan., 62 (1990) 189 (Figs. 4 and 5). ©1990, American Chemical Society.
In Figure 10.10, the profiles of increasingly large size feed samples on columns of decreasing efficiencies are compared. In this series of calculations, m was kept constant. The figure shows that the agreement between the results of the ideal and equilibrium-dispersive models remains uniformly good. It can be seen also that the combination of a high column efficiency and a low loading factor results in a markedly less diffuse band tail than that of a low column efficiency and a high loading factor.
10.4.2 Comparison of the Results of Different Calculation Methods In the previous section, we have compared the analytical solution of the ideal model and the numerical solutions given by the forward-backward finite difference scheme. We have shown excellent agreement between the two sets of profiles at high column loadings and/or high column efficiencies, as long as the apparent loading factor, m = N[k'0/(l + k'0)]2Lf is higher than 35. This agreement validates the finite difference methods discussed, assuming they still need this type of validation. It shows that the numerical solution of the ideal model (Eq. 10.72) converges toward its analytical solution, and that the difference between the true solution and the calculated profile is limited only by the degree of numerical dispersion allowed. This dispersion is controlled by the choice of the integration increment. It is negligible with the forward-backward scheme if the Courant number is set to a value very slightly higher than 1 (e.g., 1.000 001). Furthermore, this result shows that if the column efficiency and the loading factor are large enough
514
Single-Component Profiles with the Equilibrium Dispersive Model
Figure 10.12 Comparison of the numerical solutions of the equations of the equilibrium-dispersive model calculated with the forward-backward algorithm (dashed line) and the method of orthogonal collocation on finite elements (dotted line). 250 }iL injection of a 15 g/L solution of (+)-Troger's base. Column length, 25 cm; column efficiency, Np = 146 plates; F = 0.515; H = 0.076 cm/s. Isotherms in Figure 3.25
ID.0
15-0
M.O
25.0
J0.O
35.0
10.0
and m is of the order of 35 or exceeds it, the contributions of the axial dispersion and the mass transfer resistance to the band profiles are simple corrections to the band profile determined by thermodynamics. This condition (m > 35) is most often met in practical applications. Czok and Guiochon [49] have compared the profiles obtained with the three finite difference methods discussed: the forward-backward, the backward-forward, and the forward-backwardn+i schemes. There are some significant differences between the band profiles obtained at low values of the column efficiency (Figure 10.11a, N = 300), but no meaningful differences are seen at high efficiencies (Figure 10.11b, N = 5000). We compare in Figure 10.12 the band profiles calculated for the (+) isomer of Troger's base using the forward-backward numerical method and an OCFE method. To avoid a circular argument, the isotherms were obtained by frontal analysis and the column efficiency was measured under linear conditions (from very small size injections) [59]. There is a significant difference between the band profiles, because the column efficiency is poor, 110 and 150 theoretical plates for the (-) and (+) enantiomers, respectively, under analytical conditions. As expected, the finite difference method introduces significant errors even in the case of a single component profile. We have found empirically [64] that calculations of band profiles using the OCFE method give satisfactory results, with no oscillations or wiggles, and that the profiles are independent of the values chosen for the integration increments, Ah and T, provided the following empirical conditions are verified: Az < H(NP + l ) 2 / 2
(10.116)
10.4 Results Obtained with the Equilibrium Dispersive Model At
515 (10.117)
1)/UQ
As a conclusion, the choice between the different methods of calculations of band profiles depends on the purpose of these calculations. For efficiencies exceeding about 500 theoretical plates, however, the differences between the profiles calculated with the different methods (the three finite difference schemes and the OCFE method) are negligible compared with the experimental errors that always take place when recording the profiles of high concentration chromatographic bands [49,50]. When computing time is important, the finite difference method using the forward-backward difference scheme (Eq. 10.78) should always be preferred. When extreme accuracy is needed, an OCFE method is probably better than a finite difference method, although the assessment of the calculation errors made in this case is difficult and few results are available on this point.
10.4.3 Results of Computer Experiments Two types of investigations have been carried out using the equilibrium-dispersive model. First, the behavior of chromatographic columns has been investigated b
iA
2.0
2.5
3-
1
0.0
0,2
0A
0.6
0,8
1.0 ()
1.2
1.4
1.6
1.8
2.0
\ \
1 \ 1 1
\W
0.0
0.5
1.0
(J
K 2 ' 1\ \ \^ \ "\ \ \ Y
6.
l i
i
Time (min)
Figure 10.13 Equilibrium isotherms and high-concentration band profiles, (a) Isotherm models: 1, linear; 2, Langmuir (Eqs. 3.47 and 3.48); 3, bi-Langmuir (Eq. 3.53); 4, quadratic with inflection point (Eq. 3.61 with n = 2). All four isotherms have the same initial slope, hence the retention times at infinite dilution (linear chromatography) are the same for all four profiles, (b) Band profiles corresponding to a 1-mg sample with the four isotherms. Reproduced with permission from A.M. Katti, M. Diack, M.Z. El Fallah, S. Golshan-Shimzi, S.C. Jacobson, A. Seidel-Morgenstern and G. Guiochon, Ace. Chem. Res., 25 (1992) 366 (Fig. 4). ©1992, American Chemical Society.
516
Single-Component Profiles with the Equilibrium Dispersive Model
240 260 TIME.
280
300
260
2SO
TIME.
Figure 10.14 Influence of the sample size on the band profile, (a) Langmuir isotherm; column efficiency, 2500 theoretical plates. Sample size (Lf): 1, 0.1%; 2, 0.5%; 3, 1%; 4, 2%; 5, 3%; 6, 5%. (b) Sigmoidal isotherm; same column efficiency. Same sample sizes. Reproduced with permission from G. Guiochon, S. Golshan-Shirazi and A. Jaulmes, Anal. Chem., 60 (1988) 1856 (Figs. 3 and 4). ©1988, American Chemical Society.
under various sets of experimental conditions, in a variety of computer experiments [48,67-69]. The influences of the nature of the equilibrium isotherm (e.g., Langmuir, bi-Langmuir, or anti-Langmuirian), of the column efficiency, and of the sample size and volume were the main areas of study for single-component profiles [69]. Second, calculated band profiles were compared to experimental results, in order to validate calculation methods, algorithms, programs, and the procedures of measurements of the relative parameters. These latter studies are discussed in the next section. Figure 10.13 illustrates the influence of the deviation of the isotherm from linear behavior on the elution band profile [69]. With a 5000-theoretical-plate column, the elution profile obtained with a linear isotherm is a Gaussian profile (Chapter 6). A convex upward isotherm, such as a Langmuir (Eq. 3.47) or a biLangmuir isotherm (Eq. 3.53) as shown in Figure 10.13a, gives a band profile with a steep front and a diffuse rear boundary, while an initially concave-upward isotherm with an inflection point gives a band with a front and a rear shock layer (Figure 10.13b). Figure 10.14 shows the influence of the sample size on the band profiles in the cases of a convex-upward (e.g., Langmuir) and a sigmoidal isotherm (e.g., isotherm 4 in Figure 10.13). The first series of profiles (Figure 10.14a, convex- upward isotherms) is classical. The progressive change in band profiles observed in the latter case (Figure 10.14b, sigmoidal isotherm) is quite different and illustrates clearly the critical influence of an inflection point in the isotherm. Inflexion points
10.4 Results Obtained with the Equilibrium Dispersive Model
517
C mg/mL
b
8
tp=0.04 min
6
1.15
4
1.91
2
3.82 (t-tp)/to
0
40
80
120
160
200
2+0
230
320
0
1
2
3
4
5
Figure 10.15 Influence of the column saturation capacity and the sample volume on the band profile, (a) Influence of saturation capacity. Band profiles calculated for a constant amount (4.15 mmole) of a compound with Langmuir isotherm, constant a coefficient (a = 25, k'o = 6.25), and decreasing b coefficient. N = 2500 plates, L = 25 cm, Fv = 5 mL/min, f0 = 40 s. Curves 1, linear isotherm b = 0, qs infinite); 2, b = 0.02; 3, b = 0.1; 4, b = 0.25; 5, b = 2.5. Reproduced with permission from G. Guiochon, S. Golshan-Shimzi and A. Jaulmes, Anal. Chem., 60 (1988) 1856 (Fig. 6). ©1988, American Chemical Society, (b) Influence of the sample
volume. Band profiles calculated for a 15-cm-long column, with M = 1500 theoretical plates and Lf = 12%. The same sample size is injected as rectangular pulses of different widths, tv.
are often not easy to recognize from measured isotherm data, Figure 10.13a. The band profiles initially have a sharp rear boundary due to the convex-downward isotherm in the low concentration range. At higher sample loadings, the change in the curvature results in a self-sharpening front. Therefore, an unusual band profile is obtained. The effect of the column saturation capacity at constant retention factor, k'o, and constant sample size is illustrated in Figure 10.15a. In the case of a Langmuir isotherm, the column saturation capacity decreases with increasing b, and the degree of column overload increases at constant sample size. Obviously, the higher the column saturation capacity, the higher the sample size needed to reach a certain degree of column overload, hence the higher the production rate and the lower the cost of chromatographic purification. Therefore it is desirable to search for a chromatographic system that affords a large column saturation capacity, thus permitting a large feed load to correspond to a given value of the loading capacity.
518
Single-Component Profiles with the Equilibrium Dispersive Model
This allows an increase in the amount of feed loaded per unit weight of the column packing and a decrease of the production cost. This has important implications for the degree of band overlap in the separation of multicomponent mixtures (Chapter 11) and for the optimization and cost of chromatography (Chapter 18). The cost of the purification in preparative chromatography increases rapidly with decreasing saturation capacity. As long as one is not limited by the solubility of the sample in the mobile phase, it is desirable to select chromatographic systems which have a large column saturation capacity. This permits the injection of larger sample amounts and an increase in the production rate per unit time and per unit amount of column packing. The implications for the degree of overlap for binary separations will be discussed in Chapters 11 and 18. Figure 10.15b illustrates the influence of the width of the injection profile at constant sample size. When the sample width increases beyond a value corresponding to approximately half the natural bandwidth of the peak obtained under linear conditions, the overloaded band begins to widen and becomes shorter [48,70]. For a sufficiently wide injection pulse, the elution profile has a plateau at the injection concentration.
10.4.4 Comparison with Experimental Results The second type of investigation aims at comparing experimental band profiles and the profiles calculated under the same set of experimental conditions with the equilibrium-dispersive model, using independent determinations of the equilibrium isotherm and the column efficiency [59,69-84]. Most studies of this type have been carried out using programs implementing one of the finite difference methods discussed earlier. A few have used orthogonal collocation on finite elements, mainly in cases in which the result of the finite difference method was in doubt because of very low column efficiency [59]. All the work published in this area by Eble et al. [45] uses the Craig model, a special case of the finite difference method using Eq. 10.80 [49]; we have preferred the method corresponding to Eq. 10.78, which leads to extremely fast calculations [49,50]. In the next figures, we compare the results of different calculation methods of overloaded elution band profiles with the experimental profiles acquired on different systems, using a variety of experimental conditions. These figures are listed after the isotherm model that best accounts for the experimental equilibrium data, usually measured by FA. 10.4.4.1
Langmuir Isotherm
In order to calculate band profiles and compare the results with experimental profiles, we need to know the detailed experimental conditions. These conditions include (i) the adsorption isotherm, (ii) the HETP under linear conditions, (iii) the sample size, (iv) the mobile phase flow rate, (v) the hold-up volume, and (vi) the column dimensions. The column HETP can be obtained easily by injecting a very small amount of sample and measuring the band width. The band profile under overloaded conditions is very sensitive to the adsorption isotherm, which is why isotherms must be measured accurately in order to achieve good agreement
10.4 Results Obtained with the Equilibrium Dispersive Model
519
0.04
0,035
-
0.03
-
0.025
-
D.02
-
0.015
-
0.01
-
0.005 -
TIME (mtn)
D.P3
-
0.0+
\
0,03
-
0.02
-
0.01
-
\
^
*
i 1 0
-
Figure 10.16 Comparison of calculated (solid line) and experimental (symbols) band profiles. General conditions: L = 25 cm; dc = 4.6 mm; Fv = 1 mL/min (a,c,d) or 2 mL/min (b); N = 5000 plates, (a) Benzyl alcohol on silica. Mobile phase: solution of THF in n-hexane (15:85). Sample sizes (mmol): 1, 0.0025; 2, 0.00625; 3, 0.0125; 4, 0.025; 5, 0.060; 6, 0.075. (b) Acetophenone on silica. Mobile phase: mixture of ethyl acetate and n-hexane (2.5:97.5). Sample sizes (mmol): 1, 0.025; 2: 0.05; 3: 0.075; 4: 0.1; 5: 0.125. (c) Benzyl alcohol on octadecyl silica. Mobile phase, methanol/water (20:80); sample sizes (mmol): 1, 0.02; 2,0.05; 3, 0.10; 4, 0.15. (d) Phenol on octadecyl chemically bonded silica. Mobile phase: mixture of methanol and water (20:80). Sample sizes: 1, 0.015 mmol; 2: 0.03 mmol; 3: 0.045 mmol; 4: 0.06 mmole; 5: 0.075 mmol. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 60 (1988) 2634 (Figs. 7 to 10). © 1988, American Chemical Society.
between theoretical and experimental profiles [48]. We compare in Figures 10.16a to 10.16d [70] the experimental band profiles in overloaded elution (symbols) and the profiles calculated for elution performed in normal and reversed phase chromatography (solid lines). Figure 10.16a corresponds to the elution of large bands of benzyl alcohol on silica with a THF/nhexane solution. Figure 10.16b corresponds to the elution of acetophenone on silica with a (97.5:2.5) mixture of n-hexane and ethyl acetate. Figure 10.16c illustrates the profiles of bands of benzyl alcohol eluted on C18 silica by a methanol/water solution. Figure 10.16d corresponds to the elution of phenol on C18 chemically bonded silica with a (20:80) mixture of methanol and water. In all four cases,
520
Single-Component Profiles with the Equilibrium Dispersive Model
a-
ID
V
\°\a
\°
V
•
o
\ \n
\a \a \a
\a. \a
4
S
Time (min)
Figure 10.17 Comparison of calculated (solid line) and experimental (symbols) band profiles. Stationary phase, ODS-silica. Mobile phase: mixture of methanol and water (20:80). L = 25 cm; d = 4.6 mm; Fv = 1 mL/min; N = 5000 plates, (a) 3- Phenylpropanol; sample sizes: 1, 0.9 mg {Lf = 1.3%); 2,3.7 mg (Lf = 5.3%); 3, 5.5 mg (Ly = 7.8%). (b) 2,6-dimethylphenol; sample sizes: 1,0.08 mg (Lf) = 0.1%; 2,0.8 mg {Lf) = 1%; 3,3.9 mg (Lf = 4.9%); 4,7.8 mg (Lf = 9.7%). Reproduced with permission from A.M. Katti, M. Czok and G. Guiochon,}. Chromatogr., 556 (1991) 205 (Figs. 3 and 5).
there is a very good agreement between the profiles predicted by the equilibriumdispersive model and the experimental results. It must be underlined that in each case the isotherm was measured with the same column as was used for the determination of the overloaded band profiles, and on the same day. In all cases, the isotherm data were acquired by frontal analysis (Chapter 3, Section 3.5.1) and fitted to a Langmuir isotherm. Use of the ECP method was avoided in this study because it would have led to a circular argument. In practice, however, when accurate predictions of band profiles are needed, and the validity of the equilibriumdispersive model is accepted, the ECP method also gives excellent isotherm data provided efficient columns are used. A similar comparison is made in Figure 10.17 between experimental and calculated band profiles for 3-phenylpropanol and 2,6dimethylphenol on C18 chemically bonded silica [67]. In most of the cases investigated so far, as in those already discussed and as explained in Chapter 3 (Figures 3.6 to 3.8), the experimental data obtained in the determination of single-component isotherms can be fitted correctly to the Langmuir equation (q — (aC) /(1 + bC)) [77]. Occasionally, however, the isotherm data are better fitted when using other isotherm models. The bi-Langmuir isotherm
10.4 Results Obtained with the Equilibrium Dispersive Model Figure 10.18 Comparison of the experimental profiles (symbols) of overloaded bands profiles of 2-phenylethanol on a Symmetry-C18 column. All profiles were calculated with a Langmuir isotherm model. The dashed line profile was calculated with a rectangular injection profile, The other lines were profiles calculated with a realistic, quasi-Gaussian boundary condition. Solid line, profile calculated with the coefficients estimated from FACP adsorption data. Dotted line, profile calculated with the best estimates of the coefficients derived from the numerical solution of the inverse problem. Reproduced with permission from F. James, M. Sepulveda, F. Charton, I. Quinones, G. Guiochon, Chem. Eng. Sci, 54 (1999) 1677 (Fig. 4).
521
_ f 1 1 g
Time (min.)
accounts well for data obtained when there is a mixed retention mechanism, a frequent occurrence in chromatography. As shown in Figure 10.13b, the bi-Langmuir isotherm may account for band profiles that have a much longer tail than those predicted by a pure Langmuir isotherm. In the separation of enantiomers, for example, we know that there often exists two different types of sites on the stationary phase: one type being enantioselective and the other nonselective [78,85]. In such cases, the use of a bi-Langmuir isotherm (q = (« 1 C)/(1 -\-b\C) + («2C)/(1 + b^C)) permits excellent modeling of the equilibrium isotherms and achievement of excellent predictions of the experimental band profiles. James et al. measured by both FA and FACP the equilibrium adsorption isotherm of 2-phenylethanol on a 150 x 3.9 mm I. D. Symmetry C18 column (Waters) [79]. The average particle size was 5 pim; the column void fraction was 0.59, determined from the retention volume of thiourea. The mobile phase was a methanol-water (50:50, v/v) solution. Because the efficiency of the C18 column was high (ca 8000 theoretical plates) the results of FACP were in excellent agreement with those of frontal analysis. The single component profiles were derived from the detector trace converted into a concentration profile via the calibration curve determined from the FA signals fitted to a fourth-order polynomial and via the analysis of the collected fractions. An excellent agreement was obtained with the two methods. The isotherm data fitted well to a Langmuir isotherm model, in excellent agreement with the results given by the inverse method [79]. Figure 10.18 compares the experimental profile (symbols) for a 25 mg sample and the profiles calculated using the isotherm model and two different boundary conditions (see Chapter 2). The dashed line assumed an unrealistic rectangular injection profile. The solid line assumes a quasi-Gaussian injection profiles (two half-Gaussian profiles with different variances), in agreement with the record of the injection profile through a zero-volume connector. This shows how important it is to use realistic boundary conditions when comparing experimental and calculated band profiles.
Single-Component Profiles with the Equilibrium Dispersive Model
522
I
n 3
0.0
4.0
u
8
B
V:
j
M
time (minutes)
time (minutes)
Figure 10.19 Comparison of calculated (lines) and experimental (symbols) band profiles. Left N-benzoyl-L-phenylalanine on Resolvosil-BSA-7 (isotherms in Figures 3.12). Right Nbenzoyl-D-phenylalanine, same conditions. Mobile phase: 0.1 M buffer aqueous solution (pH 6.8) with 7% 1-propanol (v/v). L = 15 cm; d = 4 mm; Fv = 1 mL/min; N = 700; sample sizes: 1, 0.494 fimol; 2: 0,989 fimo\; 3: 1.48 ftmol. Bi-Langmuir isotherm: selective site: a\ = 20.1, qS/\ = 0.556 mmol/L; nonselective site: a2 = 7.09, qSi2 = 217 mmol/L. (Inset) Band profiles calculated from the Langmuir model (see Figure 3.12). S. Jacobson, S. Golshan-Shirazi and G. Guiockon, AIChE ]., 37 (1991) 836 (Fig. 2). Reproduced by permission of the American Institute of Chemical Engineers. ©1991, AIChE. All rights reserved.
10.4.4.2 Bi-Langmuir Isotherm Model In Figure 10.19 we compare the experimental band profiles of various samples of increasing sizes of IV-benzoyl-D- and L-phenylalanine on a Resolvosil-BSA-7 column [74]. The stationary phase is immobilized bovine serum albumin and the mobile phase is a 0.1 M buffer aqueous solution at pH 6.8, with 7% of 1-propanol (v/v). The single-component isotherms were measured on the same column by frontal analysis and fitted to a bi-Langmuir isotherm. These isotherms are shown in Chapter 3, Figure 3.12. The choice of this isotherm model is justified by the existence of two simultaneous retention mechanisms on this column: an achiral mechanism and a chiral selective one. The agreement between the calculated and experimental profiles is very good. In the inset, a comparison is made with the profiles calculated using a single-site Langmuir isotherm. In this case, the agreement is not good, as expected from the poor agreement between the isotherm data and the Langmuir model (Figure 3.12). These results demonstrate that the choice of the proper isotherm model is as crucial for accurate prediction of elution band profiles as the determination of accurate experimental adsorption data. In the case of single-component isotherms, however, the fitting of a spline to the experimental data is an alternative that eliminates the need for a correct isotherm model [46]. The equilibrium isotherm data of the two enantiomers of 1-indanol were measured with FA on a microbore column (i.d. 1 mm) packed with Chiralcel OB (cellulose-tribenzoate coated silica from Chiral Technologies) and eluted with a solution of w-hexane and 2-propanol (92.5:7.5) [80]. The amount of sample needed
10.4 Results Obtained with the Equilibrium Dispersive Model
523 Exp Cal -Cal
8
Exp Cal -Cal
7
5
4
6 5
C [g/L]
C [g/L]
3 4 3
2
2
1 1 0
0
-1 0 (a)
5
10
15
20
0
25
t [min]
5
10
15
20
25
t [min]
(b)
Figure 10.20 Comparison of calculated (solid lines) and experimental (symbols) elution band profiles of R-1-indanol (a), with Lf =1.0, 1.9, 2.8, 3.6, and 4.9% and S-1-indanol (b) with Lf = 0.48,1.3,2.0,2.9, and 4.2%. Reproduced with permission from D. Zhou, K. Kaczmarski, A. Cavazzini, X. Liu, G. Guiochon,}. Chromatogr. A, 1020 (2003) 199 (Fig. 3).
Figure 10.21 Comparison of the experimental profiles (symbols) of overloaded bands of pure kallidin (a) and bradykinin (b) in RPLC with the corresponding profiles (solid lines) calculated with the equilibrium-dispersive model of chromatography. Loading factors, (a): 1.4%, (b): 0.66%, apparent axial dispersion coefficient, DL = 2.74 x 10"4 cm 2 /s. Reproduced with permission from D. Zhou, X. Liu, K. Kaczmarski, A. Felinger, G. Guiochon, Biotechnol. Progr., 19 (2003) 945 (Fig. 3). ©2003 American Chemical Society.
Exp Cal_ED
E, O 0.3-
t [min]
for the isotherm data acquisition was about twenty times less than that required with a conventional column. The data obtained were fitted to different isotherm models. They fitted best to the bi-Langmuir model with the numerical coefficients of one of the Langmuir terms being nearly identical for both enantiomers and those of the other term quite different, explaining the enantioseparation. The best fitted isotherm models were used to calculate the overloaded elution profiles of the two enantiomers. All the calculated profiles were in excellent agreement with the experimental ones, as illustrated in Figure 10.20. Zhou et al. measured by FA the single-component isotherms of two related peptides, bradykinin and kallidin (MW 1060 and 1188, respectively) on a Zorbax SB Cis microbore column (Agilent Technologies), using a 20% aqueous solution of ACN (with 0.1% TFA) as the mobile phase [81]. The isotherm data were fitted to the Langmuir, the bi-Langmuir, the Langmuir-Freundlich and the Toth isotherm
524
Single-Component Profiles with the Equilibrium Dispersive Model
Figure 10.22 Comparison of theoretical (solid lines) and experimental (symbols) band profiles of chicken albumin. 50 x 1 mm column, TSK-DEAE 5PW, 5 ^m particles. Mobile phase, 50 mM Tris-acetate, 50 mM sodium acetate, pH 8.6, Fv 50 pL/min; H = 0.01 cm. Reproduced with permission ofWileyLiss Inc., a subsidiary of John Wiley & Sons, Inc. from A.M. Katti, J.-X. Huang and G. Guiochon, Biotechnol. Bioeng., 36 (1990) 288 (Fig. 3). ©1990, John Wiley & Sons
T (min)
model. These data fitted best to the bi- Langmuir model. The band profiles of large samples of either pure component were recorded. The overloaded profiles were calculated using the isotherm model. They are in rather good agreement with the experimental profiles in all cases, as shown in Figure 10.21. It is more difficult to model the elution profiles of proteins than that of lowmolecular-weight compounds for two reasons. First, it is often difficult to measure equilibrium isotherms, because this must be done under isocratic conditions whereas proteins are mostly eluted and separated under gradient elution conditions (see Chapter 15). Small changes of the composition of the mobile phase result in large changes of the initial slopes of protein isotherms, so measurements of these isotherms tend to be less precise than those of low-molecular-weight chemicals. Second, the mass transfer kinetics of proteins is often sluggish, and a kinetic model (Chapter 14) is almost always more appropriate than the equilibriumdispersive model. As an example, the equilibrium isotherm of chicken albumin on a strong ion-exchange resin was measured using frontal analysis and fit to the bi-Langmuir isotherm equation. This isotherm model and the appropriate operating parameters were used in the equilibrium-dispersive model to calculate band profiles of different size samples. Figure 10.22 compares the experimental profiles recorded on the same column and the calculated profiles. Moderate agreement between the experimental and calculated band profiles was observed [77]. The use of a kinetic model might probably have provided better agreement in this case. Further work is needed in the modeling of protein band profiles.
10.4 Results Obtained with the Equilibrium Dispersive Model
525
Figure 10.23 Comparison of calculated (dotted lines) and experimental (symbols) band profiles, (a) (+)-Enantiomer enantiomer and (b) (-)-enantiomer of Troger's base on microcrystalline cellulose triacetate (isotherm in Figure 3.25). Mobile phase: ethanol. L = 15 cm; d = 4.5 mm; Fv = mL/min; N = 140 plates; sample sizes: 0.15; 0.45; 1.25; and 3.75 mg. Reprinted from A. Seidel-Morgenstern and G. Guiochon, Chem. Eng. Sci., 48 (1993) 2787 (Figs. 8 and 9).
10.4.4.3
Anti-Langmuirian Isotherm Model
Although unusual, convex-downward adsorption isotherms sometimes occur, in such cases, the band profiles have a profoundly different shape. We show in Figure 10.23 the profiles obtained for the elution of large samples of the (-) and (+) enantiomers of Troger's base on microcrystalline cellulose triacetate with ethanol as the mobile phase. The adsorption equilibrium data for the first isomer is accounted for by a quadratic isotherm with an inflection point (isotherms in Figure 3.25), while the isotherm of the second isomer follows Langmuir behavior. Although the column efficiency is poor (about 140 theoretical plates), we observe that the band profiles calculated with the equilibrium-dispersive model are in excellent agreement with the recorded profiles [41]. In Figure 10.24, we compare the calculated and recorded profiles for samples of increasing sizes of phenyldodecane and phenylundecane on graphitized carbon. Although the surface of this adsorbent contains a rather large number of micropores, it is essentially made of large slabs of the 001 plane of graphite with an occasional step [82]. On this surface, molecules of alkylbenzenes can adsorb on the micropores, in which the alkyl chains but not the aromatic rings can penetrate, or lie flat on the surface of the 001 planes. The adsorption energy in the micropores is high, the saturation capacity is very low, and there are no adsorbateadsorbate interactions. Thus, the isotherm of adsorption in the micropores is well accounted for by the Langmuir model. The alkyl chains lying flat and parallel on the graphite surface can interact, however, and the free energy of adsorption of a second molecule is lower than that of the first one. A quadratic isotherm with an inflection point accounts well for this behavior. The composite isotherm obtained as the sum of a Langmuir and a quadratic term accounts well for the isotherm data (Figure 3.26). There is very good agreement between recorded and
526
Single-Component Profiles with the Equilibrium Dispersive Model
10 time (min)
Figure 10.24 Comparison of calculated (solid line) and experimental (symbols) band profiles. Mobile phase: acetonitrile. L = 30 cm; d = 4.5 mm; Fv = 1 mL/min; T = 50°C; N = 1400 plates, (a) Phenyldodecane on graphitized carbon (isotherm in Figure 3.26); samples of a 67 mM solution; sizes: 1, 5; 2,10; 3, 30; 4, 50; 5,100; 6,180 pL. (b) Phenylundecane under the same conditions. Sample volumes (^L): 1,10; 2,30; 3,60; 4,90; 5,129; 6,150. Reproduced with permission from (a) A.M. Katti et at, Ace. Chem. Res., 25 (1992) 366 (Fig. 2), and (b) M. Diack and G. Guiochon, Langmuir, 8 (1992) 1587 (Fig. 9). ©1992, American Chemical Society.
calculated band profiles of large size samples (Figure 10.24). Mihlbachler et al. measured the equilibrium isotherms of the two enantiomers of Troger's base on a 10 x 1 cm column packed with Chiralpak AD (a porous silica coated with amylose tri-(3,5-dimethyl carbamate)) [86]. Because the samples of Troger's base available contained an early eluting impurity with a high UV absorbance, the accurate determination of the experimental adsorption data by FA was not possible and the perturbation method was used. The adsorption isotherm data of the first eluted (—)-enantiomer fitted well to the Langmuir model, those of the second eluted (+)-enantiomer to an S-shaped isotherm model. The experimental overloaded elution profiles (solid lines) of samples of different sizes of these two compounds are compared to those calculated using the equilibrium dispersive model in Figure 10.25. The profiles calculated using two different isotherm models are also compared. The dashed lines correspond to the best isotherm models obtained for the pure enantiomers, a Langmuir and an S-shaped model for the first and the second eluted enantiomers, respectively. The dotted line correspond to the three-layer isotherm model used to account for the competitive isotherm data and applied here to the pure components. The two isotherm models give similar band profiles. The greatest differences are between the experimental and the calculated parts of the profiles of the (+)-enantiomer in a concentration range around the concentration of the inflection point of the isotherm. All the results presented here combine to demonstrate the practical usefulness of the equilibrium-dispersive model, at least in the case of single-component band profiles. Even when the column efficiency is poor {e.g., Troger's base on eel-
527
REFERENCES
8.0
8.5
8,0
9.5
10.0 10.5 11.0 11.5 12.0 12.5
Time t [min]
15
16
17
18
Time t [min]
Figure 10.25 Comparison of the experimental (solid line) and calculated (dashed and dotted lines) band profiles of the two enantiomers of Troger's base on Chiralpak AD. (Left) (^)-enantiomer; sample sizes 1.31, 2.62, 3.93, and 5.23 mg. Dashed lines, profiles calculated with an S-shaped isotherm model. (Right) (+)-enantiomer; sample sizes 3.62, 7.25, 10.87, and 14.5 mg. Dashed lines, profiles calculated with a Langmuir isotherm model. Dotted lines, profiles calculated with the three-layer competitive isotherm model. Reproduced with permission from K. Mihlbachler, K. Kaczmarski, A. Seidel-Morgenstern, G. Guiochon, ]. Chromatogr. A, 955 (2002) 35 (Figures 9 and 10).
lulose triacetate, Figure 10.23), the band profiles calculated with this model are in excellent agreement with those determined experimentally, provided accurate isotherm data are acquired, and a correct model is used to account for the adsorption data. This result is obtained because the nonlinear thermodynamic behavior controls band profiles at high concentrations, while axial dispersion and the finite rate of the kinetics of mass transfer and of adsorption-desorption act merely as a dispersive correction. The same degree of accuracy is never required for the correction term as for the main contribution.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
E. V. Dose, G. Guiochon, Anal. Chem. 62 (1990) 816. G. S. Bohart, E. Q. Adams, J. Amer. Chem. Soc. 42 (1920) 523. E. Wicke, Kolloid Z. 86 (1939) 295. E. Wicke, Kolloid Z. 90 (1940) 156. L. Lapidus, N. R. Amundson, J. Phys. Chem. 56 (1952) 984. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 603 (1992) 1. J. J. van Deemter, F. J. Zuiderweg, A. Klinkenberg, Chem. Eng. Sci. 5 (1956) 271. E. Glueckauf, Ion-Exchange and Its Applications, Metcalfe and Cooper, London, 1955. E. Glueckauf, Trans. Faraday Soc. 51 (1955) 1540. J. Giddings, Dynamics of Chromatography, M. Dekker, New York, NY, 1965. H.-K. Rhee, N. R. Amundson, Chem. Eng. Sci. 27 (1972) 199. J. Zhu, Z. Ma, G. Guiochon, Biotechnol. Progr. 9 (1993) 421.
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G. J. Houghton, J. Phys. Chem. 67 (1963) 84. P. Haarhoff, H. J. Van der Linde, Anal. Chem. 38 (1966) 573. S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 506 (1990) 495. B. Lin, S. Golshan-Shirazi, Z. Ma, G. Guiochon, J. Chromatogr. 500 (1990) 185. B. Lin, Z. Ma, G. Guiochon, J. Chromatogr. 542 (1991) 1. A. Jaulmes, M. J. Gonzalez, C. Vidal-Madjar, G. Guiochon, J. Chromatogr. 387 (1987) 41. C. A. Lucy, J. L. Wade, P. W. Carr, J. Chromatogr. 484 (1989) 61. A. Jaulmes, C. Vidal-Madjar, A. Ladurelli, G. Guiochon, J. Phys. Chem. 88 (1984) 5379. A. Jaulmes, C. Vidal-Madjar, H. Colin, G. Guiochon, J. Phys. Chem. 90 (1986) 207. B. A. Bidlingmeyer, Preparative Liquid Chromatography, Vol. 38 of Journal of Chromatography Library, Elsevier, Amsterdam, The Netherlands, 1987. H. Colin, Separat. Sci. Technol. 22 (1987) 1851. J. E. Eble, R. L. Grob, P. E. Antle, L. R. Snyder, J. Chromatogr. 384 (1987) 25. J. Eble, R. Grob, P. Antle, L. Snyder, J. Chromatogr. 384 (1987) 45. B. Lin, S. Golshan-Shirazi, G. Guiochon, J. Phys. Chem. 93 (1989) 3363. S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 61 (1989) 462. A. W. J. Dejong, H. Poppe, J. C. Kraak, J. Chromatogr. 148 (1978) 127. H. Poppe, J. C. Kraak, J. Chromatogr. 255 (1983) 395. J. H. Knox, H. M. Pyper, J. Chromatogr. 363 (1986) 1. S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 60 (1988) 2364. C. A. Lucy, P. W. Carr, J. Chromatogr. 556 (1991) 159. W.-C. Lee, J. Chromatogr. 606 (1992) 153. D. R. Jenke, J. Chromatogr. 479 (1989) 387. F. James, M. Sepulveda, P. Valentin, Math. Models and Methods in Appl. Sci. 7 (1997) 1. A. Felinger, A. Cavazzini, G. Guiochon, J. Chromatogr. A 986 (2003) 207. F. James, M. Postel, M. Sepulveda, Physica D 138 (2000) 316. C. Cercignani, R. Illner, M. Pulvirent, The mathematical theory of dilute gases, in: Appl. Math. Sci., Vol. 106, Springer, New York, 1994. R. Catflish, Comm. Pure Appl. Math. 33 (1980) 651. G. Q. Chen, D. Levermore, T. P. Liu, Comm. Pure Appl. Math. 47 (1994) 787. A. Seidel-Morgenstern, G. Guiochon, Chem. Eng. Sci. 48 (1993) 2787. G. Guiochon, L. Jacob, P. Valentin, Chromatographia 4 (1971) 6. G. Guiochon, L. Jacob, Chromatographic Review 14 (1971) 77. S. Seshadri, S. N. Deming, Anal. Chem. 56 (1984) 1567. J. E. Eble, R. L. Grob, P. E. Antle, L. R. Snyder, J. Chromatogr. 405 (1987) 1. P. Rouchon, P. Valentin, M. Schonauer, C. Vidal-Madjar, G. Guiochon, J. Phys. Chem. 88 (1985) 2709. P. Rouchon, M. Schonauer, P. Valentin, G. Guiochon, Separat. Sci. Technol. 22 (1987) 1793. G. Guiochon, S. Golshan-Shirazi, A. Jaulmes, Anal. Chem. 60 (1988) 1856. M. Czok, G. Guiochon, Anal. Chem. 62 (1990) 189. M. Czok, G. Guiochon, J. Chromatogr. 506 (1990) 303. P. D. Lax, B. Wendroff, Comm. Pure Appl. Math. 13 (1960) 217. B. Lin, S. Golshan-Shirazi, Z. Ma, G. Guiochon, J. Chromatogr. 475 (1989) 1. W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, New York, NY, 1977. A. Felinger, G. Guiochon, J. Chromatogr. 658 (1994) 511. M. Z. El Fallah, G. Guiochon, Anal. Chem. 63 (1991) 859.
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Chapter 11 Two-Component Band Profiles with the Equilibrium-Dispersive Model Contents 11.1 Numerical Analysis of the Equilibrium-Dispersive Model 11.1.1 Finite Difference Methods. Principle 11.1.2 Finite Difference Methods. Errors in the Case of Two Components 11.1.3 Finite Element Method
11.2 Applications of the Equilibrium-Dispersive Model 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5
532 533 534 539
542
Comparison of Solutions of the Ideal and the Equilibrium-Dispersive Models . . 543 The Hodograph Transform and Its Application 544 Results of Computer Experiments 546 Calculation of Multicomponent Chromatograms 554 Comparison of Calculated Band Profiles and Experimental Results 556
References
567
Introduction The success of the equilibrium-dispersive model in accounting for the profiles of single-component bands justifies our attempt to extend its use to the study of multicomponent mixtures. Conversely, the detailed investigation of the numerical solutions of the equilibrium-dispersive model for a single component which was presented in the previous chapter is justified by the possibility of extending these methods to the study of multicomponent problems. From a purely mathematical point of view, the derivation of closed-form analytical solutions for the individual band profiles in multicomponent mixtures is currently hopeless. Resolution of this problem would require the calculation of solutions of a system of partial differential equations (Chapter 1, Eq. 2.2) coupled through the competitive isotherms and possibly the dependence of the diffusion coefficients and the mass transfer rate constants on the local composition of the mobile phase stream. There are no approximate analytical solutions for problems involving more than a single component. Furthermore, the theory of systems of partial differential equations does not give any general results on which we could solidly ground the extension of the methods developed previously for single components. There are no theorems proving the existence of solutions of the system even for two components. Consequently, there are no possible studies of the convergence of the numerical algorithms toward exact solutions. Because we need to calculate numerical solutions to investigate systematically 531
532
Two-Component Band Profiles with the Equilibrium-Dispersive Model
the problems of band separation and column optimization in preparative chromatography, we shall assume that these difficulties of principle have no practical consequences, and that we may extend the equilibrium-dispersive model to multicomponent mixtures. In so doing, we assume that the band profiles calculated with the finite difference or finite element algorithms tend continuously toward the exact solutions of the system when the integration elements tend toward 0. Provided due adjustments are made to solve the technical problems that arise and which will be discussed in this chapter, the results obtained will be essentially correct. Indeed, this assumption is suggested by good chemical engineering common sense. The general agreement reported here between experimental results and the numerical calculations supports this endeavor. Nevertheless, we should remember that common sense brings no proofs in mathematics. As long as the existence and the uniqueness of the solution have not been demonstrated (which we cannot anticipate to take place in the near future), we cannot be sure that there are no sinkholes ready to open under us, even though twenty years of use of the method have not suggested any. Such are the risks taken by those who do "experimental mathematics" out of necessity!
11.1 Numerical Analysis of the Equilibrium-Dispersive Model In the case of a multicomponent mixture, the individual band profiles are given by the equations of the equilibrium-dispersive model
where the concentrations qi in the stationary phase are related to the concentrations Cj of all the compounds present in the mobile phase by the competitive isotherms, c\{ = f(Cj), and the apparent dispersion coefficients Da^ are related to the HETP for the corresponding compounds by
In order to solve the system of Eqs. 11.1, we need to specify the initial and the boundary conditions of the problem. The general boundary conditions were discussed in Chapter 6, Section 6.2.1. Simpler boundary conditions were introduced and justified in Chapter 2, Section 2.1.4, and in the discussion of the single-component problem. In isocratic elution, the column contains no sample component but is filled with the pure mobile phase (when the mobile phase is a mixture and one of its components may compete with the sample components, we have a more complex problem, discussed in Chapter 13) Q(x, t = 0) = 0
(11.3)
11.1 Numerical Analysis of the Equilibrium-Dispersive Model
533
The boundary condition expresses the continuity of the mass flux of solutes at the column inlet and outlet. The Danckwerts [1] conditions are written C(t,x = 0) ^|
D a
x = L
=
-Da/i — + C{
(11.4)
=
o
(11.5)
where
= =
Cf 0
0tp
(11.6a) (11.6b)
Finally, we need the isotherm equations that relate the concentration of each of the solutes in the stationary phase and the concentrations of all the solutes in the mobile phase. In general, these adsorption isotherms are competitive, meaning that the amount of component i adsorbed at equilibrium between phases from a solution with a constant concentration Q decreases with increasing concentrations of any one of the other components (Chapter 4). Unfortunately, there is little that the theory of partial differential equations can tell us about the properties of the system of Eqs. 11.1 to 11.6b and its solutions. It does not even give a general theorem of existence of solutions, nor any theorems of convergence for a possible numerical solution. We shall extend to this system the methods used in the case of single-component profiles. As expected, we will find that these methods do give good results. As indicated in the introduction, however, we are on shaky mathematical ground, and numerical calculations may fail under some particular sets of conditions, although that has never been reported so far provided the rules of numerical stability discussed in Chapter 10 are followed.
11.1.1 Finite Difference Methods. Principle The methods of calculation of numerical solutions discussed in Chapter 10 for a single component can easily be extended to the case of multicomponent mixtures. In this case, we have n partial differential equations similar to Eqs. 10.78,10.80, or 10.82 (i = 1,2,•••,«) +
dz
dGi(Cl,C2/..,Cu..) dt
_ ~
Da,idCi u dt2
K
'
with (n.8) In the case of the calculation of the elution band profiles for single-component samples, excellent results were obtained with the second approach discussed in
534
Two-Component Band Profiles with the Equilibrium-Dispersive Model
Chapter 10, which provides satisfactory accuracy and is the simplest and fastest calculation procedure. This method consists of neglecting the second-order term (RHS of Eq. 11.7) and calculating numerical solutions of the ideal model, using the numerical dispersion (which is equivalent to the introduction in Eq. 11.7 of a firstorder error term) to replace the neglected axial dispersion term. Since we know that any finite difference method will result in truncation errors, the most effective procedure is to control them and to use them to simplify the calculation. The results obtained are excellent, as demonstrated by the agreement between experimental band profiles recorded with single-component samples and profiles calculated [2-7]. Thus, it appears reasonable to use the same method in the calculation of solutions of multicomponent problems. However, in the multicomponent case a new source of errors appears, besides the errors discussed in detail in Chapter 10 (Section 10.3.5).
11.1.2 Finite Difference Methods. Errors in the Case of Two Components In the finite difference approach, using the equation of the ideal model and appropriate numerical dispersion, we try to compensate exactly the axial dispersion term which is dropped from Eq. 11.7 by a numerical error term. We have shown in Chapter 10 (Section 10.3.5.2) that this is possible provided we choose values of the time, h, and the space, x, integration increments such that one of the following relationships is verified:
^fL)Hi
(11.9a) H{
(11.9b)
Hi
(11.9c)
depending on whether a forward-backward (Eq. 11.9a or 10.79), a backwardforward (Eq. 11.9b or 10.81), or a forward-backward M+ i (Eq. 11.9c or 10.83) difference scheme is used. In these equations, a is the Courant number, which becomes for the component i ai=X-^
= X-^ -
h
^
(11.10)
h l + k[Q
In Eqs. 11.9a to 11.10, uz^ = w/(l + fc0(-) is the velocity associated with the concentration 0 (infinite dilution). Note, however, that these equations (e.g., Eqs. 11.9a) are exact only in linear chromatography. They are approximate in nonlinear chromatography. Obviously, the values of the time and space increments in the numerical calculations have to be the same for all the components of the system. We cannot have different values of h and T in the different finite difference equations written for the different components of the mixture. We see from Eqs. 11.9a to 11.9c that
11.1 Numerical Analysis of the Equilibrium-Dispersive Model
535
when we have chosen these values for one of the components, we have no possibilities to adjust them for any of the other components once we have decided on the value of the Courant number. We might abandon at this stage the requirement of choosing the Courant number, but elect rather to choose values of the increments T and h that satisfy Eq. 11.9a simultaneously for two compounds and thus give the exact band broadening required for both of them instead of one. For example, in the case of the forwardbackward scheme and of two components, we need to have
(11.12) The solutions are H l
-H2
~ H1(1 + h
=
-,
^1)H2{1+^) -,
(11.13b)
^0,2 ~ *0,l
The value of the former integration parameter is positive only if H decreases with increasing k0, and the value of the latter is positive only if H decreases rapidly enough with increasing k0 to compensate for the opposite variation of the term 1 + kQi. If we introduce the values of T and H given by Eq. 11.13a,b into the definition of the Courant number, we obtain
fl
2
=
—jj—
= rr „
, ,7
A
rT
, , , ,./
^
(11.13d)
A further condition applies to the value of the Courant number, which must exceed 1 with the forward-backward scheme. The requirements that both T and h are positive and that a\ and «2 are larger than unity are often difficult to satisfy. Furthermore, if x and h are small, the computation time may become excessive. Similar results are obtained with the other schemes. Accordingly, it is preferable to accept the introduction of an additional error and to choose the value of the Courant number because of its importance for the numerical stability of the calculation. Then, we can choose the value of H and, accordingly, the value of Da derived from the truncation error, for one single component. For all the other components, H and Da are fixed and there is no reason that these values are those observed in the practical chromatographic problem studied. These values may not even be realistic [6-8]. Even in linear chromatography, where Eqs. 11.9a to 11.9c are exact, if one simulates the band profiles of a series of compounds using the multicomponent scheme (although there would not be any
536
Two-Component Band Profiles with the Equilibrium-Dispersive Model
good reason to do that, as Eqs. 11.1 are not coupled when the isotherms are linear, for obvious thermodynamic reasons), one can obtain the exact band broadening for one component only, provided the space and time increments are adjusted according to the HETP of this component. For the other components, the bandwidth would be related, following a dependence on the retention factor which has nothing to do with the prediction of any realistic plate height equation. For example, let us consider a two-component problem and use the forwardbackward scheme to calculate numerical solutions, with a Courant number equal to «i for the first component. In linear chromatography, the contribution of the numerical dispersion for this first component would be exactly equivalent to the true HETP, H\, if the space and time increments are chosen according to Eqs. 10.86 and 10.96 h =
- ^ -
(11.14)
fli1 T
=
For these values of the time and space increments, the value of the Courant number for the second component is given by =
«i
and the numerical dispersion for the first component would be equivalent to
( m 7 )
As a numerical example, for a\ -1, k0 a = 3 and k0 2 = 4, we have H2 = O.6H1, quite an unreasonable change in any chromatographic method. Similarly, for the backward-forward numerical scheme we find
< m 8 )
and for the forward-backwardn+j scheme:
Equations 11.17 to 11.19 show that the efficiency of the band profile calculated for the second component would correspond to a value of its HETP which decreases with increasing retention factor (k0 2) for the first and the third numerical schemes and which increases with increasing retention factor for the second
11.1 Numerical Analysis of the Equilibrium-Dispersive Model
537
Figure 11.1 Plot of the apparent column efficiency versus the retention factor. Curves 1-3, forward-backward calculation scheme, Eq. 11.17 (a\ = 2); curves 46, backward-forward calculation scheme, Eq. 11.18 (fli = 0.5). fc'ai = 4. Reprinted from Czok and G. Guiochon, Comput. Chem. Eng., 14 (1990) 1435 (Fig. 1).
scheme. We illustrate the trends observed with the first two schemes in Figure 11.1. The dependence of the HETP on the retention factor is quite steep with the forward-backward scheme, which may generate some difficulties, but it is more limited with the other two methods. Even with the first scheme, we should note that the most important practical cases involve binary mixtures for which the separation factor is rather low, which markedly reduces the magnitude of the observed effect. This is what keeps these calculation errors reasonably small. Thus, in the calculation of the individual band profiles in the case of multicomponent mixtures, there is a third source of errors, besides the two classical error sources observed with finite difference methods, which we have discussed in the study of the single-component problem. Obviously, these two sources are also found in the calculations of solutions of the multicomponent problems. As can be seen from Eqs. 11.17 to 11.19, this new error increases with the difference between the retention factors of the two components, and it decreases with decreasing Courant number. The error would disappear with the second and third schemes (Eqs. 11.18 and 11.19) and the numerical dispersion for the two solutes would become equal and correspond to the proper value of H if a was close enough to 0. This observation is important because, for these two schemes (Eqs. 10.87 and 10.88), we can always select low values of the Courant number if needed by combining a large space increment h and a suitably small time increment T (Eq. 11.10). In this respect, the backward-forward and the forward-backwardn+i schemes both have the advantage that they permit the use of very small Courant numbers, since the conditions of stability of the numerical procedure for these two schemes are 0 < a < 1 and a > 0, respectively. For the first scheme, on the other hand, a should be greater than 1 (Eq. 10.86), which prevents any adjustment of the apparent dependence of H on kQ. This condition has important consequences in the calculation of solutions in the problem of gradient elution. In this case, the retention factor, k'o, may vary considerably during the separation and its initial and final values may differ by several orders of magnitude. When choosing the value of the Courant number, a, one should keep in mind that the value of a for both components should always satisfy the stability condition of the numerical scheme used (Eqs. 10.86 to 10.88). Otherwise, a numerical instability may occur, resulting in dramatic oscillations of the concentrations, negative values of the concentrations, or an interruption of the program during runs for overflow or underflow errors. A good solution to this difficulty consists in changing the values of T and h during the calculation while keeping a constant.
538
Two-Component Band Profiles with the Equilibrium-Dispersive Model
450 time (s)
7O0
460 time (s)
Figure 11.2 Comparison between the band profiles calculated with two different finite difference schemes. Solid line: forward-backward scheme, dashed line: backward-forward scheme. Column efficiency, 3000 theoretical plates. (Left) 1:3 mixture. (Right) 3:1 mixture. Reprinted from M. Czok and G. Guiochon, Comput. Chem. Eng., 14 (1990) 1435 (Figs. 2 and 3).
The individual band profiles that are derived from the backward-forward and the forward-backward calculation schemes when applied to multicomponent mixtures are usually quite comparable [6-8], although neither of the two numerical solutions is exact. These individual band profiles are also close to those calculated with the finite element method [9]. The differences are significant only when the ratio C^/Cf, of the feed concentrations of the second and first components is small and a strong tag-along effect takes place. We show in Figures 11.2, left and right, comparisons between band profiles calculated with these two methods, for a 1:3 and a 3:1 mixture.1 In practice, instead of choosing the value of uZ/o, hence the Courant number, for one component of a pair, the average value of uzfl = M£O[1/£R,O,I + 1AR,O,2]/2 is used for the calculations, which provides a better compromise. There is excellent agreement in Figure 11.2 left but less good agreement in Figure 11.2 right, where the tag-along effect is important. When the same sample amount is injected in a shorter time (using a more concentrated solution), the difference between the profiles is reduced [6]. Note also that, because the values used for the Courant number are 2 and 0.5, respectively, for the two calculation schemes, the amount of dispersion introduced is twice as much in the former case. In spite of some awkwardness in its formulation, the forward-backward scheme of numerical integration of the ideal model (Eq. 10.79) seems the most efficient way of calculating the band profiles of the equilibrium-dispersive model. It is particularly effective in terms of use of CPU time and is especially suitable for theoretical investigations of optimization strategies [10]. The best alternative procedure is not another finite difference scheme but one using orthogonal collocation on finite elements [9]. This procedure is more accurate but requires a much longer 1 A 1:3 mixture contains 1 part of the first component for 3 parts of the second component. Similarly, a 3:1 mixture contains 75% of the first component and 25% of the second component.
11.1 Numerical Analysis of the Equilibrium-Dispersive Model
539
C [mg/mL] [mg/mL]
10 8 6 4 2
5 length [cm' [cm]
0
10 5 10 15
time [min]
Figure 11.3 Numerical solution of the system of mass balance equation for two components. Total concentration profile. Competitive Langmuir isotherm, d\ = 12, «2 = 24,
computation time than the forward-backward finite difference scheme. Figures 11.3 and 11.4 illustrate the result obtained in the calculation of the numerical solution of the system of mass balance equations for a binary mixture. Steep fronts were obtained by strongly overloading the column (Lf = 80%). To show a separation under such conditions a large separation factor of 2 was used. Figure 11.3 displays the total chromatogram, as given by a non-selective detector. It shows how the two bands become progressively separated. The two individual profiles are shown in Figures 11.4, top and bottom. The rear part of the second component profile is not affected by the interference between the two bands. The front part recovers after the separation has taken place, but the poor efficiency does not permit a good illustration of this effect nor of the tag-along effect in Figures 11.4b. By contrast, the profile of the first component is squeezed by the displacement effect and this phenomenon is barely blurred by the strong axial dispersion. The first band profile does not begin to widen significantly until after the separation has become nearly complete.
11.1.3 Finite Element Method The finite element methods are more mathematically involved than the finite difference schemes, and a detailed explanation of these methods is not within the scope of this work. Thus, we shall briefly elaborate on the extension to multi-
540
Two-Component Band Profiles with the Equilibrium-Dispersive Model CC[mg/mL] [mg/mL]
10 8 6 4 2
5 length length[cm] [cmj
0
10 5 10 15
time [min]
C [mg/mL]
10 8 6 4 2
5 length length [cm] [cm]
0
10 1') 5 15
10 1J time time [min] [min]
Figure 11.4 Numerical solution of the system of mass balance equation for two components. Same parameters as for Figure 11.3. (Top) Profile of the first component. (Bottom) Profile of the second component.
component mixtures of the scheme developed in the previous chapter for single component band profiles. The discussion of the implementation of the method of orthogonal collocation on finite elements (OCFE) for the calculation of solutions of the mass balance equation of the equilibrium-dispersive model for a single component given in the previous chapter extends naturally to the system of Eqs. 11.1
11.1 Numerical Analysis of the Equilibrium-Dispersive Model
541
to 11.6b. No further information is needed. Although the methods of orthogonal collocation on finite elements have been developed for rapid numerical calculation of the solutions of systems of partial differential equations and are widely used for this reason in physics, mechanics, and chemical engineering, their speed advantage over finite difference methods is important only in problems that have three spatial dimensions and a complicated geometry. In practice, we always consider chromatographic problems to have only one space dimension, although it is quite probable that a more detailed investigation of the performance of large-diameter columns will require refraining of the band profile problem in at least two dimensions to take into account the lack of homogeneity of large-scale packing in columns that are barely longer than they are wide (see Chapter 2, Section 2.1.6). Even in this case, however, the geometric boundaries of the problem would remain most simple (i.e., the column wall and inlet frits). A further difference between chromatography and the classical problems of chemical engineering is that the calculation of band profiles requires the use of much smaller time and space integration increments than are typically encountered in these other problems. This explains the difficulties and peculiarities of its solution. At the end of this chapter, we compare some experimental results obtained with a low-efficiency column and the profiles calculated with a finite difference method and with orthogonal collocation on finite elements. Figure 11.5 shows chromatograms calculated in the case of a 3:1 binary mixture with an OCFE method and with the two finite difference methods, for a 3000plate column [9]. The front part of the first band and the rear of the calculated chromatograms are identical in the three cases. The only significant differences occur around the second shock layer. The rear shock layer of the first component band and the front one of the second component band are steeper in the chromatograms calculated with the OCFE method and the backward-forward numerical scheme than in the one obtained with the forward-backward scheme. There are few differences between the profiles calculated with the first two methods. In fact, the differences between the results of the three methods are insignificant, even at low column efficiencies, when the concentration of the second component is larger than in Figure 11.5, or when the separation between the two components is better, as seen in Figure 11.6. This figure illustrates a comparison of band profiles calculated for three different compositions of a binary mixture, using the forward-backward finite difference scheme and the OCFE method, for a column having only 1000 theoretical plates. The difference between the two numerical solutions increases with decreasing column efficiency. The differences between the two sets of profiles are very small in the selected cases, where the tag-along effect is small. In the case of Figure 11.6c, where the tag-along effect is important, the difference is still insignificant. Because the forward-backward method is much faster than the backwardforward method, by more than one order of magnitude, during the late 1980s and early 1990s, its use was preferred in all cases in which the column efficiency exceeded 1000 theoretical plates and the concentration of the second component exceeded half the concentration of the first one. Calculations of band profiles for columns having a few thousand theoretical plates could be performed in minutes
542
Two-Component Band Profiles with the Equilibrium-Dispersive Model
200
460
7OO
time (s)
Figure 11.5 Comparison between the results of the calculations of the individual band profiles of a 3:1 binary mixture using the OCFE method and two finite difference methods (Eqs. 10.78 and 10.80). Solid line: profile calculated with the OCFE method, with Sz = 0.050 cm, St = 0.15 s. Dotted line: profile calculated with the forward-backward method (Eq. 10.78), Sz = 0.0050 cm and St = 0.33 s. Dash-dotted line: profile calculated with the backward-forward method (Eq. 10.80), Sz = 0.0050 cm and St = 0.033 s. Calculation parameters: column length: 15 cm. Phase ratio: 0.25. Mobile phase flow velocity: 0.15 cm/s. Column capacity factors: k'Q1 = 4, k'O2 =5. Np = 3. 3:1 binary mixture. Langmuir competitive isotherms parameters: o.\ - 20, «2 = 16, b\ = 5 M^ 1 , b2 = 4 M~x. Input profile: rectangular pulse, width tp = 10 s, height Cl° = 1.5 M, d? = 0.5 M. H = 50 fim. Reprinted from Z. Ma and G. Guiochon, Comput. Chem. Eng.r 15 (1991) 415 (Fig. 5).
with an 80486-based computer, when using the forward-backward method [11]. With modern personal computers, it is rarely needed to look for an acceptable compromise between accuracy and CPU time, whatever the column efficiency and the separation scheme considered, even in complex SMB simulations.
11.2 Applications of the Equilibrium-Dispersive Model There is an abundant literature on the comparison between experimental and calculated band profiles for binary mixtures. The most popular methods used have been the forward-backward finite difference scheme and the OCFE method. The former lends itself readily to numerical calculations in many cases representative of the present preoccupations in preparative chromatography. We present first a comparison between the band profiles obtained with the ideal and the equilibrium-dispersive model to illustrate the dispersive influence of the column efficiency. Related to the comparison between these two models is the issue of the use of the hodograph transform of experimental results discussed in Section 11.2.2. Computer experiments are easy to carry out and most instructive because it is possible to show the effects of the change of a single parameter at a time. Some
11.2 Applications of the Equilibrium-Dispersive Model
IBD.HD
XH.W
Time
HOOD
543
139,00
EHLUD mtm
nun
no«
(ace)
Figure 11.6 Comparison of profiles calculated with OCFE and a finite difference method. N = 1000 plates. Line 1, forward-backward finite difference algorithm, Courant number = 2; line 2, OCFE, tp = 5 s. (a) 1:9 mixture, a = 1.5. (b) 1:1 mixture, a = 1.5. (c) 5:1 mixture, a = 1.2.
results of such calculations are presented next. Finally, experimental results are compared to band profiles calculated with the known equilibrium isotherms and column efficiencies, using first the forward-backward scheme, then other computation approaches.
11.2.1 Comparison of Solutions of the Ideal and the EquilibriumDispersive Models We compare in Figure 11.7 the individual band profiles derived from the ideal model (solid lines) and those calculated with the equilibrium-dispersive model (dotted lines) in the case of two mixtures with relative compositions 1:3 and 3:1. The loading factors for the two components are: Lf^ = 1.2% and Lf/2 = 3.8%. The calculations with the equilibrium-dispersive model were made with efficiencies of 1000 (Figures 11.7a,c) and 5000 (Figure 11.7b,d). The apparent loading factors (m = N[k / ( I + k )]2Lf, see Eq. 10.115) for the two components range between 10 and 145. The agreement between the individual band profiles calculated with the two models is uniformly very good in the case of the more efficient column (see Figures 11.7b,d), because the values of m are large. The main differences observed are the lack of the plateau on the rear diffuse boundary of the profile of the second component, replaced by an inflection point in Figure 11.7b, the replacement of the concentration discontinuities at the front of each component profile by steep shock layers (see Chapters 14 and 16), and the tails appearing at the end of both diffuse boundaries, due to the dispersion effects of a finite column efficiency. Nevertheless, there is substantial agreement between the two sets of profiles for large values of m, illustrating the comments made several times in this chapter, that thermodynamics, more than kinetics, controls band profiles at high degrees of column loading. In the case of the less efficient column (see Figures 11.7a,c), the differences between the two sets of profiles are of the same nature, but more important, as
544
Two-Component Band Profiles with the Equilibrium-Dispersive Model C mg/mL
a
6
b 6
m=10
Ideal
Ideal
m=45
N=1000 N=5000
4
4
2
2 m=145
m=30
0
0 6
9
12
6
9
12
c 12
d 12
Ideal
9
Ideal
9
N=1000
N=5000 m=140
m=30
6
6
3
3
m=50
m=10
0
0 6
9
12
6
9
t
min
12
Figure 11.7 Comparison between the individual band profiles derived from the ideal model (solid lines) and calculated with the equilibrium-dispersive model (dotted lines). Relative retention, a = 1.20; fc01 = 6.0. (a) and (b) 1:3 mixture, loading factors: Lt-y = 1.25%, Lr2 = 3.8%. (c) and (d) 3:1 mixture, loading factors: Lftl = 3.8%, Ly/2 = 1.25%. The values of the apparent loading factors, m = N[fco/(1 + ko)]2Lf, is indicated for each band.
expected from the lower values of the column efficiency and the effective loading factors. The shock layers are less steep, the individual profiles are shorter and their tips more round, the rear shock layer on the first component profile is hardly visible, the rear plateau of the second component has completely disappeared (no inflection point on the rear of the profile in Figure 11.7a, by contrast with Figure 11.7b), and the tails are more important. It remains possible, however, to predict correctly the actual profiles from those calculated from the ideal model.
11.2.2 The Hodograph Transform and Its Application When a wide rectangular injection pulse is injected in a column and the width is such that the plateau is not completely eroded when it is eluted, the solution of the system of equations of the ideal model (Eqs. 8.1a and 8.1b) includes a constant state, followed by a simple wave, as shown by the theory of partial differential equations [12,13]. The importance of this result is due to the existence of a relationship between the concentrations of the two components of the binary mixture in the simple wave region. This relationship is independent of the position of the band along the column. We have discussed the properties of the hodograph transform in the case of the ideal model (Chapter 8, Sections 8.1.2 and 8.8). In the case of the equilibrium-dispersive model (Eqs. 11.1 and 11.2), this result is no longer valid. However, the plots of Q versus C2 are often close to the simple wave solu-
11.2 Applications of the Equilibrium-Dispersive Model
0.0
02
545
0.
Figure 11.8 Illustration of the hodograph transform applied to actual chromatograms. Numerical solutions of the equilibrium-dispersive model for a 310 (Top left) and a 120 (Top center) theoretical plate column, a = 1.8, k\ = 0.7. (Bottom left and center) Hodograph plots derived from the band profiles above Reproduced with permission from Z. Ma, B.C. Lin, A.M. Katti and G. Guiochon, } . Phys. Chem., 94 (1990) 6911 (Fig. 3). ©1990 Ameri-
can Chemical Society. Determination of competitive equilibrium isotherms by the simple wave method. (Top, right) Experimental elution profiles (symbols) for a wide rectangular injection of 2-phenylethanol (•) and 3-phenylpropanol (o), and profiles calculated (lines) assuming competitive Langmuir isotherms with numerical parameters derived from the single-component isotherms measured by frontal analysis (solid lines). Sample relative composition: 1:3. (Bottom, right): Hodograph transform of these data (symbols). Solid line: prediction of the Langmuir isotherm with single component coefficients. Dot-dashed line: best fit of the experimental data. Reprinted from A.M. Katti, Z. Ma and G. Guiochon, AIChE /., 36 (1990) 1722 (Fig. 6), by permission of the American Institute of Chemical Engineers. ©1990 AIChE. All rights reserved.
tions in the case of wide band injections, because the loading factor, Lf, hence the apparent sample size, is large and thermodynamics controls the band profiles. However, in practice, the deviation between the solutions of the ideal and the equilibrium-dispersive models is small along most of the profiles. The deviations of the plot of the experimental values of C-i versus C\ from the hodograph transform of the ideal model solution are also small in most parts of the graph, see Figure 11.8, top and bottom left [5,14,15]. The only exceptions are the parts of this plot that correspond to the beginning of the second shock layer and to the
546
Two-Component Band Profiles with the Equilibrium-Dispersive Model
end of the first component diffuse profile, when the column efficiency is low (Figure 11.8, top and bottom center). In these regions the angles of the ideal model solution transform are rounded by the dispersion, and minor deviations are observed, but the difference is noticeable only at low column efficiencies, below a few hundred theoretical plates [5]. Comparison between experimental profiles for wide injection pulses of binary mixtures and calculated results (Figure 11.8, top right) shows good agreement [5,16]. The difference between these profiles originates in the isotherm model used for the calculations. Although each component follows the Langmuir model [16], the competitive Langmuir isotherm is not the best model for the binary isotherms [17]. However, using the data derived from the experimental profiles of the wide injection bands, parameters of a competitive isotherm model could be calculated (Figure 11.8, bottom right). These parameters permit the calculation of band profiles in overloaded elution which are in excellent agreement with experimental results (see below, Section 11.2.5).
11.2.3 Results of Computer Experiments The calculation procedures developed permit systematic investigation of the influence of the different parameters of a chromatographic separation on the individual band profiles and on the production rate, solvent consumption, or recovery yield (Chapter 18). The advantage of this method is in the ease with which any parameter can be isolated and its influence studied at constant value of the other parameters. This could be most difficult to achieve with actual experiments. We present in this section studies on the influence of the competitive nature of the isotherm (Figure 11.9), the column efficiency (Figure 11.10), the loading factor (Figures 11.11 and 11.14), the separation factor (Figures 11.11 and 11.12; Figures 11.14 and 11.16), the column saturation capacity (Figures 11.11 and 11.13; Figures 11.14 and 11.17), and the relative composition of the feed (Figure 11.15). Figures 11.9a,d compare the individual band profiles obtained with a competitive Langmuir isotherm and the single-component profiles for each of the two components. The single-component profiles are obtained using the singlecomponent parameters of the competitive Langmuir isotherm at the same sample injection amount. Figures 11.9a,b show the effect of competition on the band profile at relatively low loading factor, for a 1:3 and 3:1 mixture. For the 1:3 mixture (Figure 11.9a), the single-component profile of the second component is very similar to that of the second component of the binary mixture except for a slightly shorter retention time and a slight change of its top, while for a 3:1 mixture (Figure 11.9b), the single-component profile for the first component is similar to the first component of the binary mixture. However, for the 3:1 mixture the tag-along effect has already influenced the second component band quite strongly (Figure 11.9b), leaving a plateau at its top. Figure 11.9a shows that the displacement effect has shifted the first component of the 1:3 mixture markedly with a slight compression of this band increasing the concentration at the peak maximum. One can take advantage of this phenomenon for the isolation of impurities and for the large-scale production of components at minimal cost. Without competition, Fig-
11.2 Applications of the Equilibrium-Dispersive Model C mg/mL
Co1/Co2=1/3
Co1/Co2=3/1
a
6 2-
4
b
tp=0.14
Competitive Single
tp=0.14
547
6
-2
1st
Lf% mg m 2.6 6.9 39
2nd
0.8
2.3
14
-1 1-
3
2 1-
-2
-1 2-
0 5
8
0
11
5 tp=0.465
12
c
8
16
tp=0.465
Lf% 1st
Competitive
2nd
Single
1-
11 mg
8.8 2.9
m
d
23
130
7.8
45
1-2
6
8
-1
2-
-2
-2
1-
0
0 5
8
11
5
8
11 Time, min
Figure 11.9 Comparison of the individual band profiles of a binary mixture with the singlecomponent bands at the same sample size. Single-component profiles, component number in italics, for component 1 (narrow hatched line) and component 2 (narrow black line). Individual profile of a binary mixture with competitive Langmuir isotherm, component number in bold, for component 1 (thick lightly shaded line), component 2 (thick dark shaded line). Parameters: k[ = 6, k'2 = 7.2, a = 1.2; by = 0.03, b2 = 0.036, qS/1 = qS/2 = 800, L = 10 cm, e = 0.8, Fv = l mL/min, N = 2000. (a) C° = 1.7 mg/mL, C°/C° = 1:3, tp'= 0.14. (b) C° = 50 mg/mL, C°/C° = 3:1, tp = 0.15. (c) C° = 16.7 mg/mL, C°/C° = 1:3, fp = 0.465. (d) C° = 50 mg/mL, Cj/Cf = 3:1, tp = 0.465. ure 11.9a shows that the first component could not be obtained pure at that sample size, its band being entirely covered by that of the second component. By contrast, with competition for access to the adsorption sites, one can obtain the pure first component at a high yield. At larger sample sizes (Figure 11.9c), the displacement effect is further enhanced, compressing and concentrating the first component band of the 1:3 mixture. The tail of the first component band into the second has increased. Compared to Figure 11.9a, the recovery yields of both components drop markedly, the effect being more serious for the second. For the 3:1 mixture, Figure 11.9d, the tag-along effect has intensified, resulting in an elongated peak penetrating deeply into the rear boundary of the first component band. Figure 11.10 illustrates the influence of the column efficiency on the individual band profiles, using the equilibrium-dispersive model with competitive Langmuir isotherms, with two 10-cm-long columns having 2000 and 5000 theoretical plates, respectively. The primary difference in the profiles is observed in the steepness of the fronts of the second band and in the rounding of its corners and edges. Note also the changes in the apparent loading factor, m — [ko/(l + ko)]2NLj:. Figures 11.11a to 11.lid illustrate the effect of an increase of the sample size on
548
Two-Component Band Profiles with the Equilibrium-Dispersive Model C mg/mL tp=0.093 min
4
tp=0.093
a
b
N=2000 N=2000 N=5000 N=5000
m=25
4
m=30 m=8.6 m=67
m=64
m=21
2 2
m=9 m=22
0
0 4
8
12
12
4
8
c
tp=0.465
tp=0.465
m=130
16
12 d
m=320
N=2000
N=2000 N=5000 N=5000
m=43 m=107
6
8 m=135 m=340 m=45 m=110
0
0 4
8
12
4
8
12 Time, min
Figure 11.10 Individual band profiles of a binary mixture at 2000 and 5000 theoretical plates. Same parameters as for Figure 11.9, except (tp in min, Ly in %): (a) 1:3 mixture, tp = 0.093, Lfil = 0.58, Lfi2 = 1.75, sample sizes: (1) 1.55 mg, (2) 4.65 mg; (b) 3:1 mixture, tp = 0.093, Lfrl =1.75, Lfi2 = 0.58, sample sizes: (1) 4.65 mg, (2) 1.55 mg; (c) 1:3 mixture, tp = 0.465, Lf/1 = 2.9, L //2 = 8.75, sample sizes: (1) 7.8 mg, (2) 23 mg; (d) 3:1 mixture, tv = 0.465, Lfil = 8.75, Lf/2 = 2.9, sample sizes: (1) 23 mg, (2) 7.8 mg.
the individual band profiles of the components of a 1:3 binary mixture, with a = 1.2 and qSti = qSf2 = 800. The sample size is doubled from each subfigure to the next one by increasing the width of the rectangular injection pulse, i.e., the injection volume. For each figure, the injection time is given in minutes, and the sample size in loading factor, Lf (%), amount (mg), and dimensionless m value. The sample size in Figure 11.11a corresponds to a touching-bands chromatogram. Doubling the sample size (Figure 11.11b) results in displacement of the first component band by the second one. Further increases in the sample size give rise to a more intense displacement effect, leaving a more concentrated, narrower first component band. The consequences of increasing the sample size on the production rate and recovery yield of the first component become apparent here. As the sample size increases beyond touching-band conditions, the recovery yield of the first component at 99% purity decreases and its production rate increases. The production rate goes through a maximum and begins to decrease when the displacement effect becomes too intense and the band too narrow. This figure shows how the retention time of the end of the tail of the first component, £E (Eq. 8.27), does not depend on the sample size, as long as there is interference between the two bands. Since the retention time of the second component front decreases with increasing sample size, the displacement effect creates a tail whose length increases with increasing sample size.
11.2 Applications of the Equilibrium-Dispersive Model C mg/mL
4
T
tp=0.074
Lf%
mg
m
1st 0.4
1.2
6.8
2nd 1.4
3.7
22
549 b
tp=0.15
a
Lf% mg m 1st 0.9 0.9 2.5 2.5 14 7.4 43 43 2nd 2.8 7.4
4
2 2
0
0 5
8
5
11 c
tp=0.28
8 1st
Lf% mg 1.7 4.6
2nd 5.3
14
m 26
8
11 d
tp=0.465
12
Lf% mg 1st 2.9 7.8
81
2nd 8.7 23
4
m 43
135
6
0
0 5
8
11
5
8
Time, min
11
Figure 11.11 Effect of the sample size on the individual band profiles for a 1:3 binary mixture at oc = 1.2. qSri = qs^ = 800. Same parameters as for Figure 11.9. (a) tp = 0.074 min. (b) tp = 0.15 min. (c) tp = 0.28 min. (d) tp = 0.465 min.
As demonstrated in Chapter 8, the retention time of the end of the second component tail is constant and equal to t^ 2 , the retention time of the second component under linear conditions. The retention time of the band front decreases with increasing sample size. Therefore, as the sample size increases, the band spreads, and its front moves forward, driving the displacement of the first component. Consider the production of the second component. As the sample size increases, the production rate for the second component increases. With increasing sample size, the displacement effect creates a first component tail which penetrates into the front shock layer of the second component. At this point, the yield begins to decrease, while the production rate continues to increase and a mixed band is generated. When the tail penetrates too deeply into the second component band, the production rate begins to drop. Figure 11.12 illustrates the individual band profiles at different sample sizes for a separation factor of 1.6, instead of 1.2 as in Figure 11.11, with all the other parameters being kept constant. The sample sizes in Figures 11.12a and 11.11b are the same, as well as in Figures 11.12b and 11.lid. Comparison of these figures shows that increasing the separation factor at constant sample size and column saturation capacity results in a significant increase of the band resolution and a marked decrease of the degree of band overlap under overloaded conditions. For a separation factor of 1.2, the displacement effect has already begun and a mixed band formed for tp = 0.15, while for a separation factor of 1.6, the profiles are not yet in the touching-bands condition. In Figure 11.12c the sample size is double that of Figures 11.lid and 11.12b. A mixed band has formed and the onset of
550
Two-Component Band Profiles with the Equilibrium-Dispersive Model
C mg/mL
a
tp=0.15
Lf%
4
1st 0.9 2nd 2.8
mg 2.5 7.4
m
b
t t pp==00. . 4466
99 •
14 46
. 1s 1st t . 22nd nd
Lf% L f%
mg m g m m
2.9 88.7 .7
7.8 42 233 1144 2 44
66 • 2 \
3 • 3 \
0 4
8
12
16
00 • 4
\
8
1 1 12
T ^
1
16
c
18
Figure 11.12 Effect of the sample size on the individual band profiles for a 1:3 mixture at a = 1.6. qS/i = qs^ = 800. Same parameters as for Figure 11.11 except: k2 = 9.6; 62 = 0.048. (a) tp = 0.15 min. (b) tp = 0.46 min. (c) tp = 0.93 min.
\
tp=0.93 Lf% 1st 5.8
mg m 15 86
2nd
47
17
290
9
Figure 11.14abc
0 4
8
12
16 Time, min
displacement has occurred. Often, the separation factor can be increased by changing the experimental conditions of the separation, such as by decreasing the organic modifier concentration, the buffer concentration, the type of buffer used, or the pH of the mobile phase. These methods must be used to optimize the separation factor. Provided the column saturation capacity does not decrease when cc increases, which may happen with macromolecules, this means maximizing a. Figure 11.13 illustrates the effect of the sample size at a column saturation capacity four times that of Figure 11.11, all other parameters being constant. This figure shows that increasing the column saturation capacity increases the resolution and decreases the band overlap under overloaded conditions. In the laboratory an increase in the column saturation capacity can be obtained, for example, by using a packing material of similar nature, but with a larger specific surface area, i.e., a smaller average pore size, assuming that the molecules of the components to be separated are small enough to enter freely in the smaller pores. The column saturation capacity also can change with the mobile phase composition, pH, or packing type, at least for proteins whose size or even three-dimensional structure may be altered by changes in these experimental conditions. Similar calculations have been carried out for a 3:1 mixture (Figures 11.14 to 11.16). The effect of an increase in the sample size is illustrated in Figure 11.14. The sample sizes are the same as in Figure 11.11. Figure 11.14a shows a mild degree of overload and a touching-bands chromatogram. A small tag-along effect is observed. Further increase in the sample size results in the formation of a
11.2 Applications of the Equilibrium-Dispersive Model C mg/mL C mg/mL
a
99 T 1st
Lf% 0.2
mg 2.5
m 3.4
2nd
0.7
7.4
11
16
tp=0.15
551
1st
Lf% 0.7
mg 7.8
m 11
2nd
2.2
23
34
b tp=0.46
6 8 3
0
0 5
8
11
5
8
11 c 1st 2nd
tp=0.93
20
Figure 11.13 Effect of the sample size for a 1:3 mixture at u. = 1.2, for qs j = c\s^ = 3200. Same parameters as in Figure 11.9, except: bx = 0.0075, b2 = 0.009. (a) tp = 0.15 min. (b) tp = 0.46 min (c) tp = 0.93 min.
Lf%
mg
m
1.5 4.4
15 47
21 67
10
0 5
8
11 Time, min
plateau on the top of the second band, illustrating the onset of the tag-along effect (Figure 11.14b). As the sample size increases, the concentration of the first component becomes so large that the equilibrium concentration of the second component in the stationary phase decreases markedly at constant mobile phase concentration (competition for adsorption, Chapter 4). Since a greater fraction of the second component remains in the mobile phase, it elutes earlier, with the tail of the first component, forming a wider mixed band (Figure 11.14c). At larger sample sizes, the phenomenon becomes exacerbated with further penetration of the second component into the diffuse rear boundary of the first component (Figure 11.14d). This phenomenon has interesting and important consequences for the production rate. For a 3:1 mixture, increasing the sample size results in increasing penetration of the later eluting impurity band into the main component profile. This, in turn, results in a decrease in the recovery yield of the first component when a mixed band forms. The production rate increases with increasing sample size to a much lesser degree when the impurity or minor component is eluted second than when it is eluted first. The influence of the tag-along effect on the production rate and recovery of the first and second components will be discussed further in Chapter 18. The possible importance of these effects for the separation of a trace component, hence for purification, is illustrated in Figure 11.15. The amount of impurity injected, 7.8 mg, hence the loading factor, is the same for the impurity in all the four chromatograms. When the impurity is eluted first, a very strong displacement effect is observed. This could be qualified as overdisplacement, by analogy
552
Two-Component Band Profiles with the Equilibrium-Dispersive Model C mg/mL
a
tp=0.074
4
Lf%
mg
1st
1.4
3.7
21
2nd
0.5
1.2
7.2
b
m tp=0.15
6
Lf%
mg
m
1st
2.8
7.4
41
2nd
0.9
2.4
14
4 2 2
0
0 5
8
11
5 c
12
Lf%
mg
m
1st
5.3
14
77
2nd
1.7
4.6
27
tp=0.28
6
8
11 d
16 tp=0.464 Lf%
mg
m
1st
8.7
23
130
2nd
2.9
7.8
45
8
0
0 5
8
11
5
8
11 Time, min
Figure 11.14 Effect of the sample size for a 3:1 mixture at a = 1.2. qS/i = qS/2 = 800. Same parameters as Figure 11.11. (a) tp = 0.074. (b) tp = 0.15. (c) tp = 0.28. (d) tp = 0.465.
with the term used in displacement chromatography. The front of the second component follows immediately the front of the first one, and skimming the content of the first zone is difficult and requires accurate timing of the valves. Furthermore, the first component tail contains nearly half the material, so the extraction yield will be poor, although there are types of applications for which this value might be sufficient. By contrast, when the impurity is eluted second, it spreads over a large retention time and purification is practically impossible. In practice, the sample size used in the figure is too large to be representative of practical cases in which a minor component must be eliminated. Figure 11.15 also shows the importance of the feed purity on the production rate. Increasing the feed purity from 16% in a 1:6 mixture to 11% in a 1:9 mixture allows an increase in the loading of the major component from 47 mg to 70 mg, i.e., by 50%. Similarly, for the 6:1 and 9:1 mixtures the same increase in loading is observed at constant cycle time of 12 min. By increasing the purity of the feed one could also move the cutpoints to increase the yield at constant final product purity. Figure 11.16 illustrates the effect of sample size for a 3:1 binary mixture, under the same conditions as Figure 11.13, except for a larger separation factor. The sample sizes used for Figures 11.16a and 11.14b are the same. Similarly, they are the same for Figures 11.16b and 11.14d. Increasing the separation factor at constant column saturation capacity and sample size results in greatly improved resolution and a lesser degree of band overlap at high sample sizes. As a consequence, the production rate increases rapidly with increasing value of a — 1. Since the amount of feed injected during each cycle at large values of cc is larger, the effects of the nonlinear thermodynamics of equilibrium are stronger, the shock layers are steeper, and the size of the intermediate, mixed zone is smaller. Figure 11.17 shows the chromatograms obtained with a column saturation ca-
11.2 Applications of the Equilibrium-Dispersive Model m g / mL C mg/mL
1 :6 1:6
20 • 20
1st 2nd
15 • 15
I
10 10 •
a a
Lf% Lf*
mg rag
m ra
2.9
7.8
43
18
47
553
6:1 6:1
27 • 27
bb
l 1st 2nd
270
\
18 • 18
\
l
—
l
LfS Lf%
rag mg
m
18 2.9 " '
47 7.8 "
257 45 45
\
\ 9 9
•
55 •
0
-*s
0 3
9
6 6
12 12 c
1 9 1:9
30 • 30
3
Lf%
mg
m
1 « 2 2.9 . S 1st 2nd 26
,.B 7.8 70
«, 43 405
99
12 12
d
9:1 9:1
40 • 40
30 30 •
^*^66
\
Lf% Lf% 1st 26
rag mg 70
2nd 2nd
7.8 7.8
2.9 2.9
m 85 385
45 45
20 • 20
\
20 • 20
\
\
10 10 • 10 • 10
*•
0
-i—r-i-
1 12
0
6
3
9
12 12
3 3
66
99
Time, Time, min
Figure 11.15 Illustration of the effect of the feed purity on the separation at a. = 1.2. qSi\ = qS/2 = 800. Same parameters as in Figure 11.12, except feed composition. These parameters are: k't = 6; k'2 = 7.2; \>\ = 0.03; b2 = 0.036; L = 10 cm; e = 0.8; N = 2000; tp = 0.464 min. Amount of impurity, 7.8 mg.
C mg/mL
CH11_17
a tp=0.15
6
Lf% L f%
mg mg
m
t 11 ss t
2. . 88 2
. 44 7.
4411
2n n dd 2
0. . 99 0
2. . 55 2
1155
Figure 11.15 a,b,c,d
15
b
tp=0.46 Lf%
mg
m
.1st 1st
8.7 8 .7
23 2 3
130 1 30
- 22nd nd
22.9 .9
77.8 .8
448 8
10
3 5
0
0 4
8
12
4
16
28
Figure 11.16 Effect of the sample size on the band profiles for a 3:1 mixture at a = 1.6. qSi\ = CJS/2 = 800. Same parameters as Figure 11.12, except mixture composition. These parameters are : fc2 = 9.6, i>2 = 0.048. (a) tp = 0.15 min. (b) tp = 0.46 min. (c) tp = 0.93 min.
8
12
16
c
tp=0.93
st 1 1st
Lf% L f% 7 117
mg mg 7 4 47
m 2 60 260
2nd 2 nd
55.8 .8
15 1 5
96 9 6
14
0 4
8
112 2
116 6 T ime, m in Time, min
554
Two-Component Band Profiles with the Equilibrium-Dispersive Model
C m mg/mL g/mL C 1122 T
a
tp=0.15 1st 2nd
Lf% 0.7 0.2
mg 7.4 2.5
b
24 m 10 3.6
6
Lf%
mg
m
1st
2.2
23
32
2nd
0.7
7.8
11
tp=0.46
12
0
0 5
8
11
5
8
11
c
36 tp=0.93
Figure 11.17 Effect of the sample size on the band profiles at a = 1.2, for qSr\ = qS/2 = 3200. Same parameters as for Figure 11.13, except: the mixture composition, bt = 0.0075, b2 = 0.009. (a) tp = 0.15 min. (b) tp = 0.46 min. (c) tp = 0.93 min.
1st 2nd
Lf% 4.4 1.5
mg 47 15
m 64 22
18
0 5
11
8 Time, min
pacity four times larger than in Figure 11.14, at constant separation factor. The sample size in Figure 11.17a is the same as in Figure 11.14b, and it is also the same in Figures 11.17b and 11.14d. Greater resolution of the bands under overloaded conditions can be obtained by increasing the column saturation capacity at constant separation factor.
11.2.4 Calculation of Multicomponent Chromatograms The procedure described and used in the previous section for the calculation of the individual band profiles of the components of binary mixtures can easily be extended to multicomponent mixtures. Jacobson et al. [18] have calculated the profiles obtained for ternary mixtures of various compositions (Figure 11.18). In most cases, the results could be predicted from the combination of the two binary interactions, due to the displacement and the tag-along effects, between the first and the second band and between the second and third bands. This is the case in Figure 11.18, top left, where the large first band of a 9:1:1 ternary mixture spreads the second and third bands; in Figure 11.18, top right, where the large second band of a 1:9:1 mixture displaces the first band while spreading the third band; and in Figure 11.18, bottom left, where the large third band of a 1:1:9 mixture displaces strongly the first and second bands. However, when the concentration of the second component of a ternary mixture is much lower than the concentrations of the other two components, its band is squeezed between the large two bands and its profile is most unusual [18]. Figure 11.18, bottom right, shows that when the column is sufficiently overloaded for the first and third bands to interfere, the intermediate band has two very different
11.2 Applications of the Equilibrium-Dispersive Model
555
Figure 11.18 Individual band profiles of the components of a ternary mixture. &\^ — 1x2,5 = 1.10 (main figure) or 1.40 (inset), k'3 = 4.0. (Top, left) 9:1:1 mixture, total Lf = 10%. (Top, right) 1:9:1 mixture, Lf = 10%. (Bottom, left) 1:1:9 mixture, Lf = 10%. (Bottom, right) 9:1:9 mixture, Lf = 20%. Reprinted with permission from S.C. Jacobson et al, J. Chromatogr., 484 (1989) 103 (Figs. 8, 9,10a, 16).
parts. Its front part looks like a half-Gaussian profile. It ends with a rear shock layer and is followed by a long tail, making the total collection of such an impurity practically impossible.
556
Two-Component Band Profiles with the Equilibrium-Dispersive Model
11.2.5 Comparison of Calculated Band Profiles and Experimental Results Experimental band profiles have been measured and compared to profiles calculated with the equilibrium-dispersive model using various possible calculation procedures. Most often, the degree of agreement between calculated and experimental profiles depends on the accuracy with which the competitive isotherm data have been measured and modeled. 11.2.5.1 Band Profiles Calculated with the Forward-Backward Scheme In Chapter 10, we have shown excellent agreement between the recorded and the calculated band profiles for a single component. The multicomponent problems are more complicated because an accurate model of the competitive isotherm data is needed for the calculations and because the measurement of these data and their modeling are difficult, hi practice, we would like to predict the competitive equilibrium data by using models that require only the knowledge of the single (i.e., noncompetitive) adsorption data. The quest for such models has been only moderately successful (see Chapter 4). Because of these theoretical and practical difficulties, we cannot expect the same degree of agreement between calculated and measured band profiles for mixtures that we have observed for single-component profiles. The importance of a good isotherm model is illustrated in Figure 11.19. Figure 11.19 top left shows the experimental chromatogram obtained for a racemic mixture of N-benzoylphenylalanine on a column of BSA immobilized on silica, overlaid with the chromatogram calculated with the best Langmuir isotherm (Figure 3.12). Figure 11.19 top right shows the same experimental data and the chromatogram calculated using the best set of five-parameter bi-Langmuir isotherms (see Figure 3.12). There is a strong disagreement between the profiles of the second component in Figure 11.19 top left but excellent agreement for the profiles of both components in Figure 11.19 top right. Similarly, Figures 11.19 bottom left and 11.19 bottom right compare the experimental band profiles for a 1:4 mixture and the profiles calculated with the best Langmuir and bi-Langmuir isotherm, with the same conclusions. When the column is overloaded with a multicomponent mixture to such a degree that the bands of two components overlap, experimental determination of the individual band profiles from the detector response is not possible in general. There are two reasons. First, the detectors used in chromatography being highly sensitive, the concentrations that are typically used in preparative chromatography are often outside their linear range of operation and calibration is necessary. The only practical way around this serious difficulty is in the use of UV detectors having extremely thin cells, providing very short optical path length. Second, under nonlinear response conditions, the detector response for a binary solution is not the sum of the (nonlinear) responses that would be observed for each of the two components pure at the same concentration. The detector signal cannot be related simply to the composition of a binary eluent. Thus, it is impossible simply to deconvolute the response and to derive the composition of the eluent from the absorbances measured at several wavelengths and from the calibration
11.2 Applications of the Equilibrium-Dispersive Model
3
6
9
12
time (minutes)
a
s
557
a
12
is
lime (minutes)
Figure 11.19 Comparison of calculated (solid and dashed lines) and experimental (symbols) individual band profiles. Mixtures of N-benzoyl-L-(l) and D-(2)phenylalanine on a column of immobilized BSA on silica (see Figure 11.20). (Top, left) Racemic mixture, 0.966 fimole of each enantiomer. Band profiles calculated with the best Langmuir isotherm. (Top, right) Same as top left, except profiles calculated with the best five-parameter bi-Langmuir isotherm. Inset: total chromatograms comparing the concentration profile derived from the detector signal and the detector calibration curve (line) and from the analysis of collected fractions (symbols). (Bottom, left) Same as top left, except 1:4 mixture, with 0.53 fimole of D-isomer and 1.99 ^mole of L-isomer. (Bottom, right) Same as top right, except 1:4 mixture, with 0.53 ftmole of D-isomer and 1.99 ^mole of L-isomer. Reprinted from S.C. Jacobson, S. Golshan-Shirazi and G. Guiochon, AIChE }. 37 (1991) 836 (Figs. 3 and 4), by permission of the American Institute of Chemical Engineers, ©1991, AIChE. All rights reserved.
curves of the pure components. The detector response is a nonlinear function of both concentrations, and complete calibration would be time-consuming and tedious work of limited accuracy. The best solutions to this problem seem to be (i) the use of a detector with an adjustable cell length; and (ii) the use of a scavenger stream of mobile phase diluting the column effluent and carrying it through the detector cell. Furthermore, even when the detector response is linearly related to the concentrations, it is rare that two compounds whose interference on
Two-Component Band Profiles with the Equilibrium-Dispersive Model 0
558
0.10
(mM)
0.15
1
u
Ik
2
4
B
0.00
0.05
t
0
7
6
10
12
t (minutes)
Figure 11.20 Comparison of calculated (solid line) and experimental (symbols) individual band profiles. Mixtures of N-benzoyl-L-(l) and D-(2)-alanine on a Resolvosil-BSA-7 column. Mobile phase: 10 mM buffer aqueous solution (pH 6.7) with 3% 1-propanol (v/v) at Fv = 1 mL/min. L = 15 cm, d = 4 mm; N = 700 plates. Binary competitive Langmuir isotherm: selective site: a u = 14.16, ahD = 35.09, qsXh = 0.0019 mol/L, <7S;1/D = 0.0020 mol/L; nonselective site: a2 = 4-41, qSi2 = 0.01995 mol/L. (Left) Racemic mixture, sample sizes: 0.26 jimol for each isomer. (Center) 1:3 mixture, sample size: 0.105 ^mol of L- and 0.392 jimole of D-isomer. (Right) 3:1 mixture, sample size: 0.491 ^mole of L- and 0.165 jimal of D- isomer. Reprinted with permission from S.C. Jacobson, S. Golshan-Shimzi and G. Guiochon,}. Amer. Ckem. Soc, 112 (1990) 6492 (Figs. 5 and 6). ©1990, American Chemical Society.
a chromatographic column is not accidental have UV spectra that differ enough to permit the accurate detection of one in the presence of the other. Sophisticated algorithms have been suggested, but no commercial instruments are presently available. Only enantiomeric mixtures are exceptional in this respect. All physical properties of enantiomers that are not related to chirality are identical, especially their UV spectra. In this case, the total elution profiles can be derived from the detector signal, using a simple calibration curve determined with either enantiomers or any of their mixtures [19]. For all these reasons, fractions must be collected at sufficiently close intervals and analyzed in order to find the composition of the eluent as a function of time and to determine the individual elution profiles of the components [16,19,20]. When proper care is applied to the choice, the connection design, and the operation of the fraction collector, negligible extracolumn band broadening can be achieved, as demonstrated in Figure 11.19 (inset). The concentration profiles derived from the analysis of fractions collected with a Gilson (Middleton, WI) at a high enough frequency overlie exactly, except at the very top of sharp peaks, the concentration profile derived from the detector signal, and the detector calibration curve. The simplest separation problems are those involving racemic mixtures. This simplicity arises from the fact that in practice the mixture has only two components (if there are several chiral centers in the molecules, the diastereoisomers are often easier to separate than the enantiomers) and that all the parameters characterizing the behavior of the two enantiomers are identical, except those related to
11.2 Applications of the Equilibrium-Dispersive Model
559
the chiral selective mechanism. Furthermore, this separation problem, which is theoretically simple, is also highly relevant to the pharmaceutical industry. An example is the separation of mixtures of N-benzoyl-D- and L-alanine on immobilized BSA (see Figures 11.20). We have explained in Chapters 3 and 4 (i) that a competitive bi-Langmuir isotherm can be employed to account for the competitive behavior of these components (Figure 4.25c); and (ii) that, because the chiral selective retention mechanism involves adsorption of the enantiomers in the hydrophobic cavity of BSA, the column saturation capacity of the chiral selective mechanism is the same for the two enantiomers. This competitive bi-Langmuir isotherm model is simply derived from the parameters obtained from single-component isotherm measurements. There is very good agreement between the calculated and the experimental profiles in all three Figures 11.20. The consequences of the displacement and tagalong effects are clearly seen in the chromatograms. As predicted by theory, the intensity of the displacement of the first component by the second increases with increasing relative concentration of the second component (compare Figures 11.20). Figure 11.20 (right) illustrates the separation of the racemic mixture. Figure 11.20 (center) shows the individual band profiles of the two enantiomers for a 1:3 mixture. The competitive interaction of the two components results in the forward displacement, the concentration and narrowing, of the first component band by the second. Since the second component is more strongly adsorbed to the stationary phase, its presence in the mobile phase behind the first band forces earlier desorption of the molecules of the less strongly adsorbed first component. It causes it to elute as an earlier, more concentrated band. The maximum concentration of the first band is 0.17 mM in the 1:3 mixture containing 0.105 ^mol, while it is 0.08 mM under noncompetitive conditions [4] (the feed concentration is 10 mM). Because the retention mechanism is well accounted for by the bi-Langmuir model, the comparison of calculated and experimental results shows excellent agreement for the separation of all pairs of enantiomers studied on immobilized BSA [4,19]. Whereas the displacement effect is dominant in Figure 11.20 (center), a strong tag-along effect is exhibited in Figure 11.20 (right). The front of the later eluting band is dragged forward into the first component band. At the time when the tail of the first component band ends, a concentration plateau appears in the second component profile (see Figures 8.5 and 8.6). The intensity of the tag-along effect increases with increasing ratio of the concentrations of the first and second components (compare Figures 11.20 left and right). However, a slight tag-along effect can be seen already for a 1:3 mixture (Figure 11.20 center) where a concentration plateau begins at 6 minutes. Physically, this effect is explained by the reduction in the fraction of molecules of the second component adsorbed at equilibrium in the presence of a large excess of the first component. Thus, because of the competition, only a reduced fraction of its molecules are adsorbed, and the band elutes faster than if it were alone. This results in poor separation of large samples when the concentration of the second component is much lower than that of the first component. Figures 11.21a to 11.22c compare the experimental and calculated elutionband profiles for 3:1,1:1, and 1:3 binary mixtures of 2-phenylethanol and 3-phenylpropa-
560
Two-Component Band Profiles with the Equilibrium-Dispersive Model
Time (min)
Time (Min)
Figure 11.21 Comparison of experimental (symbols) and calculated (solid lines) individual elution profiles. 2-phenylethanol and 3-phenylpropanol. Calculations made with the forward-backward scheme, the coefficients of the competitive isotherm Langmuir model derived from the single-component isotherms, and a rectangular injection profile. Column: 25 cm long, packed with 10 mm Vydac ODS silica. Mobile phase: methanol-water, (50:50), 1 mL/min. (a) Sample size: 7.6 mg of 2-phenylethanol and 21.1 mg of 3-phenyl-lpropanol. (b) Sample size: 30.1 mg of 2-phenylethanol and 10.2 mg of 3-phenyl-l-propanol. Inset: Band profiles calculated for a sample twice as large, using a competitive Langmuir isotherm model. Reproduced with permission from A.M. Katti and G. Guiochon, J. Chromatogr., 499 (1990) 21 (Figs. 4, 6 and 7).
nol [21,22]. A comparison of the inset in Figure 11.21b, showing profiles calculated for a sample twice as large and with a competitive isotherm, and the experimental data in the main figure illustrates how competitive interactions between the mixture components affect the elution profiles. While the displacement effect is dominant in Figures 11.21b and 11.22b, a strong tag-along effect is exhibited in Figures 11.21a and 11.22c. The front of the later eluting band is dragged forward into the first component band. In all these figures, we used the competitive Langmuir isotherm model to calculate the band profiles. However, the coefficients of the isotherms used for Figures 11.21 are the coefficients of the single-component isotherms determined by frontal analysis, while the coefficients of the isotherms used to calculate the profiles in Figure 11.22 are measured by the simple wave method (Chapter 4, Section 4.2.4). These latter coefficients are certainly empirical coefficients, and their use would not permit an accurate prediction of single-component bands. However, they permit the calculation of band profiles that are in much better agreement
11.2 Applications of the Equilibrium-Dispersive Model
561
Figure 11.22 Comparison of experimental (symbols) and calculated individual elution profiles. 2-Phenylethanol and 3-phenylpropanol. Calculations made with the forwardbackward scheme, the coefficients of the competitive isotherm Langmuir model derived from the hodograph method and the recorded injection profile (Solid line) or a rectangular injection profile (Dashed line); Same sample sizes as in Figure 11.21. (a) 1:1 mixture. (b) 1:3 mixture, (c) 3:1 mixture. Reproduced with permission from A.M. Katti, M. Czok and G. Guiochon, ]. Chromatogr., 556 (1991) 205 (Fig. 6).
with experimental results than the coefficients of the single-component isotherms. A similar comparison, between coefficients determined by different methods, is made in Figure 11.8, in the case of a wide rectangular injection pulse, and in connection with the hodograph plot of experimental data, and the use of these plots for isotherm determinations (Chapter 4, Section 4.2.4). The differences between the calculated profiles in Figures 11.21 and 11.22 illustrate the influence of minor changes in the competitive isotherms. A comparison between these figures illustrates the need to determine the isotherm coefficients accurately or to accept appropriate compromises, as well as the need for better isotherm models. Also compared in Figures 11.22a-c are the band profiles calculated with two injection profiles, an ideal rectangular profile and the experimental profile recorded by connecting the injection valve directly to the detector [22]. The differences observed are minor, but not insignificant. The band profiles predicted with the correct experimental injection profile are wider and shorter, and the degree of separation achieved is reduced, in agreement with the experimental profiles. In Figure 11.23, we provide a qualitative comparison of the separation of cisand traws-androsterone [24]. As in the previous example (2-phenylethanol and 3-phenyl-l-propanol), the pure component adsorption data of both isomers were fitted to a Langmuir isotherm [25]. Excellent agreement was obtained between experimental and calculated single-component elution profiles. However, the column saturation capacities differed by 30%, which is explained by the very different geometric structures of these two isomers (Figure 4.9). When the column saturation capacities are different, the competitive Langmuir isotherm is no longer thermodynamically consistent (Chapter 4, Section 4.1.2). A more complex competitive isotherm model should be used.
562
Two-Component Band Profiles with the Equilibrium-Dispersive Model
Figure 11.23 Qualitative prediction of the elution order reversal of cis- and iransandrosterone. (Left) Analytical and preparative sample sizes. High concentration band profiles calculated with the LeVan-Vermeulen isotherm model. Inset: experimental data [23]. Stationary phase: silica modified with a pH 6.8 phosphate buffer. Mobile phase: (9:1) acetonitrile-dichloromethane, 0.98 mL/min. Samples, cis-androsterone: 0.026 and 5.2 mg, frans-androsterone: 0.15 and 1.8 mg. (Right) Band profiles calculated with the competitive Langmuir model. Reproduced with permission from S. Golshan-Shimzi, J.-X. Huang and G. Guiochon, Anal. Chew.., 63, (1991) 1147 (Fig. 8). ©1991, American Chemical Society.
The experimental adsorption data fit well with the prediction of the LeVanVermeulen model [24], as shown by Figures 4.10 and 4.11. Figure 11.23 (inset) illustrates the experimental elution profiles of a mixture of cis- and trans-androsterone at two different samples sizes. With the small, analytical size sample, the trans isomer elutes first, before the cis isomer. With the large sample, the peak maximum of the cis isomer elutes first and the trans second. Band profiles calculated using the proper competitive isotherm model under the same conditions as the experiment [26] are shown in Figure 11.23. At the small sample size the cis isomer elutes at 550s and the trans isomer at 510s. At the large sample size the peak maximum of the cis isomer elutes at 400s and the trans isomer at 460s. This complex phenomenon of elution order reversal at high concentrations cannot be accounted for by simple models like the competitive Langmuir and bi-Langmuir models, as shown in Figure 11.23. Finally, Figure 11.24 shows a qualitative comparison of the results of two experiments carried out with computer calculations and actually carried out with a chromatographic system, using on-line fraction collection and analysis. In the former experiment (left part of the figure), the composition of a mixture of two epimers is changed, and the individual elution band profiles are compared. The figure illustrates a strong displacement effect (top), which fades to a strong tagalong effect (bottom) as the relative concentration of the earlier eluted epimer increases. In the second experiment, large samples (100 and 300 mg) of a 1:3 mixture of the same epimers were injected on three columns of different efficiencies. Steeper shock layers are observed at the fronts of both bands when the column efficiency increases, in proportion to the reverse of the average particle size. Qual-
11.2 Applications of the Equilibrium-Dispersive Model a. Experimental Data
I \
563
b. Computer Simulations
10:90
a. Experimental Data
50:50
75:25
12 jim
8 |im
b. Computer Simulations
N - 2790
N = 5590
Figure 11.24 Qualitative demonstrations of the displacement and tag-along effects. Left set: Influence of the feed composition. Left column: Chromatograms obtained with 200 mg of a mixture of two of the epimers of a 1,1/1-trisubstituted cyclohexanone on a 250x21.4 mm column packed with 12 fim silica, with 40 mL/min of a solution of n-hexane and ethyl acetate (97.5:2.5). Composition as indicated. Right column: results of computer calculations. Right set: Influence of the column efficiency. Top two rows, experimental data under the same experimental conditions as in (a), except average particle size of silica particles, and mixture composition 1:3. Bottom row, results of computer calculations. Reproduced with permission from ]. Newburger and G. Guiochon,}. Chromatogr., 484 (1989) 153 (Figs. 6 and 8).
itative agreement between calculations and experiments is observed in the two cases. All these results demonstrate that the calculation of band profiles that are in very good agreement with experimental results is possible provided an accurate model of the competitive equilibrium isotherms is available. The main practical difficulty in the modeling of separations is in the accurate measurement and modeling of these competitive isotherms. Once such a model has been validated, it is possible to calculate the performance of a chromatographic unit and to optimize its performance [27]. 11.2.5.2 Band Profiles Calculated with the Backward-Forward Scheme We have compared in Figures 11.2 and 11.5 band profiles calculated with different numerical schemes. We compare in Figures 11.25 and 11.26 band pro-
Two-Component Band Profiles with the Equilibrium-Dispersive Model
scheme. Reproduced from S.C. Jacobson, S. Golshan-Shimzi and G. Guiochon, AIChE ]. 37 (1991) 836 (Fig. 5), by permission of the American Institute of Chemical Engineers, ©1991, AIChE. All rights reserved.
20.0
I
n ID
I •
1 a.u
o 1 i
4.U
Figure 11.25 Comparison of experimental (symbols) and calculated individual elution profiles. 4:1 mixture of 2V-benzoyl-D,Lphenylalanine on immobilized BSA. Sample size, 2.0 pmol of D-isomer and 0.56 umole of L-isomer. Same conditions as in Figure 11.19. Main figure, calculations made with the forward-backward scheme, and the coefficients of the five-parameter competitive bi-Langmuir isotherm. Inset, calculations made with the backward-forward
\
2
0.0
564
8
11
14
time (minutes)
25
Figure 11.26 Comparison of experimental (symbols) and calculated individual elution profiles. 2-Phenylethanol and 3-phenylpropanol. Calculations made with the forwardbackward (solid lines) and the backward-forward (dotted lines) schemes, and the coefficients of the competitive isotherm Langmuir model derived from the hodograph method. Same sample sizes as in Figure 11.21. (a) 1:1 mixture, (b) 1:3 mixture, (c) 3:1 mixture. Reproduced with permission from A.M. Katti, M. Czok and G. Guiochon, f. Chromatogr., 556 (1991) 205 (Fig. 7).
files calculated with the forward-backward and the backward-forward calculation schemes in two different cases. In Figure 11.25 the experimental individual elution band profiles of N-benzoyl-D- and L-phenylalanine in a 4:1 mixture are compared with the band calculated with the best competitive isotherm, using the forward-backward (main figure) and the backward-forward (inset) finite differ-
11.2 Applications of the Equilibrium-Dispersive Model
565
ence schemes [19]. The band profiles obtained for the first component are nearly identical. For the second component, the profile obtained in the latter case is in excellent agreement with the experimental data, while the profile calculated with the forward-backward scheme does not account well with experimental data. It underestimates the quality of the separation achieved. Figure 11.26 compares the individual elution profiles calculated using the same two finite difference methods, the forward-backward and the backward-forward schemes, for 2-phenylethanol and 3-phenyl-l-propanol [22]. The bands obtained with the backward-forward scheme are taller and narrower and their front sharper than those derived from the forward-backward scheme, in agreement with the result derived in Chapter 10 regarding the amount of dispersion introduced by the different numerical methods. The agreement between calculated and experimental results is slightly better with the backward-forward scheme than with the other one. Note, however, that the forward-backward calculations are carried out with a = 2, while the backward-forward calculations are done with a = 0.5. Significant or even large differences between band profiles calculated with the two finite difference methods arise for all mixture compositions at low column efficiencies. Such differences also arise at high efficiencies, when the relative concentration of the second component is low, i.e., when the tag-along effect is important [6-8]. In this last case, the only significant differences between the profiles calculated with the different methods are in the steepness of the shock layers in the mixed zone and in the retention time of the second component front [6-9,28]. The numerical problems have been discussed above, in Sections 11.1.3 and 11.1.4, and examples shown in Figures 11.5 and 11.6. 11.2.5.3 Band Profiles Calculated with Orthogonal Collocation on Finite Elements Zhou et ah measured the single-component and the competitive isotherms of the 1-indanol enantiomers by frontal analysis, using a microbore (150 x 1 mm) column [29]. Both the single-component and the binary isotherm data sets fitted well to the bi-Langmuir isotherm model. Then, the authors used these competitive isotherms to calculate the overloaded elution band profiles of mixtures of different compositions of the two enantiomers with the equilibrium-dispersive model. The computer program applied for these calculations was based on the method of orthogonal collocation on finite elements. Figures 11.27 compare the experimental (symbols) and the calculated band profiles of the racemic mixture (Figure 11.27, top left) and of two mixtures of relative compositions 3:1 (Figure 11.27, top right) and 1:3 (Figure 11.27, bottom). In the calculations of the band profiles, a number of theoretical plate 700 was used for both components. A good agreements is observed between the experimental and the calculated band profiles in all three cases. Figure 11.28 shows the band profiles obtained with a wide injection of a large sample of a racemic solution of Troger's base on microcrystalline cellulose triacetate, using ethanol as the mobile phase. The experimental profile (symbols) is overlaid on two calculated chromatograms. These profiles were calculated with
566
Two-Component Band Profiles with the Equilibrium-Dispersive Model •
(a)
Exp -Cal Cal
Exp Cal 6
5-
5 4
4
C [g/L]
C [g/L]
33 -
O 22-
3
2
IN
y\
11 -
1
0-
-2
(d)
0
0
2
4
6
8
12 14 10) 12
16 16
18 20 20 18
22 22
-2
24 26 24
tt[min] [min]
0
2
4
6
8
10 18 10 12 12 14 14 16 16 18
20 20
22 22
24 24
•
C [ g /L ]
Figure 11.27 Comparison of the experimental (symbols) and calculated (solid line) band profiles of various mixtures of the enantiomers of 1-indanol, racemic mixture (Top, left); 3:1 (Top, right) and „ 1:3 (Bottom) mixtures. The loading fac- a tors are: 2.5, 2.6, and 2.8%, respectively. Reprinted from D. Zhou, K. Kaczmarski, A. Cavazzini, X. Liu, G. Guiochon, J. Chromatogr. A, 1020 (2003) 199 (Fig. 5d, 5h, and
3.5
26
t [min] [min]
(h)
(c) c)
Exp Cal
3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -2
0
2
4
6
8
10 12 12 14 14 16 16 18 18 20 20 22 22 24 24 26 10 26 28 30 t [min] [min]
(i)
Figure 11.28 Experimental (symbols) and calculated chromatograms. 6 ml sample of a racemic solution of Troger's base, with C(+) = C(-) = 1.5 g/L. The solid and dotted lines were calculated using the competitive isotherms as predicted by the IAS theory. The dashed line was calculated using the single-component isotherms (i.e., neglecting competition). Reprinted from A. Seidel-Morgenstern and G. Guiochon, Chem. Eng. Sci., 48 (1993) 2787 (Fig. 11).
s i i \ t
1 SO
30
t (min)
the OCFE method, using the equilibrium isotherms of the pure enantiomers, determined by frontal analysis and shown in Figures 3.25. The isotherm of the (+) and more retained isomer has an inflection point and is well accounted for by a quadratic isotherm while the isotherm of the (—) isomer follows accurately the
REFERENCES
567
Figure 11.29 Comparison between calculated and experimental band profiles of the racemic mixture of Troger's base on a Chiralpak AD column. Sample sizes 3.6, 7.25, 10.85 and 14.5 mg. Solid line - experimental data, dotted line - simulation with threelayer isotherm model, and dashed line simulation with cooperative and S-shaped isotherm model Reprinted from K. Mihlbachler, K. Kaczmarski, A. Seidel-Morgenstern. G. Guiochon, ]. Chromatogr. A, 955 (2002) 35 (Fig. 11). 10
12
14
16
18
Time t [min]
Langmuir model [27]. No simple competitive model can account for such a behavior. The ideal adsorbed solution model (Chapter 4, Section 4.1.5) permits the combination of these two isotherms, and the calculation of band profiles. The dashed line in Figure 11.28 was obtained assuming that there is no competition between the two isomers, but that they "see" different adsorption sites, following the model suggested by Roussel et al. [30]. This profile accounts poorly for the experimental results because these results exhibit an obvious displacement effect of the first peak by the second, effect which is not accounted for by this model. The solid line is the total chromatogram calculated from the competitive isotherms derived from the single-component isotherms, using the IAS theory. The dotted lines are the calculated individual band profiles. These profiles are in reasonable agreement with experimental results, in spite of the very poor column efficiency which imposed the use of an OCFE numerical method for the calculation of the band profiles [27]. We note that the band of the first component in Figure 11.28 is taller and its rear profile is steeper than predicted [27]. These discrepancies between the experimental and calculated profiles suggest that the competition between the two isomers is stronger than predicted by the IAS theory [27]. Recent results obtained in a similar separation (the enantiomers of Troger's base on amylose tri-dimethylbenzocarbamate) have shown a complex, noncompetitive binary isotherm (see Chapter 4, Figure 4.28) [31,32]. The isotherm model combined with the equilibrium-dispersive model gave band profiles that were in excellent agreement with the experimental profiles recorded in overloaded elution and frontal analysis [31], and in simulated moving bed separations [32]. These results are illustrated in Figure 11.29.
References [1] P. Danckwerts, Chem. Eng. Sci. 2 (1953) 1. [2] S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 60 (1988) 2364. [3] M. Z. El Fallah, G. Guiochon, Anal. Chem. 63 (1991) 859.
568
REFERENCES
[4] [5] [6] [7] [8] [9] [10] [11] [12]
S. Jacobson, S. Golshan-Shirazi, G. Guiochon, J. Amer. Chem. Soc. 112 (1990) 6492. Z. Ma, A. M. Katti, B. Lin, G. Guiochon, J. Phys. Chem. 94 (1990) 6911. M. Czok, G. Guiochon, Anal. Chem. 62 (1990) 189. M. Czok, G. Guiochon, Comput. Chem. Eng. 14 (1990) 1435. M. Czok, G. Guiochon, J. Chromatogr. 506 (1990) 303. Z. Ma, G. Guiochon, Comput. Chem. Eng. 15 (1991) 415. A. Felinger, G. Guiochon, Biochem. Progr. 41 (1993) 134. A. Felinger, G. Guiochon, J. Chromatogr. 658 (1994) 511. R. Aris, N. R. Amundson, Mathematical Methods in Chemical Engineering, Vol. 2, Prentice Hall, Englewood Cliffs, NJ, 1973. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, Ronald Press, New York, NY, 1953. D. O. Cooney, A. Strusi, Ind. Eng. Chem. (Fundam.) 11 (1972) 123. B. Lin, S. Golshan-Shirazi, Z. Ma, G. Guiochon, J. Chromatogr. 500 (1990) 185. A. Katti, Z. Ma, G. Guiochon, AIChE J. 36 (1990) 1722. J. Zhu, A. M. Katti, G. Guiochon, J. Chromatogr. 552 (1991) 71. S. C. Jacobson, S. Golshan-Shirazi, A. M. Katti, M. Czok, Z. Ma, G. Guiochon, J. Chromatogr. 484 (1989) 103. S. Jacobson, S. Golshan-Shirazi, G. Guiochon, AIChE J. 37 (1991) 836. A. M. Katti, G. Guiochon, Amer. Lab. 21 (10) (1989) 17. A. M. Katti, G. Guiochon, J. Chromatogr. 499 (1990) 21. A. Katti, M. Czok, G. Guiochon, J. Chromatogr. 556 (1991) 205. M. J. Gonzalez, A. Jaulmes, P. Valentin, C. Vidal-Madjar, J. Chromatogr. 386 (1986) 333. S. Golshan-Shirazi, G. Guiochon, J. Phys. Chem. 95 (1991) 6390. J.-X. Huang, G. Guiochon, J. Coll. Interf. Sci. 128 (1989) 577. S. Golshan-Shirazi, J.-X. Huang, G. Guiochon, Anal. Chem. 63 (1991) 1147. A. Seidel-Morgenstern, S. C. Jacobson, G. Guiochon, Biotechnol. Progr. 8 (1993) 533. A. Felinger, G. Guiochon, J. Chromatogr. 591 (1992) 31. D. Zhou, K. Kaczmarski, A. Cavazzini, X. Liu, G. Guiochon, J. Chromatogr. A 1020 (2003) 199. C. Roussel, J. L. Stein, F. Beauvais, A. Chemlal, J. Chromatogr. 462 (1989) 95. K. Mihlbachler, K. Kaczmarski, A. Seidel-Morgenstern, G. Guiochon, J. Chromatogr. A 955 (2002) 35. K. Mihlbachler, A. Seidel-Morgenstern, G. Guiochon, AIChE J. 50 (2004) 611.
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
Chapter 12 Frontal Analysis, Displacement and the Equilibrium-Dispersive Model Contents 12.1 Displacement Chromatography with a Nonideal Column 12.1.1 12.1.2 12.1.3 12.1.4 12.1.5 12.1.6 12.1.7
Influence of the HETP Influence of the Sample Size and the Displacer Concentration Influence of the Column Length Influence of the Separation Factor Case of Trace Components Influence of the Impurities in the Displacer Solution Case of Selectivity Reversal
12.2 Applications of Displacement Chromatography 12.2.1 12.2.2 12.2.3 12.2.4
Separation of Rare Earths and Other Cations Separation of Organic Compounds Separation of Peptides and Proteins Separation of Nucleic Acid Constituents
12.3 Comparison of Calculated and Experimental Results References
570 571 572 575 578 580 583 585
587 590 590 593 598
599 603
Introduction In this chapter, we discuss the application of the equilibrium-dispersive model to the calculation of zone profiles in displacement chromatography. An introduction to displacement chromatography is provided in Chapter 9. From a mathematical point of view, the difference between the calculation of numerical solutions of the equilibrium-dispersive model in elution, gradient elution, frontal analysis, and displacement chromatography consists only in a change in the boundary conditions of the numerical problem. Instead of the pulse injection carried out in elution chromatography, the injection is a feed concentration step in frontal analysis and a feed pulse followed by a concentration step of the displacer in displacement chromatography. Accordingly, the numerical analysis done in Chapters 10 and 11 remains valid and numerical calculations are as easy to carry out in displacement chromatography as in elution. From a theoretical viewpoint, frontal analysis and displacement chromatography are important and interesting problems because there are asymptotic solutions for the breakthrough curves of frontal analysis and for the band profiles in the isotachic train in displacement chromatography. An asymptotic solution is an analytical solution obtained after an infinite migration distance. The existence 569
570
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
of these asymptotic solutions is made possible because they include a constant state [1]. In other words, because there are two plateau concentrations, one before and one after the self-sharpening front, the velocity of this front tends toward a constant limit profile, and a steady state is reached such that the front profile remains unchanged during its propagation. Such constant states do not exist in elution chromatography (the band height decays constantly), and there are no asymptotic solutions for the multicomponent elution problem either. These asymptotic solutions led to the introduction of the important concepts of shock layer and shock layer thickness, discussed in detail in Chapters 14 and 16, dealing with the kinetic models [2-6]. However, the shock layer concept is the necessary complement of the isotachic train concept brought about by the ideal model. It permits the study of the importance of the mixed zones between bands in displacement chromatography and of the influence of the various experimental parameters on the thickness of these mixed zones [7-9]. Although a shock layer theory could be developed within the framework of the equilibrium-dispersive model, this approach does not seem to be worthwhile. The result could be hardly simpler than the one obtained with the lumped kinetic model, and it would be less general. Because of the great importance of these concepts for the understanding of some practical problems encountered in the development of separations by displacement chromatography, however, we will have to anticipate and will use in some discussions the results that are presented in Chapter 16. The first part of the chapter deals with the problems which arise from the use of actual columns, having a finite efficiency. The rest of the chapter is devoted to a discussion of the practical problems encountered in developing a separation by displacement chromatography, and to a brief review of the applications of displacement chromatography.
12.1 Displacement Chromatography with a Nonideal Column We discuss in this first part the formation of the isotachic train using the equilibrium-dispersive model and the influence of the various parameters that control the characteristics of this train: the displacer concentration, the sample size, the column length, the concentration of the feed, and the column efficiency. The results differ from those reported in Chapter 9, which were obtained with the ideal model in which there is no dispersion. Because the differences observed consist essentially in the formation of mixed zones between the bands in the isotachic train, many results remain similar. We also discuss the behavior of trace components, either those contained in the sample or those contained in the displacer. We compare in Figure 12.1a the isotachic trains calculated with the ideal and the equilibrium-dispersive models under the experimental conditions given in Chapter 9. The major difference observed concerns the boundary between adjacent zones. These boundaries are no longer vertical fronts or shocks; they have become shock layers (Chapter 16, Section 16.1.4). The shock layer thickness is quite significant compared to the natural width of these zones, and it increases
12.1 Displacement Chromatography with a Nonideal Column C mg/mL C mg/mL
100 100 i
a
b
H=0.01 H=infinite
H=0.005 =0.005 H=0.001 =0.001
100
100
75 50
571
100 50
50 50
50
0
0 22
25
28
31
35
39 t, min
Figure 12.1 Comparison of the concentration profiles of the bands in an isotachic train predicted by the ideal model and the equilibrium-dispersive model, (a) Displacement chromatogram. Profiles predicted by the ideal model (black lines) and by the equilibriumdispersive model (thick shaded line). Parameters: e = 0.80;Fv = 1 mL/min; m\ = 50 mg; m2 = 75 mg; m3 = 100 mg; Q = 100 mg/mL; L = 50 cm; ax = 16; a2 = 20; a3 = 24; ad = 30; b\ = 0.016; b2 0.0171; b3 = 0.0187; bd = 0.020; H = 0.010 cm; tp = 211.4 s. (b) Band profiles in an isotachic train. Influence of the column efficiency. Black line: H = 0.001 cm; thick shaded line, H = 0.005 cm. Equilibrium-dispersive model. Parameters as in (a), except m2 = 50 mg; Q = 125 mg/mL; ax = 20; a2 = 28; a3 = 36; ad = 54; tp = 307.1 s.
with increasing column HETP. This explains why the recovery yields achieved in displacement chromatography are never 100%, but may be much lower (Chapter 18). Otherwise, the retention times of the front and the rear shock layers are the same as those of the shocks of the ideal model (Figure 12.1b), and the concentration plateaus are the same as predicted from the intersection between the operating line and the single-component isotherms (Figures 9.3, 9.5, and 12.4a). Figure 12.1b compares the band profiles for two columns of different efficiencies, with H = 0.005 and H = 0.001 cm, respectively, while all other parameters remain constant. A significant reduction in the shock layer thickness (SLT) is observed at the higher column efficiency, as a consequence of less band broadening.
12.1.1 Influence of the HETP Figure 12.1 compares the displacement chromatograms obtained with columns having different values of the HETP. As predicted by the shock layer theory, the width of the mixed zones, i.e., of the shock layers, increases with increasing height equivalent to a theoretical plate, although the relation is more complex than a proportional dependence (see Eqs. 14.32 to 14.37). Efficient columns, where fast mass transfers take place, are critical for achieving high recovery yields in preparative displacement chromatography, as in the elution mode. The yield at constant product purity decreases with increasing SLT. This result is valid only for the bands in the isotachic train. During the formation of this train, the bandwidth also depends on the sample composition and on the competitive isotherms.
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
572 C C mg/mL
a
b
Cd-125 img/mL Cd=125
Cd=125 Cd-125 mg/mL
100
100
100
200
50 50
100
50
50
100
50
0
0 31
35
39
Figure 12.2 Influence of the sample size on the band profiles in displacement chromatography. Same parameters as in Figure 12.1b, except Q = 125 mg/mL and H = 0.010 cm, and sample size, (a) Sample size 50, 50, and 100 mg of the first, second, and third components, respectively, (b) Sample size 100, 100, and 200 mg of the first, second, and third components, respectively. (c) Sample size 150,150, and 300 mg of the first, second, and third components, respectively.
22
28
34
c Cd=125 mg/mL 100
300
150
50
150
0 22
28
34
t, min
12.1.2 Influence of the Sample Size and the Displacer Concentration Figures 12.2a-c illustrate the changes in the displacement chromatogram with changes in the sample size and in the displacer concentration. In Figure 12.2a, the formation of an isotachic train has been achieved before the column exit for a sample size of 200 mg. Doubling the total sample load to 400 mg at constant composition (Figure 12.2b) results in the isotachic train being slightly incomplete, the first part of the zone profile of the first component being intermediate between an isotachic zone and an overloaded band profile in elution. A further 50% increase of the sample size to 600 mg (Figure 12.2c) results in a chromatogram that is far from the isotachic train. This trend is an illustration of how the sample size affects the column length required to achieve an isotachic train. As the sample size increases, both the thermodynamics (see Chapter 9) and the kinetics require a longer column for the formation of a constant pattern. A superficial glance at the chromatogram in this figure suggests that the sample might contain six different components. The total profile in Figure 12.2c is given by the thin black line, which shows the chromatogram recorded with a nonselective detector. This line illustrates how deceptive the response of a conventional detector can be in ascertaining whether an isotachic train has been formed during a separation by displacement chromatography. The importance of performing careful fraction collection and off-line analysis of these fractions cannot be overemphasized.
12.1 Displacement Chromatography with a Nonideal Column C mg/mL C 2 0 0 -i 200
573
aa
bb 250 • 250
Cd«225 mg/mL Cd=225
Cd=175 mg/mL Cd«175 mg/mL 200 • 200
150 150 •
200 200 150 150 • 100 • 100
100 100 100 • 100
100 100 50 • 50
50 50 •
0
0 19 19
24 24
29 29
34 34
19 19
24 24
n
29 29
34 34
c c
400 4 0 0 -i
,
Cd-400 mg/mL Cd=400
Figure 12.3 Influence of the displacer concentration on the band profiles in displacement chromatography. Same parameters as in Figure 12.1b, except H = 0.010 cm, and Q . (a) Q = 175 mg/mL. (b) Q = 225 mg/mL. (c) C^ = 400 mg/mL.
200 200 •
nil 0 19 19
J1
24
29
.
t, min
34
Raising the displacer concentration from 125 mg/mL (Figure 12.2b) to 175 mg/mL (Figure 12.3a) permits the achievement of an isotachic train with the sample size of 400 mg. At the same time the three component bands become taller and narrower, since the operating line is less steep and intersects the component isotherms at higher concentrations. Further increase of the displacer concentration to 225 (Figure 12.3b) and 400 mg/mL (Figure 12.3c) causes the zones of the chromatogram to move faster, to become increasingly taller and narrower, but it also results in mixed zones that are wider relative to the total length of the isotachic train, hence in a loss of recovery yield. As a consequence, although the isotachic train seems to have been reached in Figure 12.3c, the second band has no concentration plateau, and the first and third ones have hardly any. This phenomenon has been called overdisplacement. It takes place when the displacer front moves too fast and the SLTs are too thick for the zone profiles in the isotachic train to reach their plateau concentrations. We have seen (Figure 12.3b) that, even when the displacer concentration is above the watershed point1 (i.e., when the operating line is below the initial tangent of the isotherm of the first component, see Eq. 9.7 and Figures 9.3 and 9.5), the classical isotachic train is not necessarily achieved. Figures 12.2 and 12.3 illustrate how the sample size and the displacer concentration can be manipulated to :
See definition Chapter 9, Section 9.1.3 and Figure 9.5. The watershed point is a displacer concentration.
574
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model C mg/mL C
b b
a
50 • 50
50 • 50
Cd«85 mg/mL Cd=85
10 0 100
50 25
5C 50
25 • 25
Cd«85 mg/mL Cd=85
50 50
25 • 25
50 50
50
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0 30 30
40
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Q mg/mg packing Q mg/mg packing
30 30
40
50 50
50 50
3
, ~
7^-
800 800
2 10 • 10
|
•
1
400
\
\
50 50
00 25 5
w** W * *
50 50
C mg/mL C
C d Cd
\
0 30 30
rx 25
5 • 5
Cc
60 60
15 15
Operating Line Line Operating
0o
min min
d d
Cd«46.43 mg/mL Cd=46.43 1200 1200
, tt,
C mg/mL C mg/mL
c
Displac Displacer
K 40
50 50
, tt,
min min
60 60
Figure 12.4 Top Illustration of the watershed point. Same experimental conditions as in Figure 12.1b, except Q and sample size, (a) Displacement chromatogram with a displacer concentration at the watershed point of the first component, Q = 85 mg/mL. Sample sizes: 50, 50, and 100 mg. (b) Same conditions as in (a), except sample sizes: 50, 25, and 50 mg. Bottom Illustration of the watershed point. Same experimental conditions as in Figures 12.1b and a above, except Qj and sample size, (c) Illustration of the isotherms and the operating line for (b). (d) Displacement chromatogram with a displacer concentration at the watershed point of the second component, Qj = 46.4 mg/mL. Sample sizes: 50,25, and 50 mg.
obtain an isotachic train with a column of reasonable length. Note that the watershed point, which is calculated using Eq. 9.7, is the critical displacer concentration below which no displacement is possible. Thus, if the displacer concentration is now decreased below 125 mg/mL (Figure 12.2a) down to the watershed point for the first component, 85 mg/mL, the chromatogram in Figure 12.4a is obtained. An isotachic train is still obtained for the second and third components, but the first component is eluted as an overloaded elution band. If the amount of the first component in the sample is sufficiently small to be in the linear region of the isotherm, a Gaussian peak would be observed. Besides the smoothing of the front shock of the first band, the only difference from the profiles predicted by the ideal model is the interaction between the rear boundary of the first band and the front of the second one, leading to the formation of a mixed zone of finite width.
12.1 Displacement Chromatography with a Nonideal Column
575
Finally, we note that the fact that the displacement chromatogram is carried out at the watershed point of the first component guarantees that the profile of its band will be that of an overloaded elution band, just resolved from the isotachic band of the second component when the isotachic train is formed. As illustrated in Figure 12.4b, changing the amount of the other components in the sample, hence the width of their zones, does not affect the resolution of this overloaded elution profile with the band of the second component. Similarly, increasing the column length or the displacer concentration would not affect the resolution of the bands under isotachic conditions. The only way to affect the heights of the zone and, to some extent, the width of the mixed zones is to increase the concentration of the displacer. In Figure 12.4c, the displacer concentration has been further lowered to 46.4 mg/mL, the watershed of the second component (Figure 12.4c). Therefore, the displacer front moves too slowly for the isotachic train to include the bands of the first two components, and in the case of this figure, the train begins with the third component. The first two bands are eluted as overloaded elution bands. The plateau on the top of the second band is the result of the tag-along effect between the first and second bands, as in Figure 8.6b. It has nothing to do with the displacement or the injection plateau. Note that actual chromatograms will look like those shown in Figures 12.1 to 12.4 only if the concentrations of each of the feed components are plotted versus the volume of eluent passed. This requires either the use of a selective detector (e.g., LC/MS), or of fraction collection and analysis. The chromatogram in Figure 12.17 illustrates the difficulties encountered in developing a separation by displacement chromatography. Unless the isotachic train is fully formed and the response factors of all feed components are similar, the recording of the response of a nonselective detector is very difficult to account for.
12.1.3 Influence of the Column Length The formation of an isotachic train during the displacement of the sample through the column is progressive. Figures 12.5 and 12.6 illustrate the reorganization of the concentration distribution during the displacement of the sample and the formation of the isotachic train in two different cases, when the feed is concentrated and when it is dilute. For narrow sample pulses of concentrated feed (Figure 12.5) the initial velocity of the band fronts of the feed components is higher than that of the displacer front. However, dilution of the products takes place quickly (see the maximum concentrations of the band in Figures 12.5b-e), and the band front slows down while separation of the zones begins. The displacer front drives the formation of the isotachic train which is eventually reached in Figure 12.5f. The migration of an isotachic train is a constant pattern behavior. Any further increase in the column length has no effects on the chromatogram obtained. There is a steadystate equilibrium between the self-sharpening of the band boundaries, due to the convex-upward isotherms and the high concentrations, and the dispersive effects of mass transfer kinetics and axial dispersion.
576
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model a
C mg/mL
d
L=0 cm
200
L=15 100
100 50
0
0 0
1
2
t, min
5
b
L=5
100
100
50
50
0
10
15
e
L=20
0 0
2.5
5
5
c
10
15
f
L=50
L=10 100
100
50
50
0
0 0
5
10
25
30
t, min
35
Figure 12.5 Influence of the column length on the displacement chromatogram. I. Concentrated feed. Experimental conditions as in Figure 12.1b, sample sizes 50 mg, 50 mg, and 100 mg of the three components, tp = 30.71 s.
The chromatograms in Figures 12.2 and 12.5 are two different aspects of the same problem. If the isotachic train cannot form because the sample is too large or the column too short, one will be achieved with a longer column (Figure 12.5), or with a smaller sample size (Figure 12.2). The length of column required to achieve the isotachic train increases with increasing sample size. The loading factor is inversely proportional to the column length, and the loading factor for which the isotachic train is formed remains constant. For wide pulses of a low-concentration feed but at constant sample size, the same isotachic train is eventually formed as with a narrow pulse of a concentrated sample, all other parameters being constant. The only difference is an increase of the apparent breakthrough time of the displacer due to the longer time needed for the injection of the feed, and to the choice of the beginning of feed injection as time origin. Thus, highly dilute feed can be concentrated to a considerable extent (Figure 12.6). At the beginning of the sample injection, a classical frontal analysis pattern forms immediately at the band front. Each component breakthrough
12.1 Displacement Chromatography with a Nonideal Column
577
a a
C mg/mL
d d
L=20 L-20
L-0 cm L=0 100 100 •
100 100
50 • 50
50 50
0
0 00
8
t, «. -min
12 12
16
"
17 17
22 22
b b
L-5 L=5
e
L-22.5 L=22.5
1
100 100 •
100 100
50 • 50
50 50
/
0
rf
0
00
8 8
12 12
1G 16
17 17
22 22
ff
c c
L-15 L=15
L-25 L=25 100 100 •
100 • 100
J
50 • 50
50 50 •
0
0 00
8
1G 16
12 12
j-
17 17
rf dL)
, min t, min
t
2 2 22
Figure 12.6 Influence of the column length on the displacement chromatogram. II. Dilute feed. Same experimental conditions as in Figure 12.5, except tp = 307.1 s.
front progresses at its own velocity. The profile achieved at the end of the injection depends on the competitive equilibrium isotherms of all the feed components. However, this breakthrough behavior results in concentration of the bands at the front. While this front continues its forward migration, the displacer pushes the rear of the sample pulse, causing the increase of the concentration of the different components at the displacer front while a valley whose bottom is at the injection plateau concentration, between the rising ends, subsists (Figure 12.6c). The displacer front moves faster than the front of the first component band, however. At the moment when the front and rear concentration bulges meet, the band profiles change very rapidly, the valley disappears abruptly, and the isotachic train forms soon afterward. This phenomenon is illustrated in Figures 12.6c,d.
578
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
12.1.4 Influence of the Separation Factor The influence of the design and operating parameters just discussed—sample size, column length, HETP, displacer concentration (Figures 12.1 to 12.6) has been illustrated for a fixed set of thermodynamic conditions. The influence of these parameters has been studied experimentally, and numerous results support the validity of the theoretical conclusions presented here [10]. However, little attention has been paid so far in the literature to the influence of the separation factor. This lack of interest is probably due to the combination of two reasons. First, the boundaries between bands in the ideal model are all vertical, and the separation factor has no influence on them. Second, although the column length required to achieve the isotachic train increases rapidly with decreasing value of a — 1, there is no simple equation that relates these two parameters and that could be used to express quantitatively the practical difficulty of achieving isotachic trains with components having separation factors close to 1. The concepts used in the study of displacement originate from the ideal model, and the isotachic train obtained with actual columns is typically schematized as the same train as in the ideal model, except for very steep boundaries between pure component zones, and narrow mixed zones (see Figure 9.12). This model is valid only when the plateau concentrations of the main components are so large that their stationary phase concentrations are a significant fraction of their saturation capacities (see discussion of Figure 9.12), the separation factor is not close to 1, and there is no selectivity reversal (Section 12.1.7). Then, it is the effect of nonlinear isotherms that controls the concentration profiles, and the representation is reasonably accurate. However, displacement chromatography can be carried out successfully and results in the formation of an isotachic train either at lower concentrations, or when using compounds with slow mass transfer kinetics (see Chapterl6). This appears to be the case in many recent applications to the purification of biochemicals. The shock layer theory leads to a more realistic description of the band boundaries in isotachic trains. Although the general phenomena and the qualitative results described in this section remain valid for any isotherm model, provided they are convex upward and do not intersect, the quantitative results of the shock layer theory presented in Chaptersl4 and 16 are valid only when the adsorption behavior of the mixture components is properly described by the competitive Langmuir isotherm model. The theory shows conclusively that, when the separation factor decreases, the shock layer thickness, hence the width of the mixed zone in the isotachic train, increases in proportion to (a + l ) / ( a — 1) (Eqs. 16.27a and 16.27b). At the same time, the column length required to reach isotachic conditions increases also indefinitely, as predicted by the ideal model. As a demonstration of the validity of this theoretical result, we show in Figure 12.7 a fully developed isotachic train of inosine, deoxyinosine, adenosine, and deoxyadenosine [11]. This figure illustrates how strongly the thickness of the shock layer between bands in the isotachic train depends on the relative retention, and how widely the size of the mixed zones can vary with a — 1, as predicted by Eq. 16.27. The main features of this displacement chromatogram are (i) the very close plateau concentrations of the first two and of the last two com-
12.1 Displacement Chromatography with a Nonideal Column
579
DEOXYADENOSINE ADENOSrNE (10 tag)
- 27.5 mM BTBA
DEOXYINOSINE (10 ma)
20 21 VOLUME (ml)
22
23
Figure 12.7 Individual band profiles in the displacement chromatogram of inosine, deoxyinosine, adenosine, and deoxyadenosine. Column: 250 x 4.6 mm +150 x 4.6 mm packed with 5 Jim Supelcosil LC-18; carrier: 10 mM acetate buffer, pH 5.0; displacer: 27.5 mM benzyltributylammonium chloride in the carrier; flow rate: 0.1 mL/min; fraction volume 100 ]iL; temperature 22°C; feed: 10 mg of each component in 2.0 mL in carrier. Reproduced with permission from Cs. Horvdth,}. Frenz and Z. El Rassi, J. Chromatogr., 255 (1983) 273 (Fig. 3).
ponents, (ii) the sharp boundaries between the second and third components and between the fourth component and the displacer (benzyltributyl ammonium chloride), and (iii) the wide mixed zones between the first and second and between the third and fourth components. These results are explained by pairs of equilibrium isotherms that are close to each other in the entire concentration range, hence by values of the separation factor that are near unity, for the two pairs inosinedeoxyinosine and adenosine-deoxyadenosine. On the other hand, the separation factors for deoxyinosine-adenosine and deoxyadenosine- displacer are large and the adsorption isotherms are significantly different over the entire concentration range. The concentration plateau of a band in the isotachic train is given by the intersection of the operating line and the isotherm (see Figures 12.4 and 9.4). Thus, close values of the concentration plateaus for the components of each pair means that their isotherms are close to each other. Isotherm pairs that follow each other closely in the entire concentration range have an a value close to 1. This gives a wide shock layer in the isotachic train between the bands of the related compounds (Eq. 16.27). By contrast, the bands of deoxyinosine and adenosine have very different values of their plateau concentrations, their isotherms are far apart, their separation factor is large, and the mixed zone between their bands is narrow. The same is true for the zone between deoxyadenosine and the displacer. However, we prefer to discuss further the dependence of the thickness of the mixed zones between the bands of the isotachic train in the framework of the
580
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
lumped kinetic model, in Chapter 16 (Section 16.1.4). Although this model is more complex, and its application requires the additional determination of the mass transfer coefficient, it also provides a more detailed and correct analysis. Although possible, a study of the shock layer profile in the framework of the equilibriumdispersive model would be too approximate to be worthwhile, and its results could hardly be more simple. The two contributions to the HETP, arising from axial dispersion and mass transfer resistance, have different dependences on the concentration of the components. The approximation that neglects this dependence is not as good in frontal analysis or in displacement as in elution, because in the elution mode the concentration decreases markedly during the migration of the bands. The validity of the approximation is discussed further in Chapter 14, Section 14.1.7.
12.1.5 Case of Trace Components The ideal model predicts that the height of a band in the isotachic train is given by the intersection of the operating line and the isotherm of the component, hence is independent of its concentration in the feed (Chapter 9). The width of the band is proportional to the amount of component injected with the feed. If the component is present at trace levels, the band is very thin. With the equilibrium-dispersive model, the bandwidth of a component decreases with decreasing concentration while its band height remains constant, as long as this bandwidth is large compared to the SLT When the bandwidth becomes of the same order of magnitude as the shock layer thickness, its height begins to decrease, while its width stabilizes [12]. The elution profile of a trace component is a narrow peak located in the mixed zone between two successive major component bands. Therefore, the width of the trace component band can be much narrower than the natural bandwidth of this compound in linear chromatography [12,13]. A similar effect is observed in overloaded elution, when the peak of a trace component is eluted between the bands of two major components. The effect is less striking in elution, however, and the band profiles are much more complex, as shown in Figure 11.18 bottom right [14,15]. As a consequence, applications of this effect will be more useful in displacement chromatography than in elution. This phenomenon can be utilized in two opposite situations: first for the enrichment of trace impurities, second for the preparation of high-purity products. In the isotachic train, impurities are completely eliminated from the bands of the main components, and appear inside the mixed zones. However, it is also important in this case that the displacer be of high purity (see next section). Figure 12.8 illustrates this effect. It compares the chromatograms calculated with the equilibrium-dispersive model for the same amount of a trace compound eluted in different positions of the isotachic train in displacement chromatography. In elution (peak 1) the band would have a Gaussian profile of baseline width w = ACT = 4H^/N (see Figure 12.8c,d, inset). In displacement chromatography, the peak of a trace component is eluted at the same location as it would have in an isotachic train if its concentration were high. It can be eluted at the front of the
12.1 Displacement Chromatography with a Nonideal Column C mg/mL
581 c
a
o.os 0.05
0.05
100
100
0 17
IT 17
27
50
0
2T 27
50
H -0.001 H=0.001
H=0.01
0
, , I ,, ,
0 29
34
39
29
34
39
b
0.6 0.05
d 4
0 44
49
44
A
.IS 0.15
1 49
0
0.3 2
H=0.01
H=0.001 H=0.001
0
0 29
34
39
29
34
t, min
39
Figure 12.8 Profiles of trace components of the feed in displacement chromatography. Amounts: impurities, 0.1 mg; main components, 50, 50, and 100 mg. Injection concentrations: impurities, 0.02 mg/mL; main components 10,10,20 mg/mL; displacer, 125 mg/mL. tp = 5.1 min; void fraction, 0.8; L = 50 cm; Fv = 1 mL/min; to = 6.65 min. (a, c) Profiles of the main components and displacer; retention factors, fc^, 5, 7, 9, and 13.5; b = 0.0159, 0.0171; 0.0187; 0.02 mg/mL. (a, c, insets) Profile of impurity with an isotherm below the operating line, k'o = 2. (b, d) Profiles of the impurities eluting between the main component bands, retention factors, 6, 8, 11; respectively, (b, d, insets): Profile of impurity under elution conditions, k'o = 6; injection concentration, 0.2mg/mL, tp = 0.51min.
train, in which case the squeezing effect is already significant but remains moderate. It can be eluted between two major zones in the isotachic train, or between the last component and the displacer. In these last two cases, as seen in Figure 12.8b,d (inset), the squeezing effect can be very important. Its intensity depends on the column efficiency, but corresponds to an approximately 10- to 100-fold increase of the apparent column efficiency. For the sake of comparison, the insets in Figures 12.8b,d show the profiles of an impurity that is not part of the isotachic train because its watershed point is higher than the displacer concentration. The profile obtained is the rectangular injection profile, smoothed by the apparent dispersion of a 5000 plate column (Figure 12.8b) or a 50,000 plate column (Figure 12.8d). Thus, displacement chromatography provides an attractive procedure of trace enrichment. An enrichment ratio of several orders of magnitude can easily be achieved in practice. Such high ratios are possible because of the squeezing effect of displacement on trace bands (if the band is ten times narrower as in Fig-
582
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
100 80
20 i
Time (minutes) 40
B
i
MA
fn 1ilk
60
I 1
40 20 0
80
60
r—
1
100
"\, L V Mr
1
1 400
200 300 Scan number
-
k
'
D
•
136
61:36
5J7,45=S-«
SBJ.a'f'*
"f'
3
!
ml 75B.1
'[" 7ES.1
3.1 H^.2
61(12 * 7 '3.1
.1:00
Figure 12.9 Enrichment of trace peptides in a peptide digest by displacement chromatography. (a) Displacement chromatogram of 10 nmol of digested biosynthetic human growth hormone. A, UV at 195 nm. B, Total ion current, 12 sec scan period, (c) Mass spectra of scans 305-310 showing the boundary between the T10c! and the T13 peptide fragments, with little or no trace components in between, (d) Mass spectra of scans 288-300 (spectrum 288 is the sum of the spectra recorded during scan 288 and 289), showing the boundary between the peptides fragments T12 and TlOcj, which contains a number of minor components of the digest. Reproduced with permission from J. Frenz, J. Bourell and W.S. Hancock, J. Chromatogr., 512 (1990) 299 (Figs. 4, 6 and 7).
ures 12.8b7d, it is also ten times higher) and because a much larger sample size can be injected than is possible in conventional elution chromatography and yet the formation of an isotachic train can be achieved (which is necessary for the phenomenon described here to take place). This effect has been demonstrated and used for trace analysis [15-17]. Frenz et al. [16,17] used displacement chromatography for the analysis of the peptides obtained by tryptic digestion of recombinant proteins. This analysis was done by LC/MS, using microcolumns because of the low flow rate acceptable by the MS ion source. Displacement chromatography was the procedure adopted to
12.1 Displacement Chromatography with a Nonideal Column
583
Figure 12.10 Enrichment of ^-naphthylamine traces in methanol solutions by displacement with diethyl phthalate. Experimental (symbols) and calculated (solid lines) chromatograms. Calculations made with the equilibrium-dispersive model, a competitive Langmuir isotherm using the coefficients of the single-component isotherms, N = 3600 plates, and fluorescence quenching, (a) Displacement of 0.45 ]i% /3-naphthylamine by diethylphthalate (200 mg/mL) in methanol-water (70:30). (b) Same experiment with 275 m g / m L diethylphthalate. Reproduced with•permissionfrom, R. Ramsey, A.M. Katti and G. Guiochon, Anal. Chem., 62 (1990) 2557 (Figs. 7b and 8a). ©1990, American Chemical Society.
increase the sample size injected on the narrow-bore columns, and decrease the detection limits. On the efficient columns used, the mixed zones between successive bands of the isotachic train are narrow, and their width corresponds to the time it takes to scan only a few mass spectra (Figure 12.9). These authors showed that at the interface between successive bands, the mass spectra did not always change progressively from the spectrum of one component to the spectrum of the other one. Often, it was possible to record one or a few spectra that were entirely different from the spectra of these main components. Figures 12.9b,c illustrate the phenomenon reported. This observation permitted the identification of lowlevel trace components, possibly originating from contaminants of the analyzed proteins in these digests [16]. Ramsey et al. [13] demonstrated the nature of the phenomenon by analyzing methanol solutions containing (/3-naphthylamine) as an impurity in known concentrations and detecting it by fluorescence. There was good agreement between experimental results and the band profiles calculated with the equilibriumdispersive model and taking into account the fluorescence quenching due to the displacer (Figure 12.10).
12.1.6 Influence of the Impurities in the Displacer Solution When the displacer contains impurities, multicomponent frontal analysis of the displacer solution takes place. The impurities that are less retained than the displacer have a breakthrough curve which appears earlier and interferes with the component bands [18]. Depending on the relative retention of the feed components and the displacer impurities, various types of breakthrough profiles can be observed for these impurities. Those which are less retained than all the components are eluted as a very sharp initial peak, or staff, in the front of the first band,
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
584
krr=2.5
'£
\;
• 7.0
o-
9.0
11.0
17.5
IB
18.5
19
Retention Volumn (ml)
19.5
19
21
Retention Time (min)
Figure 12.11 Elution profiles of impurities contained in the displacer. (Left) Calculated displacement chromatograms showing the effect of the capacity factor of a displacer impurity (fc|) on the chromatogram. Impurity concentration, 0.8% of displacer, k\ = 7.5;fc^= 9.0;fc^= 12.7. (Right, top) Experimental data illustrating the influence of the displacer impurity on the band profiles. Breakthrough of 25 mg/mL phenol in water on a Vydac C18 silica column (25x0.46 cm), at 0.2 mL/min. The two traces correspond to different phenol samples. (Right, bottom) Chromatogram in (b), reconstructed from the off-line analysis of collected fractions. Reproduced from ]. Zhu, A. Katti and G. Guiochon, Anal. Chem. 63 (1991) 2183 (Figs. 6, 7a and 8a). ©1991, American Chemical Society.
followed by a plateau at the impurity concentration in the displacer solution (Figure 12.11a, 1). Those that are less retained than some components, but more retained than others, are eluted as an initial broad peak between the bands of the components that are eluted immediately before and after the impurity, followed by the concentration plateau (Figure 12.11a, 2 to 4). Experimental results (Figure 12.11b) are in agreement with the band profiles calculated using the equilibrium-dispersive model [18]. This demonstrates that displacer impurities with lower retention than the displacer will pollute the pu-
12.1 Displacement Chromatography with a Nonideal Column
585
c[mH]
0
k
b a -
X1
Column volumes
Column volumes
JO a
b
<\ I
1 A
s
\i ° B\
J\
.
Column volumes
!
D
B I
«
i 20
I-
A
/
I I, 6
4
Column voli
Figure 12.12 Crossing equilibrium isotherms and displacement profiles. (Top, left) Operating line above the separation gap. (Top, center) Operating line below the separation gap. (Top, right) Operating line inside the separation gap. (Center, left) Isotachic train corresponding to the operating line above the separation gap (top left), calculated with the LeVan-Vermeulen isotherm. (Center, center) Isotachic train corresponding to the operating line below the separation gap (top center), calculated with the competitive Langmuir isotherm. (Center, right) Isotachic train corresponding to the operating line inside the separation gap, calculated with the LeVan-Vermeulen isotherm. (Bottom, left) Isotachic train calculated with the competitive Langmuir isotherm and corresponding to the operating line in (top left). (Bottom right) Isotachic train calculated with the LeVan-Vermeulen isotherm and corresponding to the operating line in (top, center). Reproduced with permission from E Antia and Cs. HorvAth, J. Chromatogr., 556 (1991) 119 (Figs. 4 to 7).
rifled bands of the feed components. Depending on the retention factor of the impurity, the number of contaminated bands will vary. Therefore, in order to use displacement chromatography for high purity applications, it is essential to utilize high purity displacers. Similar results have been published by Jen and Pinto [19].
12.1.7 Case of Selectivity Reversal When the single-component isotherms of two compounds, A and B, cross (Figure 12.12 top left), there is selectivity reversal; i.e., compound A is the less strongly retained at low concentrations and the more strongly retained at high concentra-
586
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
tions. This case is not unusual because the adsorption energy, which controls the adsorption constant at infinite dilution, increases with increasing size of the molecular "footprint," while the stationary phase saturation capacity may vary in the opposite direction. The problem of accounting for the structure of the isotachic train becomes difficult when the isotherms cross at a concentration lower than the plateau concentration in the isotachic train, i.e., below the operating line. The competitive Langmuir model, which predicts a constant selectivity, is obviously unable to account for this effect (see Chapter 4, Section 4.1). Antia and Horvath [20] have evaluated selectivity reversal in displacement chromatography using the ideal adsorbed solution (IAS) theory (Chapter 4, Section 4.1.5) and assuming that the isotherms of the pure components follow Langmuir behavior. To account for the competitive isotherms, they used the isotherm model derived by LeVan and Vermeulen [21]. Using a stability analysis, they have shown that a region in the isotherm plane, q, C, can be mapped, called the separation gap (Figure 9.7). The position of this gap depends on the ratio of the column saturation capacities for the two components. The physical explanation for this gap arises from the hodograph plot of the separation of the two compounds, see Figure 9.7b). The tie lines in this figure are the representation of the operating lines in the hodograph plot, Figure 9.7a. The unit selectivity line (a = 1) and the two corresponding tie lines define the separation gap, as shown in Figures 9.7a and b. The stability condition shows that a completely separated pattern can form only if the concentrations C^ and Cg of the intersection of the operating line and the isotherms of the two compounds are on the same side of the unit separation line [20]. Depending on the relative position of the operating line and the separation gap, the displacement phenomenon is different, as illustrated in the different Figures 12.12top left to bottom right. Figures 12.12top, left, center and right show the adsorption isotherms of the two components and of the displacer and operating lines for three different displacer concentrations. The profiles for the isotachic trains corresponding to the three operating lines are shown in Figures 12.12center, top, center, and right, as calculated with the LeVan-Vermeulen isotherm, while those in Figures 12.12right, top and center were calculated with the Langmuir competitive isotherm, these two competitive isotherms being derived from the single-component isotherms above. No separations can take place if the operating line lies in this separation gap region (Figures 12.12 bottom, right) [20]. When the displacer concentration is high or low enough, however, and the operating line is not inside the separation gap, components A and B can form an isotachic train. The single-component isotherm of A has a lower initial slope (lesser k^) and a higher saturation capacity than the isotherm of B. If the operating line is above the separation gap (Figure 12.12 top, left), component A is eluted first in an isotachic train (Figure 12.12 top, center) and its plateau concentration can be higher or lower than that of B, depending on the coordinates of the intersection points of the two isotherms and the operating line. The same result is predicted by the competitive Langmuir isotherm, which, however, underestimates the width of the mixed zone. Separation of the two component bands and formation of an isotachic train are also possible when the operating line is below the separation gap (Figure 12.12 center, left). However, in this
12.2 Applications of Displacement Chromatography
300
350
SOU
+53
EFFLUENT V O L U M E !mL)
S00
587
2000
2400
EFFLUENTT VOLUME [ m i l
Figure 12.13 Formation of a displacement azeotrope. Separation of a-aminobutyric acid and isoleucine on Dowex 50W-X8, glass column 1.5 cm i.d., 20-40 cm long, (a) Displacer, 125 mM NaOH. (b) Displacer, 25 mM NaOH. Reprinted from G. Carta, A. Dinerman, AIChE ]., 40 (1994) 1618 (Fig. 9), by permission of the American Institute of Chemical Engineers. ©1990 AIChE. All rights reserved.
case, A is now eluted second (Figure 12.12 center, center), and the plateau concentration of A is still higher than that of B, in agreement with the relative position of the two isotherms. By contrast, the competitive Langmuir isotherm model, which predicts constant selectivity, cannot account for this azeotropic effect. When the operating line lies within the separation gap (Figure 12.12 bottom, left), the IAS theory predicts that no separation can take place (Figure 12.12 bottom right). A mixed band of A and B is formed in the isotachic train, and no separation of these compounds can take place. Carta and Dinerman [22] have observed the adsorption azeotrope experimentally for the separation of a-aminobutyric acid and isoleucine on Dowex 50W-X8 resin, a sulfonated styrene-divinylbenzene copolymer which is moderately crosslinked. Figure 12.13a shows the experimental and calculated band profiles for the case of a separation azeotrope with a 125 mM NaOH solution as the displacer. Further increase in the column length results in the same normalized profile, which means that steady state has been reached with incomplete separation. Figure 12.13b illustrates the separation with a 25 mM NaOH solution as the displacer. Complete separation occurs.
12.2 Applications of Displacement Chromatography A number of applications of displacement chromatography have been published in the literature. In most cases, the formation of an isotachic train was attempted, as this gives the highest recovery yield, although it does not permit the highest possible production rate (Chapter 18). In some publications, the challenge of finding the proper experimental conditions that result in an isotachic train is addressed. Many of the early applications deal with the separation of metal cations on ion exchange resins [23,24]. The use of large particles and the resulting poor column efficiency explain the width of the mixed zones between successive bands. Most of the recent applications to the separation of organics and biochemicals
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
588 C
g/1 R22O33 m mg/l
pH=6
300 --
200 -pH=5.5
100 -pH=5
•H
DE>H
0 25 25
35
45 45
55 65 65 Volume of of Eluate, Eluate/ liters liters
Figure 12.14 Effect of the pH of the mobile phase on the displacement profile of rare earths. Overlay of the elution of 0.01 mole of equimolar mixtures of S1T12O3 (first eluting component) and Nd2O3 (second component) at pH 6.0, 5.5, and 5.0. The vertical broken lines indicate the overlap between the bands of Sm and Nd. Reproduced with permissionfromF.H. Spedding, E.I. Fulmer, J.E. Powell and T.A. Butler, J. Am. Chem. Soc, 72 (1954) 2354 (Fig. 1). ©1954, American Chemical Society.
were developed by Horvath and the members of his group, using highly efficient columns packed with small particles [11,25,26]. The most obvious result of a literature survey is that there is a contradiction between the chromatograms published and the general comments regarding the technique [10]. Careful examination of the displacement chromatograms shows that it is more difficult to develop a successful separation by displacement chromatography than is claimed. Fully developed experimental displacement chromatograms are rarely shown. The importance of the mixed zones in the chromatogram is real but often underestimated; as a consequence, the recovery yields are often moderate. The flow rates are often low, resulting in long separation times and low production rates. Finally, in almost all these works, effective and reproducible regeneration of the column is assumed, compounding the difficulties of correctly estimating the possible production rate. The literature deals insufficiently with the means required to address this serious problem. The net effect of the trade-offs between loading, cycle time, recovery yield, and displacer cost will determine the economics of displacement chromatography. In summary, implementation of the technique at the industrial production level offers challenges and opportunities. Figures 12.14 to 12.26 below have been selected for the quality of the experimental results achieved, as illustrations of the possibilities of the method. At-
12.2 Applications of Displacement Chromatography
589
150 -
30 CM
100
50
10
i
25
150 6 0 CM
hi
100
1 50 SS
o
Figure 12.15 Effect of column length on the separation of rare earths. The elution of mixtures of Sm^Oa (1st), Nd 2 O 3 (2nd) and Pr 6 O n (3rd) on 30+40 Amberlite IR-100. 22 mm ID column, 0.1% citrate solution, pH 5.30, linear flow rate 0.5 cm/min. Vertical broken lines indicate the boundaries of the mixed zones between bands. Reproduced with permission from F.H. Spedding, E.I. Fulmer, J.E. Powell and T.A. Butler, J. Am. Chem. Soc, 72 (1954) 2354 (Fig. 8). ©1954, American Chemical Society.
150 120 CM
100
50
40
45 50 55 Volume of eluate, liters.
60
tention has been paid to select the best results published, and to show as many isotachic trains as possible. One important conclusion that arises from a careful examination of this set of figures is that the achievement of a classical isotachic train is not an easy experimental task. A list of displacers whose use has been described in the literature is given in Table 12.1. To this list, recent publications have added as potentially useful displacers for proteins in ion-exchange chromatography the following compounds: Many common amines, including Methylamine, Benzylamine, Diethylene triamine; Dimethylamine, Diethylene triamine, Pentaerythrityl tetramine, 1,4,8,11 Tetrazacyclotetradecane, Tetraethylene tetraamine, Tetraethylene pentamine Triethylamine, Triethylene tetramine as hy-
590
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
droclorides; Kanamydn A tetraspermine and polyamine derivatives; Neamine polyamine derivatives and tetraspermine; Neomycin polyamine derivatives and sulfate; Saccharin; Spermine and Spermidine; Tetraspermine; and numerous quaternary ammonium such as: Pentaerythrityl(trimethyl ammonium), Dipentaerythrityl (trimethyl ammonium), Pentaerythrityl (benzyl, dimethyl ammonium), and Tetramethyl ammonium [27,28].
12.2.1 Separation of Rare Earths and Other Cations Figures 12.14 and 12.15 show results obtained by Spedding et al. [61] in the separation of rare earths on an ion exchange resin. Note that the general appearance of the chromatograms proves that the displacer is coming from the left-hand side of the figure. The displacer concentration is at the watershed point for the first component at pH 5.0 (Figure 12.14 and 12.15 top) and its band has an overloaded elution profile (see Figure 12.4). The plateau concentrations increase from right to left. The isotachic bands form from the displacer front (far left), down to the front of the first band. Figure 12.14 illustrates the influence of the pH. The retention, hence the initial slope of the isotherm, of the rare earth ions increases with increasing pH of the mobile phase, which explains the increase of the retention volumes and the plateau concentrations with increasing pH. These effects are in agreement with the theory of isotachic train formation using the competitive Langmuir isotherm model (see Figure 12.4). Whereas the isotachic train is fully developed at pH of 5.5 and 6.0 in Figure 12.14, it is not in Figure 12.15 center and top. This figure illustrates the influence of the column length in the formation of the isotachic train, at constant sample size. With a 30-cm-long column, the first band of the isotachic train is nearly formed. Except for the rear part of its profile, in the mixed band between samarium and neodymium, this profile does not change when the column length increases from 30 cm to 60 and 120 cm. The band profiles of neodymium and praseodymium change considerably, however, and an isotachic train is nearly formed in the last figure at 120 cm.
12.2.2 Separation of Organic Compounds Figure 12.16 is reproduced from a seminal paper which marks the modern rebound in interest for displacement chromatography. Figure 12.16, left, shows the equilibrium isotherms of three hydroxyphenylacetic acids between a C18 silica and a phosphate buffer used as the mobile phase [25]. Figure 12.16, right, shows the isotachic train obtained in displacement chromatography with n-butanol as the displacer. The chromatogram shows the importance of the mixed zones and its potential influence on the recovery yield. The different units used in the two parts of the figure and the lack of an n-butanol isotherm make the figure difficult to interpret. However, simple calculations show that the isotherms do not extend to the concentration range of the plateaus. An incomplete isotachic train is shown in Figure 12.17, for a mixture of cis- and fmns-3-hexene-l-ol [62]. Comparison of the signal of a refractive index detector,
12.2 Applications of Displacement Chromatography
591
Table 12.1 List of Displacers
Displacer
Citric acid/Ammonium citrate Sodium acetate Carboxymethyldextran Carboxymethylstarch DEAE dextran Dextran sulfate «2-Globulin Heparin Nalcolyte 7105 Polyethyleneimine Polyvinylsulfate (2000 MW) Ribonuclease A Sodium hydroxide 1-Butanol Ethanol Propanol Diethylethylenediamine 2-Methyl-l-butanol 2-Methyl-2,4-pentanediol Methyl carbitol Benzylcetyldimethylammonium chloride Benzyldimethyldodecylammonium bromide Benzylhexadecylammonium chloride Benzyltributylammonium chloride 2-(2-Butoxyethoxy)ethanol 1-hexanol Carbowax-400 (polyethylene glycol) Cetrimide Cetyltrimethylammonium bromide Chondriotin sulfate Decyltrimethylammonium bromide Diethylphthalate Dipropylene glycol monomethyl ether 4-methylcatechol Octyldodecyldimethylammonium chloride Palmitic acid Phenol n-Propanol Protamine Tetrabutylammonium bromide
Application
Mode
Reference
RE L Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr AA
IX IX IX IX IX IX IX IX IX IX IX IX IX
O, Pr, P, NA
N,RP
O O O O O O O
N N N N N N RP RP
[24,29] [30] [31,32] [33] [34,35] [34-36] [37] [38] [39] [40] [41] [42] [22,43] [11,25,44] [45] [31] [46] [31] [31] [31] [47] [48,49] [50,51] [11,39,51] [40,52] [47] [44] [16,47,53] [17,52] [54,55] [52] [13] [44] [18] [56] [57] [25,44,58] [25,59] [60] [11,44]
P,Pr P
P, NA, PAA PAA,P O O,P O
PAA,P P PAA O O O O O O,S O P
P,AA
N,RP IX, N RP RP RP RP RP RP RP RP RP RP RP RP RP RP RP RP
Pr = Proteins, AA = Amino acides, PAA = Protected Amino acids, O = Organic, R = Rare Earth, L = Light metals, NA = Nucleic Acids, P = Peptides, S = Sugars. IX = Ion Exchange, N = Normal Phase, RP = Reversed Phase.
592
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
t
i
i
1
MOBILE PHASE CONCENTRATION, C M l x l O 2
FRACTION NUMBER
Figure 12.16 Adsorption isotherms and displacement chromatogram for 3,4dihydroxyphenyl, 2-hydroxyphenyl, and 4-hydroxyphenyl acetic acids. (Left) Adsorption isotherms measured by frontal analysis on a 250x4.6 mm column packed with 10 jtm Partisil ODS-2 from 0.1 M phosphate buffer, pH 2.12 at 25°C. The solid lines are a least-squares fit of the data points to the Langmuir isotherm. (Right) Displacement chromatogram, carrier: 0.1 M phosphate buffer, pH 2.12; displacer: n-butanol at 0.97 M. Flow rate: 0.05 mL/min at 25°C. Feed: 1.5 mL of 30, 35, and 45 mg of 3,4 dihydroxy-, 4-, and 2-hydroxyphenylacetic acids, respectively. Fraction size, 0.15 mL. Fraction 40 marks 12 mL of eluent volume. Reproduced with permission from Cs. Horvath, A. Nahum and J.H. Frenz, J. Chromatogr. 218 (1981) 365 (Figs. 6 and 7).
40
50 60 70 Fraction Nuiriber
Figure 12.17 Displacement chromatogram of cis- and irans-3-hexen-l-ol. (Left) Refractive index signal. (Right) Reconstructed profiles from fraction analysis. Fraction size 300 fiL. Column: two 250x4.6 mm analytical a-cyclodextrin columns; carrier: water; displacer: 8.1 mM 1-hexanol; flow rate: 1.0 mL/min; temperature: 30°C; feed: 6 mg of 3-hexen-l-ol. Reproduced with permission from Gy. Vigh, Gy. Farkas and G. Quintero, J. Chromatogr., 484 (1989) 251 (Figs. 6 and 7).
12.2 Applications of Displacement Chromatography
593
E
sI
S
16
TIME (min)
EFFLUENT VOLUME (ml)
Figure 12.18 Purification of cephalosporin C from a fermentation broth. (Left) Analysis. Same experimental conditions as in displacement. (Right) Displacement. Column: 350x4.6 mm, packed with 5 /-cm Zorbax; carrier: 1% v/v acetonitrile in 20 mM sodium acetate; pH 5.2; displacer: 40 mg/mL BEE in carrier; feed: 5 mL fermentation broth; flow rate: 1.0 mL/min; fraction volume: 300 jiL. Reproduced with permission from G. Submmanian, M. Phillips and S. Cramer, J. Chromatogr., 439 (1988) 341 (Fig. 5).
and of the individual band profiles reconstructed from the analysis of collected fractions, illustrates how poor a guide is the signal of a nonselective detector for the estimation of the degree of separation achieved in displacement chromatography. The individual band profiles and the chromatogram obtained are similar to those in Figure 12.2c. An incomplete isotachic train is illustrated in Figure 12.18, showing an attempt at the purification of cephalosporin C [40]. Clearly, the first few bands which overlap massively are not part of the train being formed (see Figures 12.4) but are a result of the operating line lying above the watershed point of these components. Strangely, the component whose peak is eluted just before cephalosporin C in the analytical chromatogram seems to have disappeared from the reconstructed displacement chromatogram.
12.2.3 Separation of Peptides and Proteins Figure 12.19 illustrates the separation of peptides by displacement chromatography and the influence of temperature on the equilibrium isotherms of peptides and on the formation of the isotachic train [11]. In Figure 12.19a, the formation of the isotachic train appears complete. The chromatogram is again quite similar to the calculated one in Figure 12.4a, with the first component band migrating through the column in the elution mode and the other two bands in the displacement mode. Again, the displacer concentration is near or at the watershed point of
594
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
VOLUME (ml)
VOLUME (ml)
Figure 12.19 Separation of dipeptides and effect of the temperature. Left: Separation of 20 mg of Val-Val, 30 mg of Gly-Leu, and 30 mg of Leu-Val in 1.0 mL carrier at 30°C. Column: 2 x (250 x 4.6mm) packed with 5 ^m Spherisorb-ODS; carrier: 50 mM phosphate buffer; pH 2.0; displacer: 150 mM tetrabutylammonium bromide in the carrier; flow rate: 0.72 mL/min; fraction volume: 72 ]iL. Center Separation of two dipeptides at 30°C; same conditions as in left figure, except one column; carrier: pH 6.0; displacer: 200 mM benzyltributylammonium chloride in carrier; flow rate: 0.5 mL/min; fraction volume: 100 jiL; feed: 25 mg Met-Pro and 225 mg Phe-Pro in 1.0 mL carrier. Right Separation of the same feed at 60°C, under the same conditions as in center figure. Reproduced from Cs. Howdth, J. Frenz and Z. El Rassi, J. Chromatogr., 255 (1983) 273 (Figs. 5 and 15).
the first component. In Figures 12.19b,c, an increase in the displacer concentration causes an isotachic train to form. However, due to the high displacer concentration relative to the amount injected, the separation factor, and the retention factors, very narrow plateaus and wide mixed zones are obtained. Under these conditions, overdisplacement threatens at 30° C. An increase of the column temperature results in a decrease of the initial slope of the adsorption isotherms and faster mass transfer kinetics. This permits an improved separation at 60°C. Overdisplacement is the phenomenon that occurs when the displacer front moves too fast, and the component bands remain entangled in the displacer front because there is insufficient space between the holdup volume and the displacer breakthrough volume to accommodate the isotachic train. Similarly, isotachic trains were reported in the separation of the reagents and intermediate products in the synthesis of the tripeptide arginine-methionine-leucine (Arg-Met-Leu) at two different stages, the preparation of Arg-Met, and that of Arg-Met-Leu [52]. Some of the problems encountered in the separation of peptides by displacement chromatography are illustrated in Figures 12.20 and 12.21 showing chromatograms obtained in the purification of a- and j6-MSH [48]. In the former case, the displacement chromatogram is unusual. The isotachic train has not formed, since there is no plateau concentration. The column used was too short and/or the displacer too dilute. However, in this particular case, the displacer solution had to be rather dilute to avoid the formation of micelles. It is interesting to observe that, in spite of the unfavorable experimental conditions, the mixed zones between the band of a-MSH and its two neighbors are very narrow. In Figure 12.21, the dis-
12.2 Applications of Displacement Chromatography
8
595
0.2
tvi
£ a
UJ
z
"1 i;
0.1
«£ 01 IE
-MSH-S.0,1
X
8
m <
4
8
12
16
8 o
MINUTES
FRACTION (O.I ml)
Figure 12.20 Chromatograms of crude «-MSH. (a) Analysis of crude a-MSH by isocratic elution. 250x4.6 mm column packed with 5 um KV-ODS. Mobile phase: 0.25% formic acid, 0.5% triethylamine, and 24% acetonitrile in water; flow rate: 1.0 mL/min; temperature: 22°C; sample size: 20 jig. Peak 1= unknown, 2= a-MSH-sulfoxide, 3= acetyl-a-MSHsulfoxide, 4= a-MSH, 5= acetyl-a-MSH. (b) Displacement of crude a-MSH. Same column and temperature as in (a); carrier: 0.25% formic acid, 0.5% triethylamine, and 19% acetonitrile in water; displacer: 50mM BMDA in an aqueous solution of 0.25% formic acid, 0.5% triethylamine, and 21% acetonitrile; flow rate: 0.1 mL/min; fraction size: 0.1 mL; feed: 35 mg crude a-MSH. Reproduced with permission from G. Viscomi, S. Lande and Cs. Horvath, } . Chromatogr., 440 (1988) 157 (Figs. 1 and 3).
placement chromatogram of jS-MSH shows bands of impurities before and after the main component band. The recovery yield of /3-MSH was 70% [48]. E 0.12 c O CVJ
0.08 UJ
I
CO
g
0.04
en
m < 4
8
MINUTES
"60
64
68
72
76
80
84
FRACTION (0.1ml)
Figure 12.21 Separation of crude /5-MSH. (a) Analysis. Same column and temperature as in Figure 12.20; Linear gradient from 25% to 43% eluent B in A over 2.5 min; eluent A: 0.25% formic acid and 0.5% triethylamine in water; eluent B: acetonitrile; flow rate: 0.1 mL/min; sample size: 20 fig of crude /5-MSH. (b) Displacement. Same column and temperature as in Figure 12.20; carrier: 0.1% trifluoroacetic acid in water; displacer: 50 mM BMDA in an aqueous solution containing 0.1% trifluoroacetic acid and 21% acetonitrile; flow rate: 0.1 mL/min; fraction size: 0.1 mL; feed: 30 mg of crude /3-MSH. Reproduced with permission from G. Viscomi, S. Lande and Cs. Horvath,}. Chromatogr., 440 (1988) 157 (Figs. 4 and 5).
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
596 0.10
< 111 0.05
0.00
0
20
40
60
SECONDS
Figure 12.22 Effect of temperature and flow rate on the separation of melittin. (a) Analysis on a 30x4.6 mm column packed with C18 pellicular silica. Gradient of acetonitrile, flow rate 3.0 mL/min, temperature: 80°C; sample: 5 jig melittin. (b) displacement chromatograms of crude synthetic P14A melittin at 23°C, 0.2 mL/min (A) and 40°C, 0.6 mL/min (B). Column: 105x4.6 mm pellicular C18 silica; carrier: 0.1% v/v TEA in water; displacer: 24 mM benzyldimethylhexadecyl ammonium chloride in 10% v/v aqueous ACN containing 1% TFA; feed: 10 mg of P14A melittin in 1.5 mL Fractions collected at 30 and 15 s, at 23°C and 40°C, respectively. Aliquots of 5 jih of each fraction were mixed with 95 fiL of 0.1% TFA and 20 }iL aliquots of the diluted samples were analyzed for melittin as in (a). Reproduced with permission from K. Kalghati, I. Fellegvdri and Cs. Horvdth,}. Chromatogr., 604 (1992) 47 (Figs. 3 and 4).
Melittin, the main component of bee venom, was analyzed by gradient elution (Figure 12.22a) and purified by displacement chromatography (Figure 12.22b) on micropellicular C18 silica [50]. An isotachic train at two different temperatures is shown. Note that the flow rate on the 30x4.6 mm analytical column was 3 mL/min, while on the 105x4.6 mm displacement column best results were obtained at only 0.6 mL/min, in spite of the fast mass transfer kinetics on pellicular sorbents [50]. The influence of the sample size and the displacer concentration on the formation of the isotachic train for j3-lactoglobulin A and B is illustrated in Figures 12.23a and b, respectively [54]. In Figure 12.23a, the isotachic train is obtained by reducing the sample size from 100 to 78 mg. In Figure 12.23b, the effect of increasing the displacer concentration is observed experimentally. The mass action model has been used to predict the elution profiles of artificial mixtures of proteins in displacement chromatography [60]. Figure 12.24 illustrates the profiles of a-chymotrypsinogen, cytochrome c, lysozyme, and sodium as the co-ion on a cation exchange column, with protamine as the displacer. Overall, the theory and experimental results are in good agreement. The experimental results show more overlap and diffuse boundaries than predicted. This is not surprising because the steric mass action isotherm model (Chapter 4, Section 4.1.12.2) is used within the context of the ideal model. It is also interesting that the experimental results show displacement of the sodium co-ion just before the breakthrough of the first band, a-chymotrypsinogen.
597
12.2 Applications of Displacement Chromatography
10
W FRACTION NUMgER
Figure 12.23 Influence of the sample size and the displacer concentration on the displacement of ^-lactoglobulins A and B. Column: TSK DEAE 5-PW, 75x7.5 mm; carrier: 25 mM phosphate, pH 7.0; displacer: 10 mg/mL of chondroitin sulfate in the carrier; flow rate: 0.1 mL/min; temperature: 22°C; fraction volume: 200 }iL; sample volume: 4 mL. (a) Feed: 100 mg LAC. (b) Same as in (a), except feed, 78 mg LAC. (c) Same as in Figure 12.22a, except: displacer concentration: 20 mg/mL and sample size: 62 mg LAC. (d) Same as in (a), except: displacer concentration: 3 mg/mL and sample size: 70 mg LAC. Reproduced with permission from A.W. Liao, Z. El Rassi, DM. LeMaster and Cs. Horvdth, Chromatogmphia, 24 (1987) 881 (Figs. 2 and 3). 3.S
90 A
Sodium [_.
Protanme —i
,-
2.5
;
2 I.S 1
:
11
^ CylC | o-Chymo
;' \ j
0.5
,
80 70 60 50 40
3.3
5 1I I I
B
3
10 I
1
2.S
Lysc
:
IS
1 a-Chymo
O.S
ft
0
2
4
Elution Volume (ml)
6
Protaame.
L-,
30 O
20 I
Sodun
0
2
4
T,
;
6
Elution Volume (ml)
Figure 12.24 Displacement chromatogram of a protein mixture. (A) Experimental results. 10x100 mm Protein-Pak SP-8HR cation exchange column; eluent, 12.5 mM Na2PO4; displacer, 15 mg/mL purified protamine. Feed, 10.78 mg a-chymotrypsinogen A, 7.86 mg cytochrome c, and 17.3 mg lysozyme in 1.7 mL. Fv 1 mL/min, 100 jtL fractions. (B) Displacement chromatogram calculated with the ideal and the steric mass action models. Reproduced with permission from } . Gerstner and S. Cramer, Biotechnol. Progr., 8 (1992) 540 (Fig. 4). ©1992, American Chemical Society.
Finally, reversed-phase is some times more useful than ion-exchange in displacement chromatography for the purification of proteins. Figure 12.25 shows the separation of recombinant brain derived neurotropic factor from its variants. This separation was an improvement over one obtained by ion-exchange, which could not remove the E. coli protein impurities. The analysis in panel (a) was performed on a column packed with 5 jim particles of C-4 bonded silica, at 65°C, using a complex TFA-water-acetonitrile gradient. The displacement separation in panel (b) was performed on a column packed with 10 fim ODS silica, with a carrier of 16% ACN and 0.15% TFA, at a flow rate of 0.1 mL/min, with a load of 79 mg of protein [63].
598
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
Figure 12.25 Displacement chromatogram recombinant brain-derived-neurotropic-factor from its variants labeled 1-5 and from £. coll protein impurities. Displaces 25mM tetrahexylammonium chloride, (a) Analytical Chromatogram of Feed, (b) Displacement Chromatogram, (c) Purity Histogram, w / w % . Reproduced with permission from K. M. Sunasam, R.G. Rupp, S. M. Cramer, Biotechnol. Progr. 17 (2001) 897 (Figs. 1, 6 and 7.).
Figure 12.26 Displacement chromatogram of 5'-AMP, 3'-AMP, 2'-AMP and adenosine. Column: 250x4.6 mm + 150x4.6 mm packed with 5 jig Supelcosil ODS; carrier: 10 mM acetate buffer, pH 5.0; displacer: 0.28 M n-butanol in carrier; flow rate: 0.1 mL/min; fraction volume: 100 yL; temperature: 22°C; feed: 15 mg 5'-AMP, 6 mg 3'-AMP, 24 mg 2'-AMP and 15 mg adenosine in 1.5 mL carrier. Reproduced with permission from Cs. Horvdth, J. Frenz and Z. El Rassi, /. Chromatogr., 255 (1983) 273 (Fig. 4).
12.2.4 Separation of Nucleic Acid Constituents A fully developed isotachic train of inosine, deoxyinosine, adenosine, and deoxyadenosine is shown in Figure 12.7. The thickness of the mixed zones between the bands in this train is in qualitative agreement with what can be inferred re-
12.3 Comparison of Calculated and Experimental Results
599
garding the relative positions of the isotherms from the heights of the concentration plateaus. Figure 12.26 shows the separation by displacement chromatography of 5'-AMP, 3'-AMP, 2'-AMP/ and adenosine [11]. The isotachic train is incomplete but the mixed zones between successive bands are relatively narrow in this case.
12.3 Comparison of Calculated and Experimental Results Although a large number of papers has been published comparing experimental results with the band profiles calculated using the equilibrium-dispersive and several kinetic models in elution chromatography and in frontal analysis (see Chapters 10,11,14, and 16), there have been relatively few studies comparing experimental band profiles in displacement chromatography with the profiles calculated using the equilibrium-dispersive model and the equilibrium isotherms obtained through independent measurements. The application of the equilibriumdispersive model in this field has been investigated in less detail than in multicomponent elution. Undoubtedly, this is because there is no need in frontal analysis for a simpler approximate model. By contrast, in displacement chromatography the equilibrium or ideal model is still the most commonly used model. The equilibrium-dispersive model that has been used most successfully in overloaded elution lumps the contribution of the resistance to mass transfers in the column with axial dispersion in an apparent diffusion term. The few results published indicate that there is a good agreement, provided the mass transfer kinetics is relatively fast. The results of numerous calculations of band profiles in displacement chromatography have been published by Katti [64] and Katti and Guiochon [65], who elucidated the details of the change in the profiles during the formation of an isotachic train in the case of a wide, dilute injection. Two such comparisons are shown in Figures 12.27 and 12.28. Horvath et al. [25] published intermediate profiles in the separation of 3,4-dihydroxyphenylacetic acid from 4-hydroxyphenylacetic acid (Figs. 12.27 top three figures), as well as the single-component isotherms. These isotherms were used in a competitive Langmuir model, and a similar isotherm was assumed for the displacer. Good qualitative agreement between experimental and calculated chromatograms is seen in Figures 12.27 bottom three figures. Similar breakthrough times, band heights, and profiles are observed. The differences were ascribed to the inaccuracy of the competitive Langmuir model [59]. Frenz [59] reported the intermediate profiles in the formation of the isotachic train in the separation of resorcinol and catechol, displaced by n-propanol on an ODS silica column (Figure 12.28, top). Similar calculations using the equilibriumdispersive model, a competitive Langmuir isotherm derived from the single-component isotherm data reported [59] lead to the profiles in Figure 12.28 (bottom), where they are compared with the experimental results. There is a very good agreement between the retention volumes and concentrations measured and cal-
600
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
I
S 8
2
S
VOLUME [ml]
Figure 12.27 Comparison of experimental and calculated band profiles using the equilibrium-dispersive model. (Top) Experimental results, same conditions as for Figure 12.16, right, but larger feeds: (Left): 114 and 57mg. (Center): 57 and 114mg. (Right): 114 and 114mg. Reproduced with permission from Cs. Horvdth, A. Nahum and J.H. Frenz, f. Chromatogr., 218 (1981) 365 (Fig. 12). (Bottom) Band
profiles calculated from the isotherms in Figure 12.16, left, and other experimental data. Reproduced with permission from A.M. Katti and G. Guiochon, J. Chromatogr., 449 (1988) 25 (Fig. 10). VOLUME [ml]
culated. For example, the concentration of the rear plateau of the first component profile (Figures 12.28a,c) is calculated as measured, at 10 mg/mL; the breakthrough time of the first component front is calculated at 3.4 mL and measured at 3.1 mL. Both the calculated and the measured profiles of the second component band in the nonisotachic train (Figures 12.28a,c) exhibit two maxima. However, the initial peak of the first component is calculated at 50 mg/mL and measured at 35 mg/mL, but the band tip was probably diluted by the large volume of the
12.3 Comparison of Calculated and Experimental Results 60
1
I
601 I i
i
i I i
i
.20
12
s-
;
d
?•
s sG-
1
<*-
I
2
J
4
5
6
VQUUUE (ml)
7
E
3
a
a
VOLUME (ml)
Figure 12.28 Comparison of experimental and calculated band profiles using the equilibrium-dispersive model. Top: Experimental results on a 0.46 cm i.d. column packed with 5}iM ODS silica, water as carrier; water with 48 mg/mL n-propanol as displacer, at 0.2 mL/min; feed, 50 mg each of resorcinol and catechol in 1 mL water; fraction volume: 250 }iL. (a) Column length, 25 cm. (b) Column length, 50 cm. Reprinted with permissionfromJ.H. Frenz, PhD Dissertation, Yale University, 1983 (Figs, lib and 12b). Bottom: band profiles
calculated from the isotherm and efficiency data available [59,66]. (c) Column length, 25 cm. (d) Column length, 50 cm. Reprinted from G. Guiochon et al., Talanta, 36 (1989) 19 (Fig. 16).
collected fractions. Also the height of the bands in the near isotachic train (Figures 12.28b,d) is slightly higher in the experimental profiles than in the calculated ones. Bellot and Condoret compared experimental results obtained in the displacement of solutions of resorcinol and catechol by phenol with those calculated using the equilibrium-dispersive model and different isotherm models [67]. Figure 12.29 compares the experimental results in the case of a 1:1 mixture with those calculated with four different isotherm models, the LeVan-Vermeulen model (Figure 12.29a), the quadratic isotherm with three floating parameters (Figure 12.29b), the classical Langmuir competitive model with coefficients derived from the singlecomponent Langmuir isotherms (Figure 12.29c), and the competitive Langmuir
602
Frontal Analysis, Displacement and the Equilibrium-Dispersive Model
Figure 12.29 Comparison of theoretical and experimental displacement separations of resorcinol and catechol by phenol. Calculations using the equilibrium-dispersive model, the LeVan- Vermeulen isotherm model, and single-component adsorption data. Experimental results on a 4.6x250 C18 Nucleosil 5 fim column, F = 0.4; carrier, water, Fv = 0.2 mL/min, T = 20°C; 1:1 mixture, V;n; = 0.5 mL; displacer, 80 g/L phenol in water; Lt\ = 30%, Le2 = 16.5%. (a) Calculation with LeVan-Vermeulen isotherm, (b) Calculation with quadratic isotherm, three floating parameters, (c) Calculation with competitive Langmuir isotherm, single-component isotherm parameters, (d) Calculation with Langmuir isotherm, best adjusted parameters. Reproduced with permission from J.C. BellotandJ.S. Condoret,J. Chromatogr., 657 (1994) (Figs. 3c, 4c, 6c, 8c) 305.
isotherm with parameters adjusted to minimize the deviations between experimental and calculated isotherms (Figure 12.29d). In general, the experimental profiles appear more diffuse than the calculated profiles, which could be explained in part by the inadequacy of the equilibrium-dispersive model at high concentrations (Chapter 14, Section 14.1.7). The different isotherm models give quite different calculated profiles. The
REFERENCES
603
LeVan-Vermeulen isotherm (Figure 12.29a) gives the best results, with the quadratic isotherm a close second. Part of the explanation for the success of the LeVanVermeulen isotherm comes from the facts that the column saturation capacity is different for the two diphenols and that the separation factor increases with increasing concentration, making the analytical separation more difficult than the preparative separation. The profiles calculated with the quadratic isotherm (Figure 12.29b) are in poor agreement with experimental results in the mixed zone between the first and second components. The competitive Langmuir isotherm (Figure 12.29d) with adjusted parameters predicts rather well the retention times and front profiles of both components but not the intermediate parts of the profiles. Finally, the classical Langmuir isotherm predicts no separation and gives poor results. This last result was expected because the competitive adsorption between two components that have different saturation capacities is too complex to be described by the Langmuir model. It should be noted that none of the isotherm models account well for the displacement profiles. This shows the complexity of competitive interactions, particularly at high concentrations.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
D. O. Cooney, E. N. Lightfoot, Ind. Eng. Chem. (Fundam.) 4 (1965) 233. H.-K. Rhee, B. F. Bodin, N. R. Amundson, Chem. Eng. Sci. 26 (1971) 1571. H.-K. Rhee, N. R. Amundson, Chem. Eng. Sci. 27 (1972) 199. H.-K. Rhee, N. R. Amundson, Chem. Eng. Sci. 28 (1973) 55. H. K. Rhee, R. Aris, N. R. Amundson, Chem. Eng. Sci. 29 (1974) 2049. W. J. Thomas, J. L. Lombardi, Trans. Inst. Chem. Eng. 49 (1971) 240. Z. Ma, G. Guiochon, J. Chromatogr. 603 (1992) 13. J. Zhu, Z. Ma, G. Guiochon, Biotechnol. Progr. 9 (1993) 421. J. Zhu, G. Guiochon, J. Chromatogr. 636 (1993) 189. J. Frenz, Cs. Horv&th, in: Cs. Horv&th (Ed.), High-Performance Liquid Chromatography — Advances and Perspectives, Vol. 5, Academic Press, New York, NY, 1988, pp. 211-314. Cs. Horvath, J. Frenz, Z. El Rassi, J. Chromatogr. 255 (1983) 273. A. M. Katti, G. Guiochon, C. R. Acad. Sci. (Paris) 309(11) (1989) 1557. R. Ramsey, A. M. Katti, G. Guiochon, Anal. Chem. 62 (1990) 2557. S. C. Jacobson, S. Golshan-Shirazi, A. M. Katti, M. Czok, Z. Ma, G. Guiochon, J. Chromatogr. 484 (1989) 103. A. M. Katti, R. Ramsey, G. Guiochon, J. Chromatogr. 477 (1989) 119. J. Frenz, J. Bourell, W. S. Hancock, J. Chromatogr. 512 (1990) 299. J. Frenz, C. P. Quan, W. S. Hancock, J. Bourell, J. Chromatogr. 557 (1991) 289. J. Zhu, A. M. Katti, G. Guiochon, Anal. Chem. 63 (1991) 2183. S. C. D. Jen, N. G. Pinto, Reactive Polymers 19 (1993) 145. F. D. Antia, Cs. Horva"th, J. Chromatogr. 556 (1991) 119. M. D. LeVan, T. Vermeulen, J. Phys. Chem. 85 (1981) 3247. G. Carta, A. Dinerman, AIChE J. 40 (1994) 1618. F. H. Spedding, A. F. Voight, E. Gladrow, N. R. Sleight, J. E. Powel, J. M. Wright, T. A. Butler, P. Figard, J. Am. Chem. Soc. 69 (1947) 2786. F. H. Spedding, E. I. Fulmer, T. A. Butler, J. E. Powel, J. Am. Chem. Soc. 72 (1950) 2349. Cs. Horvath, A. Nahum, J. H. Frenz, J. Chromatogr. 218 (1981) 365.
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[26] Cs. Horvdth, in: R Bruner (Ed.), The Science of Chromatography, Elsevier, Amsterdam, 1985, p. 179. [27] A. A. Shulda, K. A. Barnthouse, S. W. Bae, J. A. Moore, S. Cramer, J. Chromatogr. A 814 (1998) 83. [28] K. Rege, S. Hu, J. Moore, J. Dordick, S. Cramer, J. Am. Chem. Soc. 126 (2004) 12306. [29] F. H. Spedding, E. I. Fulmer, J. E. Powel, T. A. Butler, J. Am. Chem. Soc. 72 (1950) 2354. [30] B. J. Mair, A. Gaboriault, F. Rossini, Ind. Eng. Chem. 39 (1947) 1072. [31] A. Torres, S. Edberg, E. Peterson, J. Chromatogr. 389 (1987) 177. [32] A. Torres, E. Peterson, J. Chromatogr. 604 (1992) 39. [33] S. Ghose, B. Mattiasson, J. Chromatogr. 547 (1991) 145. [34] G. Jayaraman, S. Gadam, S. Cramer, J. Chromatogr. 630 (1993) 53. [35] S. D. Gadam, G. Jayaraman, S. Cramer, J. Chromatogr. 630 (1993) 37. [36] S.-C. Jen, N. G. Pinto, J. Chromatogr. Sci. 29 (1992) 478. [37] H. G. Boman, in: A. Neuberger (Ed.), Symposium on Protein Structure, IUPAC, Paris, 1957, Meuthen, London, 1958, p. 100. [38] J. Gerstner, S. Cramer, Biopharm. (1992) 42. [39] G. Subramanian, S. Cramer, J. Chromatogr. 484 (1989) 225. [40] G. Subramanian, M. Phillips, S. Cramer, J. Chromatogr. 439 (1988) 341. [41] S.-C. D. Jen, N. G. Pinto, J. Chromatogr. 519 (1990) 87. [42] Y. J. Kim, S. M. Cramer, J. Chromatogr. 549 (1990) 87. [43] G. Carta, M. S. Saunders, J. P. DeCarli, J. B. Vierow, AIChE Symp. Ser. 84 (1988) 54. [44] J. Frenz, Cs. Horvath, AIChE J. 31 (1985) 400. [45] S. Fujine, K. Saito, K. Shiba, Separat. Sci. Techol. 18 (1983) 15. [46] H. Kalasz, Cs. Horvath, J. Chromatogr. 239 (1982) 423. [47] Gy. Vigh, G. Quintero, Gy. Farkas, J. Chromatogr. 484 (1989) 237. [48] G. Viscomi, S. Lande, Cs. Horvath, J. Chromatogr. 440 (1988) 157. [49] G. Viscomi, A. Ziggiotti, A. Verdini, J. Chromatogr. 482 (1989) 99. [50] K. Kalghati, I. Fellegvari, Cs. Horvath, J. Chromatogr 604 (1992) 47. [51] F. Cardinali, A. Ziggiotti, G. C. Viscomi, J. Chromatogr. 499 (1990) 37. [52] S. Cramer, Cs. Horvath, J. Prep. Chromatogr. 1 (1988) 29. [53] Gy. Vigh, Z. Varga-Puchony, G. Szepesi, M. Gazdag, J. Chromatogr. 386 (1987) 353. [54] A. W. Liao, Z. El Rassi, D. M. LeMaster, Cs. Horvath, Chromatographia 24 (1987) 881. [55] A. W. Liao, Cs. Horvath, Ann. N. Y. Acad. Sci. 589 (1990) 182. [56] H. Kalasz, Cs. Horvath, J. Chromatogr. 215 (1981) 295. [57] K. Valko, P. Slegel, J. Bati, J. Chromatogr. 386 (1987) 345. [58] S. Claesson, Rec. Trav. Chim. 571 (1946) T65. [59] J. Frenz, Ph.D. thesis, Yale University (1983). [60] J. Gerstner, S. Cramer, Biotechnol. Progr. 8 (1992) 540. [61] F. H. Spedding, J. Powel, J. Am. Chem. Soc. 76 (1954) 2550. [62] Gy. Vigh, Gy. Farkas, G. Quintero, J. Chromatogr. 484 (1989) 251. [63] K. M. Sunasara, R. G. Rupp, S. M. Cramer, Biotechnol. Progr. 17 (2001) 897. [64] A. Katti, Z. Ma, G. Guiochon, AIChE J. 36 (1990) 1722. [65] A. M. Katti, G. Guiochon, J. Chromatogr. 449 (1988) 25. [66] G. Guiochon, S. Ghodbane, S. Golshan-Shirazi, J.-X. Huang, A. Katti, B. Lin, Z. Ma, Talanta 36 (1989) 19. [67] J. C. Bellot, J. S. Condoret, J. Chromatogr. 635 (1993) 1.
Chapter 13 System Peaks with the Equilibrium-Dispersive Model Contents 13.1 System Peaks in Linear Chromatography 13.1.1 General Experimental Results on System Peaks 13.1.2 Theory of System Peaks 13.1.3 Indirect Detection Using System Peaks 13.1.4 Application of System Peaks to Analyte Peak Compression 13.1.5 Vacancy Chromatography 13.2 High-Concentration System Peaks 13.2.1 High-Concentration System Peaks for a Single-Component Sample 13.2.2 High-Concentration System Peaks for a Two-Component Sample
606 607 609 618 622 623 626 627 638
References
647
Introduction In many cases, the mobile phase used in liquid chromatography is not a pure solvent but a mixture containing compounds that are retained and thus equilibrate between the stationary and the mobile phase. Sample injection causes a perturbation of this equilibrium when the additive concentration is finite and its isotherm is nonlinear. Because of this perturbation, a number of additive peaks is generated. These peaks can be used, for example, to detect sample components with a detector that does not respond to a change of their concentrations (Section 13.1.3). Feed injection in preparative chromatography causes major perturbations of the additive concentrations. These perturbations are coupled with the profiles of the bands of feed components. The result may be component bands exhibiting most unusual profiles (Section 13.2.2). This phenomenon may cause a considerable decrease of the production rate. Its proper use, however, may permit the enhancement of the production rate and the recovery yield. Thus, the study of system peaks, although seemingly esoteric, has direct and important applications. When some of the components of the mobile phase are adsorbed by the stationary phase, the chromatograms of even small samples are not predicted correctly by linear chromatography. Then the sample injection causes a perturbation of their equilibrium, which is not properly described by linear chromatography unless the equilibrium isotherms of these adsorbed components of the mobile phase are linear. There are basically two types of situations that may arise. They are typically exemplified by normal and reversed phase chromatography. hi the for605
606
System Peaks with the Equilibrium-Dispersive Model
mer method, a polar stationary phase (e.g., silica) is used, and the mobile phase is a low-polarity solvent (e.g., n-heptane, toluene, dichloromethane). The analytes distribute between the two phases [1]. To reduce the retention of strongly polar analytes, a certain proportion of a "stronger" solvent (e.g., acetone, ethyl acetate, butanol, methanol) is added to the low-polarity (or "weak" solvent). The concentration of the strong solvent is usually low, 0.5 to 5%. The strong solvent acts by competing with the analytes, occupying a relatively large fraction of the adsorbent surface. Because the strong solvent is adsorbed as strongly as or even, in some cases, more strongly than the analyte, the retention of the analytes is reduced. In reversed phase chromatography, a nonpolar adsorbent (e.g., silica chemically bonded to an n-octadecyl dimethyl siloxyl group) is used as the stationary phase with a polar solvent, often an aqueous solution of methanol or acetonitrile. The organic solvent is distributed between the adsorbent and the mobile phase, as shown first by Knox and Pryde [2]. The selective extraction of the organic solvent (the modifier) into the stationary phase has been demonstrated experimentally [3-7]. Although the addition of strong solvent to the mobile phase has the same effect in normal and reversed phase chromatography, i.e., the retention of the analytes decreases with increasing concentration of strong solvent, the mechanism is different. Methanol and acetonitrile are only weakly adsorbed on the stationary phase. They compete only slightly with the analytes for access to the adsorbent surface. The solubility of the analytes in the mobile phase controls their retention. This solubility is often poor in pure water and increases rapidly with increasing concentration of the organic solvent. For a more detailed discussion of the effect of the organic modifier or modulator, see Chapter 15, Section 15.2. In spite of these differences, the phenomenon of system peaks arises in all cases. The injection of the sample perturbs the equilibrium between the mobile phase (additives included) and the stationary phase. This perturbation generates a series of concentration signals, as we discuss in this chapter. There is no difference in nature but a strong difference in degree between the phenomena that take place in normal phase and in reversed phase chromatography. System peaks are much more important in the former case, in ion-pair, or in ion-exchange chromatography than in conventional reversed phase chromatography. When a small sample is injected, the problem can be considered as a mere perturbation of the phase equilibrium, and simple solutions are easily derived. When large samples are injected, the elution profiles are more complex, sometimes surprisingly so. Thus, a separate discussion of these problems in linear and nonlinear chromatography is in order. Note that system peaks arise only when chromatography is carried out under conditions that, although they may be linear for the analytes, are not linear for the additive(s).
13.1 System Peaks in Linear Chromatography When the sample size is small, and the equilibrium isotherms of the sample components can be considered linear in the range of concentrations achieved for these components, the chromatographic phenomenon remains linear regarding all the
13.1 System Peaks in Linear Chromatography
607
analytes. The behavior of the mobile phase additives is not linear but can be treated using perturbation theory.
13.1.1 General Experimental Results on System Peaks System peaks arise when one or several components of the mobile phase are adsorbed on the stationary phase, and because their concentrations are such that their adsorption isotherms are not linear. Injection of the sample in the multicomponent mobile phase perturbs the equilibrium of the strong solvent(s) or other additives of the mobile phase, which is nonlinear. As a result of this perturbation, more peaks may be recorded than there are components in the injected sample. For example, the injection of a sample of the pure weak solvent may generate as many peaks as there are additives in the mobile phase, if these additives are all adsorbed by the stationary phase. These extra peaks have received a variety of names in the early literature. They have been called system peaks, pseudopeaks, ghost peaks, eigen peaks, vacancy peaks, induced peaks, etc. We call them system peaks. Note that not all system peaks are recorded; this depends on the type of detector used. Also, system peaks may be positive or negative, depending on the experimental conditions. Because of the perturbation caused to the equilibrium of the mobile phase additives between the stationary and the mobile phases, the composition of the mobile phase that moves together with each solute differs from the composition of the mobile phase equilibrated with the column; it depends on the solute concentration. The resulting excess or deficit of each of the mobile phase components outside the sample zone migrates along the column at a velocity that is determined by the concentration of the corresponding additive and by its characteristics. Thus, if the mobile phase contains p additives (or strong solvents) dissolved in a weak solvent, and we inject a sample containing n analytes, we obtain for each additive n + p extra peaks, n of these peaks will move with the analyte zones, at their velocities, and will appear with the analyte elution peaks. They can be detected, however, only if the detector responds to changes in the concentration of the additives and not to changes in the analyte concentration. The other p peaks move at velocities determined by the concentration and properties of the additives. They can be detected only if the detector responds to changes in the concentration of the additives. Unfortunately, because of detection problems, only these p peaks are recognized in most publications and they are called the system peaks, while the other n peaks are most often ignored, except in indirect detection. In this work we call all the n + p peaks, system peaks. To distinguish between them when necessary, we call the n system peaks that move with the analyte peaks the analyte system peaks. The other p system peaks that move at velocities depending on the additive concentrations and characteristics are called the additive system peaks, or more simply the system peaks. Although a general mathematical treatment of system peaks was published very early by Helfferich and Klein [8], most early experimental work ignored these theoretical explanations. The first experimental observation of system peaks was probably reported by Fornstedt and Porath [9], while Solms et al. [10] made the
System Peaks with the Equilibrium-Dispersive Model
608
O
1 If
2
O
1
2
3O
I
2
30
1
Figure 13.1 Chromatograms illustrating displacement and vacancy peaks of ACN arising from the injection of solutes with increasing retention volumes. Mobile phase: ACN- H2O (33:67, v/v). RI detection. (A), ACN vacancy band from injection of 10 }i\ of 0.6% (v/v) MeOH dissolved in bulk mobile phase. (B), ACN displacement band from injection of 10 }i\ of 0.6% (v/v) THF in mobile phase. (C), ACN displacement band from injection of 10 fd of 0.6% (v/v) benzyl alcohol in mobile phase. (D), ACN vacancy from injection of 0.6% (v/v) benzene in mobile phase. Reproduced with -permission from R. M. McCormick and B. L. Karger,]. Chromatogr., 199 (1980) 259 (Fig. 4).
first systematic investigation. These authors used a polar adsorbent, a binary mobile phase, and a refractive index detector. Upon injection of a sample of a pure component, two peaks were recorded. The retention volume of the system peak observed was found to be equal to the retention volume of the more polar component of the mobile phase, when injected separately. Solms et al. [10] suggested that the system peak was generated by the displacement of solvent molecules from the stationary phase by the injection of the sample components. They also calculated the behavior of system peaks in the case of a binary mobile phase, using the plate model of chromatography (Chapter 6). The results of this calculation was in qualitative agreement with their experimental results. The displacement of solvent molecules from the stationary phase upon sample injection and the migration along the column of the ensuing vacancy were discussed by several authors as a reason for the occurrence of system peaks [11,12]. McCormick and Karger [12] have investigated system peaks in reversed phase liquid chromatography using a 33:67 (v/v) acetonitrile-water mixture as the mobile phase and a refractive index detector. An example of their results is shown in Figure 13.1. When a component less retained than acetonitrile {e.g., methanol) is injected, a vacancy {i.e., negative) system peak appears. When a compound more hydrophobic, and more retained, than the organic modifier {e.g., THF or benzyl alcohol) is injected, this causes a displacement of that modifier and results in a positive system peak. The surprising result in this case is the occurrence of a vacancy system peak upon injection of strongly retained compounds, such as benzene. McCormick and Karger [7] explained this unexpected phenomenon by assuming that the presence of the strongly hydrophobic solutes modifies locally
13.1 System Peaks in Linear Chromatography
609
the character of the stationary phase, favoring a net influx of the organic modifier into the layer of bonded organic phase and swelling it, possibly solvating the adsorbed analyte and the n-alkyl chains. The consequences of this swelling overshadow the displacement of the organic modifier due to sorption, and the net result is a vacancy peak. Preferential solvation of the analyte has been abundantly discussed [13,14] as another possible source of system peaks. According to Melander et al. [14], the preferential solvation of analyte molecules by one of the mobile phase components would be the first source of system peaks. Because of this preferential solvation of the analytes, the mobile phase is enriched in organic modifier in the vicinity of the analyte molecules compared to the bulk concentration. In other places, the modifier concentration becomes lower. Then, a zone of mobile phase depleted in the solvating species will appear, move down the column, and its composition will be determined by the appropriate sorption isotherm and by the magnitude of the disturbance. According to Melander et al. [14], the second cause leading to system peaks is the displacement of the solvent molecules bound to both the stationary phase and the analyte, upon binding of the analyte and the stationary phase. They developed a solvation model to relate the maximum concentration of the system peak to the retention factor and to the molecular structure of the analyte, and they used this model to gain, from the size of the system peaks, some quantitative information on the solvation of solutes in the mobile phase, as well as on the solvation effects associated with reversible binding of the analyte to the stationary phase. Ion interaction in the stationary phase [15] and intramolecular hydrophobic interactions [16] have been mentioned as other possible reasons for the occurrence of system peaks. Knox and Kaliszan [17] have discussed the origin of system peaks in their classical investigation of the methods of void volume determination.
13.1.2 Theory of System Peaks A general mathematical treatment of system peaks and of the closely related method of vacancy chromatography was given by Helfferich and Klein [8]. This work includes a detailed analysis of the phenomena that take place upon injection of a sample into a chromatographic column. It is based on the use of the solution of the ideal model of chromatography for multicomponent systems, with competitive Langmuir isotherms (see Chapters 8 and 9), and of the ft-transform. Riedo and Kovats [18,19] gave a detailed analysis of system peaks and derived equations giving the retention volume of the system peaks for isothermal, isocratic, linear chromatography with a binary or a ternary mobile phase. These equations are independent of the retention mechanism and recognize the basic thermodynamic origin of system peaks. They applied the general equation for a binary mobile phase to special cases of gas-liquid and liquid-solid chromatography and derived equations relating the retention times of pulses of the two labeled components of the mobile phase and the retention time of the system peak to the conventional chromatographic parameters. They also applied the general equation for ternary systems to special cases of gas-liquid chromatography and
610
System Peaks with the Equilibrium-Dispersive Model
derived equations giving the retention times of labeled components of the ternary mobile phase and of the two system peaks. 13.1.2.1 Theoretical Model of the System Peak Experiments We have discussed the theory of system peaks in linear1 chromatography [20]. The discussion is based on the use of the equilibrium-dispersive model. The mass balance equations are written for the n components of the sample and for the p additives2 ^
3
^
^
i = l,2,.:,n
(13.1)
d2q The concentrations in the stationary, q^, and the mobile phase, Q , are related by the equations of the competitive equilibrium isotherms of these components. In each particular case, the relevant isotherm model must be used. For a general discussion, however, it is convenient to use the competitive Langmuir isotherm model p
i+E jt=i
The initial condition is different for the sample components and for the mobile phase additives. For the former Q(t = 0 , z ) = 0
(13.4)
For the latter Cj(t = O,z) = C<j
(13.5)
where the concentrations C? are those of the mobile phase pumped into the column. Similarly, the boundary conditions are different for the components and for the additives. For the former Q(*,z = 0) = j
O'
.f
P
h < t
(13.6)
For the latter
( q C;(U = 0) = |
c
i
if 0 < t
~
where O is the additive concentration in the sample. 1 2
Linear by respect to the analyte concentrations. Same parameter definition as in the previous chapters.
(13.7)
13.1 System Peaks in Linear Chromatography
611
Other conditions can be written for other problems. For example, in vacancy chromatography there is no sample, and the initial condition (Eq. 13.5) and the boundary conditions (Eq. 13.7) with Cf — C? — 0 are applied to all the additives in the mobile phase. The system of Eqs. 13.1 to 13.7 is very similar to the other systems discussed in Chapters 10 to 12 and is solved using the same numerical methods. The only difference is in the boundary conditions (as in actual chromatographic experiments, the column can be the same, the experimental conditions are changed). 13.1.2.2 Theoretical Properties of System Peaks for Small Perturbations The model used here assumes that the system has n + p + 1 components; the sample has n components, while the mobile phase contains p additives or strong solvents in solution in a main, weak solvent. These strong solvents or additives are retained if injected in the pure weak solvent. They compete with the sample components for interaction with the stationary phase. We assume in this section that the perturbation caused by the injection is small. Under constant temperature and pressure, the sample injection causes a perturbation to the stationary phase concentration of the fcth component of the system (mobile phase additive or sample component). This perturbation is given by the following expansion: dqk = l ^ d Q + ^ d C
2
+ •••
k = 1, 2, ..., n + p.
(13.8)
There are n + p equations like Eq. 13.8, one for each of the n components of the sample and for each of the p additives. For small perturbations such as those encountered in linear or in analytical chromatography, the derivatives can be evaluated at the initial, unperturbed, steady state, that is, at zero concentration for the sample components, and at their initial concentration, C9, for the additives. The differential element in Eq. 13.8 is replaced by the finite perturbation, ACk, and the equation becomes .-.
(13.9)
Cj=c°
k
=
1, 2, • • •, n + p
(13.10)
This set of equations can be written in matrix form Aq = pAd
(13.11a)
with H
(13.11b)
612
System Peaks with the Equilibrium-Dispersive Model Figure 13.2 Matrix of the differentials of the isotherm equations. n + 1 ••• n + j
0
0
0
0
0
0
i
0
« _l_ 1
Cfa+l
»f«+l
°'/«+l
oCj
aCj
aCn
^
n + p
...
^ = ± £ •••
"It.
...
••• n + p
«C H + i
^
The matrices Ay and 4C are (n + p) x 1 column vectors. They represent the perturbations caused by the injection of the sample to the compositions of the stationary and the mobile phases at equilibrium, respectively. These perturbations are the changes of the concentrations of the n + p components of the chromatographic system. The matrix /$ is an (n + p) x {n + p) square matrix, and its general element is dq^/dQ. b is a 1 x (n + p) row vector. Its elements are the n + p partial derivatives of the stationary phase concentration q^ of thefcthcomponent of the system (sample components and additives included), by respect to the mobile phase concentrations. All the matrix elements are derived by differentiation of the isotherm equations (Eq. 13.3). They are calculated for the initial composition of the mobile phase, before the perturbation is applied. There are four different blocks in the matrix j3 (see Figure 13.2). They can be considered as two square matrices of dimensions n x n and p x p, respectively, and two rectangular matrices
23.2 System Peaks in Linear Chromatography
613
of dimensions n x p and p xn, respectively. The elements of the first square matrix (dimensions n x n) are the differentials dcji/dQ (I = 1,2, • • •, n) of the stationary phase concentration of one sample component with respect to the mobile phase concentration of another sample component. Since in analytical chromatography all the sample components are present only at infinite dilution, differentiation of Eqs. 13.3 gives cross-partial differentials that are zero. Only the diagonal elements of this matrix are different from 0. These nonzero elements are equal to «,-/(l + EjbjCj) = k'o /F, so are simply related to the apparent retention factor of each analyte. The second square matrix (dimensions p x p) contains the differentials dqj/dC^ (h = 1,2, • • • ,n) of the stationary phase concentration of an additive by respect to the mobile phase concentration of another additive. As shown by differentiation of Eq. 13.3, all these elements, diagonal or not, are different from zero. The elements of the first rectangular matrix (dimensions n x p) are the differentials dqi fdCj of the stationary phase concentration of a sample component by respect to the mobile phase concentration of an additive. As shown by differentiation of Eq. 13.3, all these differentials (which are given by dqi /dCj — —UibjCj / (1 + EjbjCj)2) are zero because the initial concentrations, Q, of the sample components are 0. Finally, the elements of the second rectangular matrix (dimensions p x n) are the differentials dqj/dCj = —ajb{Cj/(l + EjbjCj + U,-fo,-Q)2 of the stationary phase concentration of an additive by respect to the mobile phase concentration of a sample component. All these elements are different from zero. Accordingly, the perturbation caused by the injection to the concentrations of the sample components are uncoupled: Aqi=
[-J±\
\dCiJck=c1
ACt;
l
+ p;
l
(13.12)
where C£ is the initial concentration of the fcth component (e.g., 0 for all sample components). For any of the n sample components, only a single peak is observed on the chromatogram. This peak appears at the time corresponding to the retenfc' • tion factor of this compound at infinite dilution {k^ = 1 , ^.Q., where Cy is the concentration of the ;th additive). On the contrary, there is no decoupling for the additives since all the crosspartial differentials of their stationary phase concentrations are different from 0 (unless their initial concentration is negligibly small). The perturbation is given by ^=[|M-
ACk
with
l < j < p
(13.13)
fc=l \^kJck=C°k Therefore, if the chromatographic system has n + p constituents, n sample components, and p additives, the chromatogram will have n + p bands for each additive, even if the perturbation is very small and even if the additive is weakly adsorbed (e.g., methanol or acetonitrile in reversed phase HPLC). These bands are called the system peaks.
614
System Peaks with the Equilibrium-Dispersive Model
13.1.2.3 Case of a Single Additive We discuss further the chromatogram obtained for an n-component sample when using a binary mobile phase. The single additive is referred to in the following by the subscript s. Equation 13.12 applies unchanged. Accordingly, there is a single peak for each sample component. The retention factor is given by
,
_
0/a
~
K
Aqj_ _ t
AC~
fdq \ ?C
As expected in linear chromatography, there is one peak for each sample component, and this peak elutes at the same time as a pulse of the pure component in the same system. For a Langmuir isotherm, the retention factor is
Since we have only one additive, Eq. 13.13 reduces to V o c s / cs=C°
J Q=co
This equation is valid everywhere in the column, and particularly at the column exit, hence it can be used to describe the elution chromatogram. A perturbation Aqs of the stationary phase concentration of the additive takes place everywhere a perturbation of the mobile phase concentration of a sample component takes place. These perturbations migrate along the column at the constant component velocity (uZ/i = w/(l + k'Qi ), Eq. 7.3). When a sample component perturbation reaches the column exit, it is accompanied by the corresponding perturbation Aqs of the additive, which coelutes and is an analyte system peak. An additional perturbation Aqs accompanies the perturbation of the mobile phase concentration of the additive, ACS caused by the injection. However, this perturbation, which we have called the additive system peak, cannot be treated within the framework of linear chromatography unless the additive concentration is small enough and its equilibrium isotherm is linear. The additive perturbation moves at the velocity uz,s = M/(1 + k's), where k's is the apparent retention factor of the additive. k's is proportional to the slope of the additive isotherm at Cs = C° and is given by
K =F ^ \ocsyCs=cp
(13.17)
When the equilibrium isotherm of the additive is not linear, the retention factor of the additive is different from the capacity factor at infinite dilution in the weak solvent. For a Langmuir isotherm, we have (see Eq. 13.3)
= f/-,
,?^?
(13.18)
As explained in the previous subsection, there will be n + 1 system peaks. In many cases, the literature refers to the additive peak only as the system peak. However,
23.2 System Peaks in Linear Chromatography
615
all the n + 1 peaks result from the perturbation caused by the sample injection to the equilibrium of the additive between the two phases of the chromatographic system. They should all be called, collectively, system peaks. The system peaks are not detected by a detector that is selective for the sample components and does not respond to the additive, which is the general case in the analytical applications of chromatography, with the exception of indirect detection (see next subsection). This explains why they are generally forgotten. If a nonselective detector is used, we see only one extra peak, the one that is eluted at the time characteristic of the additive (retention factor k's). The n other peaks are eluted at the same time as the sample component peaks, interfere with them, and, in general, cannot be distinguished from them. In the case in which a detector selective for the single additive is used (e.g., in indirect detection), the « + 1 additive peaks are observed (see next subsection). Obviously, in the case of a detector responding to additives and sample components, the signal is the combined response, which might complicate the interpretation of the chromatograms, as illustrated later. It is possible to calculate from Eq. 13.16 the position, the sign, and the relative importance of the system peaks associated with the sample components if the competitive Langmuir model is assumed for the equilibrium isotherms of all the compounds involved, sample components and additives (Eq. 13.3). Since this isotherm model assumes that the column saturation capacity is the same for all the compounds (i.e., qs = d\/b\ = • • • = ai/ty = • • • = as/bs), the partial differentials of the additive can be written as follows: (1 + bsCs)2
KdCjJ Cs=co
1 + bsCs 1 + bsCs
F
where 9S = q/qs is the fraction of the adsorbent surface area that is covered by the adsorbed additive. Combination of Eqs. 13.16 and 13.19 gives -•••+*',
(13.20)
On the other hand, since the system peak coelutes with the ith sample component, we must have °' iap
^
(13.21)
and ACfr = 0 except for k = i Thus, there is a relationship between the concentration of the sample component and the perturbation of the additive concentration in the corresponding system peak (13.22a) or ACs
Osk
'°'iap
AC-
V —V
ZlCj
Ks
K•0,1'ap a
(13.22b)
616
System Peaks with the Equilibrium-Dispersive Model
Figure 13.3 Adsorption isotherms of resorcinol, catechol, and phenol on Lichrosorb RP-18 measured by FA and by system peak analysis. Reproduced with permission from S. Levin, S. Abu-Lofi, S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. A, 679 (1994) 213 (Fig. 3).
• •
0.6-
SPA experimental FA experimental fittoLaagnwirSPA SltoLangmuir FA
Cs0.40.2-
£*&
0.00.00
0.02
0.04
Cm
According to Eq. 13.22b, which was derived previously by Crommen et al. [21], the analyte system peak associated with a sample component corresponds to a positive fluctuation of the additive concentration if this component is eluted before the additive system peak {i.e., if k's > k'o ( ), and to a negative fluctuation of the additive concentration if the component peak is eluted after the additive system peak (if k's < k'o i ). We discuss the quantitative consequences of Eq. 13.22ab in the next subsection. Levin et al. [22] measured by FA the single-component adsorption isotherms of resorcinol, catechol, and phenol on a Lichrosorb RP-18 column from aqueous solutions. They also determined these isotherms by integration of the retention factor, k', of the system peak versus the additive concentration, C (see Eq. 13.23). A comparison of the two sets of results is shown in Figure 13.3. It shows a good agreement between the two sets of isotherms. 13.1.2.4 Numerical Calculation of System Peaks We have discussed in the previous subsections the calculation of the position and the sign of system peaks associated with analytes and additives, using the ideal model of chromatography. It has been shown that the same results are obtained when calculating numerical solutions of the equilibrium-dispersive model [20]. This calculation is carried out using the same numerical algorithms as for the calculation of the band profiles of large samples of multicomponent mixtures. It requires only the knowledge of the competitive equilibrium isotherms of the analytes and additives involved in the chromatographic system and the column efficiency. As we have already shown, these algorithms can be used for any mode of chromatography. Figures 13.4a and 13.4b compare chromatograms published by Herne et al. [23] and chromatograms calculated from the data found in their paper, assuming Langmuir competitive isotherms (Eq. 13.3) and numerical values of the other parameters that permit the best approximation of the experimental results. In this experiment, a sample containing acetonitrile, N,N-dimethylf ormamide, ethyl formate, z'so-butanol, ethyl acetate and 1-pentanol is eluted on a C18 chemically bonded silica phase (Nucleosil C18), using a water-methanol solution containing 0.00025 M salicylamide, the UV-absorbing additive. The agreement between experimental and calculated chromatograms is excellent. It demonstrates the validity of the theoretical approach. The system peaks associated with the first five components, those that are eluted before the additive system peak (peak 6), are
13.1 System Peaks in Linear Chromatography
617
Figure 13.4 Comparison between experimental and calculated chromatograms in indirect detection. (Left) Indirect UV detection. Mobile phase: 2.5 x 10~4 M salicylamide in water containing 5% v/v methanol. Solid phase: Nucleosil C18 (5 urn). Detection at 302 nm. Column length 100 mm. Sample 10-30 pg of (1) acetonitrile, (2) N,N-dimethylformamide, (3) ethyl formate, (4) /so-butanol, (5) ethyl acetate, (7) 1-pentanol. System peak: (6) salicylamide. (Right) Calculation of the system peak chromatogram in (Left). Column: length, 10 cm; dead time, 160 s; phase ratio, 0.25; efficiency, 5000 theoretical plates. Competitive Langmuir isotherms, with qs = 2, b,- = U(/qs, for all compounds. Retention factors: (1) acetonitrile, k'Q 1 = 1; (2) N,N-dimethylformamide, k'o 2 = 2; (3) ethyl formate, k'o 3 = 3; (4) iso-butanol, k'Oi =4; (5) ethyl acetate, fcg^ = 6; (6) salicylamide (additive), k'O6 = 11; (7) 1propanol, k'07 = 15. Additive concentration in the mobile phase, Cs = 0.00025 M. Amount of compounds injected, 0.835 fimol of compounds 1 to 5 and 0.417 ^mol of component 7. Figure 3a reproduced with permission from P. Herne, M. Renson, and J. Crommen, Chromatographia, 19 (1984) 274 (Fig. 1). Figure 3b reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal. Chem., 62 (1990) 923 (Fig. 6). ©1990 American Chemical Society.
positive perturbations, as predicted by Eq. 13.22b. The system peak associated with the last component (peak 7) is negative, also in agreement with Eq. 13.22b. The direction (sign) of the additive system peak is determined by the mass balance of this compound. In Figure 13.4, this peak is negative. Figure 13.5 illustrates the fact that changing the composition of the sample can change the sign of the additive peak. The chromatogram in this figure has been calculated using the same set of experimental conditions as for Figure 13.4, except for the amount of 1-pentanol injected, which has been doubled. Equation 13.22ab permits the calculation of the relative response factor of each component in indirect detection (see next section). The mass balance of the additive permits the prediction of the sign and size of the area of its system peak. Thus, theory permits a quantitative explanation of the system peak phenomenon and an accurate prediction of the chromatograms obtained, provided that an acceptable
618
System Peaks with the Equilibrium-Dispersive Model
Figure 13.5 Influence of the sample composition on the chromatogram obtained in indirect detection. Chromatogram calculated under the same conditions as Figure 13.4b, except for the amount of compounds injected, 0.835 ^mol for all compounds 1 to 5 and 7. Reproduced with permission from S. Golshun-Shimzi and G. Guiochon, Anal. Chem., 62 (1990) 923 (Fig. 7). ©1990 American Chemical Society.
model of the competitive isotherms be available. However, the relative response factors, which are important in quantitative analysis, must be determined directly from the chromatograms (see below, Eq. 13.23). Their determination does not require any prior information regarding the equilibrium isotherms.
13.1.3 Indirect Detection Using System Peaks Probably the application of system peaks of most practical importance is the detection and quantitation of analytes which cannot be detected directly. Then, an additive that is easy to detect and whose concentration can readily be monitored is added to the mobile phase. In most cases, this method is applied for the detection of compounds that have no UV chromophores in the range of conventional UV absorption photometry (e.g., triglycerides and other lipids, carbohydrates), and a UV detector is used with a conjugated aromatic compound as additive. The method has been used also with fluorescence [24] and electrochemical detection [25]. As discussed earlier, a chromatogram containing n + 1 peaks is recorded upon injection of a sample containing n nondetectable components in a chromatographic system using a binary mobile phase with a detectable additive [8,18,20]. Of these peaks, the n analyte system peaks, appear at the retention times of the nondetectable analytes. The extra system peak is characteristic of the additive. Equation 13.22ab shows that the area of an analyte system peak is proportional to the size of the perturbation, i.e., to the amount of the corresponding component in the sample injected. The response factor in the case of an additive used for indirect detection can be derived from Eq. 13.22b. In indirect detection, the detector responds to
13.1 System Peaks in Linear Chromatography
619
Figure 13.6 Indirect detection of anions and cations. Stationary phase: f/-BondapakPhenyl. Mobile phase: naphthalene-2sulfonate 4 x 10~4 M in 0.05 M phosphoric acid. Sample: (1) butyl sulfate, (2) pentyl amine, (3) hexane sulfonate, (4) heptylamine; (5) octane sulfonate; (6) octyl sulfate; (S) system peak. Reproduced with permission from J. Crommen, G. Schill, D. Westerlund and L. Hackzell, Chromatographia, 24 (1987) 252 (Fig. 1).
changes of the concentration of the single additive, not to those of the individual analytes. Thus, the ratio of the areas, A2- and Av, of the component system peaks recorded for two analytes i and i' is equal to the ratio of the fluctuations of the additive concentrations, ACj and AQi, and .
A/-,
pi
VI
1,1
A7"ZC~;"E7M-E;
(13.23)
The sign and the area of the additive system peak can be derived from a mass balance of this additive. The amount of additive injected (zero when the sample is dissolved in the mobile phase, negative when the sample is dissolved in an additive-free solvent) must be equal to the sum of the additive amounts corresponding to all the component system peaks (which are derived from Eq. 13.22b) and to the additive system peak. Indirect detection has been used extensively in ion-pair chromatography to detect and quantify the nondetectable cationic or anionic organic components by adding a detectable ionic additive to the stationary phase [26-35]. Figure 13.6 shows a chromatogram illustrating the indirect detection of butyl sulfate, pentyl amine, hexane sulfonate, heptyl amine, octane sulfonate, and octyl sulfate using as additive naphthalene-2-sulfonate (concentration 4 x 10~4 M), in a 0.05 M phosphoric acid aqueous solution as the mobile phase and ^-Bondapak-phenyl as the stationary phase [34]. In the case of ions there are two possible types of interaction [21,34]. If the analytes carry the same charge as the additive, they compete with it, and their injection decreases the amount of additive adsorbed at equilibrium. Hence, the analyte system peaks are positive when eluted before the system peak, negative when eluted after it. By contrast, analytes that carry the opposite charge coadsorb with the additive, and their injection results in an increase of the additive concentration in the stationary phase at equilibrium. Accordingly, the system peaks of analytes having a charge opposite to that of the additive are negative when eluted before the additive, positive when eluted after it. Thus, the two peak series 1, 3, 5, 6 (anions), and 2, 4 (cations) in Figure 13.6 have an opposite behavior, in agreement with their ionic nature and the use of an anionic additive.
System Peaks with the Equilibrium-Dispersive Model
620
filNUTES INJ
4
8
12
16
Figure 13.7 Indirect detection of metal ions. Stationary phase: sulfonic acid derivatized polystyrene-divinylbenzene resin particles (5 fim). Mobile phase: 0.1 M Ce(III) in water, 1 ml/min. Sample: 20 \iL containing (1) sodium, 1.6 ppm; (2) potassium, 2.1 ppm; (3) rubidium, 7.1 ppm; (4) cesium, 12 ppm; (5) magnesium, 2.5 ppm; (6) calcium, 1.6 ppm. Reproduced with permission from } . H. Sherman and N. D. Danielson, Anal. Chem., 59 (1987) 490 (Fig. 3). ©1987 American Chemical Society.
Crommen et al. [34] developed stoichiometric models for analytes and additives having charges of the same or opposite charges. Stahlberg and Almgren [36] developed a theory for system peaks and indirect detection in ion-pair chromatography using a surface potential modified Langmuir isotherm (13.24)
qa = i=\
where Za and Z,- are the charges of the additive and the fth analyte, respectively, !F is the Faraday constant, YQ is the electrostatic surface potential created by the adsorption of the ionic additive on the stationary phase, qa and Ca are the equilibrium concentrations of the additive in the stationary and mobile phase, and qa is the saturation capacity of the additive. The surface potential is set to 0 when the stationary phase is equilibrated with the mobile phase at Ca = 0, and ba = e~( 4G n/^ r ), where AG® is the free energy of adsorption of the additive on the stationary phase. Indirect detection has also been used in other chromatographic modes. In ionexchange chromatography, a UV-absorbing counterion is used as the detectable additive in solution in the mobile phase and competes with the analytes for access to the ion-exchange sites. This method is especially valuable for the detection of inorganic ions that have no chromophores. The use of a large number of anionic [37-42] and cationic [43-49] additives has been reported. Figure 13.7 shows the detection of sodium, potassium, rubidium, cesium, magnesium, and calcium separated on a sulfonic acid-derivatized polystyrene-divinylbenzene resin, using
13.1 System Peaks in Linear Chromatography
621
Figure 13.8 Indirect detection of monoglycerides. Stationary phase: Nucleosil CgHs. Mobile phase: 2 x 10^4M cholecaldferol in methanol-water (3:1). Detection at 295 nm. Sample: commercial glyceryl monostearate. S, system peak. Reproduced with
0.016 A
permission from J. Crommen, G. Schill and P. Herne, Chromatographia, 25 (1988) 397 (Fig. 5). t6
20
min
a 0.1 mM solution of Ce(III) as the mobile phase. In reversed phase liquid chromatography, the indirect detection of uncharged analytes has been carried out using a nonionic additive [21,23,50-54]. A retention model to be used to account quantitatively for the detector response was proposed by Crommen et al. [21]. For a chromatographic system containing a single additive, and following competitive Langmuir isotherm behavior, the results have been summarized in Section 13.1.2.3 above. Some experimental results [21] show that the analyte system peak is negative if it is eluted before the additive system peak and is positive if it is eluted after the additive system peak. Figure 13.8 shows the separation and indirect detection of monoglycerides on a Nucleosil-phenyl column, with water-methanol (25:75, v/v) as the mobile phase, containing 2 xlO~ 4 M cholecaldferol as the additive [21]. We note that this last experimental result is in contradiction to the predictions made on the basis of competitive Langmuir equilibrium behavior. Crommen et al. [21] suggested that complexation could take place between the analyte and one of the mobile phase components, and that the free analyte and the complex would compete for interaction with the stationary phase. If this correct, the isotherms would be written 1 + biQ + bsCs + i qs(bsCs + bisCi 1 + hQ + bsCs +
—
(13.25a) (13.25b)
where the subscripts i, s, is refer to the analyte, the additive, and the complex, respectively. Instead of the classical relation derived above as Eq. 13.22b: *0,»ap
AC;
(13.25c)
622
System Peaks with the Equilibrium-Dispersive Model
they derived the following relationship [21]: fc
ACS_( AL 1
-
K
\
fc
o,Sap\ o,iapj
kP
*«
K
0,i*v
where
When
*O*P
=
t
< 9 s
(13 28)
'
the sign of the system peak will be the same as predicted from the Langmuir isotherm (Eq. 13.22b). However, if k' "•0,S "'
ap
= ^
> 9S
(13.29)
the sign of the system peak associated with an analyte eluted before the additive system peak (i.e., when whei k's > k'Oi ) is negative, while an analyte system peak eluted later is positive.
13.1.4 Application of System Peaks to Analyte Peak Compression Consider a chromatographic system used in ion-pair chromatography. An organic ion is used as additive. When a large-volume sample is injected, a large additive system peak takes place, because of the important vacancy of additive created by injection of the weak solvent. The rear part of the profile of this system peak is a steep, positive gradient of concentration of the additive. If ionic solutes having the same charge as the additive elute during the passage of this gradient, their peaks are extremely narrow because they happen to be eluted under the conditions of a steep concentration gradient [55,56]. If an analyte peak is eluted on the front part of the system peak, its profile may be severely deformed [56]. When the analyte is eluted close to the system peak, some adjustment of the experimental conditions (e.g., change in the additive concentration in the sample solution) makes it possible to move it to a proper position, on the rear of the additive system peak to optimize its band compression. The use of different additives and their selection for matching the retention times of the additive and the analyte system peaks were also investigated. This permits the achievement of peak compression for solutes having different retention volumes [57].
13.2 System Peaks in Linear Chromatography
623
13.1.5 Vacancy Chromatography The concept of vacancy chromatography was initially introduced by Zhukhovitski and Turkel'taub [58] to describe a chromatographic method in which the mobile phase is the solution to analyze, or a dilute solution of the sample stream in a proper solvent. This solution is pumped at constant flow rate through the column, and a sample of pure solvent is injected periodically. This injection creates an analyte vacancy resulting in a series of negative peaks. Although ideally suited to the on-line analysis of pollutants, this method is rarely used in analytical applications of chromatography. Its most popular application is in the determination of equilibrium isotherm. It is a variant of the step and pulse or the elution on a plateau method of isotherm measurements [59,60] (Chapters 3 and 4). Some interesting aspects of liquid-solid vacancy chromatography have been discussed by Scott etal. [61]. From a theoretical point of view, the problem is the same as the one discussed previously, except for a change of the initial and boundary conditions, which become Cj(x,Q)
= CJ iix>0
o £?>/„-*'
(13.30a)
(13-30b)
If the vacancy pulse is small, the elution band is Gaussian. However, the system remains uncoupled only if the concentrations of the sample components in the mobile phase are so small that all their equilibrium isotherms remain truly linear. Very small deviations of these isotherms from linear behavior are sufficient to trigger nonlinear behavior of the vacancies. When the system is uncoupled, the velocity of each concentration perturbation is equal to the velocity of the band in linear chromatography. The retention time of the peak observed for each component of the mixture corresponds to its retention factor at infinite dilution. If the concentration of some of the sample components is large, and their equilibrium behavior is no longer linear, the velocity of their bands depends on the mobile phase composition, even though the vacancy injected may be very small and the signal observed remains nearly Gaussian. Figure 13.9 shows the chromatogram calculated for the injection of a small vacancy in a chromatographic system when the mobile phase contains two additives at low concentrations (linear isotherms). The two peaks are uncoupled and each one elutes at a time corresponding to its retention factor at infinite dilution. The chromatogram is the same as the one obtained with a small injection of the same mixture on the same column, with a pure mobile phase, except that here the peaks are negative. When the mobile phase concentration of an analyte is large and the isotherm is not linear, the retention time of the pulse depends on the concentration of the component in the mobile phase. The apparent column capacity factor is proportional to the slope of the isotherm at the mobile phase concentration (k' = Fdq/dC, Eq.
System Peaks with the Equilibrium-Dispersive Model
624
Figure 13.9 Vacancy chromatography. Calculation of the chromatogram obtained with a binary mixture dissolved in the mobile phase. Low additive concentrations: d = C2 = 0.0001 M. Injected amount: 4.9 ^mol of pure solvent. Competitive Langmuir isotherms: qi = (ayCy)/(l + £)fcyCy), with n\ = 12, a = 1.5, qs = 2, bj = aj/qs. Reproduced with permission from S. Golshan- Shirazi and G. Guiochon, Anal. Chem., 62 (1990) 923 (Fig. 8). ©1990 American Chemical Society. 10
15
Time (min)
7.4). Thus, increasing the mobile phase concentration of the additive and measuring the value of k' as a function of this concentration gives dq/dC, and permits the determination of the equilibrium isotherm = / k'dC
(13.31)
This is the principle of the method of elution on a plateau or step and pulse method [59,60]. When the mixture studied contains several components the isotherms of which are not linear, the problem is more complicated because the perturbations are now coupled and the number of peaks observed becomes large. For a chromatographic system in which the mobile phase contains n components (weak solvent included), the injection of a vacancy pulse of a single component causes the apparition of n — 1 peaks. The migration velocities of these peaks are related to the eigenvalues of the matrix of differentials dqj/dC^ of the competitive isotherms corresponding to the composition of the mobile phase. If vacancy pulses of several components (up to n — 1) are injected simultaneously, the negative peaks coelute with each other at the n — 1 positions and interfere. Figure 13.10 shows the chromatogram calculated for the same set of experimental conditions as used for Figure 13.9, except for the concentrations of the analytes, which are Co,i = Q,2 = 0.01 M, instead of 0.0001 M. The matrix /3 (Eq. 13.11a) of this system is a 4x4 matrix and the mobile phase concentration is represented by a 2 x 1 vector. The injection of the vacancy pulse causes two perturbations, one for each component. Each of these perturbations involves two peaks,
13.1 System Peaks in Linear Chromatography
625
Figure 13.10 Vacancy chromatography. Same as Figure 13.9, except moderate additive concentrations. C\ = C2 = 0.01 M. Reproduced with permission from S. GolshanShirazi and G. Guiochon, Anal. Chem., 62 (1990) 923 (Fig. 10). ©1990 American Chemical Society..
Time (min)
one for each component, but the chromatograph can detect only two peaks, not four. Each of the two peaks detected is the result of an interference between the positive and the negative signals arising from the perturbation of the equilibrium of each component. The migration velocities, hence the retention times of these pulses, depend on the competitive isotherms of the two components and on the mobile phase composition. These velocities are given by the eigenvalues of the /3 matrix l =coA=co
(13.32a) (13.32b)
=d>,C2=C°
If the injection pulse is small enough and the mass transfer kinetics rapid, the profiles of the peaks obtained are nearly Gaussian. The retention times of the two signals observed are not simply related to the competitive equilibrium isotherms of the two components. Furthermore, the relative area of the two peaks (of components 1 and 2) eluted at each retention time changes markedly with the composition of the mobile phase and with the isotherm coefficients. Figure 13.11 shows the chromatogram calculated for the same set of experimental conditions already used for Figures 13.9 and 13.10, except for their mobile phase concentrations, which are now C$i = CQ^ = 0.20 M. The retention times of the two pulses observed in Figures 13.9,13.10, and 13.11 are different, because of the increasing curvature of the isotherms, and the relative peak areas are also quite different. It is even possible that, under some combination of experimental conditions, the
626
System Peaks with the Equilibrium-Dispersive Model
Figure 13.11 Vacancy chromatography. Same as Figure 13.9, except high additive concentrations. Q = C2 = 0.2 M. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 62 (1990) 923 (Fig. 11). ©1990 American Chemical Society.
Time (min)
second signal disappears, because the responses of the detector to the two concentration perturbations cancel out (this is the case in Figure 13.11). The exact conditions for that effect depend on the relative response of the detector to the two components. Although for the sake of clarity the previous discussion was limited to the case of a binary mixture, these results are easily generalized to the study of an n-component mixture. Because of the coupling between the mobile phase components, the velocity eigenvalues are related to the slopes of the tangents to the ndimensional isotherm surface, in the n composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, but only if a proper isotherm model is available. Least-squares fitting of the set of slope data to the isotherm equations allows the calculation of the isotherm parameters. If an isotherm model, i.e., a set of competitive isotherm equations, is not available, the experimental data cannot be used to derive an empirical isotherm (see Chapter 4).
13.2 High-Concentration System Peaks In the abundant literature regarding either system peaks or nonlinear chromatography, we find few papers discussing system peaks under conditions of overloaded elution. Helfferich and Klein [8] discussed the phenomena that take place upon injection of a large sample of a single component in a binary mobile phase,
13.2 High-Concentration System Peaks
627
when analyte and additive follow competitive Langmuir adsorption behavior. The discussion is based on the use of the ideal model of chromatography and the /z-transform. Recently, Golshan-Shirazi and Guiochon gave a detailed analysis of system peaks in the case of single-component [62,63] and multicomponent samples [64,65] and of a multicomponent mobile phase. Theoretical studies based on the use of the competitive Langmuir model [62,64] were followed by experimental investigations which gave results in excellent qualitative agreement with theoretical predictions [63,65]. Some conclusions of this work were developed more recently [66,67] and confirmed by accurate experimental results [68,69]. Although even in the case of small injections system peaks cannot be properly discussed with a linear model of chromatography, the situation becomes much more complex when large samples are considered. This is because in the former case, the problem can be linearized by considering it as a perturbation of a nonlinear problem. The injection of a small sample in a chromatographic system using a mixed mobile phase containing additives that equilibrate between stationary and mobile phase causes a linear perturbation of these equilibria. The analyte isotherms can be considered as linear and the perturbations of the additive equilibria are related to the slope of their isotherms. The system peaks have Gaussian profiles, and the position and profiles of these system peaks are independent of the composition of the sample. In contrast, when the sample is large, the isotherms of the solutes are no longer linear and there is competition for adsorption between these solutes as well as between them and the additives during the entire migration. The profiles of the system peaks result from the integral of these effects. In some cases, they become extremely complex.
13.2.1 High-Concentration System Peaks for a Single-Component Sample From a theoretical point of view, the discussion of the profiles of the component and the additive bands at high concentrations is complicated because the perturbation due to the injection of the sample cannot be considered small and cannot be treated by assuming the system to behave linearly around the steady equilibrium point, as is done in the study of system peaks in analytical chromatography. Band profiles are accessible only through numerical calculations. The experimental results are still difficult to account for because of the scarcity of studies and data on the phenomenon, and because of the strange and unexpected shape of the profiles obtained under some sets of experimental conditions. 13.2.1.1 Theoretical Study of System Peak Profiles The mathematical approach used to account for system peaks at high concentration of the sample components is the same as that described earlier in the linear case, when the analyte concentrations are low, and the terms Yj &t'Q could be neglected in the isotherms. These terms can no longer be ignored at high concentrations, and this has a considerable influence on the shape of the solute and the system peak profiles.
System Peaks with the Equilibrium—Dispersive Model
628 C(M) C (M) 0.05 0.05-I
C (Ml) (M)
Sample -1
Additive-A Additive-A
Sample -1
Total-3
0.15 0 .15- -
ft
Additive-A Additive -A Total -3
VA
-1
0.030.03
-1
0 .050.05
0.01 • 0.01
0 -0.01
• II
C 0
2 50 250
500
v/
3-
-3
-0.01 -0.05 \
^ ^ ^
-A
-A A
-0.15-0.15
7 50 750
1000 1000 Time, sec
0
200
400 400
Time, sec
600 600
Figure 13.12 (Left) Chromatogram of a pure component with a binary mobile phase. Influence of the retention of the additive; poorly retained additive (as < < «i). Column: length, 25 cm; efficiency: 5000 theoretical plates. Flow velocity: 0.122 cm/s (to = 205 s). Strong solvent concentration: 0.25 M. Langmuir isotherm coefficients: as = 2.0; a\ = 20; bs = 1.0; b\ = 10. Sample size: 83.3 ^mol. Curve 1: elution profile of the sample (signal of a detector selective for the sample). Curve 2: elution profile of the strong solvent (signal of a detector selective for the solvent). Curve 3: sum of the two profiles (signal of a nonselective detector with equal response factors for sample and solute). (Right) Chromatogram of a pure component with a binary mobile phase. Influence of the retention of the additive; additive as strongly retained as the component (as = a{). Same experimental conditions as for Figure 13.12a, except: as = 20; bs = 10. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon,}. Chromatogr., 461 (1988) 1 (Figures 1 and 2).
In this case, the system of equations of a model of chromatography (Chapter 2) must be solved numerically. These models are very general. They apply to all modes of chromatography, independently of the model of competitive isotherms selected. Most theoretical studies used the equilibrium-dispersive model of chromatography because the mass transfer kinetics are fast under the experimental conditions employed in the study of system peaks. These theoretical studies also used the Langmuir competitive model because it is both general and convenient. Figures 13.12a to 13.13 show the calculated elution profiles for the component and the additive that are obtained upon injection of a large sample of a pure component into a column equilibrated with a solution of a retained additive. Figure 13.12a shows the chromatogram calculated for a large sample of a pure component that is more strongly retained than the strong solvent. The concentration of the strong solvent or additive is large enough to reduce the retention factor of the component, since this factor is inversely proportional to 1 + bsCs (Eq. 13.3). The component peak has a steep front and a diffuse rear profile which ends close to the retention time at infinite dilution under the same experimental conditions. This profile is similar to the classical profile observed for a component having a Langmuir equilibrium isotherm. It has been shown that when the ratio of the initial slopes of the adsorption isotherms of the component and the strong solvent in the pure weak solvent is larger than about five, the three following profiles are practically identical: (1) the experimental profile, (2) the profile calculated from the (correct) two-component model and the competitive isotherm, and
13.2 High-Concentration System Peaks C (M) C (M)
629
a
b S ample -1 -1 Sample
0 .25 0.25
0.225 5
-
Additive A dditive -A -A
*
Total otal -3 T -3
0.150 . 15 -1 0.05
0.05 --0.05
0.050 .05
-3 -A
-
-0.15 1100 00
1
0.150 .15
3 00 300
_Jk\'
--0.05 0.05 -
5 00 500
--0.150.15 1100 00
-1
-3
-A ,
,
,
1
,
-i
3 00 300
'
1
5 00 500 Time, sec
Figure 13.13 Chromatogram of a pure component with a binary mobile phase. Influence of the retention of the additive; additive more strongly retained than the component (as > a\). (a) Same experimental conditions as for Figure 13.12a, except as = 40; bs = 20. (b) Same conditions as for Figure 13.12a, except as = 200; bs = 100. Reproduced with permissionfrom.S. Golshan-Shimzi and G. Guiochon, J. Chromatogr., 461 (1988) 1 (Figs. 3 and 4).
(3) the profile calculated from the approximate one-component model (using the single-component isotherm derived from the competitive one by writing that the strong solvent concentration is constant) [70,71]. Differences appear progressively as the ratio of the initial slopes of the two isotherms tends toward unity. As long as as < «i, however, the profiles of the large bands of feed components remain quasi-Langmuirian [66]. As expected from the discussion of the linear case, there are two large strongsolvent or additive peaks in the chromatogram (Figure 13.12a). The first peak is eluted before the solute peak. A second peak appears in the same time as the sample band. It is negative because the additive concentration in the mobile phase decreases during the elution of the sample band. Accordingly, the total concentration of component and additive in the weak solvent varies less than would be indicated by the elution profile of the sample band itself. The additive band profile is proportional to the solute profile, because each molecule of sample that leaves the stationary phase is replaced by a constant number of molecules of additive that were displaced at the time of the injection, when the sample was introduced into the column. Thus, a detector that would respond to both the sample and the additive could exhibit a total lack of response during the elution of the composite band, should the two response factors be in the proper ratio (see Figure 13.12b). Obviously, the areas of the two additive bands are equal, since they correspond to the same amount of material, first displaced from the column, then replenished. Figure 13.12b shows a chromatogram calculated for the same experimental conditions as those used for Figure 13.12a, except that now the initial slopes of the isotherms of the sample component and the additive are equal. Although the sample size is the same for Figures 13.12a and 13.12b, the component band shown in Figure 13.12b is more symmetrical. In fact, it is nearly Gaussian. The first sol-
630
System Peaks with the Equilibrium-Dispersive Model
vent peak, on the other hand, exhibits a slight tailing. Its retention time increases with increasing strength of the additive. The profile of the second, negative, additive band is equal to the component elution profile. Since it is negative, the mobile phase concentration of the weak solvent remains constant during the entire elution of these bands and the only signal recorded by a nonselective detector would be the unsymmetrical first solvent band. This is a paradoxical and unexpected result which could be highly misleading if it were obtained by accident. Figure 13.13 shows the chromatograms calculated for a component less strongly adsorbed than the additive, the ratios of the initial slopes of the additive and component isotherms in the pure weak solvent being equal to 2 (Figure 13.13a) and 20 (Figure 13.13b). The increasing strength of the strong solvent from Figure 13.12a to Figure 13.13 is reflected by the decrease in the component retention at constant additive concentration. The profiles of all three bands (one for the component and two for the additive) are unsymmetrical. They are still resolved in Figure 13.13a, but no longer so in Figure 13.13b. Furthermore, the elution profile of the component has changed direction. Its front is diffuse while its rear is steep in Figure 13.13a, as if the adsorption isotherm were anti-Langmuir, although the isotherms used for the calculation are competitive Langmuir isotherms. This paradoxical phenomenon is due to the competition between the component and the additive molecules for access to the adsorbent surface. This competition explains why, in many instances, the profiles of overloaded bands obtained in reversed phase or in hydrophobic interaction chromatography seem to correspond to an antiLangmuir type of isotherm, not to a Langmuir one. This phenomenon could also explain why a change in the composition of the solvent can reverse the asymmetry direction of the band and replace an apparently anti-Langmuir band profile by a Langmuir one. In Figure 13.13b, the compound has a bimodal profile; the signal obtained with a selective detector suggests strongly that there are two partially resolved isomers. The signal of a nonselective detector exhibits a trough behind a major peak, features that are difficult to explain for an analyst who is not well aware of the riddles that system peaks may present occasionally. Finally, the two additive bands of the calculated chromatogram in Figure 13.13b are partially merged, while the negative second band is more concentrated than the component band. As a consequence, the injection of the sample results in a negative signal with a nonselective detector, whereas a much simpler, positive signal is observed with a sample-selective detector. However, even this simpler signal is not always easy to account for. Similar phenomena have been previously observed and reported in analytical chromatography [7,10,32,34] and in preparative applications [72,73]. The retention time and the profile of the component band depend on the relative strength of the adsorption of the additive and the component from their solution in the pure weak solvent, on the column efficiency, on the additive concentration in the mobile phase, on the column saturation capacity, and on the sample size. Some combinations of the values of these parameters may result in strange and unexpected elution profiles that are difficult to account for by any rationalization. A few striking examples are given now and compared to experimental
13.2 High-Concentration System Peaks
631
results. In many cases, the calculation of band profiles with series of plausible values of the parameters is the only way to understand properly the chromatograms recorded. These unusual "Shirazian" band profiles take place whenever the experimental conditions are such that the two additive peaks interfere and when the additive is more strongly retained than the feed component, but its concentration in the mobile phase is high enough for its system peak to be eluted before the feed component [66]. 13.2.1.2 Experimental Study of System Peak Profiles. Influence of Sample Size Except in connection with their use in indirect detection, system peaks have not been much studied experimentally. There are few reports on the profiles of overloaded elution bands. Kirkland [72] studied the influence of water and alcohols as organic solvent modifiers in normal phase liquid chromatography. He showed that only low concentrations of these additives were needed. Typically, concentrations between 0.05 and 0.3% of methanol, ethanol, or 2-propanol were recommended. He also reported very unusual band shapes in some instances. He noted a change in the elution profile with increasing alcohol content, the direction of asymmetry of the peak reversing, from tailing to fronting, with an intermediate stage where the band "appears as the superimposition of two different chromatographic bands" [72]. Puncocharbvci et ol. [73] reported similar phenomena in the elution profiles of bands of cyclohexanol and cyclohexanone on silica, with various binary mobile phases. A systematic investigation of the shape of experimental band profiles of 2-phenylethanol and 3-phenyl-l-propanol on silica was conducted by Golshan-Shirazi and Guiochon [63], using mixtures of dichloromethane or n- hexane as weak solvent, and methanol, ethanol, 2-propanol, and fert-butanol as strong solvent. Figure 13.14 shows the chromatograms obtained for four successive pulses of 2-phenylethanol (1, 2, 5, and 10 ]iL, respectively). For the sake of comparison, the inset shows a chromatogram calculated under linear conditions. Figure 13.14 is similar to Figure 13.13a, which shows a chromatogram calculated for a rather strongly sorbed organic modifier (ratio of the initial slopes of the equilibrium isotherms of the modifier and the component in the pure weak solvent equal to 2). The comparison of the two figures permits an easy understanding of what is happening. The calculated elution chromatogram (Figure 13.13a) exhibits one component band and two bands for the modifier, a positive additive system peak, eluted before the component band, and a negative component system peak, eluted in the same time as the component band. The areas of the two system peaks increase with increasing sample size. At large sample sizes, the positions of the first additive band and of the sample peak depend not only on the adsorption strength of the modifier (i.e., the initial slope of its adsorption isotherm from a solution in the pure weak solvent) but also on the sample size (see Figure 13.14). With increasing sample size, the modifier and component bands move closer to each other and they eventually interfere [62,63]. The experimental results (Figure 13.14, curves 1 to 4) confirm these predictions, although, at first glance, they appear to be in contradiction. A refractive index de-
System Peaks with the Equilibrium-Dispersive Model
632 RI Response RI
0.02
18 18 C (M) 4
A
15 15 -
-1
0
-3
12 12 -A 1
9-
2 3
0 8
6 -
3 --
0 0
2
4
Time, min
6
Figure 13.14 Chromatograms of 2-phenylethanol samples of increasing sizes. Main Figure Refractive index (RI) detector. Column: length, 25 cm; i.d., 4.6 mm. Mobile phase: dichloromethane with 1% 2-propanol (i.e., 0.17 M). Stationary phase: silica, 15-25 y.n\ particles. Flow rate: 2 mL/min. Sample sizes: curve 1, 1 ^L; curve 2, 2 jiL; curve 3, 5 }iL; curve 4,10 ^L. Inset Calculated chromatogram with competitive Langmuir isotherms for the additive and the sample. Langmuir coefficients: aa = 40; as = 80; ba = 8; bs = 16. Concentration of strong solvent: 0.17 M. Sample size, 1 uh. Reproduced with permissionfromS. Golshan-Shirazi and G. Guiochon, J. Chromatogr., 461 (1988) 19 (Fig. 1).
tector was used, the weak solvent was chloroform (RI, 1.4244), the modifier was 2-propanol (RI, 1.3772), and the component 2-phenylethanol (RI, 1.532). The response factor of the detector being nearly proportional to the difference in refraction indices of the solvent and the solute, a negative peak is recorded when the modifier concentration increases and positive peak when the component concentration increases. Thus, the additive system peak is recorded as a negative signal, while the effect of the component system peak (negative modifier concentration profile) adds to that of the component band, which explains the large area of the second, mixed band. When the sample size increases, the component band begins earlier and ends later. The component bands corresponding to increasingly large samples do not share a common diffuse profile as they do when the mobile phase
13.2 High-Concentration System Peaks mAU AU m
8800 00 -|
r
5 5
a
r\
"
633 b
7 00 7004
4
3
33
1 I 1 22 m.
4004 00
335050
//A
1
2
1
0-
0 7'
99
111 1
3
4
5
6 Time, min
Figure 13.15 Experimental chromatograms of 3-phenyl-l-propanol samples of increasing sizes. UV detector (no response for the solvent), (a) Same experimental conditions as for Figure 13.14, except 1% 2-propanol in w-hexane as the mobile phase, flow rate 3 mL/min. Curve 1, 1 jiL; curve 2, 5 }iL; curve 3, 10 fiL; curve 4, 15 }iL; curve 5, 25 }iL. (b) Same experimental conditions as in Figure 13.14, except 1% ethanol in dichloromethane as the mobile phase. Curve 1,0.08 jiL; curve 2,0.4 fiL; curve 3,2 /JL; curve 4,6 ffL. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon,}. Chromatogr., 461 (1988) 19 (Figs. 6 and 8).
is a pure solvent (Figures 10.12 to 10.14). Accordingly, the resolution between the modifier and the component bands decreases with increasing sample size. It becomes lower than unity for a sample larger than 5 }iL (Figure 13.14, curves 3 and 4). 13.2.1.3 Comparison of Experimental and Calculated Results. Influence of the Nature of the Modifier In the case of an additive less strongly adsorbed than the compound studied, the effect is the same as in conventional chromatography with a pure mobile phase. Quasi-Langmuirian peak profiles are observed (Figure 13.15a), although the band could possibly have a rear shock layer (compare with Figure 10.12b). When the additive is more strongly adsorbed than the compound studied, a different phenomenon is observed. This is illustrated in Figure 13.15b, in the case of samples of 3-phenyl-l-propanol and a mobile phase containing 1% ethanol. AntiLangmuirian profiles are observed. However, the diffuse front parts of the different profiles are not in coincidence as they would be in the case of a true antiLangmuir isotherm. The diffuse fronts begin earlier and earlier when the sample size increases, in the same time as the rear shock layer takes place later and later. This effect usually precedes a reversal of the band shape of the system peaks.
System Peaks with the Equilibrium-Dispersive Model
634
C ((M) M)
mAU
a
11500 500
0 .1 6 0.16
b
T
0.12 0 .12 --
11000 0007
6
0.08
6
5
5
500
4
4 3
3
0.04 0 .04 - -
2
2 1 1
0
0 0
2
4
6
8
10
0
2
4
6
8
110 0
Time, min Time, m in
Figure 13.16 Influence of the sample size on the profile of the chromatogram of a pure compound eluted with a binary mobile phase. Strongly sorbed additive, (a) Experimental Chromatograms of acetophenone samples of increasing sizes. Same experimental conditions as for Figure 13.15a, except flow rate: 1 mL/min. Curve 1, 0.02 fiL; curve 2, 1 f/L; curve 3, 2 fiL; curve 4, 5 yL; curve 5,10 ^L; curve 6,15 }iL; curve 7, 25 ^L. (b) Calculated chromatogram, with the same conditions as for Figure 13.12a, except: as = 28.8; bs = 16; aa = 14.4; ba = 8; strong solvent concentration: 0.17 M and variable sample size. Sample size: curve 1,1.66 ^mol; curve 2,16.6 ^mol; curve 3, 41.6 ^mol; curve 4, 83.3 fimol; curve 5,166.6 ^mol; curve 6,416.5 y.ma\. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, J. Chromatogr., 461 (1988) 1 (Fig. 9) and 19 (Fig. 9).
13.2.1.4 Comparison of Experimental and Calculated Results. Influence of the Nature of the Sample Figure 13.16a shows experimental chromatograms obtained with a less polar sample, acetophenone, and the same mobile phase as Figure 13.15a. Since acetophenone is less polar than 3-phenyl-l-propanol, it is less strongly adsorbed and less retained at infinite dilution in M-hexane. The adsorption strength of the modifier compared to acetophenone is higher in the case of Figure 13.16a than it was in the case of Figure 13.15a. Figure 13.16b shows chromatograms calculated under the same set of experimental conditions, but using competitive Langmuir isotherms matching approximately the experimental data. When the sample size is increased from a low value, the band profile evolves progressively. At low sample sizes (e.g., in the case 3-phenyl-l-propanol in a 1% solution of 2-propanol in dichloromethane, Figures 13.14 and 13.15b), the band profile is anti-Langmuirian (profiles 1-3 in Figure 13.16a). When the sample size becomes larger (profiles 4 in Figure 13.16a and 5 in Figure 13.16b) the front of the band becomes steeper, a hump begins to grow on its front, and the elution profile becomes a smooth rectangle (profile 5 in Figure 13.16a). Eventually, the bump turns into a sharp peak eluted at low retention times, and preceding the rest of the band. This behavior is in excellent agreement with the results of calculations performed with a modifier that is much more strongly retained than the component,
13.2 High-Concentration System Peaks
635
as illustrated in Figure 13.16b, but is used at a sufficiently high concentration for its apparent retention to be less than that of the solute [66]. The elution profiles observed in Figure 13.15b are leading peaks, exhibiting a diffuse front and a steep rear shock layer. The retention time of the band maximum increases with increasing sample size. This behavior is associated with an anti-Langmuir isotherm (see Figures 10.13b, 10.23, and 10.24). The conclusion that the adsorption isotherm of the component studied in the chromatographic system is convex downward, although tempting, would be correct only if the mobile phase were a pure solvent. It is obviously not true in the present case. The actual equilibrium isotherms may not be accurately represented by true competitive Langmuir isotherms: this theoretical model is often too crude to account quantitatively for the adsorption behavior of real components. Nevertheless, there can be no doubt that the adsorption isotherms of alcohols on silica, whether 2propanol or 2-phenylethanol, are convex upward, as in the Langmuir model. All the isotherms involved are convex upward. Furthermore, the behavior of the chromatographic system when the sample size, the organic modifier concentration (see next subsection), or its nature is changed is not consistent with a downward convexity of the isotherms. When the sample size is increased for a system with a true anti-Langmuir isotherm, all the bands start at the same time (Figures 7.9 and 10.13b). This is not so in Figures 13.14,13.15b, and 13.16. Furthermore, as shown in Figures 13.16 and 13.17, there is a reversal of the asymmetry of the band profile when the sample size and/or modifier concentration is increased to sufficiently high values. The band profiles are Langmuirian at low concentrations, become broad but steep at both ends in a narrow intermediate range, then become anti-Langmuirian at high concentrations. 13.2.1.5 Comparison of Experimental and Calculated Results. Influence of the Additive Concentration The experimental chromatograms obtained with a UV detector for pulses of increasing sizes of 2-phenyl-l-propanol on silica, using dichloromethane as weak solvent and 2- propanol as additive, are shown in Figure 13.17 [63]. The band profile is Langmuirian at low modifier (additive) concentrations, anti-Langmuirian at high modifier concentrations, and was called Shirazian [66] in the intermediate range. The band profiles depend strongly on the modifier concentration. When the additive is more strongly adsorbed than the solute, changes in the additive concentration can dramatically affect the solute band profile. As in previous figures, there is excellent qualitative agreement between the experimental and calculated profiles in Figure 13.17a,b. (No attempts were made at making quantitative predictions, which would require the accurate determination of the isotherms.) The profiles seem to be Langmuirian at low modifier concentrations and antiLangmuirian at high concentrations. They shift from one type to the other in a rather narrow concentration range, corresponding to the reversal of the elution order of the additive system peak and the solute peak in the mobile phase [66].
System Peaks with the Equilibrium-Dispersive Model
636 mAU AU m
a
0.2 0 .2 -,
b
C (M) C(M)
| 33
1 33
2 00020
00.15.15
fl 1 1 1
00 10 0-
0.1 0 .1 -
I
00.05.05 2
I I i I
1
1
0
0 0
5
110 0
115 5
20 2 0
0
A 4
1
2
5
110 0
mln Time, m in
1155
Figure 13.17 Influence of the additive concentration on the shape of the component band profile, (a) Experimental chromatograms of 2-phenylethanol samples of equal size (1 }iL), UV detector. Same experimental conditions as for Figure 13.14, except variable concentration of 2-propanol. Curve 1, 0.1%; curve 2, 0.2%; curve 3: 1%. (b) Influence of the additive concentration on the elution profile of a pure compound. Strongly retained additive. Calculated chromatograms with the same experimental conditions as for Figure 13.16b, except variable additive concentration. Strong solvent concentration: curve 1, 0.017 M; curve 2, 0.085 M; curve 3,1.0 M; curve 4, 0.17 M. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, J. Chromatogr., 461 (1988) 19 (Fig. 14), and 1 (Fig. 6).
As expected, a marked decrease in the retention time of the band with increasing modifier concentration is also observed. Similar results were also obtained with 3phenyl-1-propanol [63]. Kirkland [72] and Puncochafova et al. [73] have reported band profiles of the same type. Fornstedt and Guiochon compared theoretical and experimental results obtained for derivatives of Remoxipride for which they had previously measured the equilibrium isotherms [66]. The additive, Protriptyline, does not adsorb at 337 nm. Thus, by recording the signal at 337 nm and at one other wavelength, the concentrations of both the sample compound and the additive can be determined at any time during elution of the compound. The column was packed with Nucleosil Cig and the mobile phase was a solution of acetonitrile in a phosphate buffer (pH = 2.0, ionic strength, 0.050). Figures 13.18 and 13.19 give the chromatograms of the sample (one of two different Remoxiprides) and the additive. With O-o = k^/k^ being the separation factor of the additive and the solute at infinite dilution in the pure weak solvent3, and a the separation factor of the primary additive system peak and the solute peak, they found that: • If Ko < 1, the additive peak is eluted first, its peak is positive and it is always Gaussian, see Figure 13.18, left. A negative system peak accompanies the elution of the sample. Both bands have a strongly Langmuirian profile because a large sample size is needed to generate a significant additive perturbation. • If OCQ > 1 (a different sample is used, more retained than the previous one) and 3 In principle, for the sake of convenience, it is assumed that separation factors are larger than unity. For this discussion, however, we must consider that the separation factor of the additive and the solute studied can take any positive value.
637
13.2 High-Concentration System Peaks 3-
25
JO
Time \mh)
15
Time (min)
Figure 13.18 Comparison between experimental and calculated chromatograms. Left The adsorption strength of the additive, protriptyline (bottom profiles) is much lower than that of the solute, remoxipride (top profiles) in the pure weak solvent, CCQ = 0.62. Dotted lines: experimental profiles; solid lines: calculated profiles. Right The adsorption strength of the additive is much higher than that of the solute in the pure weak solvent, OLQ = 1.8. Top: Chromatogram of the sample compound; Bottom: Chromatogram of the additive. Reproduced with permission from T. Fornstedt and G. Guiodwn, Anal. Chem., 66 (1994) 2686 1 (Figs. 2 and 3) ©1994 American Chemical Society.
K > 1, the sample is eluted first, its profile is Langmuirian. The additive has a negative peak which has an anti-langmuirian profile, see Figure 13.18, right. • If CCQ > 1 and a = 1, the solute profile is profoundly unusual at high concentrations. At moderate concentrations, it has a Langmuirian character and a hump on the diffuse rear boundary if cc > 1 and an anti-Langmuirian character and a hump on the diffuse front boundary if a. < 1, see Figure 13.19. The sample sizes for which the profiles become most unusual decreases with decreasing value of oc-1. These results are illustrated in Figures 13.18 and 13.19. The first figure compares the profiles obtained for the additive and the sample when either one is much more retained than the other. When the additive is the least retained, its peak is positive, both bands have a strongly Langmuirian profile while the additive band is eluted first and is negative. Competition between the modifier and the component for adsorption influences considerably the shape of high-concentration elution bands. It is possible that, in some cases, the proper combination of additives at the right concentra-
System Peaks with the Equilibrium-Dispersive Model
0.6 0.0
as in Figure 13.18 Reproduced with permission from T. Fornstedt and G. Guiochon, Anal. Chem., 66 (1994) 26861 (Fig. 5) ©1994 American Chemical Society.
-1.2 -0.6
Figure 13.19 Comparison between experimental and calculated chromatograms. The retention factor of the additive is slightly lower than that of the solute, OCQ = 1.15 and oc < 1. Same conditions and conventions
Normalized Signal
2
IS
2
638
15
20
25
Time (min)
tions could enhance the symmetry of the bands. The study of numerical solutions shows that, under some well-defined set of experimental conditions, the component band can be both retained and eluted as a nearly symmetrical band (peak 3 in Figure 13.17b). Experimental results {e.g., Figure 13.17a) show that it should be possible to achieve this goal, at least in some cases.
13.2.2 High-Concentration System Peaks for a Two-Component Sample In this discussion, we assume that the mobile phase contains a single additive or modifier dissolved in a weak solvent, that the sample is a binary mixture, that the competitive isotherms of the additive and the two components are described by the Langmuir model, and that the sample is less retained than the additive, so that «o > 1/ a s stated above. The presence of an adsorbed modifier in the mobile phase (1) decreases the retention of the two components, (2) causes the elution of an additive system peak and two component system peaks, and (3) may result in considerable departure of one or both component band profiles from their expected Langmuirian behavior. 13.2.2.1 Additive System Peaks If the mobile phase additive is much less retained than the sample components, the chromatogram (Figure 13.20) exhibits a positive additive system peak which appears shortly after the column dead volume (the additive is weakly retained). The profile of this peak is nearly Gaussian, because the additive being weakly
13.2 High-Concentration System Peaks C (mM)
a
639 b
7.3
23
-1 -2
-A
14
2.9 1-A 5
-4
-1.5 0
500
1000
1500
0
500
1000
1500 Time, sec
Figure 13.20 Calculated chromatogram for a binary mixture separated with a binary mobile phase containing an additive much less retained than the first component. Column: length, 25 cm; phase ratio: 0.25; efficiency: 5000 theoretical plates; flow velocity: 0.122 cm/s t$ = 205 s; additive concentration: 0.15 M; Langmuir isotherms coefficients: U\ = 20; a2 = 24; as = 2.0; b\ = 10; b2 = 12; bs = 0.10; qs = 2.0; relative retention, a = a 2 /«i = 1-20; additive adsorption strength: aB/a\ = 0.10; sample size: 81 mmol (1:9 mixture), (a) Chromatogram of the binary mixture. Curve 1, elution profile of the first component; curve 2, elution profile of the second component; curve a, elution profile of the additive (strong solvent). (b) Chromatogram for the same amount of pure component 1, illustrating the displacement effect. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal. Chem., 61 (1989) 2373 (Fig. 1). ©1989 American Chemical Society.
adsorbed, only a small amount of additive is desorbed at injection. Later, two positive bands are eluted, one for each component. Their profiles are similar to those obtained for the overloaded elution of a binary mixture sample with a pure solvent as the mobile phase (Chapter 11). Each of these component peaks is accompanied by a component system peak, a negative peak of the additive, which corresponds to the replacement in the stationary phase of the corresponding component by the additive [63]. Comparison of Figures 13.20a and 13.20b illustrates the displacement effect that takes place also under these experimental conditions. To account for the effect of increasing concentrations of an additive that is much less strongly adsorbed than the components, we rewrite the ternary competitive Langmuir isotherm by dividing numerator and denominator by 1 + bsCs (bs, second coefficient of the additive isotherm; Cs additive concentration)
m =
kQ
b2,sC2
(13.33)
With increasing additive concentration and/or with increasing additive adsorption energy, the apparent coefficient fr;;S = b{/(l + bsCs) decreases, and the iso-
System Peaks with the Equilibrium-Dispersive Model
640 C (M)
a
b
0.16
0.09 -A
-2
0.08
1-
2-
0.00
-1
0.00 -A
-0.09 200
500
800
-0.08 200
400
600 Time, sec
Figure 13.21 Calculated chromatogram for a binary mixture separated with a binary mobile phase containing a strongly adsorbed additive, (a) The additive is as retained as the first component. Same conditions as for Figure 13.20, except Langmuir coefficients of the additive, fls = 20; bs = 12; as/a\ = 1.0. (b) The additive is more strongly retained than the second component. Same conditions as for Figure 13.20, except Langmuir coefficients of the additive, as = 40; bs = 20; as/a\ = 2.0. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal. Chem,, 61 (1989) 2373 (Figs. 2 and 3). ©1989 American Chemical Society.
therm appears less steep and less strongly curved in a given concentration range. Thus, for a given injected amount, the band becomes more symmetrical and less retained. This is why strong solvents, modifiers, or additives are most often used to elute faster compounds with high adsorption free energies. When the Langmuir coefficients of the additive and of one of the two components become equal, the band of this component is nearly Gaussian, as in the case of a single component (previous section). Figure 13.21a shows the chromatogram calculated with bs = &2- The first component peak is slightly unsymmetrical, with a diffuse front and a self-sharpening rear, as expected since its adsorption strength is now smaller than that of the strong solvent [63] (see previous section). The positive additive system peak becomes strongly unsymmetrical, exhibiting the type of profile associated with column overloading for a compound with a Langmuir isotherm, since the amount desorbed upon injection of the mixture is now large (close to the sample size). When the modifier becomes more strongly adsorbed than the components, the two component bands are highly unsymmetrical again, if the additive concentration is high enough [63]. The two component bands assume anti-Langmuir profiles, as in the single-component case (Figure 13.13a). The rear part of each component profile is very steep and its front diffuse. Accordingly, as seen in Figure 13.21b in the case of a 1:9 mixture, the displacement effect of the first component band by the second one vanishes. The two components are eluted immediately after the additive system peak and are poorly resolved because of a low retention and a reverse tag-along effect. These phenomena are related, and they all result from the competitive adsorption behavior of the three compounds involved.
13.2 High-Concentration System Peaks mAU
a
641 U) C c ((M)
b
6 6-1
On One Two
300300
al Total
4-
200 200-
-3 '
2-
100100
ll.
0(0
,
5
10
15
20
25
0 30
0 X10-3 x10-3
5
10
-2 -2
V
15
\
-1
20
25
30
T me, min Time,
Figure 13.22 Influence of the additive concentration in the mobile phase on the chromatogram of a binary mixture. Experimental conditions. Column: length, 25 cm; efficiency, 5000 theoretical plates; phase ratio, 0.215; mobile phase flow velocity, 3.0 mL/min; fo = 68.4 s. Langmuir isotherm coefficients qs = 3.0; d\ = 60; «2 = 75; as = 120; b\ = 20; t>2 = 25; bs = 40; oc = 1.25. Sample size, 16.7 mmol 2-phenylethanol (2PE, curve 1) and 60 mmol 3-phenylpropanol (3PP, curve 2) (1:3.6). Curve 3 is the sum of curves 1 and 2. Mobile phase composition: dichloromethane with 0.017 M 2-propanol. (a) Experimental chromatogram. (b) Calculated chromatogram. Reproduced with permission from S. GolshanShirazi and G. Guiochon, Anal. Chem., 61 (1989) 2380 (Fig. 3). ©1989 American Chemical
Society.
13.2.2.2 Experimental Study of the Influence of the Concentration of a Retained Additive on the Band Profiles of Binary Mixtures Important systematic experimental studies of this problem were carried out by Golshan- Shirazi and Guiochon [63,64], by Fornstedt et al. [66,68], and by Quinones et al. [74]. The former studied the separation of 2-phenylethanol and 3-phenylpropanol in normal phase chromatography on pure silica, with dichloromethane as the weak solvent and 2-propanol as the additive. As a UV detector was used to record elution chromatograms, the system peaks were not detected, and the chromatograms exhibit only the profiles of the two component bands. Experimental chromatograms are accompanied by calculated chromatograms using plausible values of Langmuir isotherm coefficients, but no efforts were made to duplicate the experimental results with numerical solutions. At low modifier concentrations {e.g., 0.0025 M 2-propanol), the influence of the competitive adsorption of the components and the modifier results in almost only the expected decrease of the retention volumes of the sample components (see Eq. 13.32). The component band profiles retain a Langmuirian shape, as if the mobile phase was a pure solvent, and the profiles calculated and observed are similar to those shown in Chapter 11. At a higher modifier concentration (0.017 M), the profiles change spectacularly [64]. Figures 13.22b and 13.23b illustrate the results of the numerical calculations. In the chromatograms of both the 1:3.6 (Figure 13.22b) and the 4.3:1 (Figure 13.23b) mixtures, the individual profile of the first component does not change much. It exhibits a front and a rear shock layer and a short tail. The profile of the second component is dramatically modified. This component
System Peaks with the Equilibrium-Dispersive Model
642 mAU mAU
a
600 n
C (M) C
b
One Two
6-
Total
\
400400 4-
\
\
\
-3
200 200-
2-
\
\
A
103 1-
2-
\
0
0 0
5
10 10
15 15
20
25
0
5 x 10-3 W
10 10
15
20
25
30
Time, min
Figure 13.23 Influence of the additive concentration in the mobile phase on the chromatogram of a binary mixture. Same experimental conditions as Figure 13.22, except sample composition: 74.3 mmol 2PE and 17.2 mmol 3PP (ratio 4.3:1); and concentration of 2-propanol in the mobile phase: 0.0133 M (0.1% v/v). (a) Experimental chromatogram. (b) Calculated chromatogram. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 2380 (Fig. 4). ©1989 American Chemical Society.
is in the transition region, and the direction of its asymmetry is reversing. The band profile begins as a dome, continues with a plateau, and ends with a steep rear shock layer. The experimental chromatograms (Figures 13.22a and 13.23a) are quite similar to the calculated ones (Figures 13.22b and 13.23b). Since these profiles are highly unusual in chromatography, this agreement suggests that the interpretation is correct and that, in this case, the Langmuir isotherm model is a close approximation of the actual isotherm. Figure 13.24a shows the experimental chromatogram of a 4.3:1 mixture, at a higher modifier concentration (0.027 M). Now the direction of asymmetry of the second component band has reversed, and its shock layers are in the rear. The first component is in the transition region. It exhibits a steep front, a round top, and a smooth decline toward a clear rear shock layer. The calculated profile (Figure 13.24b) is in good agreement with the experimental result. Figures 13.24c and 13.24d show the same comparison of experimental and calculated chromatograms in the case of a 1:3.6 mixture, all other experimental conditions remaining the same. Compare the individual profiles of the two components in Figures 13.24b and 13.24d. There are significant differences between the experimental and calculated chromatograms, especially as far as the contribution of the first component is concerned. However, both chromatograms (Figures 13.24c and 13.24d) look strikingly like three-component chromatograms. That kind of experimental result can easily fool a chromatographer who is not sufficiently informed. Finally, Figures 13.25a and 13.26a show experimental chromatograms for a modifier concentration of 0.13 M, with the feed being a 4.3:1 (Figure 13.25a) and a 1:3.6 (Figure 13.26a) mixture. Figures 13.25b and 13.26b show the chromatograms calculated for mixtures of similar compositions. In this case, both peaks exhibit the same profile associated with anti-Langmuir isotherms, although again the isotherms are Langmuirian. This demonstrates that the change in peak shape is
13.2 High-Concentration System Peaks mfcU mAU
3. a
600
643 C (M) (M) C 6fi T
r\ \
400 -
V
200 -
b b One Two Total -3
-3
4-
/I
/
•x J
/
\ 1-
/
2" \
0o 0
2-
if. . 55
10 10
15
20 20
0o 0
10
5
x10-3 C c
15 15
20
25 25
d
66 -|
400 -
300 -
100 / 101
200 -
100 -
11. .A
0 0
J
4-
10 10
/
32-
/ /
0 55
f
/
15
20 20
0 x10-3 xlO-3
,1
5
/ -2 -2
-1 10
15 15
20 25 Time, min min Time,
Figure 13.24 Influence of the additive concentration in the mobile phase on the chromatogram of a binary mixture. Same conditions as for Figure 13.23, except mobile phase concentration of 2-propanol, 0.027 M. (a) Experimental chromatogram, 4.3:1 mixture, (b) Calculated chromatogram, 4.3:1 mixture, (c) Experimental chromatogram, 1:3.6 mixture. (d) Calculated chromatogram, 1:3.6 mixture. Reproduced with permissionfromS. GolshanShirazi and G. Guiochon, Anal. Chem., 61 (1989) 2380 (Figs. 5 and 6). ©1989 American Chemical Society.
due to the competition of the solutes with the additive. The sharp valley between the two bands in the experimental chromatogram (Figure 13.25a) demonstrates the presence of a shock layer between the two bands. The differences between experimental and calculated chromatograms can be ascribed to deviations of equilibrium behavior from the Langmuir model used for the calculations [64]. The band interactions observed are similar in nature but opposite in direction to the classical displacement and tag-along effects. These reversed effects have been called retainment and pull-back effects, respectively [64]. The reasons for these names and the influence of these effects on the band profiles are selfevident. The details of the physical mechanisms involved are not entirely clear, however. The chromatograms shown in Figures 13.25 (4.3:1 mixture) and 13.26 (1:3.6 mixture) are mirror images of the profiles typically observed for overloaded elution profiles of binary mixtures in chromatography when the components have convex-upward competitive isotherms. However, in the case of the components involved in the experiments reported in Figures 13.25a and 13.26a, the singlecomponent isotherms are known to be all convex upward, while the calculations leading to the chromatograms shown in Figures 13.25b and 13.26b were carried
System Peaks with the Equilibrium-Dispersive Model
644 mAO mAU
a a
600 • 600
400 • 400
/
200 • 200
0 • 0 0
AK
/
/
C (M) (M) C
bb One
A
0 .03 0.03
-3 -3
Two Total -3
/
0.02 0.02 /
1-
0 .01 0.01
11 \
J
22-
4
00
8
4 4
88
T i M , mir Time, min
Figure 13.25 Influence of the additive concentration in the mobile phase on the chromatogram of a binary mixture. Same conditions as for Figure 13.23, except mobile phase concentration of 2-propanol, 0.133 M. (a) Experimental chromatogram, 4.3:1 mixture, (b) Calculated chromatogram, 4.3:1 mixture. Reproduced with -permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 2380 (Fig. 8). ©1989 American Chemical Society.
mAO mAU
a
800 800 -i
C C
(M) (M)
b One One Two
/
0.03 • 0.03
I
°
/
Total -3
/ 0.02 • 0.02
400 • 400
J
3-
0.01 • 0.01
0
H
I
-2
-1 h
0 0
4
8 8
0
4
1
88 Time, min
Figure 13.26 Influence of the additive concentration in the mobile phase on the chromatogram of a binary mixture. Same conditions as for Figure 13.25, except sample composition, 16.7 mmol 2PE and 60 mmol 3PP (ratio 1:3.6). (a) Experimental chromatogram. (b) Calculated chromatogram. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 2380 (Fig. 7). ©1989 American Chemical Society.
out using competitive Langmuir isotherms. The calculated chromatograms are not identical to the experimental ones because the isotherms were not measured but derived from rough estimations, and probably the competitive Langmuir model does not account exactly for the competitive isotherms of the compounds used [75], but the qualitative agreement is so good that no doubt can remain regarding the nature of the phenomenon. When the concentration of the additive increases, the competition between components and additive results in a progressive change in the direction of asymmetry of the band profiles [63,64]. This phenomenon can take place in an additive concentration range narrower than the one needed to bring down to zero the retention of the components (see Figures 13.22 to 13.26).
13.2 High-Concentration System Peaks
645 C
mAU
800
a
(M)
b
0.04
5 5
4
4
400
0.02 3 3 2 2 1
1
0
0 0
4
8
0
4
8 Time, min
Figure 13.27 Influence of the sample size and on the chromatogram of a binary mixture. Same conditions as for Figure 13.22, except 2-propanol concentration, 0.133 M and sample composition, 1:1. Sample size (^L): 1, 2; 2, 20; 3, 50; 4,100; 5,150. (a) Experimental chromatogram. (b) Calculated chromatogram. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal Chem., 61 (1989) 2380 (Fig. 13). ©1989 American Chemical Society.
13.2.2.3 Influence of the Sample Size The progressive change of the experimental elution profile of a 1:1 binary mixture is shown in Figure 13.27. Because of the combination of values of the coefficients of the Langmuir isotherms of the additive and the two components, and because of the additive concentration in the mobile phase, the band profiles seem to have an anti-Langmuir isotherm behavior. The elution profiles are unusual but characteristic of those obtained under overloading conditions when the mobile phase contains an additive that competes with the components of the mixture. The band profiles observed for the mixture are those that should be expected from the interference between two components having the profiles shown in Figure 13.15b when pure. The elution of the first band begins earlier and earlier when the sample size increases, while the elution of the second band ends later and later. The successive profiles are inside each other and have no common diffuse boundary. This behavior is profoundly different from the one observed when the equilibrium isotherms are truly anti-Langmuirian or S-shaped isotherms (Chapter 11). The intensity of the interactions between the individual bands of the binary mixture components increases with increasing sample size. The valley between the two peaks becomes narrower when the sample size is increased. Figure 13.27a shows that the intensity of the retainment effect of the second component band by the first one in the experimental chromatogram is important and increases with increasing degree of band interference. This effect keeps a degree of separation even when the bands interfere severely (profiles 5). The calculated chromatograms in Figure 13.27b show that the theory gives comparable results, the differences between the two figures resulting most probably from the deviation of the competitive Langmuir isotherm from the experimental competitive isotherm. Quifiones et al. measured by frontal analysis the single-component, binary and ternary isotherms of benzyl alcohol (B A), 2-phenyl ethanol (PE) and 2-methyl benzyl alcohol (MBA) on Symmetry-Cis, using a binary mobile phase (MeOH : H2O
System Peaks with the Equilibrium-Dispersive Model
646
a
C (g/i)
C(g/I)
\ f 30
2
3 4 Time (min)
5
6
7
Figure 13.28 Elution band profiles of the additive (a) and the solutes (b) corresponding to the separation of a binary solute mixture of BA and PE (3:1, w/w) with a total solute concentration of 40 g/1. Flow rate, 0.8 ml/min. Experimental profiles for BA (circles), PE (squares) and MBA (triangles); Calculated profiles for BA (dashed line), PE (dashed-dotted line) and MBA (solid line). Left: Volume injected 0.5 ml. Right:Volume injected 1.0 ml. Reproduced with permission from I. Quinones, } . C. Ford, G. Guiochon, Anal. Chem., 72 (2000) 1495 (Figs. 8 and 9). ©2000 American Chemical Society.
= 1:1, v/v) [74]. The adsorption equilibrium data of the multicomponent system were well accounted for by the ternary competitive Langmuir model. Then, they determined the elution profiles obtained by injecting large amounts of binary mixtures of benzyl alcohol and 2-phenyl ethanol at different relative concentrations as the feed and 2-methyl benzyl alcohol as the strong mobile-phase additive. A large number of fractions were collected during the elution and analyzed. The high concentration band profiles and the system peak profiles were calculated using the equilibrium-dispersive model of chromatography. Excellent quantitative agreement was found between the experimental and the calculated profiles, as illustrated in Figure 13.28. Extremely unusual system peak profiles were obtained even though the adsorption behavior followed very closely competitive Langmuir isotherm behavior. This work suggests that, under certain circumstances, the use of a properly chosen additive could markedly increase the separation between bands, hence the production rate, the recovery yield, and/or the purity of the fractions. Figure 13.29 illustrates the important changes in the overall elution band profile and in the individual elution profiles of the three compounds that are caused by changes in the composition of the solutions used as the mobile phase and as the sample. These changes are far more important than could be expected under
REFERENCES
647
40 a
c(g/O 30 V
J
20
34 Time (min)
5
7
0
7
0
20
b
c (g/i) <
15
10 o
y \
2
3 4 Time (min)
5
1
Figure 13.29 Same experimental and calculation conditions as in Figure 13.28, except the values of the ratio of the concentrations of BA and PE, 1:3 (left) and 2:1 (right), of the total solute concentration, 30 g/1 (left) and 40 g/1 (right), and of the MBA concentration in the mobile phase, 10 g/1 (left) and 20 g/1 (right). Volume injected 0.5 ml. Reproduced with permission from I. Quinones, ]. C. Ford, G. Guiochon, Anal. Chem., 72 (2000) 1495 (Figs. 10 and 12). ©2000 American Chemical Society. more conventional conditions (unretained additive) and often surprise those who are not used in dealing with system peaks.
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Chapter 14 Kinetic Models and Single-Component Problems Contents 14.1 Solution of the Breakthrough Curve under Constant Pattern Condition . . 653 14.1.1 14.1.2 14.1.3 14.1.4 14.1.5 14.1.6 14.1.7
Analytical Solution for the Constant Pattern Profile Numerical Solution of the Breakthrough Curve under Constant Pattern Behavior Effect of Axial Dispersion The Shock Layer Theory Shock Layer in the Case of the Langmuir Isotherm Properties of the Shock Layer Thickness in Frontal Analysis Range of Validity of the Equilibrium-Dispersive Model
14.2 Analytical and Numerical Solutions of the Kinetic Models 14.2.1 14.2.2 14.2.3 14.2.4
Solution of the Reaction-Kinetic Model in the Case of a Step Injection Numerical Solutions of the Kinetic Model for a Breakthrough Curve Analytical Solution of the Reaction-Kinetic Model in the Case of a Pulse Injection Numerical Solutions of the Reaction-Dispersive and the Transport-Dispersive Models for a Pulse Injection
14.3 Comparison Between the Various Kinetic Models 14.4 Results of Computer Experiments 14.5 Numerical Solution of the Lumped Pore Diffusion Model 14.6 The Monte Carlo Model of Nonlinear Chromatography References
654 657 657 658 661 662 668
669 670 671 671 674
680 687 689 693 695
Introduction There are a number of cases in which the equilibrium-dispersive model cannot predict accurately the band profiles observed in chromatography. The basic assumption of this model is that the kinetics of mass transfer in the chromatographic column and the kinetics of adsorption-desorption are fast. Accordingly, the band profiles are determined primarily by the nonlinear thermodynamics of equilibrium, as described by the ideal model. Thus, in the equilibrium-dispersive model, the effect of the finite column efficiency is accounted for by a finite and constant apparent dispersion coefficient. This coefficient is related through the column HETP (height equivalent to a theoretical plate) to the axial dispersion coefficient and the mass transfer kinetics [1,2]. This relationship is taken from linear chromatography and used in nonlinear chromatography. As we show in this chapter, this approximation is valid only as long as the column efficiency is high enough, and the contributions of axial dispersion and mass transfer kinetics are only cor651
652
Kinetic Models and Single-Component Problems
rections to band profiles, which remain essentially controlled by thermodynamics. The simple relationship between the apparent dispersion coefficient and the column HETP under linear conditions remains an approximation, and it does not hold for two reasons when the kinetics of one or several of the various phenomena involved in the chromatographic process becomes slow. First, and as we have shown in Chapter 6, even in linear chromatography it is not possible to lump successfully all the contributions to band broadening into a single parameter when the column efficiency is low. Second, both the apparent dispersion coefficient and the column HETP depend on the retention factor, k'. In linear chromatography there is no ambiguity regarding the value, k'Q, of the retention factor. In nonlinear chromatography, we should use instead k' = (FAq) /AC, which is concentration dependent. Thus, it is difficult to account correctly for the influence of the retention factor, k'(C), on band profiles in overloaded elution [1]. In the equilibriumdispersive model, we assume k' = k'o, which is the second approximation made in this model, and this generates significant errors at low column efficiencies. It has been shown that when the column efficiency becomes less than a hundred theoretical plates we need to take the mass transfer kinetics directly into account because its effect on the band profiles becomes comparable to or larger than the thermodynamic effect, due to the nonlinear behavior of the isotherm [1]. As discussed in Chapter 6, in the case of linear chromatography, two general approaches have been used to address the problem of the influence of the kinetics of mass transfer and of adsorption-desorption. In the first approach, all the contributions of various origins to the mass transfer resistances are lumped into a single kinetic equation, using a lumped or apparent rate constant. Three main lumped kinetic models have been used, the solid film linear driving force model, the liquid film linear driving force model, and the first order adsorption-desorption reaction model.
In linear chromatography, these models are all equivalent, as we have shown in Chapter 6, and the analytical solution derived by Lapidus and Amundson [3] applies to all of them. This is no longer true in nonlinear chromatography [1], as we show later. The second possible approach is the use of the general rate model of chromatography. Although this model is the most accurate and complete of all those used to describe the behavior of a band of solute in a chromatographic column, it is quite complex. Its formulation includes two mass balance equations for each component involved, the first in the fluid mobile phase, in the extraparticle volume, the second inside the particles of adsorbents, describing the behavior of these components in the pores. This model includes axial dispersion (i.e., molecular axial diffusion and eddy diffusion), external mass transfer, intra-particle diffusion, and the kinetics of adsorption-desorption or, more generally, the kinetics of the retention mechanism. We have given in Chapter 6 a detailed discussion of the solution of the general rate model in the case of linear chromatography. We have also presented the various simplifications that can be made when one of the various steps listed above can be considered as the rate-controlling step. For example, the pore diffusion model is a special case of the general rate model in which it is assumed that the external mass transfer resistance is negligible and the kinetics of adsorption-desorption is infinitely fast. As a result, intraparticle diffusion
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition
653
controls the mass transfer kinetics. An analytical solution in the time domain, or at least a solution in the Laplace domain, can be obtained in all cases. Finally, as we have shown in Chapter 6, the solutions of the general rate model and of the lumped kinetic model are equivalent in linear chromatography, provided that the column efficiency is not very small (i.e., is larger than about 20 theoretical plates). The same approaches that were successful in linear chromatography—the use of either one of several possible lumped kinetic models or of the general rate model—have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotherm provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. Finally, we note that the solution of the general rate model for a multicomponent mixture can be applied simply to the single component case by choosing 1 as the number of components. Thus, the general rate model and its solution are discussed in the next chapter to avoid repetitive discussions.
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition If the equilibrium isotherm is strictly linear or convex downward, the profile of a concentration step input or breakthrough curve spreads constantly during its migration. When the isotherm is convex upward, however, the profile of the breakthrough curve tends asymptotically toward a limit. This phenomenon arises because the self-sharpening effect associated with a convex isotherm is balanced by the dispersive effects of axial dispersion and of a finite rate of the mass transfer kinetics. In practice, this limit is reached after a migration distance that is often rather short, unless the mass transfer kinetics is exceedingly slow. After this constant profile has been reached, the profile migrates along the column as a mere translation, without spreading, sharpening, or otherwise changing shape. Each point of the breakthrough curve moves at the velocity of the shock predicted by the ideal model. This state is called constant pattern.
654
Kinetic Models and Single-Component Problems
14.1.1 Analytical Solution for the Constant Pattern Profile Bohart and Adams [4] were the first to recognize the existence of this constant pattern behavior. They assumed irreversible adsorption and the control of the mass transfer kinetics by the rate of adsorption. Later Wicke [5] gave an analytical solution in the case of irreversible adsorption, assuming that the mass transfer kinetics are controlled by the diffusion rate inside the particles. The asymptotic nature of constant pattern behavior has been discussed by Cooney and Lightfoot [6], who demonstrated its existence for all convex-upward isotherms. The mathematical criterion for constant pattern behavior is
^ =£
(141)
where q and C are the concentrations of the component in the stationary and mobile phase, respectively, and CJQ is the concentration in the stationary phase in equilibrium with the step concentration, Co. An analytical solution can be found for the breakthrough curve provided the following two assumptions are accepted: • Axial dispersion is negligible, and • Mass transfer kinetics are accurately accounted for either by the solid film or by the liquid film linear driving force model. Neglecting axial dispersion gives for the mass balance equation
Equation 14.2 is also the basic equation of the ideal model (Eqs. 2.13 and 7.1). In the ideal model, however, it is completed by the isotherm equation, q = /(C). By contrast, in kinetic models it is completed by a relationship between dq/dt and either local concentration, q or C. The solid film linear driving force model gives as the kinetic equation
ft=kf(q*-q)
(14.3)
where kf is the mass transfer coefficient, q is the actual stationary phase concentration, and q* is the equilibrium concentration assumed to be given by the Langmuir model: q b C
'
1 + bC
(14.4)
The solution to this problem (Eqs. 14.2 to 14.4) has been derived by Glueckauf and Coates [7], and Michaels [8] derived the solution of a similar problem (Eqs. 14.2,14.4, and 14.5) assuming as kinetic model the liquid film linear driving force model:
|f=^(C-C*)
(14.5)
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition
655
where C* is the equilibrium concentration assumed to be given by the Langmuir model (see Eqs. 2.20 and 6.45). These solutions can be written in a dimensionless form: • With the solid film linear driving force model, we obtain by combination of Eqs. 14.1 to 14.4 [7,9]: r) - 1 - 1)x = -R^eq In U(x)j 1 - In l(1 - *>-
(14.6)
— -Keq
• With the liquid film linear driving force model, we obtain by combination of Eqs. 14.1,14.2,14.4, and 14.5 [8,9]: (14.7) with the following definitions for the dimensionless parameters: X
K
=
=
Req —
C Q 1 fut
(14.8a) (14.8b)
•0
FAq
AC 1
1
l + bC0
%
Z = k1fK
Nif
=
u
Fk'ZJ
u
(14.8c)
1
=
i + r0 ^f^-Q
Z
1 + bCo u
K V
(14.8d) (14.8e) (14.8f)
According to Eqs. 14.6 and 14.7, the dimensionless plot of x versus N(T^ — 1) depends on the single parameter i?eq, which indicates the deviation of the isotherm from linear behavior. Figure 14.1 illustrates this result by showing breakthrough profiles obtained for values of Req between 0 and 0.8, with the liquid film linear driving force model, under constant pattern conditions. The comparison of Eqs. 14.6 and 14.7 shows that, in contrast to linear chromatography, the solutions obtained for the breakthrough profile are different for the liquid and the solid film linear driving force models. Whereas the number, N, of mass transfer units is independent of the concentration with the liquid film linear driving force model (Eq. 14.8f), this number depends on the concentration with the solid film linear driving force model (Eq. 14.8e). Furthermore, the profile of the breakthrough curve is no longer symmetrical as predicted by the equilibrium-dispersive model (to be discussed later, in reference to Figure 14.3). With the liquid film linear driving force model, the lower half is spreading more
Kinetic Models and Single-Component Problems
656
Figure 14.1 Constant-pattern behavior of a breakthrough profile in the case of fluidresistance-controlled kinetics. Plot of x versus Nsf at different values of R (Eqs. 14.7 and 14.8a). Reproduced from T. Vermeulen, M.D. LeVan, N.K. Hiester and G. Klein, in "Handbook of Chemical Engineering," Perry Ed., 6th ed., 1984, Chapter 16 (Fig. 16.13), with permission from McGrawHill, ©1984.
widely than the steep upper half. In contrast, with the solid film linear driving force model, the breakthrough curve is steep at the beginning and becomes smoother toward its end [9-11]. The solutions given in Eqs. 14.6 and 14.7 are not rigorous solutions of the problem, but make the following two assumptions. First, the derivation of these equations neglects the contribution of the axial dispersion. Second, these equations are asymptotic solutions, as obtained under constant pattern behavior. The length of the trajectory required for the solution to become close to the asymptotic solution depends on the degree of deviation of the isotherm from linear behavior, hence on the component concentration, and on the rate of the mass transfer kinetics. Generally, this length is small for small values of Req and increases rapidly with increasing Req. The limit number of transfer units (Ni^ required to achieve constant pattern) can be approximated [9] as 4.6
(1-Re-* ^eq)
(14.9)
When Req is very small or tends toward 0 (note that in the case of irreversible adsorption, Req = 0), the number of transfer units required to achieve constant pattern becomes very small, and the corresponding column length is very short. By contrast, in linear chromatography, Req becomes very close to 1 and N]im tends toward infinity. Thus, we cannot reach constant pattern under linear conditions. Knowing the value of Req = 1/(1 + bCo) permits the derivation of an estimate for N]im, hence (see Eqs. 14.8e and 14.8f) of an estimate of the column length at which constant pattern will be practically achieved.
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition
657
14.1.2 Numerical Solution of the Breakthrough Curve under Constant Pattern Behavior Hall et al. [12] and Garg and Ruthven [13] have given a numerical solution of constant pattern breakthrough curves, assuming the pore diffusion models, and neglecting axial dispersion. Their model includes the three following equations:
ag + ^ + l ^ g
= 0
q
=
-^ (
(14.10a)
V
qr2dr
(14.10c)
i\p J 0
where Rp is the radius of the particle, £y the interstitial or external porosity, and £p the internal porosity. The results obtained with the solid film linear driving force model, the pore diffusion model, and the micropore diffusion model were compared by Ruthven [14]. In contrast to linear chromatography, numerical solutions obtained with different models are different, especially in the initial time region. For moderate loadings (i.e., for Req > 0.5), the differences remain small. As the loading increases, however, and Req becomes lower than 0.5, the differences between the numerical solutions derived from the various models studied increase. Accordingly, differences observed between experimental results and the profiles predicted by a kinetic model are most often due to the selection of a somewhat inappropriate model.
14.1.3 Effect of Axial Dispersion The asymptotic solutions given in Eqs. 14.6 and 14.7 were derived assuming that axial dispersion is negligible. Acrivos [15] has discussed the influence on the shape of the constant pattern breakthrough curve of the combination of axial dispersion and mass transfer resistance. An exact analytical solution can be derived only in the case of an irreversible adsorption isotherm (Req = 1/(1 + bCo) = 0, or b infinite), and assuming a liquid film linear driving force model [15]. The solution shows that, in contrast to linear chromatography, the contributions of axial dispersion and of the mass transfer resistance are not simply additive. Acrivos [15] also derived an approximate solution in the case of a reversible adsorption isotherm and concluded that the "linear addition rule" is followed only when the isotherm is "almost linear." In the case of a nonlinear isotherm, the two contributions are not additive. Garg and Ruthven [16] also discussed the influence of axial dispersion on the shape of the constant pattern breakthrough curves. They used a numerical solution of the liquid film linear driving force model. They concluded that the linear addition rule is approximately valid for nonlinear isotherms. The deviation from the result of an addition of the two contributions becomes important only when
658
Kinetic Models and Single-Component Problems
S = (6FkfDi)/(dpDiU2) is large, and the isotherm deviates strongly from linear behavior.
14.1.4 The Shock Layer Theory The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amundson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as
where D^ is the axial dispersion coefficient. Equations 14.3 and 14.11 can be rewritten in dimensionless form as dC dC rdq lT + 3 - + F ; r dx dr dr | |
= =
1 9C 1TTT Pe dx2 St(f(C)-q)
,.,,,.,„ . (14.12a) (14.12b)
where q* = f(C) is the isotherm, and Pe = ^ St
=
(14.13a)
kfL -1—
(14.13b)
x
= j
(14.13c)
r
= ^
(14.13d)
The space domain is assumed to be infinite, in order to exclude any end effects, and the boundary conditions are x = -co x = +oo
C C
= =
d C
(14.14) (14.15)
Rhee et al. [17] have shown that for any isotherm, q* — f(C), there exists a unique shock layer for the system of Eqs. 14.12a and 14.12b, provided the end concentrations, C! and Cf, satisfy the shock condition, f"(0)(O — C?) > 0 for an isotherm without inflection points. Thus, if the isotherm is convex upward if" (0) < 0)/ there is a shock layer if C' < Cf. If the isotherm is convex downward, the reverse condition applies. If the isotherm has an inflection point, e.g., with f"{Cj) = 0 and f"{C) < 0 for C < Cy (where Cy corresponds to the inflection point of the isotherm, Figure 14.2), there exists a unique shock layer if the end states, C and Cf satisfy one of the two conditions
c < cvf
<
Q
<
< c/
!
c'
for
C <
(14.16)
for
C >
(14.17)
24.2 Solution of the Breakthrough Curve under Constant Pattern Condition
659
Figure 14.2 Notation for the shock layer determination in the sigmoidal isotherm case. Reproduced with permission from H.-K. Rhee, B.F. Bodin and N.R. Amundson, Chem. Eng. ScL, 26 (1971) 1571 (Fig. 1).
where Q and Q are the concentrations of the contact points between the isotherm and the tangents to the isotherm from the isotherm points of concentrations O and Cf (see Figure 14.2). Rhee et al. [17,18] have also shown that the shock layer propagates at the same velocity as the shock in the ideal model, hence at a velocity that is independent of the axial dispersion coefficient and of the mass transfer coefficient. This velocity is given by (Eq. 7.3): (14.18a)
= «A
u, = with
(14.18b)
A = and
f(Cf)-f(C)
(14.18c)
cf - o
AC
Finally, they showed that the profile of the shock layer can be obtained by integrating the nonlinear differential equation -A
d2C 1
PeStdij
1 Pe
A(l-A) Si
(14.19)
where A is the reduced migration velocity of the shock (or the shock layer), given by Eq. 14.18b. t] is a reduced coordinate describing the position of the breakthrough curve in the column
,
z
ut
(14.20)
and the function g is given by
g(C;Cf;C) =
- cf) -/(C)
(14.21)
Kinetic Models and Single-Component Problems
660
Figure 14.3 Parameters of shock layers in frontal analysis. Reprinted with permission from }. Zhu, Z. Ma and G. Guiochon, Biotechnol. Progr., 9 (1993) 421 (Fig. 1). ©1993, American Chemical Society.
1-S
v
,
c,*
/ 8 B
z;
c° 14 0
16.0
TIME
Cl and Cf are the initial and final concentrations of the breakthrough curve, respectively (i.e., C = O at t] = —oo and C = C? at t] = +oo), and AC = Cf — O,
Aq*=f(Cf)-f(O). The roles of axial dispersion (Pe) and the resistance to mass transfer (St) in Eq. 14.19 are nearly symmetrical. Their contributions to the first-order term are additive. There is, however, a coupling, second-order differential term. In practice, both the Peclet and the Stanton numbers are large, unless kf is very small. Typically, Da is lower than 1 xlO~4 cm 2 /s, and L larger than 10 cm. Thus, the product PeSt = kfL2/Da is larger than lxlO 6 , unless kf < 1. Thus, we can almost always neglect the first term. If we do so, the effects of axial dispersion and the mass transfer resistance are additive, and there is an analytical solution of Eq. 14.19. The shock layer profile is given by dC
(14.22)
where Cm is the concentration at the inflection point of the profile, at rj = 0 (see Figure 14.3). Since the integral diverges for both C = C! and C — Cf, we consider two intermediate, finite boundaries, C* and C%, associated through the parameter 6{6 < 0.5) 8=
Cf -C
7
Cf - C
cf - o
cf - a
(14.23)
If 6 = 0.05, for example, the shock layer will be the region where 90% of the total concentration variation takes place. The shock layer thickness is the distance between the two corresponding points on the breakthrough curve
i(cf)-v(c*f) 1 | A(l-A) c Pe + St M If FA
(14.24) dC
Jq g(C';Cf;O)
(14.25)
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition
661
Equations 14.22 and 14.24 are dimensionless. In the case of the elution of a breakthrough curve at the end of a column of finite length, L, we have, according to Eq. 14.20, (14.26a)
= Vi ~V} = " or
(14.26b) where Arjt is the time required for the elution of the shock layer, and (14.27) or At]z = LArjo
(14.28)
where Arjz is the actual thickness of the shock layer along the column. Equations 14.22 and 14.24 to 14.27 are valid for all isotherms and require only that the shock condition be verified, i.e., that they are applied to a self-sharpening front, and that the product Pe St is very large.
14.1.5 Shock Layer in the Case of the Langmuir Isotherm The analytical solution of Eq. 14.19 is particularly simple in the case of the Langmuir isotherm, q = («C)/(1 + bC). If we define (see Eq. 14.8ad) Rf =
R* =
-bCf
1 1 + bO
1
R=
(14.29)
bC
we have A
=
1
(14.30)
1 + k'y
(14.31) and the shock layer thickness is now given by 1
, A(l-A^
A. .
1
l-i \ Ri + Rf In 6 R'-Rf
1 + k'QRfR')
(14.32)
Equations 14.24 and 14.32 are valid for any step injection. If we consider the step injection of a concentration Co in an empty column (i.e., Q = 0, Cf = Co), and insert the values of St, Pe and A in Eq. 14.32, we obtain K Atjt
=
uL Ku1
Ku kfL{l + K)2)
foCo + 2 In 1-9 9
bC bC00
In bCo bC0
1-6
(14.34)
e In
(14.33)
l-i
(14.35)
662
Kinetic Models and Single-Component Problems
where Arjz and Arjt stand for the thickness of the shock layer in length and time units, respectively. In using these equations, we should keep in mind that they are approximate equations for several reasons. First, the shock layer is an asymptotic solution, never reached under the conditions of an actual experiment. As shown below, however, it is not difficult to find experimental conditions under which the breakthrough curve approaches the corresponding asymptotic solution within the confidence limits of the experimental results. Second, these equations have been derived by assuming that the coupling term in Eq. 14.19 is negligible. Finally, the equations have been derived with the assumption of the solid film linear driving force model and are valid only as long as this model itself is valid. We see in Eqs. 14.32 that the shock layer thickness is proportional to
h-^1 or
H=—
+2(T^1
ft
(14-37)
where K is given by Eq. 14.8ac and is a function of the concentration. This important result demonstrates that in nonlinear chromatography, as in linear chromatography, the effects of the axial dispersion and the mass transfer resistance are additive, at least as long as the efficiency is not very low. However, since A and K are concentration dependent, the combined parameter is a function of the concentration. Thus, the situation is more complex in nonlinear than in linear chromatography.
14.1.6 Properties of the Shock Layer Thickness in Frontal Analysis In this section, we discuss the consequences of Eq. 14.33a and the experimental results reported by Zhu et al. [19,20] regarding the dependence of the shock layer thickness (SLT) on the main experimental parameters, the mobile phase velocity and the height of the concentration step. As shown by Van Deemter et al. [2], the axial dispersion coefficient at large values of Pe is given by the expression 2DL = Au + 2yDm
(14.38)
where A is a numerical coefficient characterizing the packing homogeneity of the column, 7 is the tortuosity coefficient of the column packing (Chapter 5), and Dm is the molecular diffusivity (Chapter 5). This expression permits further derivations of interest. The conventional relationship suggested by Knox [21] and which is often used in HPLC (2DL = Aw4/3 + 2yDm) is more difficult to handle here. Experimental evidence regarding which expression is most appropriate in chromatography does not offer any compelling reason at this stage to prefer either
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition
663
of these two expressions. Accordingly, the former will be used in the following discussion. Combining Eqs. 14.33a and 14.38 gives K
Au + 2
uK
\ bC0 + 2
l-i
2uL
(14.39)
From this equation we can derive some interesting consequences. Before proceeding further, we emphasize that the solution of Rhee et ol. [17,18] is an asymptotic solution and that the shock layer thickness, as predicted by Eqs. 14.33a-c, is a limit that is achieved, in principle at least, only after an infinite migration distance. Theory does not permit the calculation of the actual migration distance after which the steady state or constant pattern profile is so closely achieved that it becomes difficult to distinguish the experimental profile from the asymptotic solution. It depends on the concentration height. Equation 14.9 shows that this distance is often quite short in experiments made in nonlinear chromatography with HPLC columns. As a consequence, although derived from an asymptotic solution, the shock layer thickness is relevant to practical chromatography (see Figure 14.6 below). 14.1.6.1 Influence of the Mobile Phase Flow Velocity on the SLT Differentiation of Eq. 14.39 with respect to the mobile phase flow velocity, u, shows that the SLT is minimum for the velocity (14-40) The optimum velocity for minimum SLT can be quite different from the optimum velocity for minimum HETP, u\ t [20]. We see that when the step height, CQ, tends toward 0, this equation gives as limit the optimum linear velocity for minimum plate height found in linear chromatography. Experimental results regarding the dependence of the SLT on the mobile phase velocity are in agreement with the prediction of Eq. 14.40, as shown in Figure 14.4 [20], when the contribution of the mass transfer resistance to the HETP is moderate. When this contribution is significant, the agreement is less satisfactory, suggesting that the coefficient kj is not constant, but concentration dependent [20]. The optimum linear velocity given by Eq. 14.40 is a function of the height of the concentration step [19]. It is minimum for the following value of the concentration step
So, when k'o < 1, there is no minimum, and the optimum velocity for minimum SLT increases indefinitely with increasing concentration-step height. On the contrary, at large values offcg,the optimum velocity for minimum SLT decreases with
664
Kinetic Models and Single-Component Problems
Figure 14.4 Comparison between plots of the HETP (cm) and of the SLT (cm) versus the mobile phase flow velocity. Same experimental conditions for both figures: 5 cm long Vydac column. Mobile phase: 50:50 methanol-water, monitored at 270 nm for both series of measurements. Sample: 2-phenylethanol (k'o = 0.88). Height of the concentration step in frontal analysis: 20 mg/mL. Sample size for linear elution peaks: 40 fig (0.2 jiL of a 20 mg/mL solution). Top Figure: Plot of the SLT versus the mobile phase flow velocity. Experimental data (symbols) and prediction of Eq. 14.33c (solid line). Bottom Figure: Plot of the HETP versus the mobile phase velocity under linear conditions. Experimental data (symbols) and best fit to the Van Deemter equation (solid line).
2.0
4.0
6.0
8.0
Linear Velocity (cm/min)
Reproduced with permission from J. Zhu and G. Guiochon, } . Chromatogr., 636 (1993) 189 (Fig. 2).
Figure 14.5 Plot of the optimum velocity for minimum SLT vs. the concentration step height. Curves derived from Eq. 14.40. Curve 1, conditions used for Figure 14.4. Curve 2, experimental conditions: 5 cm long home packed column; mobile phase 50:50 methanol-water, sample: 4-tert-butylphenol (k'Q = 10), sample size: 0.2 jig. Symbols: optimum velocity under linear conditions. Reproduced with permission from ]. Zhu, Z. Ma and G. Guiochon, Biotechnol. Progr., 9 (1993) 421 (Fig. 8). ©1993, American Chemical Society.
100.0
200.0
300.0
Concentration (mg/ml)
400.0
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition
665
Figure 14.6 Plot of the SLT versus the height of the concentration step. Three columns packed with Vydac C18 silica, experimental conditions as in Figure 14.4. Column length: 1 (o), 5cm; 2 (A), 15 cm; 3 (•), 25 cm. Data points (symbols) and curve calculated from Eq. 14.33a. Reproduced with permission from J. Zhu, Z. Ma and G. Guiochon, Biotechnol. Progr., 9 (1993) 421 (Fig. 9). ©1993, American Chemical Society.
-1.5
-0.5
0.6
1.6
LOG {Concentration (mg/ml))
increasing concentration-step height at first, and may become markedly lower than the optimum velocity for minimum HETP before it starts increasing with increasing step height. Figure 14.5 illustrates the dependence of the optimum mobile phase flow velocity on the height of the concentration step for two different values of k'o [20]. Thus, the column should be operated at velocities higher in frontal analysis than in elution when the step height is high or the retention factor small, while the converse is true at low step heights and high retention factors [20]. 14.1.6.2 Influence of the Height of the Concentration Step Differentiation of Eq. 14.33a with respect to Co and setting this differential to 0 gives a fourth degree equation without any simple roots [20]. A more detailed study shows that the SLT decreases with increasing concentration-step height, until impractically high steps are reached. Thus, in almost all cases, the SLT will decrease with increasing concentration-step height. Figure 14.6 illustrates the variation of the SLT with increasing step height for three columns. At high values of the step height, there is a very good agreement between the variation predicted by Eq. 14.33a and the experimental results obtained with the three columns [19], although the SLT is difficult to measure with accuracy on efficient columns. On the contrary, there is no such agreement at low concentrations. The experimental and theoretical results begin to deviate at some intermediate concentration which depends on the column length, and decreases with increasing column length. This result demonstrates that the migration length required to achieve constant pattern increases with decreasing step height. When the step height tends toward 0, the shock layer theory predicts that the SLT tends toward infinity. On the other hand, these experimental conditions correspond to linear chromatography, and we know that, in this case, the width of the breakthrough curve at the end of a column of length L tends toward the width of the
Kinetic Models and Single-Component Problems
666
100
200
300
t (min)
Figure 14.7 Staircase frontal analysis of (+)and (-)- Troger's base, (a) (+) enantiomer, 30° C, positive and negative steps, N = 16. Comparison between experimental data (symbols) and curve calculated with the equilibrium-dispersive model, (b) Effect of temperature on the frontal analysis of the (-) enantiomer; from right to left, 30, 40, and 50° C. (c) Adsorption isotherms of () enantiomer at 30, 40, and 50° C; N = 30, 43, and 60; respectively. Column: 0.46 x 10 cm, packed with 15-25pm microcrystalline cellulose triacetate; porosity, 0.602; Fv = 0.5 mL/min, mobile phase 100% ethanol. Reproduced with permission from A. SeidelMorgenstern, S. Jacobson and G. Guiochon, } . Chromatogr., 637 (1993) 19 (Figures 3, 8 and 9).
Gaussian profile of variance HL. 14.1.6.3
Staircase Frontal Analysis
Equation 14.32, derived by Rhee and Amundson [18], is a general solution of the breakthrough profile when the solute concentration rises from 0 to C^, both values being arbitrary. In conventional frontal analysis, a series of successive steps are injected and the plateau concentration is progressively built up, so the initial concentration, O is no longer 0, and Eq. 14.32 must be rewritten. Let Cn — O and Cn+\ — Cf be the concentrations of the nth and the n + lth injected plateaus, respectively, We obtain KnU
Kn
b(Cn+1 b(Cn+1 -
l - i
(14.42)
14.1 Solution of the Breakthrough Curve under Constant Pattern Condition 667 Figure 14.8 Dependence of the apparent column efficiency on the concentration. Efficiency derived from self-sharpening fronts obtained in staircase frontal analysis of ()-Troger's base, 1 at 50°C; 2 at 40°C; 3 at 30°C; and of (+)-Troger's base at 30°C. Reproduced with permission from A. Seidel-Morgenstem, S.C. Jacobson and G. Guiochon, } . Chromatogr., 637 (1993) 19 (Fig. 6)
2
3
4
5
6
number of step
with (14.43) Seidel-Morgenstern et al. [22] have fitted the breakthrough curves obtained in a series of frontal analysis experiments to the equilibrium-dispersive model (Chapter 10), and determined the best value of the apparent dispersion coefficient for minimum difference between calculated and experimental profiles. Figure 14.7 illustrates a comparison between theory and experiment for the staircase frontal analysis of (+) and (-) Troger's base. The result of the comparison is excellent, especially considering the low efficiency of the column and the use of the equilibriumdispersive model (Chapters 10 and 11). Both the adsorption and the desorption profiles are well predicted by the model (Figures 14.7a and b). The influence of the temperature on the adsorption isotherm of the (-) enantiomer is shown in Figure 14.7c. However, different values of the apparent dispersion coefficient have to be used for each concentration step in Figures 14.7a and b. It is possible also to derive from Da an apparent plate number, N = Lu/(2Da). These data are shown in Figure 14.8. From these results as well as from those of Zhu et al. [19,20] discussed above, we conclude that there is a good agreement between the predictions of the shock layer theory and the experimental measurements of the shock layer thickness, but that it seems that the less valid one of the several assumptions made in the derivation of the shock layer theory is the assumption that kf is independent of the concentration. A significant variation of kf with the component concentration would explain most of the deviations reported. These results are in agreement with the conclusions of the comparisons
668
Kinetic Models and Single-Component Problems
made between the various lumped kinetic models (Section 14.3, below).
14.1.7 Range of Validity of the Equilibrium-Dispersive Model Because it is possible to calculate the shock layer thickness with a lumped kinetic model and with the equilibrium-dispersive model, a comparison of these two expressions provides an attractive method of investigation of the range of validity of the latter model. In the equilibrium-dispersive model, the apparent dispersion coefficient is assumed to be given by the equation Da = ^
(14.44)
where H is the column HETP measured under conditions of linear chromatography. Van Deemter et al. [2] have derived from the solid film linear driving force model the following HETP equation (see Chapter 6)
Equation 14.45 applies in linear chromatography. The correct HETP equation in frontal analysis under constant pattern behavior, and with the same solid film linear driving force model is Eq. 14.36b. Comparison of these two equations shows that an error is made when the latter is used to replace the former, in the equilibrium-dispersive model. We should replace in Eq. 14.44 and 14.45 k'o by k! = (FAq)/AC. In the case of the Langmuir isotherm, this would give k' = k'0/(l + bC0) =K.
The error made remains small insofar as the second term of Eq. 14.45 is small compared to the first term, i.e., when the mass transfer resistances are small, i.e., the mass transfer coefficient, kf, is large. The error increases with increasing value of the product bQ), and would be negligible for a moderate degree of overloading, or for small values of b, i.e., for slightly curved isotherms. This, however, is also the case when the shock layer approach may not be applicable (see Figure 14.6) If the shock layer theory is applied within the framework of the equilibriumdispersive model (Eqs. 14.44 and 14.45), the shock layer thickness becomes [17,19] K
DabCo UL
[uL
2 ^ 1 - 9
KU
(
K
+
+
\ bCo kfL(l + k'0)2J bC0
l-i n
(14.47)
where AT\QI stands for the shock layer thickness calculated with the equilibriumdispersive model, using the HETP under linear conditions. The two shock layer thicknesses can be compared by calculating their relative difference -Ane,i
1 2u k'obCo(k'o2 - 1
-k'obCo)
14.2 Analytical and Numerical Solutions of the Kinetic Models
669
Note that in Eq. 14.48, for the sake of simplicity, the absolute error (ATJQ — is referred here to the approximate value (Arjgf), not to the true value (Atjg). In practice, the result is the same as long as the relative error is small. The relative error given by Eq. 14.48 is obviously small if the second term of the RHS of Eq. 14.45 is small compared to the first one, i.e., if the mass transfer resistances are small, and the column is highly efficient. If this is not true, however, and the second term in the RHS of Eq. 14.45 is dominant, Eq. 14.48 becomes SAt, An
_bCo(k'i-l-k'QbCQ) (l + ^ + bC)*
K
]
The relative error tends toward 0 with decreasing height of the concentration step, and tends toward k'o for very large heights, which might be quite large under some sets of conditions. However, if we consider a relatively high degree of concentration overload, with bCo = 1, the value for which q = 0.5^s, the error is only 11% for k'o = 1, 6% for k'Q = 2, and reaches 30% only for k'o = 4. For higher values of k'Q/ it tends toward 100%. This is still acceptable in most cases, given the very small value of the SLT observed under such a high degree of column overload, unless the rate coefficient kf is very low, and the kinetics of mass transfer very slow. If the second term of the RHS of Eq. 14.45 accounts for a fourth of the plate height, then the limit error made with a high value of k'o and bCo = 1 is only 25% under the most unfavorable set of experimental conditions. This illustrates the fact that deviations between experimental results and the prediction of the equilibrium-dispersive model will occur preferably at high mobile phase flow velocities. We see from Eq. 14.48 that the error introduced by the use of the equilibrium-dispersive model instead of a kinetic model decreases with increasing values of both Dakf/u2 = St/Pe and k'0/(l + k'Q)2. The value of Da is of the order of 1 to 5 x 10~4 cm 2 /s; that of u of the order of 3 to 10 x 10~2 cm/s. Thus, kf should be in the 10 to 100 s" 1 range. Thus, we can conclude that, as long as the mass transfer kinetics is reasonably fast, the equilibrium-dispersive model can be used as a first approximation to predict shock layer profiles. As a consequence, the results of calculations of band profiles, breakthrough curves, or displacement chromatograms made with this model can be expected to agree well with experimental results. Conclusions based on the systematic use of such calculations have good predictive value in preparative chromatography.
14.2 Analytical and Numerical Solutions of the Kinetic Models The kinetics of adsorption-desorption is rarely slow in preparative chromatography, and most examples of a slow kinetics are found in bioaffinity chromatography, or in the separation of proteins. Thus, there are few cases in which a reactionkinetic model is appropriate. This model is important, however, because there are many cases where it is convenient to model the finite rate of the mass transfer
670
Kinetic Models and Single-Component Problems
kinetics of a system by using an apparent reaction-kinetic model, with a lumped kinetic parameter. The properties of the reaction-kinetic model were studied first by Thomas [23], who neglected the contribution of axial dispersion to band broadening, a legitimate assumption when the reaction kinetics of the retention mechanism is slow. Thomas' model combines the mass balance Eq. 14.2 and a second-order Langmuir kinetics ^=kaC(qs-q)
-kdq
(14.50)
where ka and kj are the rate constants of adsorption and desorption, respectively, and qs is the saturation capacity of the adsorbent. This kinetic model leads to the Langmuir model for the equilibrium isotherm (dq/dt = 0), with b = kalk^ (Chapter 3, Eqs. 3.15 and 3.16).
14.2.1 Solution of the Reaction-Kinetic Model in the Case of a Step Injection For this model (Eqs. 14.2 and 14.50), Thomas [23] derived an analytical solution in the case of a step function input, i.e., of a breakthrough curve in frontal analysis. This solution can be written in dimensionless form using the following transformation suggested by Hiester and Vermeulen [9,23]. This transformation uses the parameters defined in Eqs. 14.8a to 14.8 d (x, Tj, K, i?eq, Nsf, Nif) and
y = f
(i4.5i)
^
(14.53)
Note that the definition of JVrea is different here and in Chapter 6 (Eq. 6.93). In the former case, Nrea corresponds to a first-order kinetics (Eq. 6.65). In the present case, Nj-ea corresponds to a Langmuir kinetics (Eq. 14.50). With these new parameters, Eqs. 14.2 and 14.36 become )
x)
(14.54)
If we let T = {t — to)/(t^o — to) be the dimensionless time, we have
bCout
l±bCo
t
M
, un , t-tR o)
^(T-1)=^r(^") =( +
T
^^=^
(
}
and we can write as follows [9] the solution in dimensionless form, at the column exit (z = L) - 1)]
14.2 Analytical and Numerical Solutions of the Kinetic Models
671
where T(u,v) = 1-J{u,v) =e~v (Ue^hily/v^dt Jo
(14.57)
1Q is the zero-order modified Bessel function of the first kind [24]. Equation 14.56 shows that a plot of x = C/Co versus T depends only on the two dimensionless parameters NTeil and Req-
14.2.2 Numerical Solutions of the Kinetic Model for a Breakthrough Curve Many other authors have reported numerical solutions for the calculation of the breakthrough curve using different kinetic models and various isotherm models. Some of these include Chen et al. [25], Hashimoto and Miura [26], Garg and Ruthven [27], Peel and Benedek [28], Yoshida [29], and Zwiebel et al. [30] who studied solutions of the liquid film linear driving force model. Lee and Weber [31], Fleck et al. [32], Neretnieks [33], Costa and Rodrigues [34], Garg and Ruthven [35], Raghaven and Ruthven [36], Rasmusson [37], Kaczmarski et al. [38], Piatkowski et al. [39], Hunter et al. [40], and Ghose et al. [41] gave numerical solutions of the various rate models.
14.2.3 Analytical Solution of the Reaction-Kinetic Model in the Case of a Pulse Injection The reaction-kinetic model is the only kinetic model for which an analytical solution can be derived for a pulse injection. The solution of Thomas' model has been derived by Goldstein [42] in the case of a rectangular pulse injection of width tp and concentration Co. It is £ = T(pra(ee0))T((i,cte) C0 E1T(7k'7e) + T^cc(e8o))T^a0)E2[T(7k'7(OeO))^V ' with the following conventions and definitions (different from those made in the previous section) 9 = — -1 k
(14.59a)
e0 = T-
(14.59b)
7 = Mo a = 7(l + bC0)
(14.59c) (14.59d)
i6 =
(14.59e)
,"
Ex = exp[(7 — a)9 — B + 7k'o] E2 = e x p [ ( 7 - a ) ( 0 - 0 o ) - J 6 + 7fco] where k i s th e dead time.
(14.59f) (14.59g)
672
Kinetic Models and Single-Component Problems
Arnold and Blanch [43] have shown that Eq. 14.57 can be modified simply. A slight change of their final result leads to the equation of the profile: T(ReqNrea, f^ [T - T0]) - T(ReC{NTea,
C
± =
^
^—
f
(14.60)
with: P =
exp[Niea(l-Req)(l-T/Req)]T{Niea,Nie!!LT) AT
-
Al
(14.61) ,(1 - Req)(l - ( T - T 0 )/K eq )] [T(NKa,NIea[x
- T0]) - 1]
and: T
=
t-f
°
(14.62a)
tR,0 - to
TO =
-—V—
(14.62b)
^ , 0 "" to
=
k'o7 = k'okdtQ = Fqskat0
I ^ f
—
=
n
(1 - e)SLqs
— bn _ bAp ^Sl¥0 = k^
=
(14.62c)
(14 62d)
-
(14.62e)
T is the dimensionless time, To is the dimensionless injection duration, and tRjo is the limit retention time under linear conditions. Nrea characterizes the contribution of the kinetics of the retention mechanism to the column efficiency. It is equal to the ratio L/Hftn, where Jijyn is the contribution of the kinetics of adsorptiondesorption to the column height equivalent to a theoretical plate, as calculated by Giddings [44]. L is the column length, S its cross-sectional area and e the packing porosity. Lf is the loading factor, a natural measure of the degree of overloading of the column [45], n is the sample size in number of moles and Ap is the area of the injection pulse. Equation 14.60 shows that a dimensionless plot of C / Q versus T = (t — to)/(tR,o — to) depends only on the three parameters N rea , Req, arid To- Req is the thermodynamic parameter, related to the loading factor Lf, which characterizes the degree of overloading of the column. In the case of an impulse input (Dirac problem, tp is infinitely small, the concentration Co is infinitely large and Ap is the area of the injection profile), Wade et al. [46] derived an analytical solution which can be written:
14.2 Analytical and Numerical Solutions of the Kinetic Models
673
where 1\ is the first-order modified Bessel function of the first kind and Q 0 is the ratio of the number of moles of solute injected to the column dead volume, Vo:
f
^ = Coff
^
to
(14.64)
to
According to Wade et al. [46], the dimensionless band profile depends on three dimensionless parameters, k'o, y and bC^ro- This last parameter characterizes the degree of column overloading. Equation 14.60 is easily rewritten as: i
C
CdiJ0
g
/
NleaLf
reaV^MreaV^ 1 _ T(N r e a , N rea T) [1 - e " N - L
(14.65)
with Cdi
<° ~W-1^-^LF-
(t^fe) " ft^fo) ~C°T°
(14 66)
-
Equation 14.65 is the limit of the general solution of Goldstein (i.e., of Eq. 14.58) for an infinitely small pulse width. As can be seen from Eq. 14.65, the dimensionless plot of C/Q^o = C(*R,o ~~ to)/Ap versus T = (t — £o)/(^R,o ~~ ^o) depends only on two dimensionless parameters, Nrea and Lf. Equation 14.57a is used to calculate T(u,v) when u < v. If u > v, however, it is more accurate to calculate first T(p, u), and to use the relationship: T{u,v) = 1 - e-("+°)I0(2Vuv) - T(v,u)
(14.67)
The equations required for the calculation of the zero- and first-order modified Bessel functions, IQ(X) and l\{x), of the first kind are available in [24], and the equation for the calculation of the integral T(u, v) by Gaussian quadrature in [24]. IQ(X) and h(x) are easily obtained from a computer software library and T(u,v) can be calculated by numerical integration of Eqs. 14.57a or 14.57b, depending on the value oiu/v. Equations 14.58 and 14.65 are the solutions of the Thomas model, depending on the width of the rectangular injection profile. We must underline the fact that these solutions are exact. No simplifications have been made in the derivation of the solution beyond the assumption made in the model itself, that Da is 0. Figure 14.9 illustrates a comparison of experimental and theoretical band profiles for proline, glutamic acid, and lysine on a Bio-Rad cation-exchange resin [47]. The theoretical profiles are calculated using rate coefficients derived from a model assuming convection and axial dispersion as the only mechanisms of mass transfer in the axial direction [48]. The rate of mass transfer in the bulk phase was estimated using conventional correlations [49]. Local equilibrium was assumed within the particles, between the stagnant fluid contained inside the pores and the adsorbent surface. Two models were used. The local equilibrium model is equivalent to the ideal model discussed in Chapter 7 in the case of a single component. The profile predicted by the rate model compares very well with the experimental
Kinetic Models and Single-Component Problems
674
0 RATE EQ. MODEL LOCAL EGM. a LVS/Na+ - 1.8
a) PRO
0
l.O
n PHOflta* "
o
LYS o
3
\
SO.O
Dimensionless
Lyg;Ma+ • 4.56
aouum* - o.r
GUI L\
rs
o
hit'i S5.O
a
\
j
RATE EU MODEL LOCAL EQM.
b)
B « 0 C 0
A /
0.5
O.O < O.O
0 M 3
o
75.0
Time
lOO.O
(t Uo/L)
.5
o o O.O
) '00 ' 5O.O
J
1OO.0 15O.O SOO.O 3SO.O
Dimensionless
Time
(t Uo/L)
Figure 14.9 Comparison of experimental data and the predictions of a kinetic and a local equilibrium model. Column 1 x 7.5 cm Bio-Rad AG 50W-X8 cation exchange resin; dp = 25 }IH\; mobile phase, 0.2 N sodium citrate buffer, pH 4.4; Fv = 2 mL/min. (a) Sample 50 mL of 0.01 N lysine and 0.01 N proline in 0.19 N sodium citrate buffer, pH 4.4. Model parameters, Pez = 672; Pep = 337; Bi = 42. (b) Same conditions, except L = 7.3 cm; sample, 40 mL of 0.02 N Lysine and 0.01 N glutamic acid, pH 3.5, at 1 mL/min. Model parameters Pez = 653, Pep = 202, Bi = 36. Reproduced from C. Lee, Q. Yu, S.U. Kim and N.-H.L. Wang, J. Chromatogr., 484 (1989) 29 (Figs, la and lb).
result. By contrast, there is a poor agreement between this experimental result and the profile of the ideal model. This is due to the assumption of an infinitely efficient column made by the ideal model, while the kinetic model accounts for the finite rate of mass transfer. This rate softens the features of the band profiles. Proline (Figure 14.9a) and glutamic acid (Figure 14.9b) are not retained, so the sides of the rectangular injection profile are merely dispersed, although the width of the profile is mainly due to the large feed volume used. In Figure 14.9b, there is retention of lysine, and some degree of overloading. Again we discuss this multicomponent system as the sum of single-component systems. One amino acid being unretained, as well as the counter-ion Na+, these two ions do not compete with the retained amino acid, lysine. Wade et al. [46] have compared the experimental band profiles of p-nitrophenyla-D-mannopyranoside on silica-bonded Concanavalin A, obtained in affinity chromatography, and the best fit parameters to their model. This model (i.e., Thomas model) uses a Langmuir kinetic and neglects the axial dispersion. The best values of the parameters are calculated using a Simplex program to minimize the sum of the residuals of the predicted and experimental band profiles. Figure 14.10 illustrates the results obtained and shows excellent agreement.
14.2.4 Numerical Solutions of the Reaction-Dispersive and the Transport-Dispersive Models for a Pulse Injection The Thomas model [23] is the only kinetic model that has an analytical solution in the single-component case, in nonlinear chromatography. In all other cases, the
14.2 Analytical and Numerical Solutions of the Kinetic Models Figure 14.10 Comparison of experimental results and the prediction of a best fit kinetic model. Column, 2.1 x 50 mm packed with immobilized Concanavalin A on silica; mobile phase, 0.02 M sodium phosphate buffer at pH 6.0, with 0.5 M NaCl, 0.01 M MgCl2 and 0.001 M CaCl2; T = 25±2°C; Fv = 1 mL/min; £Q = 9.53 s. Sample size, 719 ^mol. Model parameters: 7 = 8.203; k' = 22.37; KC0 = 1.905; W = 6824. Reproduced
675
C/Co 1200 1200
+
Experimental Calculated
800
400
0 10
20
30
t/to
from J. Wade, A.F. Bergold and P.W. Can, Anal. Chem., 59 (1987) 1286 (Fig. 7). ©1987 American Chemical Society.
solution has to be obtained by numerical methods. Then, it is not more complicated to consider the slightly more complete dispersive models, including a finite axial dispersion. The mass balance becomes 3C dq dC d2C (14.68) dt dt dz The reaction-dispersive model is an extension of the Thomas model with a Langmuir kinetics dt
= kaC(qs
- q ) - kdq
(14.69)
The transport-dispersive model assumes infinitely fast kinetics of adsorptiondesorption but a finite rate of mass transfer, following the solid film linear driving force equation
ff-*,«*-»>
(14.70)
Numerical solutions of the former [50,51] and latter [52] models have been calculated with finite difference methods. Any of the schemes discussed in Chapter 10 can be used for the calculation of solutions of these kinetic models. However, the generation of the error due to the numerical dispersion is more complex and it is more difficult to control, because two differential equations are now involved for each component, instead of one with the equilibrium-dispersive model. The numerical solutions of kinetic models predict the split peak phenomenon [50-54]. This phenomenon is observed when the mass transfer kinetics or the kinetics of adsorption-desorption is very slow. It may also occur when the column saturation capacity is very small. In all these cases, the number of transfer units is small. Then a fraction of the sample component does not have an opportunity to interact with the stationary phase. This fraction is not retained and elutes at time to, as a sharp peak, called the split peak. The rest of the sample elutes as a broad band. The effect of slow mass transfer on the band profile is illustrated in Figure 14.11 which shows a series of profiles calculated with the reaction-dispersive model, for
676
Kinetic Models and Single-Component Problems
Figure 14.11 Numerical solutions of the reaction-dispersive model obtained for various values of the rate constant of the adsorption-desorption kinetics. Calculation conditions: Lf = 1%. N ras p = 1000. Chromatogram lines: 1: N rea = 20,000; line 2: Nrea = 2000; line 3: N rea = 200; line 4: Nrea = 20; line 5: N rea = 10; line 6: Nrea = 4.; line 7: N r e a = 1. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, ]. Phys. Chem., 95 (1991) 6390 (Fig. 5). ©1991, American Chemical Society.
the same loading factor (1%), and the same number of dispersion units ( i s p (LM)/(2Dfl) = 1000). Calculations have been performed for a series of values of Nrea = ^oMo between 20 000 and 1 [50]. Similar results are obtained with the transfer-dispersive model, for which the rate-controlling step is the mass transfer kinetics [52]. These results and the change in the shape of the band profiles when Nrea decreases are explained by the progressive uncoupling between the thermodynamic and kinetic influences on the band profiles when the kinetics of the mass transfer or of the retention mechanism become very slow. The split peak phenomenon arises when Nrea becomes of the order of a few units. We see also an oscillation of the retention time with increasing value of Nrea from 1 to large values. At first, the retention time increases with increasing values of Nrea because an increasingly large fraction of the sample has access to the retention mechanism (i.e., is retained by interaction with the stationary phase). The band becomes narrower and narrower, hence taller and taller. Its high concentration part experiences stronger and stronger thermodynamic effects due to the nonlinear behavior of the isotherm. As a result, the band front becomes steeper and its retention time decreases. When both the numbers of reaction and dispersion units, Nrea and ND™, tend toward infinity, the band profile tends toward the solution of the ideal model. Cherrak et al. [55] measured by FA the single-component adsorption isotherms of the enantiomers of 3-chloro-l-phenyl-l-propanol (3CPP) on a column packed with cellulose tribenzoate coated on silica and eluted with n-hexane/ethyl acetate (95/5 v/v). The experimental data fitted well to the Langmuir model. Using these Langmuir isotherms, the single-component band profiles of S-3CPP and R-3CPP were calculated using both the equilibrium-dispersive model (ED) and the more complex transport-dispersive model based on the solid film linear driving force model of mass transfer kinetics (TD). The results are compared with the experi-
14.2 Analytical and Numerical Solutions of the Kinetic Models
677
2.0-,
(a) 1.5-
0.5-
0.0S
10 tfma (min)
8
10 time (mini
(d)
ft™ (m!n)
«m« (min)
Figure 14.12 Comparison of the experimental (symbols) overloaded elution band profiles of the two enantiomers, S (top figures) and the R (bottom figures) enantiomers of 3- chloro-lphenylpropanol on a Chiracel OB-H 250 x 4.6 mm column eluted with n- hexane/ethyl acetate, 95/5 v/v and the profiles calculated with the ED (dotted lines) and the TD model (solid lines). Sample volume, 0.25 ml (left) and 1 ml (right) of a 4.12 g/1 solution. Reproduced with permission from D. E. Cherrnk, S. Khuttabi, G. Guiochon, ]. Chromatogr. A, 877 (2000) 109 (Figures 4a,d and 5a,d).
mental band profiles in Figure 14.12 [55]. These figures show the band profiles obtained for two consecutive injections of increasing volumes of solutions of S- and R-3CPP (ca. 4.14 g/1). The numerical calculations with the ED model were carried out with an efficiency of 8000 plates (see Chapter 10). The solid lines show the best profiles calculated with the transport-dispersive model. The parameters of the Langmuir isotherm determined previously were used. A good agreement is observed between the profiles
678
Kinetic Models and Single-Component Problems
calculated with the ED model and the experimental profiles, although the rear, diffuse boundary of the experimental profiles is always more diffuse than predicted by the model. This can be explained by a mass transfer kinetics slower than included in the ED model. By contrast, there is an excellent agreement between the experimental profiles and those calculated as numerical solutions of the TD model. In this case, the calculated profiles were fitted to the experimental ones and the value of the rate coefficient, kf adjusted for best agreement. In these calculations, it was assumed that the axial dispersion coefficient accounted only for half the band variance observed (and measured in the calculation of the efficiency of 8000 plates), the rest arising from the mass resistances. They derived the coefficient of apparent axial dispersion, Dfl, from the conventional relationship Da = Luo/(2N). The best value of kf was found equal to 200 min" 1 by curve fitting. The same value of the rate coefficient was obtained for the different peak profiles corresponding to increasing sample sizes obtained by increasing the concentration or the injected volumes of the same solution. Finally, note the exact coincidence of the fronts of the experimental and calculated profiles in all cases (Figures 14.12, top). Similar results were obtained for R-3CPP (Figures 14.12, bottom). The comparison between the experimental profiles (symbols) and the profiles calculated with the equilibrium-dispersive model (dashed lines) or with the transportdispersive model (solid lines) leads to the same conclusions as in the case of the S enantiomer. There is a good agreement between the experimental profiles and those calculated with the ED model, especially as far as the fronts of the profiles are concerned, although the experimental peaks tail always more than predicted by the model. The agreement between experimental and calculated profiles becomes excellent for the profiles calculated with the TD model using the same value of the rate coefficient, fcj = 200 min^1 as for the S enantiomer. Fornstedt et al. investigated the thermodynamics and the mass transfer kinetics of the retention of the enantiomers of R- and S-Propranolol on a system using an acetic acid buffer solution as the mobile phase and the protein CBH I immobilized on silica as the stationary phase. The single-component isotherm of each isomer fitted well to the bi-Langmuir isotherm model , c + j^C
fcA^
CIA 711 (14.71)
The overloaded band profiles of the two enantiomers were recorded and compared to those calculated using the ED model. This simple model did not produced profiles in good agreement with the experimental ones. A much better agreement was observed between the experimental chromatograms and those calculated with the TD model, using the solid film linear driving force model to account for a slow mass transfer kinetics on both types of adsorption sites. In this case, the kinetic model is summarized as ^ d
=
*i0tf-9i)
~§ = hifi-K)
(14-72)
(14-73)
14.2 Analytical and Numerical Solutions of the Kinetic Models
679
Retention time (mLn]
Figure 14.13 Comparison between the experimental (symbols) overloaded elution band profiles of R- (left) and S-propranolol (right) on a 100 x 4.6 mm column packed with immobilized CBH I on silica, eluted with an acetate buffer and the profiles calculated with the TD model (lines). Samples: 100 }iL of a (1) 0.36 mM; (2) 1.02 mM; and (3) 1.55 mM solution of pure enantiomer in the eluent. Mass transfer coefficients, k\ was (1) 10,000; (2) 90; and (3) 60 min^ 1 and £2 w a $ 18 min^1 (with D^ = 0). Reproduced with permissionfromT. Fornstedt, G. Thong, Z. Bensetiti, G. Guiochon, Anal. Chew.., 68 (1996) 2370. (Figures 4 and 5) ©1996 American Chemical Society.
q = q1 + q2
(14.74)
where, for each enantiomer, q\ and qi are the stationary phase concentrations of the solute adsorbed on the sites of types 1 and 2, respectively. Assuming a single rate constant for both types of adsorption sites (k\ = ki, homogeneous kinetic), the profiles calculated with the model were in rather poor agreement with the experimental chromatograms, especially at the low concentrations. The TD model with a heterogeneous kinetic model {k\ ^ ki) gives a much better agreement, hi order to reduce the number of degrees of freedom in the case of a heterogeneous kinetics, the axial dispersion coefficient was set to 0. Figure 14.13 compare experimental chromatograms (symbols) and chromatograms calculated using the TD model with heterogenous mass transfer kinetics (solid lines) at increasingly large sample sizes of the R and the S enantiomers. In these chromatograms, both types of sites are operated in the nonlinear region; but the type-II sites are more strongly overloaded than the type-I sites. For the highest concentration (peak # 3), the value of the loading factor Lf R was 59.6%. To calculate the profiles in Figure 14.13, the axial dispersion was set to 0, ki was kept constant and equal to the optimum value derived (£2 = 18 min" 1 ) and only the value of k\ was changed (see figure). There is an excellent agreement between the calculated and the experimental tails of the chromatograms, but, at the highest concentrations, the experimental chromatograms exhibit a somewhat narrower shock layer than the calculated ones. It is probable that this disagreement arise from a concentration dependence of the mass transfer coefficients for both types of sites. Similar effects have been reported previously [56].
680
Kinetic Models and Single-Component Problems
14.3 Comparison Between the Various Kinetic Models of Nonlinear Chromatography We have shown in Chapter 6 that, in linear chromatography, the band profiles predicted by the general rate model, the various lumped kinetic models, and the equilibrium-dispersive model are equivalent when the number of mass transfer or reaction units is not very small, and provided corresponding values of the parameters are used in the different models. An apparent number of transfer units can be derived from the addition of the variance contributions of each independent source of band broadening. This general result cannot be extended simply to the case of nonlinear chromatography. In the case of frontal analysis, under constant-pattern conditions, the analysis of Rhee and Amundson [57] based on the shock layer theory shows that the contributions to the shock layer thickness of axial dispersion and of a solid film mass transfer resistance are additive. However, according to Eq. 14.36b, the apparent number of transfer units is now concentration dependent. Furthermore, as we have seen in the discussion of the band profile under constant-pattern behavior, the number of mass transfer units with a solid film linear driving force model is concentration dependent, unlike in linear chromatography. Thus, the situation is more complex in nonlinear than in linear chromatography. The solid and liquid film linear driving force models can be written under the same general form of a second order Langmuir kinetic model [1]. We can insert the Langmuir isotherm equation (q* = (qsbC)/(l + bC)) in the partial differential equation of the solid film linear driving force model (Eq. 14.3)
we obtain dq _ k.t dt kd(l
[kaC(qs-q)-kdq]
(14.76)
Similarly, insertion in the equation of the liquid film linear driving force model (Eq. 14.5): ^=k'f(C-C*)
(14.77)
of the Langmuir isotherm equation (now q = (qsbC*) /(1 + bC*), or C* = q/[b(qs — gives Fkr (1478)
In linear chromatography, bC and q/qs are negligible. Then, the three Eqs. 14.50 (reaction-kinetic model), 14.5 (liquid film linear driving force model), and 14.3
14.3 Comparison Between the Various Kinetic Models
681
(solid film linear driving force model) become identical, with kf
= kd
(14.79)
%
= ^
(14.80)
Provided that the apparent equivalent parameters given by Eqs. 14.79 and 14.80 are used, the three models give the same result. In nonlinear chromatography, however, this equivalence is impossible. The rate constants of the kinetic equations are concentration-dependent. The relationships between the three kinetic equations (Eqs. 14.3,14.5, and 14.50), stating the solid film linear driving force model, the liquid film linear driving force model, and the second-order Langmuir kinetics, respectively, have been discussed by Hiester and Vermeulen [58] in the case of the breakthrough curve. These authors concluded that the rate parameter is concentration dependent. They suggested as a first approximation for the breakthrough curve using the average value of the concentration, i.e., Q)/2 in the case of a step injection of concentration Co in an empty column. The average value of q is <jo/2 = (qsbCo)/ [2(1 + bCo)]. Using this approximation in Eq. 14.76 gives
If we compare Eqs. 14.50 (Langmuir kinetics) and 14.81, we see that they are equivalent provided that 2k1f
=1
(14.82)
or 2k f k
2kfRpa
= 1T+Tl<¥eq^
(1483)
1 + l< where Req = 1/(1 + bC0). This result means that even if the kinetics follows the solid film linear driving force model the breakthrough curve can be fitted successfully to the Thomas model, provided that the apparent desorption rate constant, kd, given by Eq. 14.82b is used. Introducing in Eq. 14.78 for the liquid film linear driving force model the Hiester and Vermeulen approximation, and writing that q = qo/2 = qsbCo/2(l + bCo), we obtain 2Fk'(l dt
[kaC(qs-q)-kdq]
kdk'0(2 +
(14.84)
Comparison between Eqs. 14.50 and 14.84 shows that they are equivalent provided that kk'(2 + bC)
= 1
(1485)
682
Kinetic Models and Single-Component Problems
or IFk'Al bCo) 2Fkf(l + bC 0) =
2FEf 2Fk =
k'(l + R)
(1486)
This result means that if the mass transfer kinetics follows the liquid film linear driving force model, the breakthrough curve can be fitted to the Thomas model [23], provided that the apparent parameter k^ given by Eq. 14.85 be used. Again, the apparent rate parameter k^ is concentration dependent. All these results demonstrate that • The rate parameter or number of transfer units in nonlinear chromatography is concentration dependent. • The band profiles for overloaded elution calculated with the reaction-dispersive, transport, and transport-dispersive models can be fitted to Thomas' model [1]. This last result has been verified by calculating numerical solutions of these models, fitting them to the equation of Wade et al. [46] for the Thomas model, and determining the best values of the three parameters k'Q, Lf, and Nrea- A modified Simplex algorithm [1,59-61] was designed for the successive estimation of these three parameters. Typical results are shown in Figures 14.14 and 14.15, and in Table 14.1. If the effect of dispersion is not taken into account in the apparent number of reaction units (N^a), there will be a large difference between the solutions of the Thomas model and the transport-dispersive or the reaction-dispersive models. This is illustrated in Figure 14.14, which compares the analytical solution of the Thomas model and the numerical solution of the reaction-dispersive model. The front of the latter solution is less steep than that of the former because the Thomas model does not take into account axial dispersion, but only the kinetics of adsorption-desorption. Figure 14.14b demonstrates that it is possible to lump together the two independent contributions to band broadening arising from the kinetics of adsorptiondesorption and from axial dispersion into a single apparent adsorption-desorption parameter. In this figure, the band profiles obtained as numerical solutions of the reaction-dispersive model for four different values of the loading factor are overlaid with the best Thomas-model profiles obtained by fitting the numerical solutions to Eq. 14.65, using a Simplex algorithm. The best parameters of this latter model (fcg, Lf, and N^.) are listed in Table 14.1. N^a is the apparent number of reaction units which corresponds to an apparent rate parameter including the effects of a finite rate of adsorption-desorption and a finite axial dispersion. There is excellent agreement between the two sets of profiles, and it would be impossible to select one model or the other as the one accounting best for experimental band profiles. As a consequence, the parameters obtained by fitting experimental results to the equations of a model are empirical, and the agreement between the experimental profile and the best profile derived from the model is no more than circumstantial evidence of the validity of the model. This validity can be proven only by agreement between the experimental profiles and those supplied by the model, using parameters obtained by direct measurements.
14.3 Comparison Between the Various Kinetic Models
683
t (min)
Figure 14.14 Comparison of the numerical solution of the reaction-dispersive model (solid line) and the analytical solution of the Thomas model (dotted line). Both models: k'o = 5; Nrea = k'QkdL/u = 2000. Thomas model: N Disp Reaction-dispersive model: 2NDisp[fc(,/(l + k'Q)}2 = 2000. (Left: Lf =1%. (Right) Lf = 1, 1%; 2, 5%; 3, 10%; 4, 20%. The values of the parameters of the Thomas model (see Table 14.1) were obtained by fitting the profiles obtained as numerical solutions of the reaction-dispersive model (solid lines) to Eq. 14.65, for the different values of the loading factor. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, ]. Chromatogr., 603 (1992) 1 (Figs. 1 and 2).
The best values of the parameters obtained when fitting the numerical solution of the reaction-dispersive model to Eq. 14.65 for the Thomas model are given in Table 14.1. The results in Table 14.1 show a very small error, of the order of 0.1% for the retention factor. For the loading factor, the error is approximately 1%. Both errors are independent of the loading factor. On the other hand, the value obtained for N^ja is not constant. It is significantly lower than 1000 and it decreases with increasing loading factor. In Figure 14.15a, we compare two series of band profiles. The first series was calculated as numerical solutions of the transport model (no axial dispersion, solid film linear driving force kinetics), with a number of transfer units, Nm — k^k^L/u = 1000 and k'o = 5 for four different values of the loading factor. The second series of profiles was calculated using the Wade et al. equation solution of the Thomas model (Eq. 14.65). The three coefficients of these Thomas profiles (k'o, Lf, NieSL) are determined so as to minimize the difference between the two profiles, using a Simplex program. These profiles are overlaid, and they are practically impossible to distinguish.
Kinetic Models and Single-Component Problems
684
Table 14.1 Best Values of the Parameters Obtained by Fitting the Band Profiles Calculated with the Reaction-Dispersive Model8 to the Thomas Model*
K
o Lf <%)
5 5 5 5 5
1 5 5 10 20
U d 0
K
5.0 4.996 4.998 4.993 4.995
T d (Ol \
Lf
(/o) 0.991 4.952 4.87 9.91 19.82
iv
rea 954 s 886 a 823fo 828 fl 727a
a
Parameters of the generated profiles: 2 ^ / ( 1 + k'Q)]2 N = 2000; Nrea = k'QkdL/u = 2000; hence, N ^ p = 1000. b Parameters of the Thomas model: Afaisp = °o. c Best values of the parameters obtained by curve fitting. NrgaP apparent number of reaction units assuming a linear isotherm.
Table 14.2 Best Values of the Parameters Calculated by Fitting the Transport" and the Transport Dispersive*1 Models to the Thomase Model
Uc K
K
Q
5 5 5 5 5 5 a
L (Ol
1 5 5 5 10 20
4.994 4.992 0.993 19.98 4.986 4.968
/ \
0.98 4.91 4.93 4.92 9.81 19.54
954 865 865 865 798 690
Parameters of the generated profiles: Npisp = °°; N m = k'okfL/u = 1000; hence,
N^a P = 1000. Parameters of the generated profiles: 2[k'0/(l + k'o)}2 NDisp = 2000; Nm
b
=
P
k'okfL/u = 2000; hence, Afr°^ = 1000. c Parameters of the Thomas model: Noisp = °°A Best values of the parameters obtained by curve fitting. A similar comparison is shown in Figure 14.15b, using the transport-dispersive model with a number of transfer units, Nm, equal to 2000, and a finite coefficient of axial dispersion, Di, such that uL
K
="Mi+fe
= 2000
(14.87)
14.3 Comparison Between the Various Kinetic Models
685
Figure 14.15 (Left) Comparison between band profiles obtained as numerical solutions of the transport model of chromatography (solid lines) and profiles given by the analytical solution of the Thomas model (dotted lines). These profiles are overlaid, and cannot be distinguished. Calculation conditions for the linear driving force model: limit retention factor, k'o = 5; number of mass transfer stages, Nm = k'akfL/u = 1000; coefficient of axial dispersion, DL = 0, N Disp = oo. Values of the loading factor, Lf. 1,1%; 2,5%; 3,10%; 4,20%. (Right) Comparison between a profile obtained as numerical solution of the transportdispersive model of chromatography (solid line) and a profile given by the analytical solution of the Thomas model (dotted line). Same conditions as Figure 14.15a for Lj = 5%, except Nm = k'okfL/u = 2000, and 2NDisp[fcg/(l + k'Q)]2 = 2000. In linear chromatography, the overall band broadening would be the same in Figures 14.15a and b. For both figures, the parameters of the Thomas model are obtained by fitting Eq. 14.65 to the profile obtained by numerical integration of the linear driving force model (see best values of the parameters in Table 14.2). Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, ]. Chromatogr., 603 (1992) 1 (Figs. 3 and 4).
where Afoisp is the number of dispersion units. Thus, in linear chromatography the column efficiency would be 1000 theoretical plates and the band profiles would be the same as for the conditions selected for Figure 14.15a. There is a very little difference between the two profiles of each corresponding pair in Figure 14.15a, and little difference between the two chromatograms in Figure 14.15b. The parameters used to calculate the band profiles of the transport-dispersive and the transport models and the parameters of the Thomas model giving the best fit to these calculated profiles are compared in Table 14.2. As for the data in Table 14.1, there are negligible differences between the values of the retention factor and
686
Kinetic Models and Single-Component Problems
Figure 14.16 Elution peaks of 3-phenyl-lpropanol for various loadings. Experimental data (symbols) and best fit of the kinetic model (line). Stationary phase: Vydac 201THP ODS silica (10 fim particles, 300 Apores, 56 m 2 /g); column dimensions: 150 x 4.6 mm. Mobile phase: Fv = 1 mL/min; methanol-water (25:75); 30°C; sample volume 100 ftL. The figure on each curve is the sample size (jimol). Reproduced with permission from C.A. Lucy, J.L. Wade and P.W. Carr,}. Chromatogr., 484 (1989) 61 (Fig. 1).
Figure 14.17 Effect of the isotherm curvature for two different values of the rate constant. k' = 8.0. Curves at KC0 = 0.01; 0.05; 0.10; 0.25; 0.50; 1.0. (a) 7 = 100. (b) 7 = 5. Reproduced from ]. Wade, A. Bergold and P.W. Carr, Anal. Chem., 59 (1987) 1266 (Figs. 1 and 3). ©1987 American Chemical Society.
the loading factor. On the other hand, there are important differences for which is not constant, but decreases with increasing loading factor. We conclude from these comparisons that it will be possible to fit any set of experimental data to any of the simple kinetic models. Different models will give approximately the same values for the limit retention factor at infinite dilution, k'0/ and for the loading factor, Lf. However, the lumped kinetic parameters derived in the process will not be constant. These parameters will appear to be concentration dependent and to decrease with increasing concentration or increasing sample size. These results agree well with those of Lucy et al. [62] obtained in a few specific experimental cases. Band profiles were recorded at increasing sample sizes for a series of compounds, and fitted to the Thomas model. The values obtained for the limit retention factor, k'Q, and the column saturation capacity were nearly independent of the sample size, while the kinetic parameter was concentration dependent (Figure 14.16). Thus, it will be extremely difficult at best to separate the influences of the various phenomena that may be responsible for the effects of a slow kinetics of mass transfer and a slow kinetics of the retention mechanism. The fitting of experimental data obtained in overloaded elution chromatography to various models of chromatography will not permit the choice of a best model, nor the identification of the slowest step in the chromatographic process. Independent measurements of the kinetic parameters are necessary.
14.4 Results of Computer Experiments
687
Figure 14.18 Concentration profile of an adsorbed solute along the column at different values of the rate constant, k' = 10; D' = 0.001cm2/s. Solid line: ka = 0.1 s" 1 , number of mass transfer units: 250. Dotted line: ka = 0.01 s" 1 , number of mass transfer units: 25. Variable sample size. Reproduced from A. jaulmes, C. VidalMadjar, Anal. Chem., 63 (1991) 1165 (Fig. 4). ©1991 American Chemical Society.
.5
i/L
14.4 Results of Computer Experiments A number of theoretical investigations regarding the influence of various parameters on the band profiles have been performed using different approaches and several of the kinetic models discussed above. The most important are those focused on the influence of the rate constant. For example, Wade et al. [46] have studied the influence of the curvature of the Langmuir isotherm on the band profiles at two different values of the rate constant. They assumed a Langmuir kinetics and neglected axial dispersion. Figure 14.17 illustrates these results. Each profile corresponds to a different value of the parameter b, i.e., since a is kept constant, to a different value of the saturation capacity. For a high value of the rate constant (Figure 14.17a), the result is classical and has already been illustrated in Chapters 7 and 10 (Figure 10.14a). The peak front moves toward shorter retention times, the peak height decreases, and the band broadens. For a slow kinetics, the change in the band shape is less dramatic, and significantly different. The band is much broader, to start with. Although the retention time still decreases with decreasing saturation capacity, and the band broadens significantly, the peak height also increases. It is unfortunate that no experimental demonstration of this effect has been published yet, to the extent of our knowledge. Jaulmes and Vidal-Madjar [51] studied the influence of the mass transfer kinetics on band profiles, using a Langmuir second-order kinetics, and a constant axial dispersion coefficient, D'. They derived numerical solutions using a finite difference algorithm. The influence of the rate constant on the band profile at various sample sizes is illustrated in Figure 14.18. As the mass transfer kinetics slows down, the band broadens and the shock layer becomes thicker. When the sample size increases, however, the influence of thermodynamics on the profile becomes more dominant, as shown by the change in shock layer thickness which decreases with increasing sample size. The effects of the mass transfer kinetics on band profiles in linear and nonlinear chromatography are compared in Figure 14.19. As the mass transfer co-
Kinetic Models and Single-Component Problems
688
50
100
150
200 250 300 TIME (sec)
350
400
450
500
0
50
100
150
200 250 300 TIME ( s e c )
350
400
150
500
Figure 14.19 Effect of mass transfer kinetics on the band profile. Constant sample size, 4.15 mmol, Fv = 1 mL/min; F = 0.2; L = 25 cm; 4.6 mm i.d. (Top left) Linear chromatography, a = 25. kf = 1, 0.004; 2, 0.01; 3, 0.1; 4, 0.5; 5,1; 6,5 s" 1 . (Top, right) Nonlinear chromatography, Langmuir isotherm, a = 25; b = 0.25. kf = 1, 0.004; 2, 0.01; 3, 0.1; 4, 0.5; 5, 1; 6, 50 s" 1 . (Bottom) Effect of diffusion coefficient, a = 4; fe = 1; Values of D: 1, 0.005; 2, 0.01; 3, 0.02; 4, 0.03; 5, 0.05; 6, 0.1 cm 2 /s. Reproduced from B. Lin, Z. Ma, S. Golshan-Shirazi and G. Guiochon, Anal. Chem., 60 (1988) 2647 (Figs. 3 to 5). copyright 1988 American Chemical Society.
TIME
efficient decreases, the band broadens, and its height decreases. When this coefficient becomes very small, the phenomenon known as split peak occurs. A narrow band containing the molecules that have not interacted with the stationary phase is eluted at the hold-up time, followed by a broad, tailing band containing the molecules that have been adsorbed (Figures 14.19a,b). In linear chromatography (Figure 14.19a), the band remains Gaussian until the rate constant becomes very low. In nonlinear chromatography (Figure 14.19b), the Langmuir profile is formed, with a sharp front, and the retention time of the peak maximum increases at first when the rate constant decreases, because the front shock layer
14.5 Numerical Solution of the Lumped Pore Diffusion Model Figure 14.20 Comparison between an experimental overloaded elution profile (symbols) and the profiles calculated with the anti-Langmuir isotherm and the POR and the ED models. Butyl-benzene on a monolithic Cis silica column. Sample: concentration, 6.0 g/L; injection volume, 0.3 mL. Reproduced with permission from A. Cavazzini, G. Bardin, K. Kaczmarski, P. Szabelski, M. Al-Bokari, G. Guiochon, J. Chromatogr. A, 957 (2002) 111 (Fig. 4).
689
4.S 4 3.5
I* I r 1
4.4 f
3 f 3*
2.5
! t
EipBrimenlal
i
ED model -»--—
3.6
/ 1.5
46
4.8
/
1
0,
becomes thicker and the peak height being lower, the high concentrations of the band move at a lower velocity (profiles 1^4). The effect of the dispersion coefficient is shown in Figure 14.19c, for a Langmuirian band. When the dispersion coefficient increases, the band becomes broader and shorter, but, in contrast to the linear chromatography case, it remains unsymmetrical and its retention time increases slightly.
14.5 Numerical Solution of the Lumped Pore Diffusion Model (or POR Model) for Single Component Systems The derivation of numerical solutions of the general rate model (GR) for singlecomponent problems is a special case of that of numerical solutions of this model for multi-component systems, which will be discussed in Chapter 16 and that there is no need to reproduce here. However, since several authors have used this model to compare the experimental overloaded profiles of pure-component bands with those calculated with this and other models, we discuss briefly the numerical solutions of the general rate model for single-components. Cavazzini et al. measured by FA the equilibrium isotherm data of butyl- and amyl-benzene on a monolithic Chromolith Performance Cis silica column from Merck (Darmstadt, Germany) using a methanol-water solution (80:20, v/v) as the mobile phase [63]. The column porosities were measured using the reversed size exclusion method. The adsorption isotherms were found to be concave upward in the entire concentration range investigated. This range is limited by the rather poor solubility of the alkyl aromatic hydrocarbons in the mobile phase, so it was not possible to explore the high concentration range of the isotherm. The data fitted well to the anti-Langmuir model, suggesting significant interactions between the adsorbed molecules. Elution band profiles were recorded under overloaded conditions. These profiles were in satisfactory agreement with the profiles calculated using either the equilibrium-dispersive model or the lumped pore diffusion (POR) model of chromatography and these adsorption isotherms. The numerical parameters were obtained by parameter estimation [63]. Significant differences are found at the beginning of the diffuse front boundary, around the top of the
Kinetic Models and Single-Component Problems
690
100
200
300 t [s]
200 300 t [s]
Figure 14.21 Comparison between calculated and experimental band profiles for pure phenetole. Profiles calculated with the GR model. Dm = 5.46 x 1CT6 cm 2 /s. Left Co = 3.755 g/cm 3 ; tp = 30 s; Lf = 0.91%. Right Co = 14.496 g/cm 3 ; tp = 120 s; Lf = 14.12%. Insert: enlarged tops of the profiles. Solid line, profile calculated in taking into account the concentration dependence of the total porosity; dashed line, profile calculated with a constant total porosity. W. Piqtkowski, D. Antos, F. Gritti, G. Guiochon, J. Chromatogr. A, 1003 (2003) 73 (Fig. 2).
band, and at the bottom of the rear shock layer (see Figure 14.20). The experimental band profiles returns to the baseline more slowly than the calculated ones. Later, Piatkowski et al. [64] used the general rate model to predict the singlecomponent band profiles of ethoxy-benzene (see Figure 14.21) and propyl benzoate (see Figure 14.22) in a reversed-phase HPLC system. Numerical solutions of general rate model were calculated using a program based on the method of orthogonal collocation on finite elements. Single-component adsorption equilibrium data were measured by frontal analysis. The BET isotherm model (see Chapter 3, section 3.2.3.4), which assumes multilayer adsorption was found to provide the best fit to the adsorption data of both solutes. This model of gas-solid adsorption was developed to describe adsorption phenomena in which successive molecular layers of adsorbate molecules form at pressures well below the pressure required for completion of the monolayer. The BET isotherm equation can be written as: with
i = 1,2, • • •, n
(14.88)
where qs is the monolayer saturation capacity of the adsorbent for component i; b$ is the equilibrium constant for surface adsorption-desorption (i.e., over the free surface of the adsorbent), and bi is the equilibrium constant for surface adsorptiondesorption over a layer of adsorbate molecules. Series of overloaded band profiles for single-component samples were acquired, the experimental conditions being varied systematically. These profiles were compared to those calculated using the GR or the TD models and the BET isotherm. The lumped mass transfer rate coefficient, fcW/I was found to be practically propor-
24.5 Numerical Solution of the Lumped Pore Diffusion Model
propyl
benzoate
E a-
100
200
300 400 t [S]
691
n 500
600
b
700
Figure 14.22 Comparison between calculated and experimental band profiles for pure propyl benzoate. Profiles calculated with the GR model. Dm = 4.87 x 10~6 cm 2 /s. Left Co = 3.755 g/cm 3 ; tp = 30 s; Lf = 1.24%. Right Co = 14.496 g/cm 3 ; tp = 120 s; Lf = 19.18%. Insert: enlarged tops of the profiles. Solid line, profile calculated in taking into account the concentration dependence of the total porosity; dashed line, profile calculated with a constant total porosity. W. Pigtkowski, D. Antos, F. Gritti, G. Guiochon, ]. Chromatogr. A, 1003 (2003) 73 (Fig. 3).
tional to Deg for both compounds. The molecular diffusivities, Dm,-, of the two compounds were calculated from the Wilke and Chang equation. Some examples of band profiles are given in Figures 14.21 for phenetole and 14.22 for propyl benzoate. For both compounds, the single-component BET isotherm model, with a constant effective diffusion coefficient, Dm, given by Eq. 14.88 allows the accurate prediction of the band profiles over the range of loading factors that was investigated, 0.002 < Lf < 0.20 (or between 0.2 and 20%). The loading factor is defined as the ratio of the amount injected to the saturation capacity of the column. In the Figure 14.21b, the overloaded elution band profiles were calculated in two different ways, taking the concentration dependence of the internal porosity into account or assuming this porosity to be constant. Because the amount adsorbed under BET adsorption behavior is very large, the volume available for the stagnant mobile phase in the particles decreases significantly at high concentrations [64]. The agreement between the calculated and the experimental profiles is excellent under all conditions, but slightly better if the concentration dependence of the porosity is taken into account. Liu et al. calculated the profiles of the overloaded elution bands of three insulin variants, human, porcine, and lispro, using the equilibrium-dispersive and the POR models [65]. The single-component equilibrium isotherms of these compounds were measured by FA on a YMC-ODS Cig column with an aqueous mobile phase containing 31% ACN and 0.1% TFA. These isotherms were fitted to the Langmuir, the Langmuir Freundlich and the Toth models, the later providing the best fit. The calculated profiles do not agree well with the experimental band profiles when the former are calculated using the equilibrium-dispersive
Kinetic Models and Single-Component Problems
692
l.S
0.8 0.7 0,6
ias
A
1.4
I Vt Zf V V
0.6
0.3 •
0
\
0.4
0.2 0.1
1
O 0.8
O 0.4
B
i
\
0.2
0
A
| \ \
MlB
J ^^_
t(min)
Figure 14.23 Comparison of overloaded band profiles of Lispro calculated using the ED and POR models. Solid lines: experimental data, + ED model, A POR model. Top left: (A) injection volume: 1 ml, concentration: 1 g/L, Lf = 7.82%, N = 120, fc; = 1.67 x 10~3 cm min^ 1 . (B). injection volume: 0.5 ml, concentration: 0.4 g/L, he = 1.56%, N =150, kj = 2.17 x 10~3 cm min" 1 . Top right: (A) Lf = 5.1% , N = 100, k{ = 1.33 x 10~3 cm min" 1 . (B). Lf = 2.03% , N = 200, kt = 2.66 x 10~3 cm min" 1 .. Bottom: (A) Lf = 11.43%. N = 100. kj = 1.33 x 10" 3 cm min" 1 . (B) Lf = 5.71%, N = 150, k{ = 1.67 x 10" 3 cm min" 1 . Reproduced with permission from X. Liu, K. Kaczmarski, A. Cavazzini, P. Szabelski, D. Zhou, G. Guiochon, Biotechnol. Progr., 18 (2002) 796 (Figs. 7, 8, and 9). ©2002 American Chemical Society.
model and the apparent axial dispersion coefficient derived from the efficiency of the column for insulin at infinite dilution (Da = LUQ/(2N)). The POR model does not give quite satisfactory results either, if the parameters used are derived from the molecular diffusivity calculated using the conventional correlation for proteins [66] and the effective pore diffusion coefficient obtained from the empirical correlations described by Suzuki [67] and Satterfield [68]. Markedly better fits were achieved by using either the apparent axial dispersion coefficient of the equilibrium-dispersive model or the mass transfer coefficient, fc,, of the POR model obtained by parameter estimation, adjusting the calculated breakthrough curves to those obtained in the FA measurements of isotherm data. Figures 14.23 compare the overloaded band profiles of human insulin, porcine insulin, and lispro to those so calculated. When the values of Da and fc; determined by parameter estimation of the breakthrough curves are used in the equilibriumdispersive or the POR models, respectively, there are no significant differences
14.6 The Monte Carlo Model of Nonlinear Chromatography adsorptions
693
*-(^desorption
02
tp*n
*p*
time Figure 14.24 Schematic representation of the "list of events" on the axis time. Event ep (for instance, an adsorption) takes place at the time tp (present) and it is processed. The corresponding random adsorption time TS is generated and a new desorption event is scheduled in the future at the proper time tp + TS. Since other n events (ep+i, e p+ 2,..., deriving from previous adsorption-desorption processes) are already scheduled in the time interval [tp, tp + TS], such new desorption will become the ep+n event and its corresponding occurrence time (tp + TS) referred as tp+n. Reproduced from F. Dondi et ah, Anal. Chem., 72 (2000) 4353 (Fig. 1). ©2000 American Chemical Society.
between the experimental profiles and those calculated with either model.
14.6 The Monte Carlo Model of Nonlinear Chromatography Dondi et al. developed a stochastic approach to nonlinear chromatography based on the Monte Carlo method [69]. The Monte Carlo method consists in simulating the migration of an ensemble of molecules through the chromatographic column that contains a finite number of adsorption sites. The random sequence of adsorption-desorption events is modeled at the molecular level with the stochastic terms and concepts discussed in Chapter 6, Section 6.5. The linear chromatographic process is modeled as a composite Poisson process: a chain of exponentially distributed fly times, followed by exponentially distributed adsorption times are generated. When adsorbed, the molecule is stationary; the mean adsorption time is TS. When desorbed, the molecule travels with the velocity of the mobile phase; the mean fly time—residence time between a desorption and the subsequent adsorption—in the mobile phase is xm. The elution time of the molecule is recorded when it reaches the end of the column after n adsorption-desorption events on the average. The distribution of the single molecule elution times gives the band profile. In nonlinear chromatography, each adsorption site should be labeled and continuously monitored since the behavior of the molecule will depend on the presence of other molecules. The discrete event simulation technique which processes a list of events ordered along the time axis can be used for the simulation of this process. This technique is illustrated in Figure 14.24. Two types of events are possible: adsorption and desorption. To characterize
Kinetic Models and Single-Component Problems
694
dumber of Molecules 20000 18000 • 16000 14000
\
•
2000
•
1000 0 400 600 800 1000 1200 1400 1600
-
4000 2000
3000
V
•
12000 10000 8000 8000
a.
Musec)
•
•
•
•
-
Lp25% Lp12.5% Lp7.5% Lp5.0% Lp3.75% L,=2.5%
Figure 14.25 Comparison of the Monte Carlo microscopic model with the Thomas macroscopic kinetic model, at different loading factor values. Symbols: results of the Monte Carlo simulations. Lines: solutions of the Thomas model. ReproducedfromA. Cavazzini el al., Anal. Chem., 74 (2002) 6269 (Fig. 1). ©2002, American Chemical Society.
one event, we have to specify the type of the event, as well as the molecule and the site involved in the event. The processing of the event includes: 1. 2. 3. 4.
Specifying the next event; Generating the corresponding random time; Scheduling the event at the proper time; Updating the system status.
Nonlinearity is taken into account by generating the adsorption time as a function of local coverage. For instance, in the case of Langmuir kinetics, the site coverage factor is either 1 or 0, depending on whether the site is free or already occupied. The molecule will stop and adsorb only if the site is free. In Figure 14.25, the onset of nonlinear behavior can be observed when the loading factor is increased. The loading factor is defined here as the number of molecules injected relative to the number of adsorption sites in the column. Cavazzini et al. showed that the above Monte Carlo model of nonlinear chromatography is equivalent to the Thomas kinetic model of second order Langmuir kinetics [70]. The solution of the Thomas model for a Dirac impulse injection is given by Eq. 14.65. When the chromatographic process is modeled at the molecular level with the stochastic model, the Thomas model becomes [70]:
C(ts,Lf) =
L - e -" f)
(14.89)
REFERENCES
695
where ts = t — to, rs and rm, are the mean adsorption and fly times, respectively, n is the average number of adsorption-desorption events, Lf is the loading factor, and T is given by Eq. 14.57. The comparison of Eqs. 14.65 and 14.89 confirms that n = Nrea, i.e. the average number of adsorption-desorption steps (in the microscopic model) is identical to the number of mass transfer units (in the macroscopic model). When the loading factor tends toward zero and the condition of infinite dilution is approached, the solution of the Thomas model becomes identical to the Giddings-Eyring model of linear chromatography: (14.90) »
i s /
The above equation is identical to Eq. 6.116. Overloaded chromatograms calculated with the Monte Carlo model and with the microscopic Thomas model (Eq. 14.89) are compared in Figure 14.25 in the range of Lf = 2.5-25%. The band profiles completely agree, confirming that the Monte Carlo model described above and the Thomas' kinetic model are equivalent.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 603 (1992) 1. J. J. van Deemter, F. J. Zuiderweg, A. Klinkenberg, Chem. Eng. Sci. 5 (1956) 271. L. Lapidus, N. R. Amundson, J. Phys. Chem. 56 (1952) 984. G. S. Bohart, E. Q. Adams, J. Amer. Chem. Soc. 42 (1920) 523. E. Wicke, Kolloid Z. 86 (1939) 295. D. O. Cooney, E. N. Lightfoot, Ind. Eng. Chem. (Fundam.) 4 (1965) 233. E. Glueckauf, J. Chem. Soc. (1947) 1302. A. S. Michaels, Ind. Eng. Chem. (Fundam.) 44 (1952) 1922. T. Vermeulen, M. D. LeVan, N. K. Hiester, G. Klein, in: R. H. Perry (Ed.), Handbook of Chemical Engineering, 6th Edition, McGraw-Hill, New York, NY, 1984, Ch. 16. E. Glueckauf, in: Ion Exchange and its Applications, Soc. Chem. Ind., London, UK, 1954, p. 34. F. G. Helfferich, P. W. Carr, J. Chromatogr. 629 (1993) 97. K. R. Hall, L. C. Eagleton, A. Acrivos, T. Vermeulen, Ind. Eng. Chem. (Fundam.) 5 (1966) 212. D. R. Garg, D. M. Ruthven, Chem. Eng. Sci. 28 (1973) 791. D. M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, NY, 1984. A. Acrivos, Chem. Eng. Sci. 13 (1960) 1. D. R. Garg, D. M. Ruthven, Chem. Eng. Sci. 30 (1975) 1192. H.-K. Rhee, B. F. Bodin, N. R. Amundson, Chem. Eng. Sci. 26 (1971) 1571. H.-K. Rhee, N. R. Amundson, Chem. Eng. Sci. 27 (1972) 199. J. Zhu, Z. Ma, G. Guiochon, Biotechnol. Progr. 9 (1993) 421. J. Zhu, G. Guiochon, J. Chromatogr. 636 (1993) 189. J. H. Knox, J. Chromatogr. Sci. 15 (1977) 352. A. Seidel-Morgenstern, S. C. Jacobson, G. Guiochon, J. Chromatogr. 637 (1993) 19.
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[23] H. Thomas, J. Am. Chem. Soc. 66 (1944) 1664. [24] Z. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, 1972. [25] J. W. Chen, J. A. Burge, F. L. Cunningham, J. J. Northam, Ind. Eng. Chem. (Process Des. Dev.) 7 (1968) 26. [26] K. Hashimoto, K. Miura, J. Chem. Eng. Jpn 9 (1976) 388. [27] D. R. Garg, D. M. Ruthven, AIChE J. 21 (1975) 200. [28] R. G. Peel, A. Benedek, Can. Ind. Chem. Eng. 59 (1981) 688. [29] H. Yoshida, T. Kataoka, D. M. Ruthven, Chem. Eng. Sci. 39 (1984) 1489. [30] J. Zwiebel, R. L. Gariepy, J. J. Schnitzer, AIChE J. 18 (1972) 1139. [31] R. G. Lee, T. W. Weber, Can. J. Chem. Eng. 47 (1969) 60. [32] R. D. Fleck, D. J. Kirwan, K. R. Hall, Ind. Eng. Chem. (Fundam.) 12 (1973) 95. [33] I. Neretnieks, Chem. Eng. Sci. 31 (1976) 107. [34] C. Costa, A. Rodrigues, Chem. Eng. Sci. 6 (1985) 983. [35] D. R. Garg, D. M. Ruthven, Chem. Eng. Sci. 29 (1974) 1961. [36] N. S. Raghaven, D. M. Ruthven, Chem. Eng. Sci. 39 (1984) 1201. [37] A. Rasmuson, Chem. Eng. Sci. 40 (1985) 621. [38] K. Kaczmarski, D. Antos, H. Sajonz, P. Sajonz, G. Guiochon, J. Chromatogr. A 925 (2001) 1. [39] W. Pi§tkowski, E Gritti, K. Kaczmarski, G. Guiochon, J. Chromatogr. A 989 (2003) 207. [40] A. K. Hunter, G. Carta, J. Chromatogr. A 897 (2000) 81. [41] S. Ghose, D. Nagrath, B. Hubbard, C. Brooks, S. M. Cramer, Biotecnol. Progr. 20 (2004) 830. [42] S. Goldstein, Proc. Roy. Soc. London A219 (1953) 151. [43] F. H. Arnold, H. W. Blanch, J. Chromatogr. 355 (1986) 13. [44] J. Giddings, Dynamics of Chromatography, M. Dekker, New York, NY, 1965. [45] S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 60 (1988) 2364. [46] J. L. Wade, A. F. Bergold, P. W. Carr, Anal. Chem. 59 (1987) 1286. [47] C. K. Lee, Q. Yu, S. U. Kim, N.-H. L. Wang, J. Chromatogr. 484 (1989) 29. [48] S. F. Chung, C. Y. Wen, AIChE J. 14 (1968) 857. [49] E. J. Wilson, C. J. Geankopolis, Ind. Eng. Chem. (Fundam.) 5 (1966) 9. [50] S. Golshan-Shirazi, G. Guiochon, J. Phys. Chem. 95 (1991) 6390. [51] A. Jaulmes, C. Vidal-Madjar, Anal. Chem. 63 (1991) 1165. [52] B. Lin, S. Golshan-Shirazi, G. Guiochon, J. Phys. Chem. 93 (1989) 3363. [53] D. S. Hage, R. R. Walters, J. Chromatogr. 436 (1988) 111. [54] J. L. Wade, P. W. Carr, J. Chromatogr. 449 (1988) 53. [55] D. E. Cherrak, S. Khattabi, G. Guiochon, J. Chromatogr. A 877 (2000) 109. [56] P. Sajonz, H. Guan-Sajonz, G. Zhong, G. Guiochon, Biotechnol. Progr. 13 (1997) 170. [57] H.-K. Rhee, R. Aris, N. R. Amundson, Phil. Trans. Roy. Soc. London A269 (1971) 187. [58] N. K. Hiester, T. Vermeulen, Chem. Eng. Progr. 48 (1952) 505. [59] J. A. Nelder, R. Mead, Comput. J. 7 (1965) 308. [60] E. R. Aberg, G. T. Gustavsson, Anal. Chim. Acta 144 (1982) 39. [61] E. V. Dose, Anal. Chem. 59 (1987) 2420. [62] C. A. Lucy, J. L. Wade, P. W. Carr, J. Chromatogr. 484 (1989) 61. [63] A. Cavazzini, G. Bardin, K. Kaczmarski, P. Szabelski, M. Al-Bokari, G. Guiochon, J. Chromatogr. A 957 (2002) 111. [64] W. Piaticowski, D. Antos, F. Gritti, G. Guiochon, J. Chromatogr. A 1003 (2003) 73. [65] X. Liu, K. Kaczmarski, A. Cavazzini, P. Szabelski, D. Zhou, J. V. Horn, G. Guiochon, Biotechnol. Progr. 18 (2002) 796. [66] M. E. Young, P. A. Carroad, R. L. Bell, Biotechnol. Bioeng. 22 (1980) 947.
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Chapter 15 Gradient Elution Chromatography under Nonlinear Conditions Contents 15.1 Retention Times and Band Profiles in Linear Chromatography 15.1.1 Analytical Chromatography 15.1.2 Analytical Solution for Volume-overloaded Gradient Elution Chromatography . 15.2 Retention of the Organic Modifier or Modulator 15.2.1 Representation of the Isotherm in Gradient Elution 15.2.2 Effect of the Adsorption of the Organic Modifier on the Band Profiles in KPLC . 15.3 Numerical Solutions of Nonlinear Gradient Elution 15.3.1 Experimental and Calculated Band Profiles under Nonlinear Conditions 15.3.2 Experimental and Computational Study of Preparative Gradient Elution of Peptides Using the Solid Film Linear Driving Force Model 15.3.3 Experimental and Computational Study of Preparative Gradient Elution of Peptides Using the General Rate Model 15.3.4 Extension of the VERSE Model to Nonlinear Gradient Elution Chromatography 15.4 Gradient Elution in Ion-Exchange Chromatography References
701 701 703 705 708 709 711 711 719 721 724 726 731
Introduction In gradient elution, the eluotropic strength of the mobile phase is progressively increased during the separation. This procedure was introduced 50 years ago by Alan et al. [1]. It permits achievement of the proper separation of certain mixtures in the many cases in which this cannot be done under isocratic conditions. Two important types of applications explain why gradient elution is widely used in analytical chromatography. In many complex mixtures, some components are unretained or too weakly retained by the stationary phase when the mobile phase used has an elution strength high enough to elute all the mixture components in a reasonable time. Conversely, some components would never be eluted with a mobile phase weak enough for the early eluting components to be retained and resolved. By increasing the concentration of a strong solvent in the mobile phase as a function of time, it is possible to achieve the separation of most components while keeping a reasonable analysis time. Because of the cost associated with the recovery and recycling of the components of the mobile phase and because the extraction, enrichment, or purification of a single component rather than of several is usually the goal in preparative chromatography, step gradient operations are usually preferred to continuous gradients in preparative chromatography. Whether a 699
700
Gradient Elution Chromatography under Nonlinear Conditions
continuous or a step gradient is used, the cycle time should include the regeneration time. As in displacement chromatography, this time is difficult to estimate with any accuracy on a general basis. There are other cases, however, where continuous gradients (most often linear ones) have to be used. One of the most common is the separation of proteins in reversed phase chromatography (RPLC). Because of the large size of protein molecules, their surface area of interaction with the stationary phase, often referred to as their "footprint/' is large, and their retention factor varies very rapidly with increasing eluotropic strength of the mobile phase. Often, the retention factor decreases from a very large value to almost 0 for an increase in the mobile phase concentration of the solvent of a few tenths of one percent. To achieve the separation of closely related proteins by HPLC, and to do so in a reproducible fashion, a linear composition gradient is most often more reliable than the preparation of mobile phase solutions with a composition accurate to within better than 0.1%. The major difference between isocratic and gradient separations is that in the former case the equilibrium isotherm is the same in all points of the column during the entire separation, while in the latter case the isotherm depends on time and location. Thus, the velocity associated with a concentration in the ideal model varies along the column, and at a given point in the column, it depends on the time elapsed since the beginning of the gradient, hi practice, we assume that the isotherm function, i.e., the functional dependence of the equilibrium concentration in the stationary phase on the mobile phase composition, remains unchanged. Only the numerical coefficients of the isotherm depend on the mobile phase concentration. This assumption is supported by numerous experimental results. hi principle, there are no restrictions regarding the gradient profiles, or time dependence of the mobile phase composition used in gradient elution. Step gradients and linear composition gradients are the simplest and most popular composition variations used, but any gradient profile that enhances the resolution between two critical components and reduces the analysis time is acceptable. In practice, the only gradient profile for which a simple algebraic solution to the problem of the elution of an analytical pulse can be written is a linear eluotropic-strength gradient, hi RPLC, this is practically a linear composition gradient [2]. We consider only this type of gradient here. Because the velocity associated with a concentration in the ideal model (see Eq. 7.3) is a function of the isotherm coefficients, the retention time of a concentration can be calculated only in analytical chromatography because of the simplification introduced by the fact that C is negligibly small and dq/dC is constant [3]. When gradient elution chromatography is conducted under nonlinear conditions, the velocity associated with a concentration varies during the entire separation, due to the constant changes in the solute and the modifier concentrations. Accordingly, there are no analytical solutions for the ideal model in gradient elution, even for the band profile of a pure component. A numerical solution is necessary.
15.1 Retention Times and Band Profiles in Linear Chromatography
701
15.1 Retention Times and Band Profiles in Linear Chromatography Two important solutions have been derived for gradient elution in linear chromatography: one pertaining merely to analytical applications, the other addressing the wider problem of the periodic injection of pulses of any width in a gradient of any profile.
15.1.1 Analytical Chromatography The behavior of chromatographic columns operated in gradient elution, under linear conditions (i.e., assuming linear isotherms for all the solutes) has been studied theoretically by numerous authors [2,4-10] . The most comprehensive treatment is that based on the linear solvent strength (LSS) theory of Snyder et ah [5,6]. This theory has found widespread acceptance [7,8] and has been extended to include the contributions of the various mass transfer resistances to band broadening [9-11]. It assumes the injection of infinitesimal pulses of a feed and a linear gradient of the volume fraction of a mobile phase modifier, (p. (15.1) where <po is the initial modifier concentration, /3 the gradient slope, and T the gradient delay time or time required for the gradient to be delivered to the column inlet, due to the dwelling volume of the HPLC instrument. If the column is operated in the isocratic mode, the retention behavior of a given solute in RPLC for a series of successive runs carried out at increasing concentration of the organic modifier can be described through an empirical relationship such as a polynomial relationship between the logarithm of the retention factor k\ and the volume fraction of modulator in the mobile phase, (p In k • = fl0 + ai
(15.2)
where ao, a.\, and ai are numerical coefficients that are obtained by regression of experimental data. These coefficients have values that are characteristic of each given solute and column-solvent system. In a narrow concentration range of the modifier, a linear dependence can be valid: Ink\ = a0 + a1(p = Ink'iw - Stf
(15.3)
where k\ and k\w are the capacity factors of the compound considered when the volume fractions of the organic modifier in the mobile phase are (p and 0 (i.e., in the pure weak solvent, which is usually water in RPLC), respectively, and S,- is a constant characteristic of the elution strength of the solute and the modifier. Assuming that Eq. 15.3 is locally valid in gradient elution, the elution time of a component at any position, z, along the column is given by
tR,i = T + tp + I + ^ In [l + Si J8 (^(4>o) " T)]
(15.4)
702
Gradient Elution Chromatography under Nonlinear Conditions
where
]
(15.5)
At the end of column, z — L and L/u — to, therefore t
+ t + t + l [ln[l l +S S;j3£ofc^o - T)]
(15.6)
and, ignoring tp and x, *R,i = fo + A l n t 1 + siPo*i(o)] (15-7) 3 P Note that these equations assume that to is independent of
where A is a band compression factor given by
p is given by
(mo)
« T T W
At the end of the column, z = L and z/u = to- Therefore, Eqs. 15.8 and 15.10 can be written as
^ [
fc/( 0)
^
1
(1511)
p = A linear concentration gradient is the most common gradient used in practice as it is the simplest gradient to implement. However, step gradients and nonlinear gradients have the advantage of sometimes providing better separations [10]. In specific cases, they may allow the separation of compounds that coelute with a linear gradient.
15.1 Retention Times and Band Profiles in Linear Chromatography
703
15.1.2 Analytical Solution for Volume-overloaded Gradient Elution Chromatography Carta et ol. [15] derived an analytical solution in the form of a convergent series for periodic, rectangular feed pulses, in the case of volume-overloaded gradient elution chromatography, with an arbitrary gradient shape and type (e.g., linear, nonlinear, single-step or multiple step gradients). The model assumes linear isotherm behavior, plug flow (i.e., no axial diffusion), and treats the effects of a finite rate of the mass transfer kinetics by using the linear driving force approximation. The equations of the model are
|)
(15.14)
and uses boundary conditions given by: C(0,t)
= CF,
C(0,0
=
0,
(i-l)tp
+ tF
+ tF
(15.15) (15.16)
with; = 1,2,3, •••. Note that, in contrast with the symbols used in most parts of this book, we use K to stand for the initial slope of the isotherm, instead of a, in order to be consistent with the symbols used by Carta et al. [15]. The slope of isotherm is a function of tp, K = K(
x = me KQ
Y
(15,7a) LF
=
-?--
(15.17b)
n =
-^—
(15.17c)
ft = ( I ^ K
with0
=Oforf =°
(15-17d)
where X and Y are dimensionless mobile and stationary phase concentrations, n is the number of transfer units in the bed, and 9 is a dimensionless time. Since K decreases with increasing time during a gradient separation, 9 runs slowly at the beginning of the separation and accelerates toward the end. Under dimensionless form, the equations of the model become: -^
=
-X + Y
-Z = X-Y X(0,9) X(O,0)
= =
1,2nr(j - 1) < 9 < 2nr{j -l) + 7trF 0,27rr(; - 1) + nrF < 9 < 2nrj (j = 1,2,3, • • •)
(15.18)
(15.19) (15.20) (15.21)
704
Gradient Elution Chromatogmphy tinder Nonlinear Conditions
The quantities
are the dimensionless durations of the total period and of each successive feed slug. A time periodic solution of the set of equations 15.17a to 15.17d with a period 2rcr is easily derived by applying the residue theorem to the general inversion integral of the Laplace transform solution of these equations [15]. It can be written as:
rF
2 ~ 1
/
"
7i k ,
\
K
k2n \ . K
-\-r / cos
fknrF\ \ z.r j
(7 - ^
- i r 1 ^j
(15-24)
This solution consists of a series of terms that have a sinusoidal component that determines the location of the band and an exponential decay that determines the extent of band spreading due to the mass transfer resistances. Four of five terms may suffice when these resistances are important but many more terms, up to 100 or more, are needed if the column is highly efficient. Equation 15.24 is general: It is valid whatever the gradient shape and time, whether the gradient is continuous or stepwise. Only the definition of the dimensionless time, 6, depends upon the integral of 1/K(t) (see Eq. 15.17d) [15]. The time-average effluent concentration, C, between two times, t\ and ti, is of interest in preparative chromatography (see Chapter 18) as it permits the calculation of the purity of the fraction collected between these two times. This concentration can be derived from Eq. 15.24 [15]. It is given by a series that converges much faster than the one in Eq. 15.24. Although this solution was derived for plug flow (Da = 0), it can be generalized by using an apparent mass transfer coefficient to account for axial dispersion, a coefficient that is related to axial dispersion, the external film mass transfer coefficient and intraparticle pore diffusion through the following equation [16]: (15.25) where Di is the axial dispersion coefficient, dp the particle diameter, kr the external film mass transfer coefficient, and Dp the pore diffusivity of the compound considered. The number of transfer units, n the plate number of the column, N, and the HETP are related through N = L/H ~ = n/2 = koaL/2u. Therefore, k^a can be derived directly from IV, obtained itself from an isocratic run. Therefore, the effect of axial dispersion and of the mass transfer resistances can be combined into an apparent mass transfer coefficient which is related to the number of theoretical plates measured for the component of major interest.
15.2 Retention of the Organic Modifier or Modulator
705
Figure 15.1 Comparison of experimental (solid lines) and predicted (dotted lines) gradient elution profiles for ethyl paraben. Gradient from 30 to 60% (v/v) methanol in water, in 15 min., at 2.5 mL/min. Vp is the volume of sample injected and Cf the feed concentration (0.008 g/1). Reproduced with permission from G. Carta, W. B. Stringfteld, } . Chromatogr., 605 (1992) 151 (Fig. 8).
The solution derived by Carta et al. [15] is explicit and allows the direct calculation of the effluent concentration. It is given as a series, which makes it is easy to use, although a programmable calculator or a personal computer may be needed to compute a sufficiently large number of terms of the series and achieve an adequate precision. The solution is valid for the injection of small, narrow sample pulses as well as for that of large, wide rectangular feed slugs. The first moment and the standard deviation of the calculated band converge toward those that are predicted by the LSS theory in the limit of small feed pulses and a Gaussian peak, as afforded by Eqs. 15.5 and 15.8. For larger feed pulses, the solution predicts asymmetric bands with a maximum concentration that exceeds that of the feed sample. Actually, the solution provides the periodic response of the column that corresponds to the injection of periodic feed pulses. When the response to the injection of a single, isolated pulse is desired, it may be obtained by selecting a sufficiently large total period which avoids the interference of repeated injections. In order to compare experimental results and theoretical predictions, gradient elution experiments were carried out using large sample loops in order to check the ability of the theory to predict correctly the band profiles in gradient elution from isocratic elution data and to assess the effects of volume overloading. The actual injection of the sample into the column is delayed by the equipment dwell time, so the gradient was started immediately after the introduction of the feed slug into the column. Samples of ethylparaben (concentration Cf = 0.008 mg/mL) were injected in an HPLC instrument and analyzed with a concentration gradient of methanol in water, at program rates of 30 to 60% in 15 min, with a flow rate of 2.5 ml/min. Experimental results illustrating the influence of the sample volume (Vp) are shown in Figure 15.1. They are compared with the calculated profiles. As the sample size increases, the profile becomes more and more unsymmetrical. Its maximum concentration increases and eventually exceeds the feed concentration.
15.2 Retention of the Organic Modifier or Modulator The modulator (in NPLC, a strong, polar solvent, or in IXC, a buffer) or the organic modifier (in RPLC, methanol, acetonitrile, or THF) may affect the retention of the components of the sample in different possible ways. In the most classical case, such as in ion-exchange chromatography or in normal phase HPLC, the
706
Gradient Elution Chromatography under Nonlinear Conditions
modulator competes directly with the solutes for binding to the stationary phase. In such cases, the modulator can be treated as the competing component in a multicomponent isotherm, involving it together with the other solutes. In other cases, the modulator affects the retention of the solutes primarily by changing the parameters of the adsorption equilibrium without competing directly with them for adsorption [17] (or the competition is weak and its effects small or moderate compared to the influence of the modifier on the solubility of the solutes). This is typically the case in reversed-phase liquid chromatography, when aqueous solutions of methanol or acetonitrile are used as the mobile phase. Increasing the concentration of the strong solvent increases the solubility of the feed components in the mobile phase, without the modifier competing with the solutes for adsorption. This is the essential mechanism found in RPLC. Note, however, that there are more complex cases. For example, it has been shown that, although the adsorption of acetonitrile on Cis silica is not very much stronger than that of methanol, the behavior of the bonded layer is quite different in both cases [18,19]. Methanol and water molecules participate in hydrogen bonds in which they can be either donor or acceptor of hydrogen. This is not the case of acetonitrile, which can be only acceptor and is more hydrophobic. Agglomerates of acetonitrile molecules form in aqueous solutions and a layer of pure acetonitrile forms over or mixes into the layer of alkyl chains bonded to silica and against this layer. As a consequence, the nature of the adsorption isotherm of solutes may be quite different from methanol or acetonitrile solutions [18]. All the consequences of this phenomenon have not been clarified yet. Although the primary result is the same, i.e., the retention time of the component band decreases with increasing mobile phase concentration of the strong solvent, there are important differences in the behavior of overloaded columns in normal and in reversed phase chromatography. The competitive retention mechanism, its behavior in nonlinear chromatography, and its consequences when the strong solvent is adsorbed are discussed in Chapter 13. The solubility mechanism is often much simpler. The isotherm coefficients merely depend on the concentration of the strong solvent in the mobile phase. In practice, the situation may be more complex in reversed phase chromatography because the strong solvent is often slightly but significantly adsorbed. For example, the retention factors at infinite dilution of methanol and acetonitrile in the system water-octadecyl silica are relatively small. Their effect could be neglected if the stationary phase is close to saturation at the initial concentration of strong solvent. This is not the case with acetonitrile, for example, since it takes a concentration of 6.5 M (or close to 25% w/w) to reach half the saturation capacity of the stationary phase [20]. As a consequence, the breakthrough curve of the strong solvent is not the same as the gradient profile injected at column inlet and delayed by a constant hold-up time (to = L/u), but it exhibits a marked degree of self-sharpening [20,21], as illustrated in Figure 15.2. Furthermore, there is a possibility of competition between the strong solvent (if it is markedly stronger than methanol) and the feed components, which is discussed below (see Subsection 15.2.2). The competitive interactions between the feed components and the strong sol-
15.2 Retention of the Organic Modifier or Modulator
707
Figure 15.2 Comparison between the output concentration profile of ACN if there were no retention (solid lines), profiles calculated using the Langmuir model (dashed lines), and recorded profiles (dotted lines). 1: 1% ACN/min; 2: 2% ACN/min; 3: 4% ACN/min. Reproduced with permission from M.Z. El Fallah and G. Guiochon, Anal. Chem., 63 (1991) 2244 (Fig. 7). ©1991, American Chemical Society.
vent can be neglected in most cases because the initial slopes of the adsorption isotherms of the feed components and the strong solvent in the pure weak solvent are too different [22]. However, because of the self-sharpening effect associated with a convex-upward isotherm, the breakthrough profile of the strong solvent has a steep front followed by a curved ramp which tends slowly toward the linear input [21,23]. The calculation of the elution band profiles in gradient elution under overloaded conditions can take into account this modified dependence of the mobile phase composition on the time and location [24]. Decoupling of the competitive adsorption is legitimate when there is a very large difference in initial slopes for the isotherms of the strong solvent and the sample components in the pure weak solvent [22]. Results obtained with acetonitrile, however, suggest that a more complex physico-chemical situation can often take place in RPLC [18,19]. For macromolecules, changing the modulator concentration may result in important changes of the molecular structure such as in changes of the conformation of a protein chain or of its degree of ionization, or in changes of the population of accessible binding sites [25,26]. For example, increasing the modulator concentration in ion-exchange chromatography may change the interaction mechanism with the stationary phase from an electrostatic to a hydrophobic interaction. A popular relationship between solute retention and modulator concentration that is suitable for both electrostatic and hydrophobic interactions was proposed by Melander et al. [26] log k • = a. - ft log Cm +
(15.26)
708
Gradient Elution Chromatography under Nonlinear Conditions
15.2.1 Representation of the Isotherm in Gradient Elution The mobile phase is a solution of a weak solvent and of one or several additives, often referred to as the strong solvent. The retention factors of the sample components decrease with increasing concentration of the strong solvent. We assume that the mechanism of adsorption is not so affected by the change in mobile phase composition that the isotherm model of any one component itself would vary from one composition to another. Accordingly we assume that the coefficients of the isotherm are functions of the mobile phase concentration. If the equilibrium isotherm is accounted for with the Langmuir model, for example, we can write the isotherm as
where
(15.28)
Knowing the characteristics of the gradient, which are part of the boundary conditions of the problem, we can calculate the local isotherm at any time during the separation. If we assume that the strong solvent is not adsorbed (which is often an acceptable assumption in reversed phase chromatography), its breakthrough profile is given by the gradient equation. If the strong solvent is adsorbed, its breakthrough curve can be calculated (see Chapter 7, Sections 7.2.5.2 and 7.4.2). Depending on the circumstances, the problems of the breakthrough of the strong solvent and of the elution of the feed components can be treated separately, handling the breakthrough of the solvent as a boundary condition, or these two problems can be treated together, as a ternary competitive problem with an unusual boundary condition. The former approach is the simplest and the most popular. The latter was used successfully in cases in which the multicomponent isotherms had been measured (see Chapter 13, Figure 13.28). The innumerable studies reporting on the dependence of the retention factors of countless compounds on the modifier concentration in reversed phase chromatography [2,3] indicate that within a reasonable range, the first coefficient of the isotherm is given by «*(#) = a0/ie-s'
0
(15.29)
where «o,i = ai{^ = 0) i s the coefficient of the adsorption isotherm with the pure weak eluent and S{ the slope of a plot of the logarithm of the retention factor versus the modifier concentration. In a wide range of modifier concentrations, a quadratic relationship (i.e., Eq. 15.2) may account better for the dependence of In k'o i on the modifier concentration. Also, Eq. 15.29 is not valid at low concentrations of the organic modifier [14]. There are rather few studies on the mobile phase composition dependence of the second coefficient, fc;(3>). Jacobson et al. [27,28] suggested that the saturation capacity of the adsorbent remains constant, making £>,(<£>) proportional to
15.2 Retention of the Organic Modifier or Modulator
709
This has also been observed by El Fallah and Guiochon [29] for small molecules. The 10% variation reported for lysozyme in reversed phase chromatography is probably of marginal significance [24]. However, because of possible changes in the tertiary structure of proteins with solvent composition, even small changes in the mobile phase composition may result in significant changes of the saturation capacity and this relationship cannot be expected to hold. The recent results of Gritti on the dependence of the isotherm coefficients of phenol, caffeine, and propranolol in RPLC on the methanol concentration of the mobile phase suggest, however, that the saturation capacity may vary significantly in many cases [30]. Accurate determination of the equilibrium isotherm in a wide concentration range is necessary [20], because the dilution of the band during its migration is much less important in gradient than in isocratic elution. As a consequence, the maximum band concentration remains high, the band velocity is controlled by the high concentration range of the isotherm, and the band behavior is strongly nonlinear. Furthermore, classical methods of isotherm determination require isocratic experiments. In most cases, this is not a severe constraint, but it becomes so in the case of proteins, whose retention time under isocratic conditions varies very rapidly with the mobile phase composition. Nevertheless, it is necessary to carry out at least a few ECP experiments under isocratic conditions using different mobile phase compositions, while remembering that ECP is not accurate since it is based on an equation that was derived using the ideal model [24]. The dependence of the initial slope of the isotherm on the mobile phase composition can be derived in two different ways [29]. The retention time, tRj, of analytical pulses eluted under isocratic conditions is a function of the mobile phase composition. Since f^, — to is proportional to a;,1 Eq. 15.29 permits the determination of Sj as the slope of a plot of log(^- — to)/to versus
15.2.2 Effect of the Adsorption of the Organic Modifier on the Band Profiles in RPLC Empirical and semi-empirical approaches dominate the modeling of chromatographic band profiles under nonlinear conditions, in the presence of a multicomponent mobile phase. This holds particularly true in gradient chromatography, in which case the solvent composition is changed progressively or abruptly (step gradient) during the separation. The method that is typically employed consists in measuring the adsorption equilibrium isotherms of the feed components at various concentrations of the organic modifier. Then, the isotherm coefficients are correlated with the organic modifier concentration, using one of several theoretical models developed originally for linear chromatography [14]. These simplified models are efficient and adequate for optimization purposes and offer some 'Accurate determinations require the measurement of to for each solvent composition [12,31,32].
Gradient Elution Chromatography under Nonlinear Conditions
710
0.0D007 0.00006 •E 0.0D005 o
\v-1ctn 3 /min \
RP 18e
>!/=2cm3/min
>2 0.00004RP 18 £
J
0.0D003 -
\
D.OD0Q2 I 0.00001 D.ODOQO 800
\ \
1V
\
C
1000 1200 1400
161
2000
2200
2400
2600
200 300 400 500 600 700 600 900 1000 1100 1200 1300 1400 1500 Time [s]
Figure 15.3 Band profiles of cyclopentanone on Lichrosphere Cig in pure water (Left), and in methanol-water (10, 30, and 50% MeOH, v/v) (Right). Reprinted with permissionfromI. Poplewska, D. Antos, Chem. Eng. Sci.r 60 (2005) 1411 (Figs. 1 and 5).
valuable insights into the trend of equilibrium changes due to the presence of the organic solvent. However, they do not take properly into account the competitive adsorption equilibrium of the sample components and the organic modifier(s) contained in the mobile phase. Therefore, they cannot elaborate on the adsorption mechanisms operating under nonlinear conditions. Poplewska et al. [33] proposed an isotherm model that describes quantitatively the adsorption of organic modifiers by reversed-phase adsorbents and takes into account the nonideal behavior of the mobile phase by introducing the activity coefficients. The nonideal behavior of the adsorbed phase was assumed to originate from multilayer adsorption on the heterogeneous surface, in agreement with the model suggested by Kazakevich et al. [34] for retention in acetonitrile/water solutions. Also, a competitive heterogeneous adsorption model for solutes dissolved in a binary mobile phase was proposed. These models account for the partition and for the displacement mechanisms of adsorption of compounds dissolved in complex mobile phases and for the influence on the separation of the adsorption equilibrium of organic solvents. Unfortunately, the isotherm models presented require knowledge of a number of critical parameters such as the loading capacities for the polar and non-polar types of sites, equilibrium constants on these sites and for multilayer adsorption, and of the number of the adsorbed layers, parameters that cannot be determined independently. Also the activity coefficients of the compounds and of the organic modifiers are needed. These coefficients were determined using the UNIVAC method [35]. For experimental verification of these models, Poplewska et al. [33] used binary mixtures of methanol-water and acetonitrile-water as the mobile phases and measured the adsorption equilibrium isotherms of cyclopentanone on two similar adsorbents having different degrees of surface heterogeneity, a Cig nonendcapped and a Cis endcapped silica. Due to its structure, cyclopentanone exhibits affinity for adsorption on the bonded alkyl chains and for the polar, uncovered silica surface of the adsorbent. Overloaded elution bands of cyclopentanone in pure water were recorded (Figure 15.3) and the isotherms were derived using an inverse method (see Chapter 3). Five independent parameters (the excess coefficients and the equilibrium constants for partition-adsorption and for
15.3 Numerical Solutions of Nonlinear Gradient Elution
711
displacement-adsorption, and the column efficiency) were determined by minimizing the sum of the squares of the differences between the experimental and the calculated band profiles. The excess isotherms of acetonitrile in acetonitrile-water and of methanol in methanol-water were determined by the perturbation method that gives the experimental isotherms. These isotherms were fitted to the excess isotherm model proposed, and the best values of the isotherm parameters of acetonitrile and of methanol were obtained by minimizing the sum of the squared differences between the experimental results and the model prediction. The independent isotherms of the solute and the solvents were coupled in the adsorption model which was then used for the calculation of band profiles at different mobile phase compositions, using equilibrium-dispersive model (see Chapters 10 and 11). Numerical solutions of this model were calculated using the method of orthogonal collocation. The proposed competitive adsorption model was implemented and used to calculate the band profiles of cyclopentanone with mobile phases of different compositions. One more adjustable parameter, an equilibrium constant for additional interactions, was introduced in order to match calculated and experimental retention of cyclopentanone. Figure 15.3 compares some calculated and experimental band profiles of cyclopentanone for mobile phases containing different concentrations of methanol in the mobile phase. In general, the agreement observed with either methanol or acetonitrile is good. However, like as any other complicated model, there are many parameters which must be determined by fitting the experimental data to the model and still, at the end, the calculated retention times of the solute(s) must be adjusted using a last empirical parameter in order to match the experimental retention times. There are no independent ways to verify that these parameters are correct. Therefore, in practice, the use of a more simplified model remains preferable.
15.3 Numerical Solutions of Nonlinear Gradient Elution Band profiles obtained in gradient elution can be calculated using the same models of chromatography, implemented by the same computer programs as used under isocratic conditions.
15.3.1 Experimental and Calculated Band Profiles under Nonlinear Conditions While analyses made in gradient elution often involve the use of small and dilute samples, the column is often overloaded in preparative gradient elution chromatography. This causes the adsorption isotherms to be nonlinear and competitive. Therefore, interference effects become important. Furthermore, the mass transfer resistances can be very significant, especially for macromolecules. Various dispersive effects, such as axial dispersion and the mass transfer resistances often counter-balance the thermodynamic effects of adsorption and desorption
Gradient Elution Chromatography under Nonlinear Conditions
712
o •H
+>
1.0 mg
2.5 mg
2.5/10 mg
2.5/25 mg
|
| •a
Figure 15.4 Experimental band profiles in isocratic (top) and gradient elution (bottom). Experimental conditions: mobile phase water and 5-100% organic solvent in 12 min. Separation of two xanthines, HET and HPT. Sample sizes as indicated below each figure. Reproduced with permission from J.E Eble, R.L. Grob, RE. Anile and L.R. Snyder, ]. Chromatogr., 405 (1987) 51 (Fig. 3).
which depend on the gradient slope, the initial modulator concentration, and the adsorption properties of the solutes and the modulator. Mathematical modeling and theoretical analysis should play an important role in the scale-up of separation processes. Because of the complexity of preparative chromatography, no analytical solution is available for overloaded gradient elution chromatography. Only numerical solutions are possible. Snyder was the first to attempt the calculation of the elution profiles of highconcentration bands in gradient elution, using the Craig model and some semiempirical relationships [36-38]. Eble et al. [36] compared the experimental and calculated width of overloaded bands in gradient elution. They studied the influence of the sample size for small and large molecules, using semiempirical relationships and using the Craig model for the calculation of profiles. Figure 15.4 compares the experimental profiles of two xanthines (HET and HPT) injected as a mixture in isocratic and gradient elution and the profiles of the same amounts of these components injected individually. The top series of chromatograms shows the band profiles with increasing sample sizes for an isocratic separation. A slight displacement of the first component begins for the injection of 2.5 mg of each component, with stronger displacement appearing with 2.5/10 mg and 2.5/25 mg samples of the two components. In the bottom series of chromatograms in Figure 15.4, the profiles of the individual components and of the binary mixtures are shown
15.3 Numerical Solutions of Nonlinear Gradient Elution
713
in gradient elution, with a steep gradient. Again, displacement of the first component band by the second can be seen in the 2.5/10 mg and 2.5/25 mg cases. Comparing the isocratic and gradient elution separations illustrates the influence of the gradient of the organic solvent modifier. The end of the second component profile is eluted much earlier under gradient conditions, and the band has the convex-upward rear profile characteristic of this elution mode, which is often referred to in the literature as the shark fin profile. Later Antia and Horvath [39] discussed the numerical solution of overloaded linear gradient elution. They calculated numerical solutions of the mass balance equations for single components and binary mixtures. These authors assumed: (1) The classical one-dimensional differential mass balance dQ
~dt + F
dqi(C1,---,Cn,4>)
dt
dQ + U o
3Q
^ = Dfl -w
(15 30)
-
(2) A linear gradient, therefore ^
(15.31)
Note that the delay time needed to account for the system dwell volume was ignored, although it can readily be introduced by adding it to t^. (3) And a Langmuir adsorption isotherm with parameters such that ki = lnito,,- - Stf
(15.32)
Therefore, the Langmuir isotherm can be written as: S^)Q ;=1
A
.
^ ^ . . . ^
(1533)
j
The set of Eqs. 15.30 to 15.33, with the proper initial and boundary conditions was solved numerically, using orthogonal collocation on finite elements. Their results are illustrated in Figures 15.5 (single component) and 15.6 (binary mixture). The band profiles for single components depend considerably on the sample size and the gradient steepness (G = Stpto). At low values of G, the band of a component with a Langmuir isotherm has a sharp maximum at the top of the shock layer (Figure 15.5, G = 0.15). At high values of G, the band profile assumes the shape of a shark fin (Figure 15.5, G = 0.6). This is because the initial slope of the isotherm decreases exponentially with increasing strong solvent concentration, so the elution of the rear part of the high-concentration profile is much more accelerated during the gradient than the elution of the low-concentration part. Antia and Horvath [39] also calculated the individual band profiles of a binary mixture with competitive Langmuir isotherm. Their results are illustrated in Figure 15.6, showing that the first component tends to exhibit very sharp front and
714
Gradient Elution Chromatography under Nonlinear Conditions
J. G
2
4
a 0.6
6
Column Volumes Figure 15.5 Calculated single-component band profiles in gradient elution. Effect of the sample size and the gradient steepness. Reprinted with -permission from F.D. Antia and Cs. Horvith,]. Chromatogr., 484 (1989) 1 (Fig. 2).
rear boundaries, and the second, a sharp front boundary and a convex-upward rear boundary. They compared the separations obtained in gradient and isocratic elution and concluded that the production rates and the purities of both components of a binary mixture are better under isocratic conditions, provided that the separation factor is independent of the mobile phase composition. The recovery yields achieved with both methods are comparable. The integration of the mass balance equation of the equilibrium-dispersive model could also be done using a finite difference method. In this case, however, we have an additional constraint compared to the isocratic case. Because of the strong dependence of the retention factor on the mobile phase composition, the retention factor of the eluate varies within a large range. This eliminates the forward-backward calculation scheme with which the dependence of the column efficiency on the retention factor is too strong and which converges in too narrow a range of retention factors. A backward-forward calculation scheme should be used. Satisfactory results have been obtained with that scheme [20,29]. The most serious problem encountered in the calculation of band profiles in gradient elution stems from the fact that, in contrast to isocratic elution, the maximum band concentration decreases only slowly during the migration. If the feed concentration is not too large, the maximum concentration of the band may even increase during migration, when a steep gradient is selected. Accordingly, the problem becomes more complex. Factors that do not even arise in the isocratic
15.3 Numerical Solutions of Nonlinear Gradient Elution
715
> 0.3
S
10
15
Column Volumes
Figure 15.6 Individual band profiles of the components of a binary mixture in gradient elution. Effect of the sample size (C, loading factor). Reprinted with-permissionfromF. D. Antia and Cs. Horvdth, J. Chromatogr., 484 (1989) 1 (Fig. 3).
case, such as the retention of the modifier and the effect of its retention on its breakthrough curve (see previous section), have to be taken into account. Others must be accounted for more accurately in the calculations carried out for gradient elution than in those made for isocratic elution. The most important of the latter factors are the profile of the feed injection band and the fact that the high concentration range of the isotherm plays a critical role, since the maximum concentration of the bands changes relatively little. Accurate modeling of the highconcentration region of the isotherm becomes necessary [20]. There have been few other comparative studies of calculated and experimental band profiles in gradient elution. These studies have illustrated the need for very accurate modeling of the equilibrium isotherm. The lesser degree of band dilution in gradient elution makes the profiles more sensitive to the isotherm in the high concentration range. This has been carefully documented in three reports regarding the band profiles of 2-phenylethanol on a C18 column with acetonitrile-water as the mobile phase [20,29] and of lysozyme under similar conditions [24]. The experimental band profiles of 2-phenylethanol were in excellent agreement with those calculated when a gradient of acetonitrile (ACN) in water at 1%/min and loading factors of 1 to 10% were used. A Langmuir isotherm model accounts for the adsorption data in the concentration range 0 to 120 mM, and the dependence of the isotherm parameters on the ACN concentration was derived from the plot of the retention time of an analytical pulse versus the gradient slope (see Section 15.2). The agreement was still satisfactory for these loading factors and a program rate of 2%/min, but it was poor at all sample loadings for 4%/min [29]. At the lowest program rate, the agreement was less satisfactory
Gradient Elution Chromatography under Nonlinear Conditions
716
0.15
o"
0.12
\
O
E8.
\
nc.
-" o
\
88d
\
. \ \
o d
'An
%
001
\
1
T
8
10
12
14
16
(t-tp-to) (min)
18 20
2
4
6
8
10 12 14 16 18 20
(t-tp-tO) (min)
Figure 15.7 Comparison between experimental and calculated band profiles of 2phenylethanol in gradient elution. Experimental conditions: mobile phase flow rate: 1 mL/min.; initial mobile phase composition: 10% ACN in water; column temperature: 40°C; column dead volume: VQ = 2.71 mL. Concentration of the sample solution: Co = 132.5 mmol/L. Gradient, 2% ACN/min. Experimental band profiles (square symbols) and profiles calculated using a composite isotherm (linear plus Langmuir terms), taking into account (solid line) or not (dotted line) the adsorption of ACN on the stationary phase, (a) Injection volume, V{ = 4 mL; sample size, 0.53 mmol, loading ratio, 29 molecule per 100 bonded groups, (b) Injection volume, Vj = 6 mL; sample size, 0.80 mmol, loading ratio, 43 molecule per 100 bonded groups. Reproduced with permission from M.Z. El Fallah and G. Guiochon, Anal. Chem., 63 (1991) 2244 (Fig. 8). ©1991, American Chemical Society.
when the profiles were calculated using for the isotherm coefficients the ACN concentration dependence derived from the plot of the isocratic retention time of analytical pulses versus the ACN concentration. The profile shape was correct, but a shift of 40 s was observed, although proper corrections for extracolumn volumes were determined. With the same system, using the sum of a linear and a Langmuir term (i.e., a quadratic isotherm) as an empirical model to represent the adsorption isotherm in the concentration range 0 to 8 M, and taking into account the retention of ACN on the stationary phase (see Figure 15.2), the same authors [20] achieved excellent agreement between experimental and calculated band profiles with loading factors of 30% and program rates up to 2%/min (Figure 15.7). For a loading factor of 43%, the agreement is still very good except for the top of the band, which exhibits an unexplained rear shock layer. The agreement is only qualitative for faster gradient programs.
15.3 Numerical Solutions of Nonlinear Gradient Elution
8
9
10
(i-tp-tO) min.
717
19
21
(t-tp-tO) min.
Figure 15.8 Comparison between experimental (symbols) and calculated (solid lines) band profiles of lysozyme in gradient elution. Experimental conditions: column dimensions: 150 x 4.6 mm; packing: 5 pm particles of C18 silica (Vydac)/ 300 Apores; VQ = 1.81 mL; temperature: 30°C; initial mobile phase composition: 10% ACN. (D): 1 mL injection (0.40 mg, Lf = 2.9%); (x): 2 mL injection (0.81 mg, Lf = 5.8%); (+): 3 mL injection (1.21 mg, Lf = 8.7%); and (o): 4 mL injection (1.62 mg, Lf = 11.6%). Solution concentration: 0.405 mg/mL. (a) Gradient rate: 1% ACN/min. (b) Gradient rate: 0.5% ACN/min. Reproduced with permission ofWiley-Liss Inc., a subsidiary of John Wiley & Sons, Inc. from M.Z. El Fallah and G. Guiochon, Biotechnol. Bioeng., 39 (1992) 877 (Fig. 6). ©1992, John Wiley & Sons.
The adsorption isotherms of lysozyme on C18 silica in mixtures of water and acetonitrile were measured by ECP. Isotherm measurements have to be performed under isocratic conditions, which is difficult for proteins in RPLC. The retention time of lysozyme decreases by a factor of 2 when the concentration of acetonitrile is increased from 31.6 to 32.2%. Thus, an extremely stable pumping system is required (such as the one of the Agilent HP 1090). The adsorption data were accurately modeled by a bi-Langmuir isotherm. There was excellent agreement between the experimental and calculated band profiles (Figure 15.8) for program rates of 0.5 and 1% ACN/min and loading factors up to 11%, the only significant difference between these profiles being found around the top of the band, which is rounder in the experimental profiles than in the calculated ones [24]. The optimization of the experimental conditions of the separation of this mixture was discussed under two different types of constraints, with baseline resolution of the proteins and under induced displacement of some proteins by their neighbors in the chromatogram. The effects of the adsorption properties of the
718
Gradient Elution Chromatography under Nonlinear Conditions
12-
Figure 15.9 Calculated (solid lines) and experimental (symbols) band profiles of phenol on Kromasil CJS. Gradient of methanol in water (0 to 50%) in time tg, as shown. Sample size 80 mg. Reprinted with permission from F. Gritti, G. Guiochon, J. Chromatogr. A, 2027 (2003) 45 (Figs. 8 and 9).
u
O
820 min 30 min
jr
4-
1
fe.
I 40 min / 60 min
vLJ
0-
!
'
r 10
[
'
20
, 1
30
'
1
40
'
1
50
Time [min]
feed components, of the feed volume, of the gradient slope, of the feed purity, and of the feed concentration on the separation performance were considered using the numerical solution. The results obtained indicate that, under appropriate conditions, sample displacement can be employed dramatically to improve the production rate with minor losses in the yield and/or the purity of the separation products. Recently, Gritti et ah [14] calculated the overloaded band profiles of phenol eluted on a Cis-Kromasil column in gradient elution, with aqueous solutions of methanol and a gradient from 0 to 50% methanol (v/v). Because pure water does not wet the packing material, these authors used an empirical adsorption model, which relates the amount of phenol adsorbed to its concentration and to the concentration of methanol in the mobile phase. This model, derived from frontal analysis measurements, is an extension of the bi-Langmuir model to non-isocratic conditions. The low-energy sites followed the classical linear solvent strength model (LSSM), not the high-energy sites, the saturation capacity of which decreases linearly with increasing q>. This model was validated by comparing the experimental and calculated band profiles in gradient elution, under linear and nonlinear conditions. The band profiles were calculated by means of the equilibrium-dispersive model, with a finite difference algorithm. A very good agreement was observed using steps gradients {Aq>) from 0% to 50% methanol and continuous gradients of methanol in water, with times tg of 20 to 100 min, as shown in Figure 15.9. The agreement was excellent for steps gradient from 5% to 45% and with tg from 25 to 50 min. Significant differences between experimental and calculated profiles were observed for gradient slopes larger than 3.3% methanol min^ 1 . These differences were attributed to the kinetic of rearrangement of the Qg-bonded chains when the methanol concentration increases in the mobile phase, particularly at values of
15.3 Numerical Solutions of Nonlinear Gradient Elution
15.3.2
719
Experimental and Computational Study of Preparative Gradient Elution of Peptides Using the Solid Film Linear Driving Force Model
Experimental and computational studies were carried out by Kim and Velayudhan [40] on the separation of mixtures of the chemotactic peptides, n-formyl-MetPhe (P) and n-formyl-Met-Trp (T) by reversed-phase linear gradient elution on a Cjg-bonded silica column, with acetonitrile as the organic modifier. Large feed volumes were used, at the maximum feed concentration allowed by the solubility of the peptides. The separations were also carried out with stepwise gradients and with isocratic elution. The yields and productivities were compared to those obtained with continuous gradient elution. Computer simulations of all these separations were run using a linear driving force kinetic model and assuming plug flow along the column. A simple kinetic model was used with a differential mass balance and a kinetic equation:
||
-q)
(15-35)
Single component isotherm of T and P were measured using the ECP method and fitted to single component Langmuir isotherms in order to determine the column saturation capacity of each solute, qsj and qSip. The binary Langmuir isotherms were written as: k'
HP
=
v
^CP
u
(15-36)
(15.37) where F is the phase ratio, qS)p and qsj are the column saturation capacities for the two components P and T, which are determined from the single component Langmuir isotherms, k'T and k'p are the retention factors under linear conditions of T and P, respectively. The authors showed that the retention factors of these two components as function of acetonitrile do not follow the linear solvent strength theory that predicts fc'=fc'oexp(-S0)
(15.38)
Instead, the data for both P and T fit well to the sum of two exponential terms, according to the following equation: k' = Ax exp(-Bi^) + A2 exp(-B2<£>)
(15.39)
where A\, B\, Aj, and B2 are numerical coefficients that were obtained for P and T by the experimental measurement of their retention factors as functions of the
720
Gradient Elution Chromatography under Nonlinear Conditions -CO
«•
,S
2.0 -
(a)
1 \
§ IS -
U
§
1|
1.0-
1 W
J
0.5 -
1, i_ L
0.0 -
10
15
20
Time (minutes)
Figure Figure 15.10 15.10 Gradient elution elution of 1.70 1.70 mg mg of of n-formyl-Met-Phe n-formyl-Met-Phe (P) (P) and and 1.63 1.63 mg mg of of n-formyl-Met-Trp n-formyl-Met-Trp (T). (T). Gradient Gradient from from 10 10 to to 40% 40% ACN ACN in in 20 20 min. min. (a) (a) Experimental Experimental results results (composition (composition of of collected collected fractions). fractions), (b) (b) Calculation Calculation of of band band profiles profiles with with isotherms isotherms in in Eqs. Eqs. 15.36 15.36 and and 15.37. 15.37. Reprinted Reprinted with with permission permission from K. K. Billy, Billy, A. A. Velayudhan, Velayudhan, J.]. Chromatogr. Chromatogr. A, A, 796 796 (1998) (1998) 195 195 (Fig. (Fig.4). 4).
2.0 -
11
(b)
I]
\ \ - IS
I
1.0-
I u <
0.5-
1
I i
0.0 -
10
- 0
15 Time (minutes)
organic modifier concentration (0) and the fitting of these data to the above equation. The set of equations 15.34 and 15.35, along with the equilibrium isotherms given by Eqs. 15.36, 15.37, and 15.39 for the two compounds was solved by the method of characteristics, using a method previously developed and applied to nonlinear isocratic elution, gradient elution and displacement chromatography [41,42]. Good agreement was achieved between the experimental results and the solutions calculated with the above empirical isotherm equations. The simultaneous concentration and separation of the two feed components was achieved by gradient elution. Figure 15.10 compares the experimental and calculated results obtained with a linear gradient of 10 to 40% ACN in 20 minutes, with a 2.4 mL feed
15.3 Numerical Solutions of Nonlinear Gradient Elution
721
volume of a P + T mixture prepared in 10% ACN. Since the detector (operated at 214 run) is overloaded with preparative runs, fractions were collected every 15 seconds and reinjected (after dilution if necessary) into the analytical column for analysis and quantitation. Figure 15.10a shows the experimental results. Figure 15.10b shows the numerical solution obtained using the model in Eqs. 15.34 to 15.37 and 15.39. A reasonable agreement with the experimental results is observed. The maximum concentration of P is over-predicted and that of T underpredicted. There is also a discrepancy in the retention times of P and T. There are several possible reasons for these differences, such as neglecting the influence of the dwell time, of the limited accuracy of the gradient formation, and of the approximate character of the binary isotherm. Calculations, however, demonstrate that the combination of the focusing power of the gradient with the multicomponent feed interactions may be most useful in preparative separations, even at high feed loadings.
15.3.3 Experimental and Computational Study of Preparative Gradient Elution of Peptides Using the General Rate Model Tingyue et al. [43] presented a general rate model for the study of gradient elution in multicomponent nonlinear chromatography. The model considers axial dispersion, film mass transfer, intraparticle diffusion, and a second-order kinetic of adsorption-desorption. This model can simulate various gradient operations including linear, nonlinear, and stepwise gradients. The mass balance equations for each solute in the mobile and in the stationary phases are given by:
The adsorption/desorption process was assumed to be relatively slow compared with the mass transfer and the assumption of local equilibrium is no longer valid. Consequently, the solid phase concentration must be related to the adsorption and the desorption rates, via a kinetic equation. The second-order kinetic is accounted for by the following equation 9C*, LL
/ — k .r
^
•
N
\
C°° — V C* • \ — kj C* • \ /=1 /
C\ 5 A7\
The boundary conditions of these three equations are: i = 0
Q
=
Q(0,z)
(15.43)
Cp/i
=
Cp/i(0,R,z)
(15.44)
C*-
=
C*-(0,R /Z )
(15.45)
722
Gradient Elution Chromatography under Nonlinear Conditions
z =L R=0
^
=
0
(15.47)
—M. oK
=
0
(15.48)
These three partial differential equations can be rewritten in dimensionless form. The dimensionless quantities used are: Ci = Ci/Qjo, y,i ~ ^-pj'^-ifi' x = z/L, n T/n 1 C(
—
tji =
Daf
=
ULi/
LJ^f
epDpiiL/R2pu
L(ka/iCi/0)/u,
cPri = Cf/i/( ' — R/Rp T = ut/L, Dx- D / Dli
—
li
=
Daf =
&i±\.r) /
n
(15.50)
t-pL/pj,
3Biim(l-eT)/eT,
Lfc^/w
and the dimensionless mass balance equations are
i a2^
-° =
0
(15.53)
If the adsorption and desorption Damkohler numbers (Daf and Daf) are sufficiently large (i.e., if the adsorption and desorption rates, ka,- and kdp are large), the partial differential term in the LHS of Eq. 15.53 can be set to zero and the equation reduces to the multicomponent Langmuir isotherm if the saturation capacities of all the components are the same [43], with
C* =
?'*
^
or c*. = !
1 + L^bf
P'
J ^
(15.54)
1+ ^ b
00
where ty = k^i/k^i and a,- = fe/C . The last component (i.e., component N) is the gradient modifier and the following solute-modifier relationship, derived from a correlation proposed by Melander et al. [26], was used: log bi = at - fa log Cp>N + 7;C P/N
(15.55)
The set of equations listed above was solved numerically by discretizing Eqs. 15.51 and 15.52 and using the finite-element and orthogonal-collocation methods,
15.3 Numerical Solutions of Nonlinear Gradient Elution Figure 15.11 Relationship between eluite retention and modifier concentration for the system studied. Reprinted with permission from T. Gu, Y.-H. Truei, G.-J. Tsai, G. T. Tsao, Chem. Eng. Sci., 47 (1992) 253 (Fig. 1).
723
« p y - -3.0 -•- -U)
t
3J 3j
3.2 12
4
2 / 0
me.
Dlmenslonless Tine
Dimensionless Time
Figure 15.12 Isocratic elution (a, left) and linear gradient elution (b, right) of the three eluites studied.. Reprinted with permission from T. Gu, Y.-H. Truei, G.-J. Tsai, G. T. Tsao, Chem. Eng. Sci., 47 (1992) 253 (Figs. 2 and 3).
respectively [44]. Equation 15.53 is already an ordinary differential equation and does not need discretization. The system of equations is then solved with the IVPAG subroutine from IMSL, which uses Gear's stiff method for initial value ODE problems. To show the advantage of gradient elution over isocratic elution, a system of four arbitrary components was chosen as an example. Components 1 to 3 are the solutes and component 4 is the modifier. The relationship between the adsorption equilibrium constant of each solute and the modifier concentration (see Eq. 15.55) is shown in Figure 15.11. In the system studied, the separation has to be carried out in the low modifier concentration range because the affinities of the three solutes are too close in the high modifier concentration range. Figure 15.12a shows the dimensionless effluent history of the four components with an isocratic run. This run is very long. It is not possible to use a higher modifier concentration to reduce the separation time because the first and second peaks would overlap. With a linear gradient of the same modifier, the separation of the three solutes can be improved considerably. Figure 15.12b shows that the base line separation of the three solutes can be carried out within a dimensionless time T = 8 instead of T = 120, which was needed with the isocratic run in Figure 15.12a. hi the same
724
Gradient Elution Chromatography under Nonlinear Conditions
time, the heights of the second and third peaks are increased several fold. The peak width of the third solute has a dimensionless time ST = 1 in Figure 15.12b instead of Sr = 60 in Figure 15.12a. The ratio of the two band widths indicates that the average concentration of the third solute is 60 times higher in gradient elution than under isocratic conditions. The gradient serves to reduce the retention of the second and third solutes and to increase their concentrations. Numerical calculations showed also that nonlinear or step gradients provide better separations than a linear gradient when there are more than two solutes present in the mixture [43]. Finally, the authors suggested that, under certain circumstances, the use of a negative gradient could be helpful in improving the profile of tailing peaks.
15.3.4 Extension of the VERSE Model to Nonlinear Gradient Elution Chromatography The VERSE method was extended to describe the consequences of protein denaturation on breakthrough curves in frontal analysis and on elution band profiles in nonlinear isocratic and gradient elution chromatography [45]. Its authors assumed that a unimolecular and irreversible reaction taking place in the adsorbed phase accounts properly for the denaturation and that the rate of adsorption/desorption is relatively small compared with the rates of the mass transfer kinetics and of the reaction. Thus, the assumption of local equilibrium is no longer valid. Consequently, the solid phase concentration must then be related to the adsorption and the desorption rates, via a kinetic equation. A second-order kinetics very similar to the one in Eq. 15.42 is used. N
(15.56)
where Y^,- represents the loss of component i in the pore (stagnant) phase by adsorption. If the adsorption and the desorption rates {i.e., kaj and fc
15.3 Numerical Solutions of Nonlinear Gradient Elution 5 10"'
725 05
Cai
4 W-
1 10"0 101 10 IS t [rain]
20
10 IS t [min]
20
25
10
15
SO
25
20
25
t III11-1,
SltT
O
2ft
10 15 t Lminj
Figure 15.13 Gradient elution of papain (G = 0.0388) [47]. Dotted lines, experimental data, solid line, solution of VERSE-LC Initial concentrations of propanol: (a) 0; (b) 0.04; (c) 0.08; (d) 0.12. Reprinted from R. D. Whitley, X. Zhang, N.-H. L. Wang, AIChE J., 40 (1994) 1067 (Fig. 1), by permission of the American Institute of Chemical Engineers. ©1990 AIChE. All rights reserved.
perature and decreasing pH. Figure 15.13 shows the chromatograms of papain obtained for different starting mobile phase concentrations in propanol, the organic modifier. For a given contact time, the size of the native protein peak increases with increasing initial concentration of 1-propanol in the mobile phase. The presence of the organic solvent in the mobile phase decreases the denaturation rate in the solid phase. The solid line in the figures shows the chromatograms calculated with the VERSE-LC model for papain. The four calculated profiles closely approximate the experimental records. The retention times, the peak widths, and the relative peak heights are in excellent agreement. This indicates that the model accounts quantitatively for the denaturation process [45]. Systematic numerical calculations showed that the elution order, the peak sizes, their resolution, and their relative heights depend greatly on the gradient conditions and on the rate of the denaturation reaction [45]. In contrast to the case of systems in which there is no reaction, reducing the gradient slope can actually re-
726
Gradient Elution Chromatography under Nonlinear Conditions
duce the resolution observed in denaturing systems. Using the numerical solution of the model, the authors showed that, in frontal analysis, denaturation results in multiple, unsymmetrical waves in the breakthrough curves, waves that can be mistakenly attributed to the presence of impurities. However, by recording the breakthrough curves at increasing flow rates it becomes possible to differentiate between the effects due to impurities and those due to denaturation. In isocratic elution, peaks of the native and the denatured forms cannot be fully separated because the denaturation reaction proceeds during the elution of the band of protein.
15.4 Gradient Elution in Ion-Exchange ChromatograIon exchange is a well established method in analytical chemistry. The earliest reports on its use date back to 1850 when Thompson studied the adsorption of ammonium ions onto soils [48]. Today, ion-exchange is, besides reversed-phase liquid chromatography, the most popular mode of separation in liquid chromatography. At first, ion-exchange was applied for analytical purposes. During the last few decades it has become well established as a purification method at the preparative scale. The preparative purification of biomolecules from complex mixtures is now emerging as an important challenge for the pharmaceutical and biotechnology industries. Among the various methods, ion-exchange chromatography is one of the most effective. This technique is nowadays widely used for the analysis, the extraction, and the purification of peptides, proteins and polynucleotides [49]. However, despite its numerous applications, this useful technique suffers from a serious lack of the theoretical bases that would be necessary for a good understanding of the retention of polymeric ions such as those given by proteins in solution. A significant amount of work has been published, dealing with the study of the phenomena that are encountered with small organic or inorganic ions [50-53]. Saunders et al. [54] and Dye [55] have contributed much to improve our understanding of the adsorption and the retention of amino-acids on ion-exchange resins. However, an extension of these models to larger molecules, such as peptides and proteins, is not straightforward. This is due to the great difficulties encountered in attempting to evaluate the number and the spatial distribution of the charges of these molecules that interact with the stationary phase. A huge effort has been made to provide a theoretical description of the ionexchange process. The literature in this area is vast but the approaches focus essentially on two theories. Of these two, the stoichiometric theory has become the most popular because of its simplicity and its familiarity. From a physical point of view, the main problem of the stoichiometric theory is that it considers only the exchange of two ions and neglects the contributions of the activity coefficients of the solute and of the counter-ion. These activities are determined by long-range electrostatic forces. This implies that more than two ions are involved in the ionexchange process, which, therefore, cannot be described by a mere stoichiometric
15.4 Gradient Elution in Ion-Exchange Chromatography
727
approach. In addition, biopolymers such as peptides and proteins show multipoint interactions that arise from the complex spatial distribution of their charges. The second theory used to describe the ion-exchange process is based on the Donnan potential concept. It was introduced by Mattson in his investigation of ionexchange equilibrium in soils [56]. Detailed theories of the interactions between a charged surface and ions swimming in the surrounding electrolyte solution are still active subjects in surface and physical chemistry. In the Gouy-Chapman theory the electrostatic potential in the diffuse double layer is a function of the concentration and of the net charge density. Separations of proteins by ion-exchange chromatography are often carried out with a pH, a pi, or a counter-ion concentration gradient. These variations of the mobile phase composition during their elution may affect the tertiary structure of proteins, and, therefore, the distribution of electrical charges on their surface, which significantly changes the affinity of the proteins for the stationary phase. These changes, however, are little known and poorly predictable [57]. As pointed out by many authors, protein denaturation is often another result of these changes in the protein conformation and in the distribution of electrical charges on their surface. In addition, certain characteristics of the stationary phase, notably its charge density and its physical structure, are modified by changes in the mobile phase composition and play an important role in protein retention. A model taking into account all these effects and parameters would be extremely complex. Its application would require a considerable amount of information on the proteins studied, the stationary phase used, and their interactions. There are no proper methods to acquire this information through measurements that are independent of the chromatographic process. Accordingly, it is more appropriate to develop simplified, empirical models that may neglect some relatively less important features of the process but the use of which requires far less information to give predictions that are correct, even if it is for a limited range of operating conditions. Precisely with this purpose in mind, the stoichiometric displacement model (SDM) has been developed during the last decade to describe the retention of proteins in ion-exchange chromatography. This model has been successfully used to predict the chromatographic behavior of proteins under analytical conditions. Cysewski et dl. [58] developed a numerical method using SDM to calculate the isocratic and gradient elution profiles of a single protein in ion-exchange chromatography under mass overloaded conditions. Bellot et al. [59] extended their work for the theoretical study of the separation of a mixture of two proteins. They compared the preparative separations obtained in isocratic and gradient elution. A description of the SDM model is given in Chapter 4, Section 4.1.12.1. The SDM isotherm is given by
C- -
Hi
(
mACA
V
(15 57)
where K; is the equilibrium constant of component i, m^ the valency of the counterion, CA its concentration, Qx the loading capacity of the exchanger, Nc the number of components, m;- the characteristic charge of the protein i, and z = m{/m/y. This
Gradient Elution Chromatography under Nonlinear Conditions
728
Figure 15.14 Gradient elution of a 1:1 mixture in ion-exchange chromatography. Influence of the loading factor and the gradient steepness (/5). (a) /} = 0.025 mM s" 1 . (b) j6 = 0.55 mM s" 1 . Qx = 11 mM; L = 5 cm; u = 0.01 cm s" 1 ; HETP = 50 fim; CA/i = 20 mM. Loading factors (If) for compounds 1 and 2: (1): 2.61 and 1.44; (2): 12.96 and 8.64; (3): 21.6 and 14.4; (4): 30.24 and 20.16. Reprinted with permission from J. C. Bellot, ]. S. Condoret, ]. Chromatogr., 635 (1995) 1 (Fig. 4).
isotherm equation has the disadvantage of leading to an implicit form of the competitive isotherm which must be used when several proteins are separated. This leads to difficult numerical calculations to solve the set of several nonlinear partial differential equations, one differential mass balance for each protein. For very dilute solutions, the total concentration in the adsorbed phase is negligible and Eq. 15.57 becomes the following explicit equation —r
(15.58)
Bellot et al. [59] combined the use of the equilibrium-dispersive model (see Chapter 10) and the SDM isotherm (Eq. 15.57) to calculate numerical solutions of the elution profiles of binary mixtures of proteins. A linear gradient profile was used in these calculations. The counter-ion is small and the gradient started immediately after the feed injection. Figure 15.14 compares the band profiles obtained for various loading factors, using two gradient separations having different slopes. It shows the progressive evolution of the composition profile of the mixing zones formed between two bands in gradient elution chromatography. Figure 15.15 illustrates qualitatively the influence of the loading factor and of the counter-ion concentration on the chromatograms obtained in isocratic elution. They compare the productivity, the yield, and the enrichment factor obtained in isocratic and gradient elution. The authors concluded that these three parameters of a preparative separation are higher in gradient elution than in an isocratic run. This improved performance is due to the displacement power of the counter-ion gradient, which squeezes the peaks, concentrates the feed components, and enhances the effects of their competition for occupation of the adsorption sites. The use of gradient elution also has the great advantage of reducing the cycle time by reducing the duration of the tail of the last peak. This time gain more than compensates for the time loss caused by the need to flush the column for a certain time after the end of each gradient run with a stream of the mobile phase having the initial composition of the gradient
15.4 Gradient Elution in Ion-Exchange Chromatography
729
Figure 15.15 Gradient elution of a 1:1 mixture in ion-exchange chromatography. Influence of the loading factor and the counter-ion concentration, (a) CA - 25 mM; (b) CA = 40 mM. Qx = 11 mM; L = 5 cm; u = 0.01 cm s" 1 ; HETP = 50 }im; CA/i = 20 mM. Loading factors (Lf) for compounds 1 and 2 : (1): 2.16 and 1.44; (2): 12.96 and 8.64; (3): 21.6 and 14.4; (4): 30.24 and 20.16. Reprinted with permission from } . C. Bellot, J. S. Condoret,}. Chromatogr., 635 (1995) 1 (Fig. 6).
to regenerate this column. The Steric Mass Action (SMA) model accounts for the multicomponent, competitive adsorption of proteins in ion-exchange chromatography (see Chapter 4, Section 4.1.12.2). It was used by Gallant et al. to predict the behavior of mixtures in preparative chromatography, with a step gradient [60] or a linear gradient [61]. These authors used the equilibrium-dispersive model (see Chapter 10) combined with the SMA isotherm to calculate numerical solutions. Figure 15.16 compares calculated and experimental band profiles. An eight-column volume sample of a solution containing 0.2 mM of a-chymotrypsinogen A, 0.2 mM of Cytochrome C, and 0.2 mM of lysozyme in a buffer (50 mM sodium phosphate at pH = 2.2) was injected into the column, after which a linear gradient of [Na + ], with a slope of 10 mM per column volume was started. Figure 15.16a shows the experimental chromatogram. The three protein peaks are quite broad and their profiles exhibit substantial nonlinear effects. Figure 15.16b shows the calculated profiles, in rather good agreement with the experimental data. This result shows that the model predicts well the nonlinear adsorption behavior of the different components and the displacement of the first component by the second. Finally, Figure 15.16c shows the calculated profile of the composition of the fractions collected successively, which illustrates the effect of the pooling of these fractions. Wiesel et al. [62] published a theoretical model of the ion-exchange behavior of biopolymers in multicomponent, nonlinear ion-exchange chromatography. This theory is based on a discrete distribution of the counter-ions in the double layer and a mass action isotherm that considers the specific adsorption of the solute counter-ions onto the charged surface. The distinction of the two layers is based on their distance from the surface. This model allows neutral components to bind to a charged surface by simultaneous ionization. This model can describe the development of elution profiles for single and multicomponent samples in isocratic or in gradient elution. A version of this model was proposed and experimentally verified for bioseparations by ion-exchange chromatography. It could be applied to anion and cation exchange, whether the ion exchanger resins used are weak or
730
Gradient Elution Chromatography under Nonlinear Conditions
0
S
10
15
JO
25
M
3S
«
Figure 15.16 Comparison of experimental and calculated chromatograms. (a) Reconstructed chromatogram from analyses of collected fractions, (b) Calculated chromatogram. (c) Calculated profile of the composition of the collected fractions. Reprinted with permission from S. R. Gallant. S. Vunnum, S. M. Cramer, J. Chromatogr A, 725 (1996) 295 (Fig. 3).
strong. After suitable modifications, this approach is applicable to amino acids, which have a double charge, to polysaccharides, peptides and proteins which have a three-dimensional charge distribution. The modifications include a further selectivity in the diffuse layer, which describes the selectivity reversals that are observed for amino acids or the size exclusion effects taking place in the diffuse layer and arise from the size of the protein molecules usually exceeding the dimensions of the double layer. The influence of the eluent concentration on the retention and separation in ion-exchange and that of the characteristic charge of the protein was considered. Experimental measurements showed that a strong nonlinear ion-exchange equilibrium takes place, with a transition from a Langmuirian to a sigmoidal isotherm at high eluent concentrations. The compound binds to the surface even though it is not ionic. Therefore, the model can take into account both possibilities of retention by ion-exchange and by adsorption. A simplified distribution of the counter-ions based on the Gouy-Chapman theory with a discrete distribution of
REFERENCES
731
^r ^-K !•
•5 1, 5
7f
V
•
I •/
MI
/ /l
A* / |\ *
\\ e
: I
e
...l-l-l-
!1 :
10
12
u
1/t0 [-] (since start of elution)
Figure 15.17 Comparison of experimental (symbols) and calculated (solid lines) chromatograms of a 3-component mixture. Dimensionless concentrations, relative to the feed concentrations, dimensionless time relative to the hold-up time. Reprinted with permission from A. Wiesel, H. Schmidt-Traub, J. Lenz, J. Strube, J. Chromatogr. A, 1006 (2003) 101 (Fig. 10).
the counter-ions was used. Figure 15.17 compares the experimental and calculated band profiles of a multicomponent mixture. The column was loaded with the feed solution at a neutral pH. After a rinsing step, three successive gradient elution steps were carried out, each one by flushing through the column a solution of constant concentration of HC1 (at 10, 20, and 30 mM). The experimental profile of component P (A) is the same, whether injected pure or in the mixture. The sharp bend in its profile is due to the second step of the gradient elution. Note that this double peak is well predicted by the model. The shapes of the elution bands of the last two components are totally different. The third component (•) is the strongest retained. It is eluted in the desorption front of the second component. Its profile is determined by both the tag-along effect caused by the second component and the third gradient step. This combination gives it a nearly Gaussian peak shape. Note also that the maximum band concentrations of the first two components are nearly two and three times higher than their feed concentrations, respectively. This is due to the concentration effect of gradient elution but also, in the case of the first component, by its displacement by the second one (see Chapters 8 and 11).
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This Page is Intentionally Left Blank
Chapter 16 Kinetic Models and Multicomponent Problems Contents 16.1 Analytical Solution for Binary Mixture; Constant Pattern Behavior 16.1.1 16.1.2 16.1.3 16.1.4
The Shock Layer Theory for a Binary Mixture Shock Layer in the Case of Competitive Langmuir Isotherms Shock Layer Thickness in Binary Frontal Analysis Shock Layer Thickness in Displacement Chromatography
736 737 738 740 742
16.2 L i n e a r D r i v i n g Force M o d e l A p p r o a c h 747 16.3 N u m e r i c a l S o l u t i o n of T h e G e n e r a l R a t e M o d e l of C h r o m a t o g r a p h y . . . . 754 16.3.1 Formulation of the General Rate Model with the Pore Diffusion Model 755 16.3.2 Solutions of the General Rate Model with Pore Diffusion Model 757 16.3.3 The GRM Formulated with the Maxwell-Stefan Surface Diffusion Model . . . . 765 16.3.4 The VERSE Model 769
References
775
Introduction In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measurements improve and more interest is paid to complex 735
736
Kinetic Models and Multicomponent Problems
molecules and systems.
16.1 Analytical Solution for a Binary Mixture under Constant Pattern Behavior We discussed the origin of constant pattern behavior in Chapter 14. We have shown that an asymptotic solution exists and that in practice this solution accounts for the features observed for the breakthrough curves obtained in singlecomponent frontal analysis, when the adsorption isotherm is convex upward. Under such conditions, the shock of the ideal model is replaced by a shock layer, i.e., a region where the concentration varies very steeply and which propagates at the same velocity as the shock. The band profile in the shock layer region is obtained by integration of the rate expression, subject to the constant pattern condition. Cooney and Strusi [1] have derived an approximate analytical solution in the case of a binary system, assuming competitive Langmuir adsorption behavior and that both mass transfer zones propagate under the constant pattern condition. They ignored axial dispersion and assumed that the mass transfer kinetics follows the solid film linear driving force model. This gives the following system of equations:
(16.3) with i = 1, 2. Cooney and Strusi [1] have argued that, in the case of a binary mixture, the relationship between the concentrations, C\ and Ci, of the two components in the intermediate mixed zone between the bands of the pure components should be essentially the same during the migration of the constant pattern breakthrough curve under nonequilibrium conditions and in the equilibrium case. Especially when the two components are chemically similar and have close values for their diffusion and mass transfer coefficients, the differences should be small. The relationship between C\ and Ci derived with the ideal model can then be substituted into the competitive Langmuir isotherm, giving instead two pseudo- single-component isotherms. Thus, this transformation reduces the twocomponent competitive Langmuir system into two single-component, independent Langmuir systems [2]. In the general case, the solutions of this model must still be calculated numerically. When the mass transfer coefficients of the two components are equal, however, an analytical expression has been derived for the constant pattern breakthrough curve [1], using the pseudo-single-component (i.e., decoupled) isotherms obtained in the solution of the ideal model (Chapter 8).
16.1 Analytical Solution for Binary Mixture; Constant Pattern Behavior
737
16.1.1 The Shock Layer Theory for a Binary Mixture A more comprehensive analysis of constant pattern behavior for a binary system has been given by Rhee and Amundson [3]. In this work, these authors have extended to binary systems the analysis of the combined effects of mass transfer resistance and axial dispersion that they had previously made in the case of single-component bands [4]. Rhee and Amundson [3] assumed the solid film linear driving force model, finite axial dispersion, and no particular isotherm model. The system of equations becomes
»§ + § - f - °»% ^
=
qf
kfM-1i)
= MCVC2)
(16.4b)
(16.4c)
where q* = fi{C\, C2) is the competitive isotherm equation. Equations 16.4a and 16.4b can be rewritten in dimensionless form
dc aq a^+a7 +
F
% 1 a2q ^ = p^a^ §£ = Sti{fi{CXlC2)-qi)
(16 4d)
(16.4e)
The equilibrium isotherm must satisfy the following condition:
(16.5) The solution of this system of equations is a mathematical problem similar to the one encountered in the study of the propagation of shock layers in compressible fluids. The shock layer theory developed by von Mises [5] and by Gilbarg [6] can be applied. The treatment is similar to the one previously discussed in Chapter 14, in the case of a single component. The concentration profiles in the shock are given by the system of two nonlinear differential equations (i = 1,2)
The propagation velocity of the shock layer is the same as the shock velocity of the ideal model
Since A is independent of the subscript, we must have
738
Kinetic Models and Multicomponent Problems
with Aq\ = q*/ - q*/, Aq* = q*/ - q*/, AC, = c{ - C\, and AC2 = Cf2 - C\. Equation 15.10 is the compatibility condition of the ideal model. This condition must be fulfilled by the component concentrations at both ends; if it is, the velocity of the shock layer, given by Eq. 16.7, is independent of the axial dispersion and the mass transfer coefficients. Rhee and Amundson [3] concluded also that • There exists a shock layer only if C2 > C\. • For a given state (Cj, Cj), there exist two shock layers. For one of them, the end states are on a F + characteristic line; for the other one they are on a T~ characteristic line. • If A or C2 is specified, the shock layers of either kind are unique. • If C£ < C2 — C{/r+, where r+ is the slope of the F + characteristic line, the shock layer having its end state along this line does not exist.
16.1.2 Shock Layer in the Case of Competitive Langmuir Isotherms In the case of two components, the shock layer thickness is defined as in the singlecomponent case (Figure 14.3), with the help of threshold concentrations (C* and Cj*, j = 1, 2), and the auxiliary parameter 8 (Eq. 14.21). In displacement chromatography, there is a shock layer on each side of the isotachic bands, and we need to distinguish C-* (band rear) and CJ* (band front), C- and Q'-. In the case of competitive Langmuir isotherms,1 it is possible to derive an approximate solution of Eq. 16.6. As in the case of the single-component shock layers (Chapter 14), we neglect the coupling term in this equation. We must also assume that either the Peclet numbers are the same for the two components and their mass transfers are infinitely fast or that the Stanton numbers are the same for the two components and their axial dispersion are negligible. If neither of these two conditions is fulfilled, only numerical solutions can be calculated. 16.1.2.1 Shock Layer Thickness Controlled by Axial Dispersion For equal Peclet numbers and infinitely fast mass transfers, the profile of the shock layer thickness is given by the following equation [3]: L And = — (1 + ^—] . ' " In /y Pe\ FA2 Rf R'J Rl- Rf The shock layer migrates at the velocity Aw, with
A=
- ^^ 1 + F A2 Rf Rl
(16.9)
(16.10)
16.2 Analytical Solution for Binary Mixture; Constant Pattern Behavior
739
while the parameters are defined as follows: Rf
=
Ri
B2
(16.11a)
7
1
BC{
(16 nb)
= i^ko =
A =
-
- ^
,-b2
,
—
——
(1
^ ~^
r±
(16.11e)
where a = a2/a\ and r± stands for either the positive (+) or the negative (-) root of the following algebraic equation 2 r ~ (—iT" + C l
?~C2)r
ETC1 = °
(16.12)
The positive root is not bound, but the negative root is larger than —1. Equation 16.9 is general and is applicable to the profile of each step in the "staircase" profiles obtained in conventional binary frontal analysis. When a binary mixture is introduced as a step into an empty column (C- = 0 and CJ = C9, j = 1,2), Eq. 16.9 becomes l-i
(16.13)
16.1.2.2 Shock Layer Thickness Controlled by Mass Transfer Resistance The other limiting case is when the axial dispersion is negligible (i.e., Pe «s oo) and the thickness of the shock layer is controlled by the mass transfer resistances. Furthermore, the two components must have the same Stanton number [3]. The equations obtained are the same as Eqs. 16.9 and 16.13, except for the replacement ofl/Peby^^. The analytical solution derived for the band profile under the constant pattern condition [1] and the equation giving the shock layer thickness (Eqs. 16.9 and 16.13) are valid only provided that the mass transfer zone behaves as a shock. If the mass transfer resistance is too high, constant pattern behavior has no time to develop during the migration along a column of realistic length. Finally, the solution is not valid in elution where there is a constant erosion of the shock height and the shock velocity decreases regularly. In such cases, the solution can be calculated only by numerical methods, using the set of mass balance equations coupled by the competitive isotherms and the set of coupled kinetic equations.
Kinetic Models and Multicomponent Problems
740
20
40
60 SO t-Time (min)
Figure 16.1 Comparison between the experimental breakthrough curves obtained with benzene (•) and toluene (o) vapors on activated charcoal and the prediction of constant pattern. Reproduced with permission from W.J. Thomas and ].L. Lombardi, Trans. Chem. Eng., 49 (1971) 240 (Fig. 23).
16.1.2.3 General Case. Neither St nor Pe is Negligible In this case, there are no analytical solutions, even under constant pattern conditions [3]. Numerical solutions can be obtained. We can just speculate that, since in the single-component case the two effects are additive under almost all cases of practical significance, we may use the same approach in the multicomponent case. This assumption is supported by its agreement with the results of experimental determinations. Experimental results [7,8] obtained in the case of the breakthrough curves of binary mixtures under constant pattern condition have been compared with the analytical solution. Figure 16.1 compares the experimental breakthrough curves obtained in the case of the vapor phase adsorption of benzene and toluene carried by nitrogen through a bed of activated carbon [8] with the analytical solution calculated from the binary adsorption data and under the assumption of constant pattern behavior [1,3]. The agreement achieved is excellent.
16.1.3 Shock Layer Thickness in Binary Frontal Analysis As shown by Rhee and Amundson [3], the profile of the breakthrough curves of the two components can be accounted for in the following way [9,10]: • Assuming that the contributions of axial dispersion and mass transfer resistances are additive, the breakthrough profile of the first component is given by the same equation as in the single-component case, but with a higher plateau concentration, Cf (Eq. 8.12a). Accordingly, the shock layer thickness for this
16.1 Analytical Solution for Binary Mixture; Constant Pattern Behavior
741
breakthrough is given by Eq. 14.28a (16.14)
The breakthrough curve of the second component, which is accompanied by a decrease of the first component concentration from Cf to C°, is given by the same equation as the breakthrough profile of a single component (Eq. 14.28a), but with a decoupled isotherm
whose coefficients A2 and B2, related to the coefficients of the competitive Langmuir isotherms of the two components, are given by Eqs. 16.11c to 16.11e. The thickness of the second shock layer is given by K2u
B2C2fi
1-1 ln
(16.16)
with K2 = k'2fi/{l + B2C2/0) and k'2c = FA2 (instead of k'lf) = Fa2, F phase ratio). Since Eqs. 14.33 to 14.35 (SLT for a single component) and 16.14 (SLT for the first component) are identical, the conclusions of the previous section regarding the influence of the experimental parameters on the SLT apply directly. There is an optimum velocity for minimum SLT of the first component breakthrough curve [11]. This optimum velocity is =
This optimum velocity depends on the nature and concentration of the second component through C^. Similarly, the SLT of the second component is minimum for the following value of the mobile phase velocity, different from the former given by Eq. 16.17: s,2
°P4 ~
The simplification is a result of the assumption that the coefficients of axial dispersion and of mass transfer kinetics are equal for the two components. The ratio of the two optimum velocities is K2) }
Kinetic Models and Multicomponent Problems
742
Figure 16.2 Definition of the parameters of the shock layer and the SLT in displacement chromatography, in an isotachic train. Reproduced with permission from ]. Zhu and G. Guiochon, } . Chromatogr. A, 659 (1994) 15 (Fig. 2).
Concen tration Profile ( 20 30 10
/
\
;
cPi r
* " •
•' \ \
V
A '\
; CPH-, 60.0 50.0
*
•
An,
!
\
72
Elution Time (V100 min)
The intermediate plateau of the first component between the two shocks, at the concentration CjS does not form instantaneously. It takes some time to grow. Ma and Guiochon [9] have calculated an estimate of the time needed for the formation of this plateau At = 0.5
(16.20)
This value is in agreement with the one derived from band profiles calculated with the equilibrium-dispersive model [9]. The time given by Eq. 16.20 provides useful information regarding the specifications for the experimental conditions under which staircase binary frontal analysis must be carried out to give correct results in the determination of competitive isotherms. The concentration of the intermediate plateau is needed to calculate the integral mass balances of the two components, a critical step in the application of the method (Chapter 4). This does not apply to single-pulse frontal analysis in which series of wide rectangular pulses are injected into the column which is washed of solute between successive pulses.
16.1.4 Shock Layer Thickness in Displacement Chromatography In displacement chromatography, an isotachic train forms after a certain period of time. The ideal model permits an investigation of the band profiles during the formation of this train (Chapter 9). Assuming infinite column efficiency, we can find analytical expressions giving the intermediate band profiles. The numerical solutions of the equilibrium-dispersive model (Chapter 12) give the band profiles under any set of experimental conditions. However, since the isocratic train is an asymptotic solution, an analytical expression can be derived that gives the concentration profiles in the intermediate, mixed zones, if we assume competitive Langmuir behavior. The definitions of the parameters used in the study of the SLT in displacement are given in Figure 16.2.
16.2 Analytical Solution for Binary Mixture; Constant Pattern Behavior
743
In this case, as shown above, a constant pattern is reached, and a shock layer propagates at the velocity predicted by the ideal model [1,3]. The velocity of the isotachic train is the velocity of the front shock of the displacer breakthrough curve US4 = ^
(16.21)
where Td = bdCd is the reduced concentration of the displacer. In the isotachic train, the front breakthrough curves of each component of the mixture move at the same velocity. Thus, the plateau concentration of each band is given by p,i — T
(1 + Id) ~ 1
(16.22)
Each pair of bands is separated by a shock layer in which the concentration of the first component of the pair deceases rapidly, from Cp, to 0, while the concentration of the second one increases from 0 to Cn.,+i. We define the thickness of the shock layer as in the previous sections [3,11]
where Cp^=eCp4
(16.24a)
Cp^
(16.24b)
= eCN
As discussed in Section 1, Rhee and Amundson [3] have shown that we can obtain an asymptotic solution for the concentration profile in the shock layer. This solution is written A d2Q
(1 +
+
A(1-A)WQ )
=XVT{Ci
C
^
Q 1
C
l
)
(16 25)
"
where i is the rank of the component in the train, between 1 and n. The function ^"(Q/Cp^Q+i/Cp^+i) depends on the parameters of the multicomponent isotherm. Pe and St are the Peclet and Stanton numbers, respectively. Using the hodograph transform, Rhee and Amundson [3] have also shown that a plot of Q versus Q+i is a straight line (solid line in Figure 16.3), provided that these two components have the same axial dispersion coefficient (D^) and mass transfer rate constant (kf), in addition to the competitive Langmuir isotherm behavior. The equation of this straight line is C ^
+ i
+i
(1626)
Therefore, the function of two variables ^ ( Q , Cpj, Q + i , CP/;+i) can be decoupled into two simpler functions T{C{, Cpj, Cp ! + i) and J-(Cj+\, Cpj, Cp^+i), and the following equations give the SLT in displacement chromatography [11]
*
=
~^~ {TL + {l + KdYkfL)
l^ln
(16.27a)
Kinetic Models and Multicomponent Problems
744
8-
Figure 16.3 Hodograph transform of the concentration profiles of two components in the shock layer between their zones. Solid line, competitive Langmuir isotherms, Dj,i = D^fc/,1 = ^fX dotted line, same conditions except fy-1 = 20, kf2 = 200 s" 1 . Reproduced with permission from }. Zhu and G. Guiochon, ]. Chrotnatogr. A, 659 (1994) 15 (Fig. 3).
1+Kd
Atjt
=
Thus, the SLT between two successive zones in the isotachic train depends on the axial dispersion coefficient and the mass transfer coefficient of the two components (the theory is valid only if DL/i = DL/i+1 and kfj = fc/,;+i)/ on their separation factor, and on the concentration and retention factor of the displacer [11]. The SLT does not depend on the retention factor or the feed concentration of the components. Obviously, when the isotachic train is formed, the concentration of each band reaches a plateau concentration which is determined by the concentration of the displacer. Numerical calculations performed using the transport-dispersive model give values of the SLT between successive bands in an isotachic train that are in close agreement with those calculated from Eq. 16.27a [11]. This agreement merely proves that the equation is correct and that isotachic train conditions were achieved in the numerical calculations. In a given isotachic train, each successive SLT depends only on the value of tty. If the two adjacent components do not have the same axial dispersion and kinetic coefficients, the plot of Q versus Q + j in the shock layer region is no longer a straight line (see chain line in Figure 16.3), and the decoupling of F(Q,C p/ ,,C, + i,Cp / j + i) is no longer possible. In this case, no simple analytical solution can be derived for the concentration profiles and for the SLT. However, numerical calculations with the transport-dispersive model remain possible, and their results show that the SLT is between the two values calculated using Eq. 16.27c with the smaller and the larger kf. Thus, Eqs. 16.27a to 16.27c still provide a good approximation of the SLT in this case.
16.2 Analytical Solution for Binary Mixture; Constant Pattern Behavior
745
16.1.4.1 Dependence of the SLT on the Displacer Parameters Differentiation of Eq. 16.27c with respect to K^ shows that there is an optimum value of this parameter for which Arjt is minimum [11]. This optimum corresponds to Kd = 1 or Jfd
=
l + bdCd
(16.28a)
Q
=
^ ^
(16.28b)
The SLT is very large at low displacer retention or high displacer concentrations,, when k'd 1 + &dQ, the SLT tends to increase linearly with increasing k'd. Thus, there are no optimum displacer retention factor and optimum displacer concentration, but a combined optimum given by Eqs. 16.28a or 16.28b. 16.1.4.2 Dependence of the SLT on the Mobile Phase Linear Velocity There is a minimum of the SLT for an intermediate value of the mobile phase velocity, as in linear chromatography [11]. At low velocities, axial dispersion is large due to the long migration time during which axial diffusion proceeds constantly to relax the concentration gradients, while at high velocities, the finite rate of the mass transfer kinetics causes the SLT to increase in proportion to the velocity. If we assume as above the Van Deemter equation for the axial dispersion term (Eq. 14.302), we obtain for the optimum velocity for minimum SLT in displacement (under isotachic conditions) Kd)2kf
This equation is identical to the one derived in single-component frontal analysis, if applied to the displacer [12]. Figure 16.4 illustrates the dependence of the SLT on the flow velocity. It shows that the SLT increases rapidly with increasing flow velocity above the minimum. This phenomenon explains the sharpness of the production rate optimum reported by Horvath and associates [13,14]. When the flow velocity exceeds the optimum, the SLT broadens, and the volume of pure fractions that can be collected drops rapidly. Differentiation of Eq. 16.29 shows that the optimum mobile phase velocity, which is a function of the displacer retention factor and concentration, is minimum for K
2DL = Au + 2yDm
746
Kinetic Models and Multicomponent Problems
Figure 16.4 Plot of the SLT in displacement chromatography versus the mobile phase flow velocity. Reproduced with permission from ]. Zhu and G. Guiochon, ]. Chromatogr. A, 659 (1994) 15 (Fig. 8). 0.0
0.5
10
1.6
2.0
Mobile Phase Velocity (cm/min)
O_
Figure 16.5 Plot of the SLT in displacement chromatography versus the separation factor between the components of two successive zones. Reproduced with permission from } . Zhu and G. Guiochon, ]. Chromatogr. A, 659 (1994) 15 (Fig. 12).
f
8-
o
if
\ s-
\V 1.0
1.5 SELECTMTY
2.0
conditions. These two velocities could be equal only at unusually low values of k'^. These results are in agreement with experimental reports [13-15]. 16.1.4.3 Dependence of the SLT on the Separation Factor of the Two Components Figure 16.5 shows the dependence of the SLT on the separation factor a = k'2/k[ of the two components [11]. The SLT increases dramatically as a decreases toward unity. It tends slowly toward 0 with increasing value of oc. Thus, it is no more possible to separate a binary mixture with a very small value of a (e.g., components differing by an isotopic substitution) by displacement than by overloaded elution. The shock layer thickness in the isotachic train would be so large that the shock layer would encompass the whole bands, and the time required to achieve isotachic conditions would result in a very low throughput. The isotachic train
16.2 Linear Driving Force Model Approach
747
shown in Figure 12.7 (Chapter 12) illustrates most clearly the dependence of the shock layer thickness on the separation factor. The mixed zone between closely related compounds, those having a value of K close to unity, is wider than between unrelated compounds with a large separation factor.
16.2 Linear Driving Force Model Approach In these kinetic models of chromatography, all the sources of mass transfer resistance are lumped into a single equation. In the case of the solid film linear driving force model, we have for each component i d
3i=kffi(qt-qi)
(16.30)
wherefc^,-is the lumped rate coefficient or apparent mass transfer coefficient of component i and q* is its stationary phase concentration at equilibrium with the concentrations C\, Ci, • • •, Q, • • • in the mobile phase, q* is related to these mobile phase concentrations through the competitive equilibrium isotherm, q* = /(Cy). The mass balance of each component is written
In the case of a step input, the numerical solution of the system of Eqs. 16.30 and 16.31 has been discussed in the literature for multicomponent mixtures [16]. The numerical solution of Eqs. 16.30 and 16.31 without an axial dispersion term (i.e., with D; = 0) has been described by Wang and Tien [17] and by Moon and Lee [18], in the case of a step input. These authors used a finite difference method. A solution of Eq. 16.31 with D,- — 0, combined with a liquid film linear driving force model, has also been described for a step input [19,20]. The numerical solution of the same kinetic model (Eqs. 16.30 and 16.31) has been discussed by Phillips et al. [21] in the case of displacement chromatography, using a finite difference method, and by Golshan-Shirazi et al. \¥L, 23] in the case of overloaded elution and displacement, also using finite difference methods. When the number of transfer units is large, the band profiles calculated in overloaded elution with the solid film driving force model are very similar or identical to those calculated with the equilibrium-dispersive model [22]. By contrast, when the number of mass transfer units is small (i.e., at very low values of kf), very different profiles can be obtained. The series of figures 16.6 to 16.9 illustrates the influence on their band profiles of the number of mass transfer units (k'okf/to) of the column for the two components of a 1:9 binary mixture [22]. In Figure 16.6, this number is large (kf \ = kf^ = 50 s^1). The profiles obtained for the two components are the same as those obtained by numerical solution of the equilibrium-dispersive model (Chapter 11). Figure 16.7 shows the profiles obtained under the same conditions, except for the value of the mass transfer coefficient {kft\ — kf/2 - 0.05 s^1), resulting in 1,000 times fewer mass transfer units for the column. The change in band profiles from
Kinetic Models and Multicomponent Problems
748 C m/L
0.1
-3
0.05
-2
-1 0 100
200
300
400
Time, sec
Figure 16.6 Chromatogram calculated for a binary mixture with large mass transfer coefficients: kf:i = kft2 - 50 s" 1 . Solid gray line: elution profile of the first component; solid black line: elution profile of the second component. The dashed line is the total chromatogram. Characteristics of the separation: Langmuir competitive isotherms, with a\ = 24, Hi = 29; b\ = 2.5; b2 = 3.0; a = 1.20. Phase ratio: F = 0.25; k'2° = 6.0. Column length: 25 cm; flow velocity: 0.625 cm/s; feed composition: 1:9. Sample size: 4.15 mmol (Lf^ = 4.5% of column saturation capacity). Reproduced with permission from S. Golshan-Shimzi, B.C. Lin and G. Guiochon, J. Phys. Chem., 93 (1989) 6871 (Fig. 1). ©1989, American Chemical Society.
C m/L
Figure 16.7 Chromatogram calculated for a binary mixture with low mass transfer coefficients: k:fA = kfi2 = 0.050 s" 1 . Same as Figure 16.6, except for the values of kf^ and kf^- Reproduced with permission from S. Golshan-Shirazi, B.C. Lin and G. Guiochon,}. Phys. Chem., 93 (1989) 6871 (Fig. 5). ©1989, American Chemical Society.
0.02 -3
-2
0.01
-1
0 0
200
400
Time, sec
600
Figures 16.6 to 16.7 is dramatic. The profiles are much broader, the influence of the nonlinear behavior of the phase equilibrium is considerably reduced, and the intensity of the displacement effect of the first component by the second one is much reduced. As a consequence, there is almost no separation. Unusual band shapes are obtained when the number of mass transfer units is markedly different for the two components [22]. For example, Figure 16.8 shows the chromatogram calculated with kjt\ = 50 and kfr2 = 0.05 s" 1 and Figure 16.9 the
16.2 Linear Driving Force Model Approach
749
C m/L
Figure 16.8 Chromatogram calculated for a binary mixture with a large and a low mass transfer coefficients: fc^i = 50 s^1 and kf% = 0.050 s"1. Same as Figure 16.6, except for the value of kf^ = 0.050 s"1. Reproduced
0.04
-3
0.02
with permission from S. Golshan-Shirazi, B.C. Lin and G. Guiochon, J. Phys. Chem., 93 (1989) 6871 (Fig. 11). ©1989, American Chemical Society.
-2
1-
0 0
200
400
Time, sec
600
C m/L
0.1
Figure 16.9 Chromatogram calculated for a binary mixture with a large and a low and a large mass transfer coefficients: kt \ = 0.20 s^ 1 and kt 2 = 50 s^1. Same as Figure 16.6, except for k^x = 0.20 s" 1 . Reproduced with permission from S. GolshanShirazi, B.C. Lin and G. Guiochon, J. Phys. Chem., 93 (1989) 6871 (Fig. 15). ©1989, American Chemical Society.
-3 0.08
Page 1 2-
0.06
0.04
0.02
-1 0 100
200
1
300
Time, sec
400
chromatogram calculated for kji = 0.20 and kf^ = 50 s . In the former case, there are practically no band interactions. The elution profile of the first component exhibits the thin front shock layer characterizing components having a Langmuir isotherm and a fast rate of mass transfer, while the profile of the second component is considerably broadened and smoothed by the slow rate of mass transfer and exhibits no shock layers. Note that the second component begins to elute well before the first one. Depending on the relative concentration of the two components, the chromatogram of the mixture may appear as a narrow peak on a drifting baseline or as a narrow peak on the top of a diffuse front, and followed by a long tail. By contrast, in Figure 16.9, the profile of the second component band, which has a high mass transfer coefficient, is typically Langmuirian, while the first component profile results from the combination of its slow mass transfer kinetics and of the displacement effect due to the second component. The coupling effect remains strong in this case. If the concentration discontinuity of the first component has disappeared, a discontinuity exists for its derivative, which abruptly changes sign on the front shock of the second component. The profiles of individual zones in displacement chromatography have also been calculated using the solid film linear driving force model [23]. Again, when the number of mass transfer units of the column is high, the results are very similar to those obtained with the equilibrium-dispersive model (Chapter 12). As an example, Figure 16.10 shows the displacement chromatogram calculated with kfi = kft2 = kf4 = 50 s" 1 . The bands in the isotachic train are clearly formed
Kinetic Models and Multicomponent Problems
750 C m/L
0.04 3 3-
2-
o.o2. 0.02
11-
J
0 35
40
45
Time, min
50
Figure 16.10 Calculated zone profiles in displacement chromatography. kfy = kj^ = kj^ = 50 s^1. Solid gray line: first component; thin solid black line: second component; fat solid black line: displacer. Characteristic of the separation: Column length: 25 cm. Phase ratio: 0.25. Flow velocity: 0.05 cm/s. Competitive Langmuir isotherm with coefficients ay = 18.75; a2 = 22.5; ad = 27; by = 12.5; b2 = 15; bd = 18; «2,i = 1-20. Loading factors: LfA = 1.333%, Lj2 = 2.666%. Displacer concentration: Q = 0.01 mM. Mass transfer coefficients: kfy = kf2 = kf4 = 50 s" 1 . Apparent dispersion coefficient: Da = 0.005 cm 2 /s. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal. Chem., 61 (1989) 1960 (Fig. 6). ©1989, American Chemical Society.
C m/L
0.04
3-
0.02
2-
1-
0 35
40
45
Time, min
50
Figure 16.11 Calculated zone profiles in displacement chromatography. Same as in Figure 16.10, except kt\ = kt^ = ktj = 0.2 s 1 . Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal. Chem., 61 (1989) 1960 (Fig. 11). ©1989, American Chemical Society.
and separated. When the mass transfer rate constant decreases, the widths of the plateaus in the isotachic train decrease and the shock layers become thicker. If the mass transfer coefficients become very small, the plateaus disappear, and the
16.2 Linear Driving Force Model Approach
20
30
751
40
volume, ml
Figure 16.12 Comparison of the experimental (symbols) overloaded elution band profiles of a mixture of «-chymotrypsinogen and horse-cytochrom c on a monolithic column of ion- exchange resin and calculated profiles. Top: mobile phase : 50 mM buffer, pH = 6.0; Bottom: same mobile phase + 80 mM Na + . Left calculations made with the RD model. Right calculations made with the TD model. Reproduced with permission from S. Ghose, S. M. Cramer, f. Chromatogr. 928 (2001) 13 (Figures 5 and 6).
band heights begin decreasing [23]. As an example of this last case/ the profiles shown in Figure 16.11 are obtained with kt\ = kt^ = kt^ = 0.2 s" 1 . Although this is an isotachic train in the sense that the chromatogram exhibits constant pattern behavior, there are no plateaus and the resolution between successive bands is poor. Ghose and Cramer [24] studied the separation by displacement chromatography of a-chymotrypsinogen A and horse cytochrom c on a monolithic ion exchange column, using a 4 mM solution of neomycin sulphate in a phosphate buffer solution as the displacer and investigated the influence of the salt solution. These authors compared the experimental chromatograms and those calculated using the reaction-dispersive (RD) and the transport-dispersive (TD) models (see Figure 16.12). They used the competitive protein-salt multicomponent equilibrium isotherm afforded by the Steric Mass Action Model. The addition of a salt caused a dramatic improvement of the separation performance. The RD model accounts reasonably well for all the experimental results while the TD model does so only at low salt concentration but gives poor results at high salt concentrations. The adsorption isotherms of the pure enantiomers of 3-chloro-l-phenyl-l-propanol were measured by FA on a cellulose tribenzoate coated on silica, eluted with a 95/5 mixture of n-hexane and ethyl acetate. These data were well accounted for by a simple Langmuir isotherm model. The adsorption data measured fitted well to the Langmuir isotherm model. The elution band profiles of large amounts of
752
Kinetic Models and Multicomponent Problems
Figure 16.13 Comparison of the experimental (symbols) overloaded elution band profiles of the racemic mixture of the R and S enantiomers of 3-chloro-l-phenyl-lpropanol on a Chiracel OB-H 250 x 4.6 mm column eluted with n-hexane/ethyl acetate, 95/5 v/v and the profiles calculated with the equilibrium-dispersive (dotted lines) and the transport-dispersive models (solid lines). Sample volume, 1 ml, loading factor Lc = 5%. Reproduced with permission from D. Cherrctk, S. Khctttabi, G. Guiochon, J. Chromatogr. 877 (2000) 109, (Figure 7d).
3.5-
time(min)
Figure 16.14 Comparison between calculated and experimental band profiles for a 1:1 mixture of phenetole and n-propyl benzoate on a Symmetry-Cig column. Calculations made with the TD model. Solid symbols (•), experimental data; solid line, calculated profile. The empty triangular symbols show the single-component band profile of n-propyl benzoate, for the sake of comparison. The inset shows the corresponding calculated single- component profiles. Ca = 9.28 g/dm 3 and C2 = 9.28 g / d m 3 ; tjnj = 60 s. Reprinted from W. Piqtkowski, D. Antos, F. Gritti, G. Guiochon, ]. Chromatogr., 1003 (2003) 73 (Fig. 7).
the single components were calculated using the equilibrium-dispersive model. These profiles were found to match satisfactorily the experimental band profiles. The experimental band profiles of large samples of binary mixtures agreed reasonably well with those calculated with the ED model, showing the competitive Langmuir isotherm accounts well for the binary adsorption data in this case. This agreement between calculated and experimental band profiles was significantly improved when the calculations were made using the transport-dispersive model (see Figure 16.13), in an effort to take into account the mass transfer kinetics. The mass transfer rate coefficients, kp for both pure components were determined by parameter adjustment. These coefficients were used to predict the band profiles of mixtures of the two enantiomers to fairly good agreement [25]. Piatkowski et al. measured the single-component and the competitive equilibrium isotherms of phenetole (ethoxy-benzene) and n-propyl benzoate on a 150 x 3.9 mm Symmetry -Cis (endcapped) column (Waters), using a methanol/water (65:35, v/v) as the mobile phase [26]. The adsorption equilibrium data of the single-component systems were acquired by frontal analysis. For both compounds,
16.2 Linear Driving Force Model Approach
753
0 0 0-
A 8 0 0 -
600
n
-
•
•••
4 0 0-
2 0 0 -
0 -
100
•••
fl L j\
"•.[.?•
200
•••
•
300
400
500
600
t [S]
Figure 16.15 Comparison of the experimental (solid symbols) overloaded elution band profiles of mixtures of phenetole and propyl benzoate on a Symmetry-C18 150 x 3.9 mm column eluted with MeOH/H 2 O, 65/35 v/v and the profiles calculated with the IAS theory of competitive adsorption and the transport-dispersive model (solid lines). The open symbols are the single component profiles of each component in (a) and of component 2 in (b). (a) 1:2 mixture, Q = 3.4 g/1 and C2 = 7.28 g/1, tinj = 30 s. (b) 1:1 mixture Q = C2 = 12.08 g/L, t{nj = 120 s. The inset shows the calculated profiles of the two individual bands in this binary mixture. Reproduced with permission from W. Piatkowski, D. Antos, E Gritti, and G. Guiochon, J. Chromatogr. 1003 (2003) 73 (Figures 6 and 8).
these single-component isotherm data fit best to the multilayer BET model (see Chapter 3, Section 3.2.3.4). The competitive equilibrium data for the binary mixture were acquired by the perturbation method. These competitive adsorption data were modeled by applying the IAS theory (see Chapter 4). The numerical values of the coefficients were obtained by fitting the retention times of the perturbation pulses to those calculated using the IAS theory compiled with the coherence conditions. The experimental overloaded band profiles of either compound are in excellent agreement with the profiles calculated with the general rate model [26]. Finally, the elution profiles of binary mixtures were recorded. The transport-dispersive and the general rate models were used to calculate the band profiles of binary mixtures and of single-components, respectively. Figures 16.14 and 16.15 show that the elution profiles of the binary mixtures compare very well with those calculated using the transport dispersive model. The bands exhibit two unusual features when they interfere. First, we observe an unusual retainment effect of the chromatographic band of the more retained component by the less retained one, consistent with the shape of the equilibrium isotherms. The characteristic displacement effect that takes place in the case of a binary mixture exhibiting Langmuirian isotherm behavior does not take place here. The component that is first eluted exhibits a self-sharpening rear front. Its presence ahead of the band of the second component inhibits or rather delays the migration of this second band. This effect is similar to the well-known displacement effect but acts in the opposite direction. It has rarely been observed. The progressive evolution of this phenomenon with
754
Kinetic Models and Multicomponent Problems
increasing loading factor can be observed in Figures 16.15 left, 16.14 and 16.15 right. The position of the first eluted band remains unchanged, with a retention time similar to the band of the single solute. The second remarkable feature of these bands is the formation of two concentration steps in the first component band, the first one having a constant height, once it is formed. Slight deviations of the calculated profiles from the experimental data were explained by the complexity of the multi-layer adsorption process for the mixture, compounded by the difficulties of the comparison of the experimental profiles using the UV responses for these complex mixtures.
16.3 Numerical Solution of The General Rate Model of Chromatography We have listed the equations of the general rate model (GRM) in Chapter 2 and discussed this model in detail in Chapter 6, in the context of linear chromatography. It uses two different mass balance equations for each of the mixture components: one written for the mobile phase percolating through the chromatographic bed, the other one for the stagnant fluid contained inside the pores of the particles. Among the phenomena controlling the band profiles, the GRM includes axial dispersion (i.e., axial and eddy diffusion), external mass transfer, intraparticle diffusion, and the kinetics of adsorption-desorption. In principle, the GRM is the most complete model of chromatography. As a matter of fact, more than a model, the GRM is a family of models: additional mass transfer steps, valid under particular conditions, may be added at will, provided a suitable equation is available to account for them. The VERSE model (see later) is probably the most comprehensive version of the GRM. The GRM is also the most realistic model, since it takes into account all the phenomena that may have any influence on the band profiles. It may turn out to be too complicated in many cases and its use requires the independent determination of too many parameters. This explains why it is not as popular as the equilibrium-dispersive or the lumped kinetic models. The major difference between the various GRM models is due to the mechanism of intraparticle diffusion that they propose, namely pore diffusion, surface diffusion or a combination of both, independent or competitive diffusion. The pore diffusion model assumes that the solute diffuses into the pore of the adsorbent mainly or only in the free mobile phase that impregnates the pores of the particles. The surface diffusion model considers that the intraparticle resistance that slows the mass transfer into and out of the pores proceeds mainly through surface diffusion. In the GRM, diffusion within the mobile phase filling the pores is usually assumed to control intraparticle diffusion (pore diffusion model or PDM). This kind of model often fits the experimental data quite well, so it can be used for the calculation of the effective diffusivity. If this model fails to fit the data satisfactorily, other transport formulations such as the Homogeneous Surface Diffusion Model (HSDM) [27] or a model that allows for simultaneous pore and surface diffusion may be more successful [28,29]. However, how accurately any transport model can reflect the actual physical events that take place within the porous
16.3 Numerical Solution of The General Rate Model of Chromatography
755
particles remains open to question. However, we still have to make the following assumptions. First, we assume that the column packing is entirely homogeneous; the permeability of the packing is the same everywhere, along and across the column; the distribution of the mobile phase velocity is homogeneous; radial concentration gradients are negligible; and a linear, unidimensional model of the column is satisfactory. Second/ convection and axial dispersion are the only mechanisms of mass transfer in the axial direction. Third, the physical properties of the components are constant and independent of the concentrations. Note that, as explained in Chapter 5, the physical properties of the solutes, such as their diffusivities, are not constant but are concentration dependent and are often affected by the presence and the concentrations of other solutes, as well as by the fraction of the surface that is occupied by the solutes in the cases in which surface diffusion is a prominent contribution of the mechanism of intraparticle diffusion. Fourth, the column is isothermal. Compared to our discussion of the general rate model in Chapter 6 for linear chromatography, the only important changes are the need to use nonlinear competitive isotherms and to consider the possible dependence of diffusion and the mass transfer rate coefficients on the local concentrations of the feed components. These changes increase dramatically the mathematical complexity of the problem and only numerical solutions are possible.
16.3.1 Formulation of the General Rate Model with the Pore Diffusion Model The classical formulation of the GRM includes a mass balance equation in each of the two fractions of the mobile phase, axial dispersion, intraparticle diffusion, and the kinetics of adsorption-desorption. In dimensionless form, the following set of equations is written for each component of the system, as follows. 16.3.1.1 Mass Balance in the Bulk Mobile Phase This mass balance is written
In this equation, we define the following reduced parameters and concentrations: x
=
j
(16.33a)
T
=
—
(16.33b)
xp
=
—^
(16.33c)
cbii
=
^
(16.33d)
Q•0,i
Kinetic Models and Multicomponent Problems
756
C
f,i =
Peb,i = £-
(16.33g)
3(1-€h)Lkfi (16.33h) — —'— ebRpu where Q , , Cp/l- and C ^ ( T ) are the concentrations of component z in the bulk mobile phase, in the pore volume, and in the feed, respectively, Qy is the maximum concentration of i in the feed, eb is the volume fraction occupied by the bulk mobile phase (i.e., external porosity), Rp is the particle radius, Dj,^ is the axial dispersion coefficient, and kfj is the external film mass transfer coefficient. The initial condition is Nfi
=
(16.34)
T= 0
In most cases, Cjy(0, x) = 0. The boundary conditions depend on the nature of the experiment performed. We always have as a first boundary condition =
PebJ(cbj-cfJ)
(16.35a)
= 0
x=
(16.35b)
There is a second condition which depends on the mode of chromatography used: • For frontal analysis x=0
cfii(x) = 1
(16.36a)
• For elution
x =0
if
= 0
T
if
(16.36b) (16.36c)
For displacement
For the feed components: x=0
Cfri(r)
= 1
if
T
Cfit(T)
= 0
if
T>T p
(16.37b)
< Tp T>Tp
(16.37c) (16.37d)
< Tp
(16.37a)
and, for the displacer x=0
cd(r) = 0 cd(r) = 1
if if
T
16.3 Numerical Solution of The General Rate Model of Chromatography
757
16.3.1.2 Diffusion in the Pores This mass balance is written = 0
(16.38)
In this equation, we have:
cpi
=
evvDnv ,-L '
(16.39a)
-^
(16.39b)
where ep is the particle porosity and Dpi is the effective diffusion coefficient in the pores. Equation 16.38 is the same in linear and nonlinear chromatography, the only difference being in the relationship between the concentrations cs i and cP/,-. In the former case, it is linear, in the latter it is given by the competitive isotherm model, for example, the Langmuir model cp,i =
1 + E
/^'
C o
,
^
c
(16.40)
with a, =fe;C™,where C?° is the saturation capacity of the adsorbent. The initial condition for the integration of Eq. 16.38 is T
= 0
cPii = cpj{Q,x)
(16.41)
and the boundary conditions are r=0
dcvi -I±
=
0
(16.42a)
r=\
dcpi - ^ dr
=
Bii{cbi-cpir=1) ' '
(16.42b)
with Bii = ^ e
-
(16.42c)
Pup,i
This set of equations constitutes the general rate model of chromatography.
16.3.2 Solutions of the General Rate Model with Pore Diffusion Model Many authors have described procedures for the calculation of numerical solutions of the general rate model of chromatography with a variety of initial and boundary conditions corresponding to practically all the modes of chromatography (with the notable exception of system peaks). Orthogonal collocation on finite elements seems to be the most popular approach for these calculations.
758
Kinetic Models and Multicomponent Problems
Figure 16.16 Comparison between an experimental band profile for BSA and the profile calculated with the general rate model. Experimental elution profile obtained for BSA on DEAE sepharose ion exchanger, with an NaCl solution at pH 8 and Fv = 0.84 mL/min (symbols). Profile calculated with a finite element method (solid line). Reproduced with permission from Yu and Wang, Comput. Chem. Eng., 13 (1989) 915 (Fig. 17).
10.0
SO 0
TIME
30.0
CmirO
Figure 16.17 Simultaneous breakthrough profiles of 2-butanol and tert-amyl alcohol on a carbon column. Comparison of experimental (symbols) and calculated (lines) profiles, (a) Column length, 82 cm; CQ^ = CQ^ = 1 mg/mL. Calculated profiles with solid lines, kfZ = 1.95 x 10~3 cm/s; dashed lines, kf2 = 1.68 x 10" 3 cm/s. (b) Column length; 41 cm, Q),l = Q),2 = 1 m g / m L - Calculated profiles with solid lines, D p l = 7.40 x 10" 6 cm 2 /s; dotted lines, D p l = 7.77 x 10" 6 cm 2 /s; dashed lines, D p l = 7.03 x 10"6 cm 2 /s. Reproduced with permission from A.I. Liapis and D.W.T. Rippin, Chem. Eng. Set, 33 (1978) 593 (Figs. 2 and 5).
The major difficulty encountered in attempts at comparing experimental results with the predictions of the general rate model of chromatography comes from the large number of parameters involved and the need for independent methods of measurement or evaluation. In most cases, there are no satisfactory methods for their experimental determination. They are often evaluated from empirical expressions that have a limited accuracy and have often been developed in areas and with aims foreign to chromatography. Furthermore, because of the large number of these parameters and the ambiguities discussed in the previous chapters, it is illusory to try and fit experimental band profiles to the general rate model of chromatography. As shown by Golshan-Shirazi and Guiochon [30], curve fitting does not permit the simple selection of the best lumped kinetic model. Figure 16.16 compares the calculated and the experimental band profiles calculated by Yu and Wang [31] for the overloaded elution of bovine serum albumin on a DEAE sepharose ion-exchanger, by a saline solution at pH = 8. The band profiles were calculated using the GRM model. In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and
16.3 Numerical Solution of The General Rate Model of Chromatography
759
dispersion. The Fritz-Schlunder isotherms [34] were used with orthogonal collocation method. The film transfer coefficients were taken from the literature and the pore diffusivities were derived from independent batch measurements. As shown in Figure 16.17, the agreement between the calculated profiles and the experimental results of Balzli [35] (Figure 16.17) is excellent. Still better agreement is observed when small changes are made to the values of the film mass transfer coefficient, Kf2, and the effective diffusivity, D p l , while keeping them within the range of confidence of the correlation used for the estimation of these parameters [32]. Mansour [36] solved a similar model for a breakthrough curve using a finite difference method. Morbidelli et al. solved the same problem for a step injection (breakthrough curve) [37] and a pulse injection (elution) [38]. Yu and Wang [31] and Lee et al. [39] used orthogonal collocation on finite elements for the fluid phase equation and an orthogonal collocation method for the particle phase equation to calculate numerical solutions of the same model in the cases of a breakthrough curve and of overloaded elution. Many authors have successfully modeled breakthrough curves and this problem will not be elaborated here in more detail. Gu et al. [40] examined the displacement effect using the numerical solution of the general rate model of multicomponent chromatography. These authors included in their model axial dispersion, the external mass transfer kinetics, and intraparticle diffusion. As for the multicomponent isotherm, they assumed that the system follows competitive Langmuir isotherm behavior. They studied the importance of the displacement effect in the three major modes of chromatography, elution, displacement, and frontal analysis. They showed that the displacement effect is the most important of the sources of multicomponent interference or interactions between bands that are directly attributed to the competition for binding sites among the different components. This effect depends on the degree of competition among the feed components (i.e., on the separation factor between two successive components) and on the degree of the nonlinear behavior of the system (i.e., on the highest values offc,-Qachieved during the separation). 16.3.2.1 Numerical Solution of a Simplified General Rate Model, the Lumped Pore Diffusion Model (POR) Morbidelli et al. [41] discussed a numerical procedure for the calculation of numerical solutions of the GRM model in the case of an isothermal, fixed-bed chromatographic column with a multicomponent isotherm. These authors considered two different models for the inter- and intra-particle mass transfers. These models can either take into account the internal porosity of the particles or neglect it. They include the effects of axial dispersion, the inter- and intra-particle mass transfer resistances, and a variable linear mobile phase velocity. A generalized multicomponent isotherm, initially proposed by Fritz and Schliider [34] was also used: a C d
qi =
i i i0
i + EUWu
760
Kinetic Models and Multicomponent Problems
where d^ and d^ are adjustable numerical coefficients. In the particular case of d^o — d{tj — 1, this isotherm equation reduces to that of the competitive Langmuir isotherm. Morbidelli et al. [41] argued that the solution of the GRM requires a greater numerical effort than that of their model, because the GRM requires the numerical solution at each node in the calculation grid of a second-order differential equation with different boundary conditions instead of that of an algebraic equation. Note that this point was much stronger in 1982 than it is today, in view of the considerable increase in computing power that has taken place since 1982. To simplify the calculations, a lumping procedure was applied to Eq. 16.38, as follows. The equation is integrated inside a particle, giving ep—f1 = fc/,*-J-[Q - Cp,i{Rp)} - -^p/0s(l - ep)
(16.44)
where Cpj and Cp are the volume-averaged values of the concentrations in the solution impregnating the pores and in the adsorbed phase, respectively. The first term in the RHS of Eq. 16.44 is derived from the Laplacian term in Eq. 16.38 by using the boundary condition in Eq. 16.42b. Using the linear driving force approximation, this term can be approximated in terms of the average composition, i - Cp,i(Rp)] = ku(Ci - C~)
(16.45)
where fc/ ,• is the global mass transfer rate coefficient, which can be evaluated in terms of the external mass transfer rate coefficient, kj,- and the internal mass transfer rate coefficient, ks/. T- = T~ + T^— K
l,i
K
f,i
K
( 16 - 46 )
£
s,i p
The internal mass transfer rate coefficient, fcS/l- is difficult to evaluate. It can be approximated as the time-averaged value proposed by Glueckauf [42]. This approximation model (POR) is less accurate than the general rate model but it is more accurate than kinetic models using the linear driving force equations. The numerical procedure developed for the calculation of numerical solutions of this model derives from a method previously applied by Lax and Wendroff for conservation-type equations [43]. Morbidelli et al. [41] used numerical solutions of the POR model to study the dynamics of fixed-bed absorbers, in response to a step (frontal analysis) or a pulse (elution chromatography) change of the feed composition. The influence of the operating conditions and of the relevant physico-chemical parameters on the efficiency of separation processes on fixed-bed absorbers were investigated through the systematic use of numerical solutions. Zhou et al. [44] used this model to compare the experimental band profiles of mixtures of the enantiomers of 1-indanol on a cellulose derivative and the profiles calculated using the POR model. The single component data fitted very closely to the Bilangmuir model. A competitive Bilangmuir model was derived from these
16.3 Numerical Solution of The General Rate Model of Chromatography
761
5
•
12
•
8
Exp Cal
Exp Cal
Exp Cal
4
7 '
•
10
I »• 4 I 4.
\ 6
V
A
0
5
|
1 3-
\ 4
2
Concentration (g/L)
3
Concentration (g/L)
Concentration (g/L)
6 8
2
1^
1 0
3
2
1
0
-1 -2
(E)
0
2
4
6
8
10
12
Retention time (min)
14
16
18
20
-2
(F)
0
2
4
6
8
10
12
Retention time (min)
14
16
18
-2
20
(G)
0
2
4
6
8
10
12
14
16
18
20
Retentton time (min) Retention
Figure 16.18 Comparison of calculated (solid lines) and experimental (symbols) profiles for different R- and S-1-indanol mixtures. E 1:1 mixture, Lf = 15%. F 3:1 mixture, Lf = 8.9%. G 1:3 mixture, Lf = 8.9%. Reproduced with permission from D. Zhou, D. E. Chermk, A. Cavazzini, K. Kaczmarski, G. Guiochon, Chem. Eng. Sci., 58 (2003) 3257 (Figs. WE, F, and G).
single component isotherms. It accounted well for the binary adsorption data. Good agreement between the two sets of profiles, experimental and calculated, was observed for a variety of relative composition of the feed as shown in Figure 16.18. Hong et al. [45] used numerical solutions of the POR model, under linear conditions, to determine the internal diffusion coefficient of rubrene in the particles of Symmetry-Cis, with methanol/water solutions (90 to 100% methanol) as the mobile phase. The results derived from the analytical solution of the model in the Laplace domain and from the first and second order moments were in excellent agreement. Zhou et al. measured the competitive adsorption isotherms of the closely related peptides bradykinin and kallidin on a Zorbax SB-Cis microbore column from an aqueous solution of ACN by FA [46]. These isotherms fitted best to the competitive bi-Langmuir isotherm model. These authors compared the experimental overloaded elution band profiles of various mixtures of these two peptides with the profiles calculated from these isotherms, using either the equilibriumdispersive (ED) or the pore diffusion (POR) model. The predicted band profiles obtained with the two models are similar (see Figure 16.19) and relatively close to the experimental profile. Note that the POR model requires more numerical coefficients than the ED model. The overall rate of mass transfer is related to the external and the internal coefficients of mass transfer which both depend on the molecular diffusivity. The correlation of Young, Carroad and Bell [47] (see Chapter 5) gave values of 2.33 and 2.42 x 10~6 cm 2 /s for bradykinin and kallidin, respectively. The average of value of 2.38 x 10~6 cm 2 /s was used in the calculation of the overall mass transfer coefficient. The difference between the experimental and the calculated band profiles are probably due to errors made in the estimation of the kinetic parameters. On the other hand, there is only one parameter in the ED model, the apparent axial dispersion coefficient (D^ = UQL/(2N)), which is derived directly from the experimental estimate of the column efficiency at infinite dilution. As observed in numerous occasions, although the General Rate model, the POR model, and some other kinetic models are more accurate than
762
Kinetic Models and Multicomponent Problems Exp Cal_ED Cal_POR
0.12-.
0.10-
0.08-
^
0.06-
0.04-
0.02-
0.00 10
12
14
16
t [min] Figure 16.19 Comparison of the experimental (symbols) overloaded elutionband profiles of a mixture of the closely related peptides bradykinin and kallidin on a Zorbax SB-C18 150 x 0.5 mm column eluted with ACN/H 2 O, 20/80 v/v (+ 0.5% TFA), for a loading factor Lf = 0.29% and the profiles calculated with the ED (solid lines) and the FOR models (dashed lines), (a) Parameters of the POR model derived from conventional correlations, (b) Dm in the POR model optimized numerically at 1 x 10~6 cm 2 /s. Reproduced with permission from D. Zhou, X. Liu, K. Kaczmarski, A. Felinger, G. Guiochon, Biotech. Prog. 19 (2003) 945 (Figure 4.) ©2003, American Chemical Society.
the equilibrium-dispersive model and account better for chromatographic effects as a matter of principle, they require the use of several kinetic parameters that are often most difficult or impossible to measure independently and are not always easy to determine with a sufficient accuracy. Accordingly, the band profiles calculated with these sophisticated models do not necessarily fit better to the experimental profiles than those obtained with the equilibrium-dispersive model. So, in the case of bradykinin and kallidin [46], the agreement between the experimental profiles of the two bands and those calculated with the POR model was improved by using a molecular diffusivity of 1 x 1CT6 cm 2 /s instead of the average value afforded by the conventional correlation in order to calculate the overall mass transfer coefficient (see Figure 16.19). This suggests that the conventional correlations may slightly overestimate the diffusivity in the pores. The POR and GR rated models were used to predict experimental band profiles from the experimental adsorption data of (R) and (S) 1-phenyl-l-propanol on
16.3 Numerical Solution of The General Rate Model of Chromatography
763
a Q.35-
i
0.300.25-
i \ A I\ J V \.
0.200.150.10-
•
0.050.00-
t [min]
t [min]
Figure 16.20 Comparison of the experimental (symbols) overloaded elution band profiles of racemic mixtures of (R) and (S) 1-phenyl-l-propanol on a Chiracel OB 200 x 10 mm column eluted with n-hexane/2-propanol, 97/3 v / v and the profiles calculated for a loading factor Lf = 0.29% and the profiles calculated with the POR model (solid line). (Top left) Lf = 0.37%, Dm = 0.0017 cnvVmin; (Top right) Lf = 0.37, DmA = Dra,2 = 1.0 x 10"4 cm 2 /min; (Bottom left) Lf = 0.89%, DeffA = 0.9 x 10" 4 cm 2 /min, Deff/2 = 1.0 x 10" 4 cm 2 /min; (Bottom right) Lf = 3.75%, Deffl = 0.77 x 10" 4 cm 2 /min, Deff;2 = 1.5 x 10" 4 cm 2 /s. Reproduced with permission from A. Cavazzini, K. Kaczmarski, P. Szabelski, D. Zhou, Z. Liu, G. Guiochon, Anal. Chem. 73 (2001) 5704 (Figure 3a, 3b, 4a, 5b). ©2001, American Chemical Society.
a Chiracel OB, acquired by FA [48]. Calculations of band profiles using the GR or the POR model require the advanced knowledge of several important parameters, such as the molecular diffusivity, Dm, the effective diffusion coefficient, Deff, the external mass transfer coefficient, kextr the external, the internal and the total porosity of the column, and the axial dispersion coefficient, DL. Most of these parameters are very difficult to estimate. Figure 16.20a compares the experimental (symbols) and the calculated (solid line) elution profiles for the racemic mixture of 1-phenyl-l-propanol. For this calculation, the molecular diffusivity, Dm = 0.0017 cm 2 /min, was calculated using the Wilke-Chang equation. It was also used to calculate Deff and kext- The agreement between the experimental and the calculated profiles is poor with these values of Dm, Deff and kext. The agreement between the
764
Kinetic Models and Multicomponent Problems
experimental and the calculated profiles is considerably improved when a value of Dm of the order of 1 x 10~4 cm 2 /min is used instead (see Figure 16.20b). Figures 16.20c and 16.20d show similar profiles obtained with a higher loading factor. In order to obtain a good agreement between the experimental and the calculated profiles in these figures/ different values of D<>// had to be used for the S- and the R-enantiomers. The differences between the profiles calculated with the two models, the GM and POR models, are too small to be seen and, thus, the GR and the POR models are interchangeable at St/Bi > 5. In the case illustrated in Figure 16.20, the St/Bi ratio is of the order of 100. The results of this work can be used to observe the relative intensity of the external and internal mass transfer resistances. The value of was equal to kext = 3.8 cm/min based on a Dm of 0.0017 cm2 /min. This value is a bulk property and can not be applied to the stagnant solution inside the pore to measure the internal mass transfer resistance. Assuming a Dm = 0.0001 cm 2 /min an internal mass transfer coefficient, k{nt = 0.076 cm/min is obtained. Thus the ratio of the external and internal mass transfer is of the order 50 meaning the external resistance can be neglected compared to the internal one. 16.3.2.2 The GRM Formulated with the Homogenous Surface Diffusion Model (HSDM) In the HSDM model, pore diffusion is neglected and it is assumed that surface diffusion is the dominant mechanism of intraparticle mass transfer. The unsteadystate diffusion process within the particle is taken into account by a diffusion process model assuming a constant matrix of Fick diffusivity. The variation of qj with the distance along the column and the time is governed by the diffusion equation [49-51]
Using a dynamic model comprised of the mass balance equations of all the solutes in the liquid phase percolating through the bed, employing the HSDM model for intraparticle mass transfer and the IAS theory for the multicomponent adsorption isotherms, and assuming that the surface diffusion coefficient, Ds, is constant, Lee et al. [52] successfully calculated numerous breakthrough curves. All these curves fit well to the experimental breakthrough curves of single components, and of binary and ternary mixtures of various mono-saccharides (fructose, galactose, and glucose), di-saccharides (sucrose, maltose, and lactose) and malto-oligosaccharides (malto-triose, malto-tetraose, and malto-pentaose) dissolved in aqueous solutions, on activated carbon as the stationary phase. Examples are shown in Figures 16.21a and 16.21b. As we discussed in Chapter 5, the surface diffusivity depends on the adsorbed phase concentration, therefore using the HSDM with a constant surface diffusivity is an approximation. Yet, the agreement observed is quite satisfactory. Many macromolecular compounds exhibit very complex mass transport mechanisms. It was suggested that this phenomenon originates from the contribution
16.3 Numerical Solution of The General Rate Model of Chromatography
765
1.5
1.5
1
Ci /Ci0
Ci/Ci0
1
0.5
0.5
Sucrose
Glucose
Maltose
Maltose
Lactose
Predicted
0
0
10
20
Time, hr
30
Predicted
40
0
0
10
20 Time, hr
30
40
Figure 16.21 Left: Binary breakthrough curve of glucose and maltose in water on a 1 x 15 cm column packed with activated charcoal. Left: Ternary breakthrough curve of sucrose, maltose, and lactose on the same column. Reproduced with permission from J.-W. Lee, T.-O. Kwon, I.-S. Moon, Carbon 42 (2004) 371 (Figs. 8 and 9).
of the concentration dependence of surface diffusion in the overall mass transport mechanism [27]. The dependence of intraparticle mass transfer on the concentration can be explained by assuming that surface diffusion plays a major role in their overall mass transfer kinetics. An alternative method to take surface diffusion into account consists in lumping pore diffusion and concentration-dependent surface diffusion together, thus creating an apparent effective diffusion coefficient, Deff, which is concentration dependent. This approach was used by Ma et al [53], by Piatkowski et al. [28] and by Zhou et al. [10]. This method is also an approximation, but it is still an improvement over the simpler HSDM model.
16.3.3 The GRM Formulated with the Maxwell-Stefan Surface Diffusion Model In order to account correctly for surface diffusion in the GRM, we should consider the concentration-dependent generalized Maxwell-Stefan (GMS) matrix diffusivities (see Chapter 5). Originally, the GMS approach was used to investigate gas adsorption on zeolites [54,55]. The same approach, however, can be used to describe surface diffusion in the packing materials used in HPLC, especially for the separation of macromolecules. Van Den Broeke and Krishna [56] compared the calculated and the experimental breakthrough curves of single components and of mixtures containing methane, carbon dioxide, propane, and propene on microporous activated carbon and on carbon molecular sieves. They ignored the external mass transfer kinetics and assumed that there is local equilibrium for each component between the pore surface and the stagnant fluid phase in the macropores. They also assumed that the surface-diffusion contribution is much larger than that of pore diffusion and they neglected pore diffusion. They used in their calculations three different
766
Kinetic Models and Multicomponent Problems
models to describe intraparticle diffusion. 1. The linear driving force (LDF) model with a constant diffusivities
f D, is assumed to be equal to the Maxwell-Stefan single component micropore diffusivity at zero coverage, D; = £>, (0). 2. The homogenous surface diffusion model (HSDM)
In this case, a diffusion process with a constant matrix of Fick diffusivities describes the unsteady-state diffusion process within the particles. The matrix of diffusivities is assumed to be diagonal and the diagonal elements to be equal to the Maxwell-Stefan diffusivities at zero coverage, D, = £>;(0). 3. The solid-diffusion model with a concentration-dependent Fick matrix of diffusivities:
dt
r dr
In this case, the matrix of Fick diffusivities depends on the surface coverage and is given by the single file Maxwell-Stefan relationship. [D] = [B]~1[r]
(16.51)
For single file diffusion mechanism, it is assumed that no counter-exchange is possible between the adsorbed solutes i and /. For single file diffusion mechanism the Fick surface diffusivity matrix is given by: Pi 0
0 P2
0 0
0 0
0 0
0 0
"-. 0
0 Vn
Figure 16.22 shows the breakthrough curves obtained as results of a co-sorption experiment made with a mixture of methane and carbon dioxide, using helium as an inert or carrier gas on a column packed with a microporous carbon (Kureha MAC). In this figure, the three models used for the calculation of the breakthrough curves are compared with the experimental results. From Figure 16.22, we conclude that the single file Maxwell-Stefan single diffusion model is definitely superior to the other two models.
16.3 Numerical Solution of The General Rate Model of Chromatography iCH4:CO2:He = 1:1:2 | Kureha MAC lr=345K = 200kPa Up = 0.01 m/s
1.5
767
! ! ! j j
Maxwell-Stefan SFD model LDF constant [D] model - Constant [D\ model
100
Figure 16.22 Breakthrough curve normalized to the inlet concentration of CH4 and C)2 carried by helium through a 1 x 30 cm column packed with a microporous carbon (Kureha MAC). Dimensionless time, T = tUg/L. Symbols show experimental results, the lines the calculated profiles with the models indicated. The variation of the gas velocity along the column was accounted for. Reproduced with permission from L. ]. P. van den Broeke, R. Krishna, Chan. Eng. Sci, 50 (1995) 2507 (Fig 12).
Kaczmarski et al. used a similar model for the calculation of the band profiles of the enantiomers of 1-indanol on a chiral phase in HPLC [29,57]. These authors ignored the external mass transfer and assumed that local equilibrium takes place for each component between the pore surface and the stagnant fluid phase in the macropores (infinite fast kinetics of adsorption-desorption). They also assumed that surface diffusion contribution is much faster than pore diffusion and neglected pore diffusion entirely. Instead of the single file Maxwell-Stefan diffusion, these authors used the generalized Maxwell-Stefan diffusion (see Chapter 5).The calculation (see below) requires first the selection of equations to calculate the surface molecular flux [29,57,58]. The surface diffusion flux is given by the following two equations [57]:
hi
dCn
= -(
dr 'C P/ i
Ds
dCn
dr
(16.52)
(16.53) dr These equations can be derived by assuming that the surface chemical potential gradient is the driving force of surface diffusion, see Krishna [55] and the discussion presented by Karger and Ruthven [59]. The coefficients Ds y in these equations are the Fickian diffusivities of the generalized Maxwell-Stefan equations (GMS). According to the formulationof the Maxwell-Stefan model, these coefficients are given by the following set of equations:
hi
= "(
D,s,22~
D, D,
Dlr2 C,
p2
0iD 2 D! i D 2 + 922D D1
(16.54) (16.55)
Kinetic Models and Multicomponent Problems
768
t [ min ]
Figure 16.23 Comparison of calculated (solid lines) and experimental (symbols) profiles for different mixtures of (S)- and (R)-l-indanol on a column packed with cellulose tribenzoate. Left to right and top to bottom, relative composition: 1:1, 1:1, 1:3, 3:1; Lj: 4.5, 8.8, 5.8, 5.9%, respectively. Reproduced with permission from K. Kaczmarski, M. Gubernaz, D. Zhou, G. Guiochon, Chem. Eng. ScL, 58 (2003) 2325 (Figs, lib, d,f, andg).
(16.56)
Ds,2,2
=
e2Dx + D 1 ; 2 0i D 2
D1/2
(16.57)
The counter-exchange diffusion coefficient, Dj 2 , was derived from the equation suggested by Krishna [60] based on the generalization of Vignes [61] relationship for diffusion in bulk liquid mixtures: Di 2 = (Di ^i/^i+^c^u^/CSi+fe)
(16 58)
The GRM using the generalized Maxwell-Stefan equations has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on finite elements [29,62,63]. The set of discretized ordinary differential equations was solved with the Adams-Moulton method, implemented in the VODE procedure [64]. The relative and absolute errors of the numerical calculations were 1 x 10~6 and 1 x 10~s, respectively.
16.3 Numerical Solution of The General Rate Model of Chromatography
769
t [min]
Figure 16.24 Comparison of the experimental (symbols) overloaded elution band profiles of the racemic mixture of the R and S enantiomers of 1-phenyl-l-propanol on a Chiracel OB 200 x 10 mm column eluted with n-hexane/2-propanol, 97/3 v/v and the profiles calculated with the GR-GMS model (solid lines). (Left) Sample volume 2 ml, loading factor Lf = 1.87%. (Right) Sample volume, 4 ml, loading factor L^ = 3.75%. Reproduced with permission from K. Kaczmarski, A. Cavazzini, P. Szabelski, D. Zhou, X. Liu, G. Guiochon,}. Chrom. 962 (2002) 57 (Figures 4 and 6).
Figures 16.23a to d compare experimental profiles of mixtures of the enantiomers of 1-indanol on cellulose tribenzoate with those calculated with the GMSGRM model of these authors [57]. For the numerical calculations, they assumed that surface diffusion plays the dominant role in mass transfer across the particles and neglected the contribution of pore diffusion to the fluxes. Unfortunately, it was impossible independently to measure or even estimate the surface diffusion parameters. So, the numerical values of the surface diffusion coefficients needed for the calculation were estimated by minimizing the discrepancies between the measured and the calculated band profiles (i.e., by parameter adjustment). Yet, it is impressive that, using a unique set of diffusion coefficients, it was possible to calculate band profiles of single components of binary mixtures in the whole range of relative composition, for loading factors between 0 and 10%. The competitive equilibrium isotherm model best fitting the FA experimental data for the R and S enantiomers of 1-phenyl-l-propanol on cellulose tribenzoate was the Toth model. This model was used to calculate the elution profiles of samples of mixtures of the two enantiomers [29]. The General Rate model combined with the Generalized Maxwell-Stefan equation (GR-GMS) was used to model and describe surface diffusion (see Chapter 5). The mass transfer kinetics is slow and this model fits the experimental data well over a wide concentration range with one single set of numerical parameters to account for the band profiles in a wide range of concentrations, as shown in Figure 16.24.
16.3.4 The VERSE Model Recently, Wang and associates [65] have modified the general rate model to include the possibility that chemical reactions take place in either the solution (mo-
770
Kinetic Models and Multicomponent Problems
bile phase), or the stationary phase, or in both. They called this new model the VErsatile Reaction-SEparation model (VERSE). They described a numerical method for the calculation of its solutions for step input, elution, and displacement. In the VERSE model, modules are added as needed to a general rate model to allow for the generation or the consumption of each species, based on sets of kinetic expressions selected by the user. This design permits a careful adaptation of the model used to the problem studied. In the development of the VERSE model, Whitley et ol. included provisions to calculate not only the behavior of any chromatographic separation system but also that of a chromatographic system coupled with a reaction. They considered a detailed mass transfer kinetics and a complex reaction mechanism. The VERSE model includes the following equations, reported in dimensionless form [65]. 16.3.4.1 Mass Balance in the Bulk Mobile Phase This balance is written
This equation is very similar to Eq. 16.32. The only difference is the new generation term, v\,, which accounts for the generation or destruction of component i due to possible chemical reactions in the mobile phase, including dimerization, or aggregation. The initial condition for the integration of Eq. 16.59 is T= 0
cb4 = cb4(0fx)
(16.60)
The boundary conditions are x=0 x =
l
^
=
Peb,i(cb,i-Cf,i)
(16.61a)
^M
=
o
(16.61b)
and • for frontal analysis x =0
(16.61c)
CfAr) = 1
• for elution
= 1
x =0
c/,i(r)
x =0
c/Ar) =
0
if
T
(16.61d)
if
T >Tp
(16.61e)
• for displacement x =0
cf,i{r) cf,i(r)
= =
1 0
if
T < Tp
(16.61f)
if
T > Tp
(16.61g)
16.3 Numerical Solution of The General Rate Model of Chromatography
771
and, for the displacer cd(r)
=
0
if
T
cd{x)
=
1
if
T>TP
(16.61h)
16.3.4.2 Mass Balance in the Pores This mass balance equation is written
3cP.i ,„ xr ^ - V,j + (1 - ^ I
n a - * [ ? S ( ^ j j =0
(16-62)
This equation is similar to Eq. 16.38. The differences are the introduction of the size exclusion factor, Ke, and the dimensionless generation term, vp, which accounts for the possible changes in the amount of compound i. The term (1 — ep)viCT/Qy accounts for adsorption-desorption. If the kinetics of adsorptiondesorption is infinitely fast and the concentrations in the stationary phase and in the mobile phase inside the pores are in equilibrium, this term is equivalent to the term (1 — ep)dcs ; /3f found in the equation of the general rate model. If the kinetics of adsorption-desorption is slow, the appropriate kinetic model for these kinetics should be used to calculate V\. The initial condition for the integration of Eq. 16.62 is T= 0
cpJ = cP/i (0,r,x)
(16.63)
and the boundary conditions are r=0
dcvi —^
=
0
(16.64a)
r=l
- ^
=
Bii(cbi-cpir=1)
(16.64b)
16.3.4.3 Reaction Equations Aggregation of proteins is not an uncommon problem in the handling of their concentrated solutions. An aggregating system is more complex than a multicomponent system without reaction, due to the interconversion that takes place between the various stages of aggregation of the components considered. Each aggregate of a particular protein should be treated as an individual component, with the caution that aggregates can convert back into free individual proteins when conditions change, or can form in larger concentrations, or can form higher, more complex aggregates. Accordingly, each aggregate has its own isotherm. Each aggregate of a solute can compete with the other forms; the sorbent may also have a different saturation capacity and a different affinity for each of these forms. The aggregation kinetics may vary widely among protein systems. The kinetic expressions, however, are similar, regardless of the particular solute under consideration.
772
Kinetic Models and Multicomponent Problems
To allow for a wide variety of aggregation reactions, the VERSE model [65] uses the following set of basic reactions from which more complex sequences may be built if necessary. A 2A 3A
A +B A + 2B
# ^ ^ ^ #
P P P P P
(16.65a) (16.65b) (16.65c) (16.65d) (16.65e)
For a protein that interconverts between monomer and dimer forms. Eq. 16.65b is applicable. It can be written as: 2M ^ D
(16.66)
with k+m and fc_m being the rate constants of the forward and the reverse reactions. The kinetic expression for this reaction can be written as:
vbM
= -2[k+mCiM-k-mCbD\
(16.67a)
vhD
= \k+mClu-k^mCbD]
(16.67b)
where Q, ^ and Cj, o are the local concentrations of M and D. The equilibrium distribution is defined by:
The above reaction expressions are written for the reaction that takes place in the mobile phase; similar expressions must be written for the reactions that take place in the pore-phase by substituting vp,- for vjy and Cpj for Cjy. 16.3.4.4 Numerical Solution of the VERSE Model Only numerical solutions of the VERSE model can be obtained [65]. The partial differential equations are discretized by application of the method of orthogonal collocation on fixed finite elements. Equation 16.59 is divided into 50 or 60 elements, each with four interior collocation points. Legendre polynomials are used for each element. For Eq. 16.62, only one element is required. It is described by a Jacobi polynomial with two interior collocation points. The resulting set of ordinary differential equations, with their initial and boundary conditions and the chemical equations, are solved using a differential algebraic system solver (DASSL) [65,66]. 16.3.4.5 Verification of the VERSE Model Experimental data on the aggregation of myoglobin and /Hactoglobulin A previously obtained were used to verify the validity of the VERSE model. The first set
16.3 Numerical Solution of The General Rate Model of Chromatography
773
Figure 16.25 Experimental breakthrough curves of myoglobin on an immobilized metal affinity column (dotted line) and model calculations as a dimerizing system (solid line). Reproduced with permission from R. D. Whitley, K. E. van Cott, J. A. Berninger, N.-H. L. Wang, AIChE ]., 37 (1991) 555 (Fig. 2).
of experimental data used for this purpose was the experimental breakthrough curves of myoglobin on an immobilized metal affinity chromatography (IMAC) column. The experimental data are shown in Figure 16.25. The two-plateau breakthrough curve results from the injection of what appears to have been a monomerdimer mixture at equilibrium. The shape of this curve is similar to that of a twocomponent breakthrough curve, but as concentrations change, the relative heights of the two plateaus change, suggesting an aggregation reaction. For the calculations, it was assumed that a simple reaction of dimerization was taking place and the extinction coefficient of the dimer was assumed to be twice that of the monomer. The results of the calculations are also shown in Figure 16.25. They closely match the experimental results. The second set of experimental data used for verification of the VERSE model was the experimental data of j6-lactoglobulin A reported by Karger and Blanco [67]. The injection of pure j3-lactoglobulin A resulted in three peaks which are, in the order of their elution, a dodecamer, a tetramer, and an octamer. In the calculations, the following stoichiometric equations developed by Grinberg et al. [68] were used. 2P4 # Pi + P8 #
P8, P12,
Km = (2.4±0.5)x 3
Kn = (3.3 ± 0.8) x 10 M"
1
(16.69a) (16.69b)
The kinetic expressions for these reactions can be written as: 3CP4 dt
=
—2(fc + m C p 4 — k^
=
(k+mCP4:
(16.70a)
RXN
dCP8 RXN
— k-mCp$)
—
(16.70b)
Kinetic Models and Multicomponent Problems
774
Figure 16.26 Calculation results of the separation of a /3-lactoglobulin A system, based on the experimental results of Karger and Blanco [67]. CM = 2.717 x 1(T4 (a) and 1.359 x 1(T5 M. Reproduced with permission from R. D. Whitley, K. E. van Cott, J. A. Berninger, N.-H. L. Wang, AIChE }., 37 (1991) 555 (Fig. 3).
dt
tiwi.it1
— k-nCpi2)
(16.70c)
RXN
Figure 16.26 shows that the calculated results closely match the relative peak area ratio and the retention times reported by Karger and Blanco [67]. These authors had also shown that, at higher concentrations, the formation of the octamer and the dodecamer is favored over that of the tetramer and that their corresponding peaks are more pronounced. This behavior is well predicted by the VERSE model (see Figure 16.26) Using systematic numerical calculations, Whitley et al. [65] also investigated, the effects of the sample concentration, the equilibrium distribution, the reaction rate, the convection rate, the particle radius, and the relative affinity on the band profiles, for the elution, frontal analysis, and displacement modes of chromatography. They concluded that, when the reaction rates are slow compared to those of convection and mass transfer, the system behaves like a multi-component system. Frontal analysis breakthrough curves and displacement chromatograms exhibit multiple plateaus, and the elution bands exhibit peak splitting as in the elution of multicomponent samples. When the aggregation rates are high, the peaks are generally merged. The degree of band asymmetry and their spreading due to aggregation depend on the relative affinity differences. If aggregation is overlooked, serious errors may result in the values of the parameters estimated by curve fitting, using either frontal or pulse analysis. The dimensionless group principles developed in this work are useful in scaling-up separations and in predicting when peak or band splitting or merging will occur in reaction chromatography systems. 16.3.4.6
Extension of VERSE Model to Nonlinear Gradient Elution Chromatography
The VERSE method was extended to describe the consequences of protein denaturation on breakthrough curves in frontal analysis and on elution band profiles in nonlinear isocratic and gradient elution chromatography [69]. These authors assumed that a unimolecular and irreversible reaction taking place in the adsorbed phase accounts properly for the denaturation. The choice of this reaction was based on numerous literature observations of the irreversible adsorption of proteins on hydrophobic surfaces [70]. This model (VERSE-LC) was verified using
REFERENCES
775
two sets of literature data, for a-chymotrypsinogen and for papain, for the gradient elution of these proteins under denaturing conditions. The results of the calculations gave chromatograms that account closely for the experimental chromatograms obtained under different gradient conditions. Through systematic numerical calculations, Whitley et al. [69] showed that the elution order, the peak resolution, and the relative peak heights in gradient elution depend all greatly on the gradient conditions and on the rate of the denaturation reaction. In contrast with the case of nonreacting systems, reducing the gradient slope can actually reduce the resolution observed in denaturating systems. Using numerical solution, they showed that, in frontal analysis, denaturation results in multiple, unsymmetrical waves in the breakthrough curves, waves that can be mistakenly attributed to the presence of impurities. However, curves recorded at increasing flow rates permit the differentiation between the effects due to impurities and those due to denaturation. In isocratic elution, peaks of the native and the denatured forms cannot be fully separated because of the denaturation reaction.
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[29] K. Kaczmarski, A. Cavazzini, P. Szabelski, D. Zhou, X. Liu, G. Guiochon, J. Chromatogr. A 962 (2002) 57. [30] S. Golshan-Shirazi, G. Guiochon, J. Chromatogr. 603 (1992) 1. [31] Q. Yu, N.-H. L. Wang, Computers Chem. Eng. 13 (1989) 915. [32] A. I. Liapis, D. W. T. Rippin, Chem. Eng. Sri. 33 (1978) 593. [33] A. I. Liapis, R. J. Litchfield, Chem. Eng. Sci. 35 (1980) 2366. [34] W. Fritz, E. U. Schliider, Chem. Eng. Sci. 29 (1974) 1279. [35] M. Balzli, Int. rep. systems engineering group, Tech. rep., Technical Chemistry Laboratory, E.T.H., Zurich (1976). [36] A. R. Mansour, Separat. Sci. Technol. 24 (1989) 1047. [37] E. Santacesaria, M. Morbidelli, A. Servida, G. Storti, S. Carra, Ind. Eng. Chem.(Proc. Des. Dev.) 21 (1982) 446. [38] S. Carra, E. Santacesaria, M. Morbidelli, G. Storti, D. Gelosa, Ind. Eng. Chem. (Proc. Des. Dev.) 21 (1982) 451. [39] C. K. Lee, Q. Yu, S. U. Kim, N.-H. L. Wang, J. Chromatogr. 484 (1989) 29. [40] T. Gu, G. Tsai, G. Tsao, AIChE J. 36 (1990) 784. [41] M. Morbidelli, A. Servida, G. Storti, S. Carra, Ind. Eng. Chem. Fundam. 21 (1982) 123. [42] E. Glueckauf, Trans. Faraday Soc. 51 (1955) 1540. [43] R. Aris, N. R. Amundson, Mathematical Methods in Chemical Engineering, Vol. 2, Prentice Hall, Englewood Cliffs, NJ, 1973. [44] D. Zhou, D. E. Cherrak, A. Cavazzini, K. Kaczmarski, G. Guiochon, Chem. Eng. Sci. 58 (2003) 3257. [45] L. Hong, A. Felinger, K. Kaczmarski, G. Guiochon, Chem. Eng. Sci. 59 (2004) 3399. [46] D. Zhou, X. Liu, K. Kaczmarski, A. Felinger, G. Guiochon, Biotechnol. Prog. 19 (2003) 945. [47] M. E. Young, P. A. Carroad, R. L. Bell, Biotechnol. Bioeng. 22 (1980) 947. [48] A. Cavazzini, K. Kaczmarski, P. Szabelski, D. Zhou, X. Liu, G. Guiochon, Anal. Chem. 73 (2001) 5704. [49] G. McKay, Chem. Eng. J. 81 (2001) 213. [50] C. Chang, A. M. Lenhoff, J. Chromatogr. A 827 (1998) 281. [51] V. K. C. Lee, G. McKay, Chem. Eng. J. 98 (2004) 255. [52] H. J. Lee, Y. Xie, Y. M. Koo, Biotechnol. Progr. 20 (2004) 179. [53] Z. Ma, R. D. Whitley, N.-H. L. Wang, AIChE J. 42 (1996) 1244. [54] E. L. Cussler, Multicomponent Diffusion, Elsevier, Amsterdam, 1976. [55] R. Krishna, J. Wesselingh, Chem. Eng. Sci. 52 (1997) 861. [56] L. J. P. Van Den Broeke, R. Krishna, Chem. Eng. Sci. 50 (1995) 2507. [57] K. Kaczmarski, M. Gubernak, D. Zhou, G. Guiochon, Chem. Eng. Sci. 58 (2003) 2325. [58] F. Kapteijn, A. J. Moulijn, R. Krishna, Chem. Eng. Sci. 55 (2000) 2923. [59] J. Karger, D. M. Ruthven, Progress in Zeolite and Microporous Materials, Pts A-C 105 (1997) 1843. [60] R. Krishna, Chem. Eng. Sci. 45 (1990) 1779. [61] A. Vignes, Ind. Eng. Chem. (Fundam.) 5 (1966) 189. [62] K. Kaczmarski, M. Mazzotti, G. Storti, M. Morbidelli, Comput. Chem. Eng. 21 (1997) 641. [63] J. Berninger, R. D. Whitley, X. Zhang, N.-H. L. Wang, Comput. Chem. Eng. 15 (1991) 749. [64] P. N. Brown, A. C. Hindmarsh, G. D. Byrne, Variable coefficient ordinary differential equation solver, procedure available on http://www.netlib.org (1989). [65] R. D. Whitley, K. E. Van Cott, J. A. Berninger, N.-H. L. Wang, AIChE J. 37 (1991) 555. [66] L. R. Petzold, A description of DASSL: A differential algebraic equation system solver,
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Tech. Rep. STR, SAND 82-637, Livermore, CA (1982). B. L. Karger, R. Blanco, Talanta 36 (1989) 243. N. R. Grinberg, D. M. Blanco, D. M. Yarmush, B. L. Karger, Anal. Chem. 61 (1989) 514. R. D. Whitley, X. Zhang, N.-H. L. Wang, AIChE J. 40 (1994) 1067. J. D. Andrade, V. Hladly, Annal. New York Acad. Sci. 516 (1987) 158.
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Chapter 17 Simulated Moving Bed Chromatography Contents 17.1 Introduction 780 17.1.1 True Moving Bed Chromatography (TMB) 780 17.1.2 Simulated Moving Bed Process (SMB) 781 17.1.3 Applications of SMB Technology 781 17.2 Modeling of Simulated Moving Bed (SMB) Separations 783 17.3 Analytical Solution of the Linear, Ideal Model of SMB 785 17.3.1 ColumnModel 786 17.3.2 The Node Model 787 17.3.3 True Moving Bed Chromatography 788 17.3.4 Simulated Moving Bed Chromatography 789 17.3.5 Comparison of Ideal, Linear SMB and TMB Chromatography 802 17.3.6 Conclusion on the Solution of the Ideal Linear Model 803 17.3.7 The Zone Boundaries in Linear, Ideal SMB are True Shocks 804 17.4 Analytical Solution of the Linear, Nonideal Model of SMB 806 17A.I First Particular Case, Axial Dispersion is Negligible 807 17.4.2 Second Particular Case, Mass Transfer Resistances are Small (Equilibrium Limit) 808 17.5 McCabe-Thiele Analysis 808 17.6 Optimization of the SMB Process 809 17.6.1 Optimization of an SMB System with a Linear Isotherm, Using the Safety Margin Approach 809 17.6.2 Standing Wave Design of SMB for Linear Isotherm 810 17.6.3 Optimization of an SMB System with the Triangle Method 812 17.6.4 Optimization of the SMB Process with a Linear Isotherm Using the Triangle Method 814 17.6.5 Comparison of the Safety Margin and the Triangle Method for Linear Isotherms 815 17.7 Nonlinear, Ideal Model of SMB 816 17.7.1 Optimization of the Operating Conditions for a Nonlinear Isotherm Using the Triangle Method 817 17.8 Recent Improvements in SMB Performance with New Operating Modes . . 826 17.8.1 Supercritical Fluid Simulated Moving Bed (SF-SMB) 826 17.8.2 Temperature Gradient SMB in Simulated Moving Bed 827 17.8.3 Operation of a Simulated Moving Bed Unit under a Solvent Gradient 827 17.8.4 Improving SMB Performance by Operating Under Complex Dynamic Conditions 830 17.8.5 Multicomponent Separations in SMB 833 17.9 Numerical Solutions for Nonlinear, Nonideal SMB 836 17.9.1 Numerical Solutions of the SMB Model 837 17.9.2 Numerical Solutions of the TMB Model Equivalent to an SMB 837 17.9.3 Numerical Solutions of the Equivalent Mixing Cell TMB Model 844 17.9.4 Numerical Solutions of the Equivalent Mixing Cell SMB Model 845
References
845
779
780
Simulated Moving Bed Chromatography
17.1 Introduction There are two important, practical modes of carrying out industrial purifications by preparative chromatography. The most straightforward and the most frequently used of these is the traditional mode of operation of adsorption separation processes: a cyclic batch operation mode. The two classical variants of this mode are overloaded elution chromatography (see Chapters 11 and 14 to 17 and displacement chromatography (see Chapters 12 and 17). In elution, a sample pulse of appropriate size is injected into the stream of mobile phase, just upstream of the column inlet, selected fractions are collected at the column outlet, the column is regenerated then equilibrated again with the mobile phase, and the process is repeated as often as needed to process the whole amount of feed. In displacement chromatography, the injection of the feed is followed by a switch of the mobile phase stream to a solution of a compound more strongly retained on the stationary phase than the most retained feed component. Fractions are collected at column outlet, the column is regenerated by purging it with an appropriate solution, then equilibrated with the mobile phase and the process is repeated until the whole feed amount has been processed. The second mode available is counter-current chromatography, a mode in which the two phases of the chromatographic system, the fluid and the solid phases, flow through the column in opposite directions. In principle, there are two possible implementations of this mode, the continuous true moving bed and the cyclic simulated bed method. In practice, only the second method is widely used.
17.1.1 True Moving Bed Chromatography (TMB) The simplest design consists in a vertical column in which the fluid phase percolates upward while the solid phase is poured into the column in the opposite direction (i.e., downward, from the column outlet to its inlet). A feed solution is continuously injected into the stream of the fluid phase, in the middle of the column. Both phases are collected when they reach the end of the column opposite the one through which they entered; these phases are stripped of the feed components, and they are recycled into the column. The main advantage of this continuous mode over the former one are that (1) as in a heat exchanger, the counter-current flow of the two phases of the separation system maximizes the mass-transfer driving force and thus increases the efficiency with which the adsorbent capacity is utilized, leading to a significant reduction in the mobile phase consumption and increase in the stationary phase utilization when compared with elution chromatography; (2) the process is continuous, i.e., the feed is injected into the separation instrument as a continuous stream of constant composition while two streams, each one of nearly constant composition, are continuously collected; and (3) as a consequence, the dilution of the separation products in the fluid phase stream is less than in the batch process, which results in considerable economical advantages. However, enormous practical difficulties have been encountered in the implementation of this mode, called the true moving bed process of separation (TMB), because this counter-current
17.1 Introduction
781
operation requires the continuous circulation of the two phases in the opposite direction, and that has proven too difficult to achieve in practice. The main drawbacks of the TMB process are the near impossibility of achieving plug flow of the solid phase, important backmixing resulting from the radially heterogeneous flow of this phase, and the significant rate of particle breakage. The TMB process has been replaced by another implementation of countercurrent chromatography, the SMB process.
17.1.2 Simulated Moving Bed Process (SMB) The drawbacks of the TMB process have been alleviated by replacing the continuous flow of the solid by a discontinuous one. The solid phase does not actually move. Its movement is simulated by periodically moving the inlet and outlet ports of the system, hence the name of the new process, Simulated Moving Bed separation (SMB). Instead of using one long column in which solid and liquid phases move in opposite directions, it uses a series of shorter columns, connected through complex ports. Instead of moving the solid phase, the apparent movement of the solid phase in respect to these ports is caused by the periodic switch of the positions of these ports (liquid phase and feed inlets, low- and high-retention fraction outlets) along the column train. The SMB process is attractive for its efficient use of the packing material and of the eluent and for its high productivity. It is particularly suitable for the separation of binary mixtures and is ideal for the separation of enantiomers. The yield and the purity achieved are often very high. The main disadvantages of the SMB process are the long time needed to achieve steady state operation, the complexity of the switching systems and the difficulties of separating more than two fractions. SMB has recently become an important method of application of preparative chromatography.
17.1.3 Applications of SMB Technology The SMB technology was developed by UOP and its major field of application is in the area of binary separations. For example, SMB has been used in the chemical industry for several separations known as SORBEX processes [1-3], which include, among others, the PAREX process for p-xylene separation from a Cs aromatic fraction [4], the OLEX process for the separation of olefins from paraffins, the SAREX process to separate fructose from glucose [4] and the MOLEX process [5]. Simulated moving bed is being used particularly for separation of enantiomers from racemic mixtures or from the products of enantioselective synthesis [6,7]. It has been used for the production of fine chemicals, and petrochemical intermediates, such as Cg-hydrocarbons [8], food chemistry such as fatty acids [2], or certain sugars from carbohydrate mixtures [8] and protein desalination [9]. New applications are envisaged in the near future, particularly in the emerging area of bio-separation such as purification of enzymes, peptides, antibiotics and natural extracts. Numerous companies offer SMB equipments for the pharmaceutical and the fine chemical industries [2,8]. This technology covers a broad range of production scales from the laboratory units, which use chromatographic
782
Simulated Moving Bed Chromatography
Raffinate
Figure 17.1 Schematics of a SMB unit. Reproduced with -permission from G. Zhong and G. Guiochon, Chem. Eng. Scl, 51 (1996) 4307 (Fig. 1)
Extract
Desorbent
column with a 4.6 mm internal diameter to the multi-ton production unit licensed by Novasep for chiral separations with column diameter between 20 and 100 cm, to the largest SMB unit licensed recently in South Korea by the Institut Francais du Petrole with a column diameter of 8 meter for the production of 2000 tons/day of p-xylene. Although SMB can also be used to separate multi-component mixtures, it is far more economical when the desired product is the least or most retained component of the feed. Multi-component extraction or purification by SMB is complex. It seems that the use of a cascade of SMB units would be more appropriate (see Section 17.8). A schematic of a SMB unit is shown in Figure 17.1. There are two inlet ports, for the feed and the desorbent or mobile phase, respectively, and two withdrawal ports, for the streams of the raffinate and the extract, respectively. The raffinate is the early eluting compound, the one with the lower retention factor, the extract is the late eluting compound, the one with the higher retention factor on an analytical column with the same packing properties. Such a separation principle can be implemented in several different ways. Constant flow rates in the two inlet and the two outlet streams must be achieved together with constant mass of fluid in the unit, which requires proper control of the pumps. The system can operate in closed loop, with the desorbent pump operating in closed circuit, which is reasonable with small units but may cause technical problems in larger ones (e.g., ram shocks). Then, an open circuit design using a ballast tank for the mobile phase may be preferred. It is possible to reduce the number of pumps by installing flow meters and pressure control devices to control the flow of the extract and raffinate streams. According to the position of the columns relative to the feed and the draw-off nodes, the process can be divided into four different zones, each with a specific role in the separation process:
17.2 Modeling of Simulated Moving Bed (SMB) Separations
783
1. Zone I, between the eluent and the extract ports where desorption of the more retained component takes place, 2. Zone II between the extract and the feed ports where desorption of the less retained component takes place, 3. Zone III between the feed and the raffinate ports where adsorption of the more retained component takes place, 4. Zone IV between the raffinate and the eluent ports where adsorption of the less retained component takes place. Each zone may have a different number of columns. Zones II and III contain the overlapping region of the two migrating bands. These two zones are needed for the separation and are called the separation zones. Zones I and IV are used to recover the extract and the raffinate products and to prevent cross contamination. They are called the buffer zones. Design procedures of continuous liquid-solid absorption systems are basically similar to the procedures used to design any steady-state counter-current mass transfer operation such as gas-solid adsorption [10]. This chapter deals essentially with the applications of the theory of chromatography to the calculation of solutions of the SMB model in different cases of general interest. The theoretical tools required are a general model of the SMB process and a model for its columns. The former is an integral mass balance that is easy to write. The possible column models were described in the previous chapters. Finally, an accurate model of the competitive isotherms of the feed components is necessary.
17.2 Modeling of Simulated Moving Bed (SMB) Separations Because the physical configuration of an SMB unit is complex, it is necessary to model it in order to investigate its operation and to optimize its design and the experimental conditions under which it is used. This modeling is not straightforward but it is necessary. In the design of a new SMB separation, the key issue is the determination of the optimum values of the operating parameters which are the four zone flow rates and the switching time. In terms of the mathematical models used in design, there are two basic approaches for this modeling: 1. The Fixed Bed approach, which considers the SMB process as a series of fixed beds or columns and incorporates the actual periodic flow switching of the process. In this approach, the actual behavior of a SMB unit configuration, with the periodic change of its boundary conditions will be calculated. 2. The True Moving Bed (TMB) approach, assumes that the SMB process is equivalent to the true counter-current moving bed (TMB). This TMB model neglects the dynamics associated with the periodic switching of the columns and gives the mean concentration profiles over a switching period.
784
Simulated Moving Bed Chromatography
The moving bed approach, although less realistic, gives results that are very similar to those obtained from the fixed-bed approach for the description of the final steady state behavior of a four zone SMB process [11]. In turn, each of these two approaches can follow one of the following two paths: a. A continuous process. b. A plate or stage process. In this approach, each column (or bed element) of volume V is considered to be equivalent to a certain number of theoretical stages (n), with each stage being considered as an ideal mixing cell of volume V/n, distributed between the fluid and the adsorbed phase of respective volumes: Vt Vs
= =
eV/n (l-e)V/n
(17.1a) (17.1b)
The differential mass balance for the z'th stages gives Q_!) = Q(Q) + V^
+ Vs ^
(17.2)
The main difference between the SMB and TMB is that the steady state of an SMB unit can be obtained only as an asymptotic limit while it is a definite state in the equivalent TMB unit [12]. In an SMB, even under steady state, the raffinate and the extract concentrations vary during a cycle. However, the average concentrations of these streams over a whole period are practically the same as those obtained for a true TMB. Therefore, even though small differences exist between the results obtained with these two strategies, the prediction of the performance of an SMB operation can be done using the equivalent TMB approach. Theoretically, the TMB model describes the situation existing in an SMB having an infinite number of columns in each zone [13]. However, experience shows that the simplifying TMB assumption holds true even in the case of a relatively small number of columns (e.g., two to four) in each of these sections [14]. Finding the optimum flow rate in each zone, the optimum zone length and the optimum port switching time that guarantee the required purity and a high yield is a major challenge in designing an SMB. Since the TMB and SMB configurations are nearly equivalent, i.e., since they achieve the same separation performance provided geometric and kinematic conversion rules are fulfilled, the simpler model of the equivalent TMB unit can be used to predict the steady state separation performance of SMB units. The conversion rules are given by the following relationships: Vj =
tijV
(17.3a)
V
J&.
(17.3b)
-
SMB Q^SMB
= =
Q TMB +
^^
(1Z3C)
17.3 Analytical Solution of the Linear, Ideal Model of SMB
785
where Vj and V are the volumes of section j of the TMB unit and of one column in the SMB unit respectively, tij is the number of columns in section j section of the SMB unit, t* is the period or switching time of the SMB unit, e is the void fraction of the bed, Qs is the volumetric flow rate of the solid phase in the TMB unit, Q?MB and Q?MB are the volumetric flow rates of the liquid phase in the SMB and the equivalent TMB units, respectively. Dunnebier et al. [15] presented a thorough review of the current design approaches. In terms of the complexity of the mathematical model used, two approaches are available for the design of the four models listed above (la, lb, 2a, and 2 b). These models can be studied and sometimes solved using either 1. Equilibrium designs (ideal model), in which the mass-transfer and the axial dispersion effects are neglected. Assuming the validity of the ideal model, analytical solutions are available for both linear and nonlinear isotherms. For the equilibrium designs, the operating parameters can be obtained either by including a safety factor or by finding the triangular separation region, using the triangle method [16,17]. 2. Nonequilibrium designs, in which mass transfer and axial dispersion effects are considered. For a single-zone equivalent TMB model, an analytical solution is available for a linear isotherm, considering both axial mixing and a finite rate of mass transfer which is accounted for with the linear driving force (LDF) model (i.e., model 2a) [18]. For a single-zone equivalent TMB model, an analytical solution is also available for a binary nonlinear isotherm, ignoring axial dispersion but considering mass transfer resistance. We will present these solutions successively. The general case of nonlinear, nonideal chromatography has only numerical solutions, and optimization must be carried out using numerical solutions of mathematical models to simulate the performance of SMB and to search for the flow rates and switching times that give the desired separation. This approach is faster and more effective than proceeding by trial and error but, in addition to other parameters, the mass transfer parameters are needed to estimate the operating parameters [19-21]
17.3 Analytical Solution of the Linear, Ideal Model of SMB An algebraic solution of the SMB can be derived for the ideal linear model, i.e., for the model assuming a linear isotherm and instantaneous equilibrium everywhere in the column, and neglecting axial dispersion and the resistances to mass transfer. Such analytical solutions were derived by Rhee et al. [22,23] and by Zhong et al. [24]. For the sake of simplicity, we will consider here a simple SMB unit consisting of only four identical columns, one column for each section (see Figure 17.1). The
786
Simulated Moving Bed Chromatography
counter-current movement of the stationary solid phase is achieved by switching the positions of the feed and the draw-off ports in the direction of the liquid phase flow. The solid phase velocity is the same in each column because the different columns in a SMB train are supposed to be identical (the influence on the performance of a unit of small variations of the properties of the different columns of an SMB train was studied by Mihlbachler [25,26]). It is inversely proportional to the switching time or period of the SMB unit, with us = hit*. As in all models of SMB, we need to consider successively the column model and the conditions at the nodes.
17.3.1 Column Model There are N columns in the SMB unit (here, we will simplify the presentation by considering the simple case when N = 4). These columns are numbered 1,- ••,},••• ,N. We consider for these columns the ideal model (see Chapter 7). However, we need to account for the movement of the solid phase, so the differential mass balance in column j , for component i is written: -M
dt
+ H/^-H
S
' dz
(17.4)
- ^ + F-^=0
dz
dt
where Cy and qq are the liquid and the solid phase concentrations of component i in column /, respectively, Uj and us are the liquid and the solid phase velocities, respectively, F = (1 — e)/e is the phase ratio, e is the column total porosity, and DL is the axial dispersion coefficient. As we consider the separation of a binary mixture and a four column SMB system, i = 1,2 and j = I, II, III, IV. In the ideal model, the solid phase concentration (q in Eq. 17.4) is related to the mobile phase concentration by the isotherm equation, which, in the general case, will be nonlinear and competitive. Thus: C 2 , ; )
(17.5)
By definition, component 1 is less retained than component 2. The initial condition of the problem corresponds to a system of columns that are all initially empty of feed but contain only the liquid and the solid phases in equilibrium. The boundary conditions result from the SMB design, i.e., what flows into a column depends on the composition of the fluid phase that is eluting from the previous column. These conditions are Cy(x,0) = 0, Ci,j(p,t) = Cg,
qirj(x,0) = 0 qi,j(O,t) =
where C™- is the concentration of component i, related to the node flow and to the mass balance equations. In SMB, the feed and draw-off nodes are shifted at the end of the period, t*, to the next position along the fluid flow direction. This creates the apparent counter-current motion of the solid phase characteristic of the SMB process (Figure 17.1). In the calculation of the concentration profiles along each column and of the concentration histories at each node, the boundary conditions of each column must be updated at the beginning of each cycle. The set
17.3 Analytical Solution of the Linear, Ideal Model of SMB
787
of Eqs. 17.4 to 17.6 can represent the behavior of a column in either a TMB system, in which case H J M B ^ 0 is the true velocity of the solid phase, or an SMB system, in which case wfMB = 0 but the apparent velocity of the liquid phase is SMB TMB TMB H = w + M [ 2 7 / 2 8 ] _ In the case of a linear isotherm, the solution of the linear ideal model is trivial for the boundary conditions of elution or frontal analysis. This solution is the boundary condition transported along the column at a velocity that is constant. In the case of SMB, the cyclic nature of the process makes the solution more complex to derive but also most useful as it predicts with a reasonable precision the concentration profiles and concentration histories.
17.3.2 The Node Model Let the volumetric liquid phase flow rates through the four columns be Qj, Qu, Qm, and Qw, the flow rates at each node be QD for the desorbent flow rate, QE for the extract flow rate, Qp for the feed flow rate, and QK for the raffinate flow rate. These flow rates are given by the following integral mass balance equations, written for both components at each node [27]. 17.3.2.1 Desorbent (eluent) node
q?vQiv
QIV + QD =
Qi
(17.7a)
+ Ci,DQD =
<%Qi
(17.7b)
17.3.2.2 Extract draw-off node Qi
=
-QE
(17.7c)
QII
17.3.2.3 Feed node QII + QF
ii
QII
+ <~i,F
=
QF
(17.7e)
Qm =
^m
Qm
(17.7f)
17.3.2.4 Raffinate draw-off node QUI-QR
^•tfii
=
Qiv
— H/v —H #
(17.7g) (17-Th)
In these equations, C°^ and C^ are the o and the inlet of column j , respectively. Qj is the actual volumetric flow rate of mobile phase through column /'. It is related to the liquid phase velocity, Uj, by Qj = eAuj, where A is the column cross-section area, which is assumed, without loss of generality, to be the same for all the columns including the feed and drawoff columns.
788
Simulated Moving Bed Chromatography
17.3.3 True Moving Bed Chromatography The most important parameter in the operation of either TMB or SMB is the net ratio between the mass flow rates of the fluid and the solid phase in each column [27]. This ratio is Oc >ii>Qii 11,]
n,
(1\ — e) )Us sQ;; ii,j
For a linear isotherm, qq/C^j = a;/ therefore
In order to achieve the complete separation of a binary mixture, we must adjust the value of 7 so that it fulfills eight inequalities [27]. These relationships can be reduced to four because the more strongly adsorbed component is the critical one in columns I and III while it is the less adsorbed component which is the critical component in columns II and IV. So, if the condition is fulfilled for the more strongly adsorbed component in columns I and III, it will also be fulfilled for the other compound in these columns while, conversely, if the condition is fulfilled for the less strongly adsorbed component in columns 71 and IV, it will also be fulfilled for the more strongly adsorbed component in these columns. If we assume [27] that the four critical flow constraints are all satisfied with the same safety coefficient, we obtain the following set of equations (17.10a) (QI -QE)
Qs (Qi -QE
+
-
QF)
QE + QF -QR)
Qs
(17.10b)
Qs
Qs (QI
Qn QIII
Qs Qiv Qs
I
(17.10c) (17.10d)
where /3, called the safety factor, is a dimensionless number. It is obvious that the optimal design would correspond to the limit value, /3 = 1. In practice, however, to achieve the complete separation of a binary mixture within the framework of the ideal model, the safety margin, /3, must be larger than 1 [27]. The system of algebraic linear Eqs. 17.10a to 17.10d can be easily solved by eliminating Q; (17.11a)
(i7.no
17.3 Analytical Solution of the Linear, Ideal Model ofSMB
789
Knowing Q$, we can derive the value of the switching time, t* in an SMB system equivalent to a given TMB one. This time is given by
f = ± =
( 1
"e)LA
(17.12)
us Qs since Qs = (1 — e)Aus. It is obvious that /} must be smaller than y ^ / f l i in order to obtain positive values for all the flow rates. This result demonstrates why it is difficult with an SMB unit to separate two compounds that are closely eluted {i.e., a pair such that ail'a\ = a is close to 1), because the smaller a, the lesser the flexibility in choosing /3. The actual flow rates in the different columns are easily derived from the sets of Eqs. 17.10a and 17.11a. The following relationships are always valid: Qj >
Qiu > Qn > Qiv [24,27]. In a TMB under steady-state conditions, the mass balances over each column reduce to ^ Ci4-^
=
F^(%j-<§)
(17.13a) (17.13b)
If we consider an ideal situation in which the separation achieved is such that there is no component 1 in column I and no component 2 in column IV, we can easily obtain the concentrations at the raffinate and extract withdrawal ports of the system by combining Eqs. 17.7 and 17.13a, which gives (17.14a) C2,E
=
/!2!1f!2C2,F
(17.14b)
Equation 17.14 shows clearly that, since jS is larger than unity, the raffinate and the extract concentrations can never exceed the concentrations of the corresponding compounds in the feed, a result that was previously demonstrated by Ruthven and Ching [18], using the McCabe and Thiele diagram. Figure 17.2 illustrates the concentration profiles along each column of a TMB unit, with the following values of the parameters, a\ = 5, a% = 10 and /3 = 1.05. The two feed concentrations, Qyr, were chosen arbitrarily.
17.3.4 Simulated Moving Bed Chromatography In an SMB unit, we have wfMB = 0 and M| M B = ujMB + MJ MB in Eq. 17.4. For the sake of simplicity, we write this equation more simply, as u?MB = Uj + us. From Eq. 17.4, the propagation velocity associated with a concentration can easily be obtained (see Chapter 7, Eq. 7.3). It is given here by Vi
U; + Us =
(1715)
Simulated Moving Bed Chromatography
790
IV
ffl
D
D
Figure 17.2 Concentration profiles along the column train of a TMB unit. Reproduced with permission from G. Zhong and G. Guiochon, Chem. Eng. Set, 51(1996) 4307 (Fig. 2).
Since, in this section, we consider only a linear isotherm, this velocity is independent of the concentration and equal to (UJ + M S ) / ( 1 + k^), where k\ = Fa, is the retention factor of component i. For their separation to be complete, the two components must propagate in opposite directions, the more retained one in the direction of the solid phase, the less retained in the direction of the liquid phase. Thus, in column III, for example, we must have irm >us> V2iui
(17.16)
This relationship is exactly the same as the one that was found in the TMB case. Similar conditions apply to the two compounds in the other three columns. As a consequence, Eqs. 17.10a and 17.11a apply as well. Combining the node equations and the propagation equation (Eq. 17.15), we can derive the concentration profiles of both components in each column. This calculation must be done at the end of each period, as is explained in the next few subsections. These concentration profiles are complex functions of the experimental conditions that depend on the rank of the period considered. One interesting result is that these functions have asymptotic limits that are easily derived from simple theorems on the infinite limits of suites, series, and products. This allows the calculation of the steady-state profiles. 17.3.4.1
Concentration Profiles at the End of the First Period (time t*)
At the end of the first period of the SMB, t*, the positions of the inlet and outlet streams are shifted by one column length in the direction of the mobile phase stream, which is equivalent to a backward jump of equal length of the solid phase bed. What happens during a period is defined by the initial and the boundary
17.3 Analytical Solution of the Linear, Ideal Model ofSMB
791
conditions. At the beginning of the first period, all the columns are empty and the initial condition is Q^ = 0 for all i and ;. During the whole period, the feed is injected into the system through the feed node, at a constant composition. Those conditions are the initial and the boundary conditions of frontal analysis. So, a concentration plateau of each feed component migrates along column III. Depending on the experimental conditions, one or both plateaus can possibly reach into column IV. t* should be selected in such a way that the concentration plateau of component 1 enters into column IV before the end of the period but that the concentration plateau of component 2 does not. Using the propagation velocity associated with a concentration and the node equations, it is easy to calculate the concentration plateau of each component in the columns III and IV and the location of the concentration shocks at both ends of these plateaus [24]: Cf = (l-KA)CirF
(17.17)
where K
_ A
l + Ffli/3
— -I .
F
.
in
(1/.18)
1 + Jfl/P
The locations of the two front shocks1 limiting these concentration plateaus at the end of the period are, respectively ia
_ (1 + F a ^ m ^ - a O ,
(m9a)
L
L (17 19b) ° " 1 + F« 2 ' where L is the column length. By definition, the distance Lg that determines the position of the raffinate front extends from the outlet of the column IV and the distance LQ that characterizes the position of the extract front extends from the inlet of the column III. From the definition of K& (Eq. 17.18) and from Eqs. 17.10a, 17.11a, and 17.15, K^ represents the ratio of the flow rates in columns II and III. Since the feed flow rate is positive, the flow rate is higher in column III than in column II and K^ is lower than unity. Figure 17.3 illustrates the concentration profiles in the column train at the time immediately before the moment when the columns are switched. KA is positive, so the concentrations of both components in the columns III and IV are lower than the feed concentrations, as shown in Eq. 17.17. This results from the dilution of the feed, at the feed node, by the mobile phase stream entering into column III. From the raffinate node mass balance equation (Eq. 17.7g), we know that the concentration of the first component in the raffinate stream is exactly the same as its concentration at the inlet of column IV. At this stage, a stream of raffinate begins to be collected, at the end of this period. By contrast, the second component does not appear at the extract node during this first cycle and the fraction collected there is pure mobile phase.
*We demonstrate in the next subsection that, although we are under linear conditions, the boundaries of the concentration plateaus opposed to the feed port are true, stable concentration shocks.
Simulated Moving Bed Chromatography
792
IV
ni
-O D
R
-0-
•o
E
D
Figure 17.3 Concentration profiles along the column train of a SMB unit at the end of the first period. Reproduced with permission from G. Zhong and G. Guiochon, Chem. Eng. Sci., 51 (1996) 4307 (Fig. 3).
The condition for complete separation given in Eq. 17.16 is clearly illustrated in Figure 17.3. The switching time, t*, (or, alternately, the solid phase velocity) must be selected in such a way that the first component front passes the raffinate node but that the second component front does not reach it. This latter condition is equivalent to j6 > 1. When j6 = 1, the extract reaches just the raffinate node at time t*. This is the limit situation. We can see from Eq. 17.19 that, in this case, Lg is equal to 0. 17.3.4.2 Concentration Profiles at the End of the Second Period (time It*) At the beginning of the second period (time t* + St), the feed and the draw-off points are switched simultaneously to their next positions in the direction of the liquid phase flow. The evolution of the concentration profiles during this period depends on the new initial and boundary conditions for each column. These conditions are illustrated in Figure 17.4a, which shows the concentration profiles just after the columns have been switched (i.e., at the very beginning of the third period). In the following we describe qualitatively the phenomenon and then we derive the equations giving the new positions of the shocks (see later, Eqs. 17.20a to 17.20e) and the heights of the new concentration plateaus. Profiles in Column I As during the first period, nothing happens in column I during the second period because there was no feed component in column II at the end of the previous period. Thus, there are no feed components in column I during the period considered.
17.3 Analytical Solution of the Linear, Ideal Model ofSMB
793
(a)
IV
in
-O-
O-
D
-O £
D
E
D
(h)
IV
in
r L»,L»0
D
Figure 17.4 Concentration profiles at the beginning (left) and at the end (right) of the second period. Reproduced with permission from G. Zhong and G. Guiochon, Chem. Eng. Set., 51 (1996) 4307 (Fig. 4).
Profiles in Column II The new set of initial conditions (see Figure 17.4a) includes the two concentration plateaus that were in columns III and IV at time t* — St at the end of the first period and are now in columns II and III, respectively. The plateau of component 1, at concentration C^, extends over the entire length of column II and until a distance Lg in column 11/ (see subsection below). The plateau of component 2, at concentration Cj*, extends from the inlet of column II to a distance L — LQ into this column. Both plateaus migrate downstream. However, they do so more slowly than they were migrating along column III, during the first period, because U\\ < uui- The boundary condition for column II remains the input of pure mobile phase {i.e., of the desorbent). Thus, the two plateaus migrate toward column III, without any change in their composition.
794
Simulated Moving Bed Chromatography
The stream reaching the feed node contains at first a concentration C| of component 1 and no component 2, then, after a time equal to t* + LQ/V^U, that same concentration of component 1 and a concentration C£ of component 2. Finally, after time L/VIJJ, this stream contains only the concentration Cf* of component 2 and no component 1. At the end of the second period, the column contains only a concentration plateau of component 2, over a length extending between a distance Ljj (to be calculated later) and the column exit (Figure 17.4b). Profiles in Column III The new set of initial conditions for this column (Figure 17.4a) includes a concentration plateau of the first component, at C[*, extending from the column inlet to a distance LQ and no component 2. The boundary condition remains unchanged. It is the introduction of the feed that is then diluted into the stream exiting from column II into column III. However, this stream is no longer the pure desorbent (see subsection above). The plateau at Cj* initially existing in column III migrates through this column and enters into column IV. At the end of the second period, the front of this plateau is at the distance LQ + L\ from the column inlet (Figure 17.4b). The stream entering column III contains initially a concentration C\(*, larger than Cf because the stream exiting from column II contains already a concentration Cf, during most of the time (but not at the end of the period, see subsection above). The plateau of the first component has a concentration C^*, it migrates along column III, and it enters into column IV. However, before the end of this period, the concentration of component 1 in the stream exiting column II falls to 0. Then only the feed contributes to the component 1 entering column III and a second concentration plateau at C^ ends the profile of component 1 when the period ends (Figure 17.4b). The width of this second plateau is LQ. For component 2, the boundary condition of column II includes at first the feed injection during the beginning of the period, until the concentration shock at C£ reaches the exit of column II. Then, the concentration of component 2 increases to C~2* and it remains constant until the end of the period (Figure 17.4b and previous subsection). Accordingly, two concentration shocks for component 2 exist in column III at the end of the period. The first shock (0 to Cj ) has migrated over the distance L — LQ (the concentration velocities are independent of the concentrations with a linear, noncompetitive isotherm) and the second shock (C|* to C%*) over the distance L-LhQ-L\ (Figure 17.4b). Profiles in Column IV The initial condition for column IV is a column empty of feed components. The boundary condition is a stream of desorbent containing no component 1 during the initial part of the period. Then, a concentration plateau at Cf appears, followed by a second shock and a plateau at C\l*. This second plateau migrates over a distance equal to LQ at time It* because concentration velocities are independent of the concentrations with a linear, noncompetitive isotherm. The first shock, at C[ migrates an extra distance L" (Figure 17.4b).
17.3 Analytical Solution of the Linear, Ideal Model of SMB
795
Quantitative Characteristics of the Concentration Profiles At the end of the second period (t = It* — 6t), the concentration profiles of the feed components are concentration plateaus limited by shocks. The plateau heights and the locations of their fronts are given by the following equations [24]: Cf
=
(l-K2A)Ci/F
(17.20a)
L\
= KBLaQ
(17.20b)
L\
=
(17.20c)
^L\ L
(17.20d)
where Lg and Lg are given by Eq. 17.19, KA is given by Eq. 17.18, and
Similar to KA, K$ represents the ratio of the flow rates in columns IV and III. In fact, the factors Kg and K^ in Lf and h\ are directly related to the ratios of the migration velocities in the different columns. A comparison between the values of the heights of the two concentrations shocks leads to C2t* _ C f ' rt* l
= KA
(17.20g)
This equation means that the relative increase of the concentration of the highest plateau from step 1 to step 2 is equal to KA. 173A3
Profiles at the End of the Third Period (time 3t*)
The calculation of the position of the concentration shocks and of the plateau concentration follows a simple iteration process. The column switching at the beginning of the new period changes the initial conditions in the four columns and their boundary conditions. This means that, at each period, each actual column exchanges position and role in the SMB with its neighbor in the same order, being successively in the positions I, II, III, IV, and again, I, II, etc. The new initial and boundary conditions are illustrated in Figure 17.5a (to be compared with Figure 17.4b). The concentration profiles in the SMB train, at the end of the third period, are shown in Figure 17.5b. The pattern is clear and requires little further explanation. Profiles in Column I This column contains now a concentration plateau of component 2 at Cj , plateau that was left in column II at the end of the previous period (Figure 17.4b). This plateau is entirely eluted out of column I during the third period (see Figure 17.5b). During the first part of the third period, the extract
Simulated Moving Bed Chromatography
796 (a)
TV
m
rl
iT
-O
L a ,L a 0
D
R
E
(b)
Figure 17.5 Concentration profiles at the beginning (left) and at the end (right) of the third period. Reproduced with permission from G. Thong and G. Guiochon, Chem. Eng. Sci., 51 (1996) 4307 (Fig. 5).
is a stream of desorbent that contains component 2 at the concentration Cj* and which lasts during the time LQ/V^I- This stream is followed by a stream of the pure desorbent. Profiles in Column II The initial condition of column II (Figure 17.5a) includes two concentration plateaus at concentration C\l* and C^*, respectively, ending as a concentration staircase at the inlet of the column for component 1 and at its outlet for component 2. Both profiles migrate along column II and into column III through the feed node, where they mix with the feed. Component 1 is entirely eluted from column II into columns III and then IV, but not component 2 (Figure 17.5). The profile of component 2 includes two concentration plateaus and
17.3 Analytical Solution of the Linear, Ideal Model ofSMB
797
two rear shocks. The first part of column II, the one on the inlet side of that shock, is free of component 2 at the end of the third period. On the outlet side, the concentration is C|f* until distance LQ. Then, the concentration is C^ until distance
Profiles in Column 71/ Elution of the concentration plateaus at C^1* and C|f* from column II into column 77/ after mixing with the feed stream causes the formation of plateaus at higher concentrations, C\l* and C^*, respectively, for the two components. For component 1, this plateau extends down to column IV. However, at the end of the period, the concentration of component 1 in the stream exiting from column 77 into column 777 jumps down from C^* to Cf and then to 0. Accordingly, plateaus at C^** and Cf appear at the column inlet. The migration distances of the corresponding shocks are LJ and Lg, respectively (Figure 17.5b). For component 2, the concentration of the stream that exits column 77 increases successively from 0 to Cl2* and from C2 to C|f*. Accordingly, two successive plateaus, at concentrations C^ and C|f*, appear in front of the main plateau, at C|f*. The corresponding shocks are located at the distances Lg, Lg + h\, and Lg + h\ + L|, respectively. The end of the column, between its outlet and the distance Lg7, contains no component 2 (Figure 17.5b). Profiles in Column IV Finally, column IV is initially empty of feed components (Figure 17.5a). It receives a stream of carrier coming from column 777, a stream that is initially pure and in which the concentration of component 1 jumps successively to Cf, to C\l*, and to Cf* (see previous subsection). Three concentration shocks are successively formed. They all migrate to distances of L^ + U[ + Lg, L\ + Lg, and Lg, respectively (Figure 17.5b). Quantitative Characteristics of the Concentration Profiles
The height of the
different concentration plateaus and the location of the concentration shocks in all these profiles are obtained as follows [24] Cf
=
(l-K3A)d,F
La2
a
=
K\L
L\
=
-i-Lg
0
(17.21a) (17.21b)
(17.21c)
K
A
L\
=
-^-Lc0
(17.21d)
h\
=
Kchl
(17.21e)
where -
1 + Fa
^
Kc represents the ratio of the flow rates in the columns II and I.
(17.22)
Simulated Moving Bed Chromatography
798
Figure 17.6 Concentration profiles at the end of the nth period. Reproduced with permission from G. Zhong and G. Guiochon, Chem. Eng. Sci., 51 (2996) 4307 (Fig. 6).
The relative concentration increase during the third period is ft*
~
A
(17.23)
Because K^ < 1, this relative concentration increase is smaller than it was during the second period. 17.3.4.4 Profiles at the End of the nth Period (time nt*) A clear iteration pattern has now emerged. Figure 17.6 shows that, from one period to the next, two series of higher and higher concentration plateaus are formed for components 1 and 2 in column III. These plateaus are transferred at the beginning of the following period into column II. They move along this column and into column III during the period. These plateaus become narrower and narrower with increasing number of periods. During each successive period, higher concentration zones replace lower ones, lower concentration zones are compressed and pushed progressively along columns IV and II, farther and farther away from the corresponding end of column III. At the end of each period, the raffinate and the extract bands extend into columns IV and II, respectively. The procedure followed above for the calculation of the concentration profiles at the end of the first, second, and third periods can be iterated as needed. It is easily shown that, if we assume that the concentrations of the highest concentration plateaus of the two components at the end of period n, at time nt*, and the locations of the new concentration discontinuities formed are given by the following equations at the end of period n, ^nt*
=
{l-K\)CiiF
(17.24a)
17.3 Analytical Solution of the Linear, Ideal Model of SMB
799
L«_x
=
K*" 1 ^
(17.24b)
4-1
=
^-fLo
(17.24c)
L
n-2
K
A
=
4-2
these equations are also valid at the end of period n — n + 1. The relative concentration increase is nt* _
r(n-l)t*
=KAn~1
^ — ^ C
(17.25)
J
The positions of the four discontinuities that define the widths of the bands of the raffinate and the extract in the system can be calculated from the following sums [24] i
Aan
=
vn
La0 + La1 + ... + Lan_1 = —^La0
(17.26a)
1 — 1 / Kn b
r l i i l i
i
i r k
A
-1/KA
%
(17.26b)
%
(17.26c)
{n
A* =
LaQ + Li + ... + L«_1 = ^—^La0
(17.26d)
Because the three constants KA, -KB and KQ are all smaller than 1, both Aan and have finite limits when n tends toward oo. These limits are I-
At
KB
= ^V
(17.27b)
This result has several important consequences: • The behavior of a SMB unit is unconditionally stable, at least under linear, ideal conditions, provided that /3 meets the requirements 1 < jS < \fcc. • The operational steady state of a SMB unit can be reached only in the mathematically defined asymptotic sense. In practice, however, the number of cycles needed to approach steady state closely is realistic. • The raffinate component cannot pass through the asymptotic point, A%,. • The extract component cannot move behind the asymptotic point, A^,. In fact these two limits represent somehow the equilibrium points of the flow rates between the solid and the fluid phases in columns IV and III, and in columns II and I, respectively.
Simulated Moving Bed Chromatography
800
IV
m
-O
O
A&,
D
D
R
Figure 17.7 Concentration plateaus of the extract in column III at the end of periods n and n + 1. Reproduced with permission from G. Zhong and G. Guiochon, Chem. Eng. Sci, 51 (1996) 4307 (Fig. 7).
By contrast, A\ and Acn axe increasing series and their values have no limits. Thus, there are values of n for which An and/or Acn become equal to or larger than L. Obviously, there is no reason for the same value of n to satisfy both relationships A^_j < L < A\ and A^_1 < L < Acn. Let nj, and nc be the values of n which satisfy these two relationships, respectively. Since Lc0 < LhQ (Eqs. 17.19b and 17.20d), nc will obviously be larger than nj,. When n becomes larger than nj,, a new plateau can no longer form on the ascending staircase at the front of the band profile of component 2 in column III before the end of the Mf,th period. So, this plateau ceases to grow in column III. This case is illustrated in Figure 17.7. It is easy to show that, from now on, the concentration profile of the extract will remain the same at the end of each new period. A steady state of the concentration pattern of component 2 in column III has thus been achieved. Similarly, whence n reaches the value nc(> tit,), the concentration plateau of the raffinate at "c cannot form in column III. Thus: The extract and the raffinate concentration profiles may reach steady state at the end of different periods. When this happens, the profile of the raffinate reaches a steady state later than that of the extract. When both Abn and Acn have reached a value larger than L, the steady concentration profiles are given by the equations Cl,R
=
C2,E
=
(17.28a)
a2 C2,F
(17.28b)
17.3 Analytical Solution of the Linear, Ideal Model of SMB
801
where C\^, C\f, Ci£, and Cif are the concentrations of the first and second component in the raffinate, the extract, and the feed, respectively. • A numerical example illustrates this discussion. Assuming the following values for the parameters, with equilibrium constants a\ = 5, aj = 10, and a = 2.0, safety margin j3 = 1.05, and phase ratio F = 0.5, we obtain nb = nc = 5 periods before both the raffinate and the extract concentration profiles reach steady state in column III. In this case, Lb5 < L < Lb6 and Ac5 < L < Ac6. However if ai = 7 and a = 1.40, we have A\Q < L < Ahn and Ac9 < L < AC1Q, so n& = 10 and nc = 9. Figure 17.6 illustrates the former situation, when the concentration profiles of both the raffinate and the extract reach steady state in column III for the same value, n — 5, and A\ — L — Ac5. In this case, when n becomes larger than 5, the concentration profiles of component 1 in column IV and component 2 in column II will still change progressively, their fronts will tend toward their respective asymptotic positions, A%, and A%,, never to be reached exactly. Figure 17.7 shows the concentration profiles achieved under the same conditions as in Figure 17.6 but after 10 periods. Although steady state can be reached only in the asymptotic sense, this figure illustrates the trend towards the slow build up of a unique concentration shock for each component. In the general case, there are no integers for which A\ = L and Acn = L. The value of «{, and n are derived from the inequalities. In this general case, the system undergoesc oscillations around the steady state instead of drifting monotonically towards it. Denote 1? = L- l\_x < A^/K^1 c>
L = L- L£_a < Al/KX
1
(17.29a)
(17.29b)
Without entering into greater detail, it is possible to show that in the next step, i.e., for n — nc +1, the raffinate concentration plateau between the inlet of column IV and the distance Lg will be split into two different plateaus, as shown in Figure 17.8. One of these plateaus, located near the column inlet, is at concentration C"f and has a width Lg — LC'K$ /K^. The other plateau is at concentration CJ"+ ' and has a width of Lc Kg/K^. For the next step, n = nc + 2, this concentration band will be pushed forward into column IV and compressed between locations Lg and L^ while their concentrations are unchanged and their lengths are reduced by the same factor, Kg, becoming Kg(Lg — Lb Kg/K^) and Kg(Lb Kg/K^), respectively. This same concentration patterns will be repeated at step nc + 3 and will form again between the column inlet and location LQ. SO, the only difference between this case and the previous one is that in the former case, Lb' — 0 and there are no oscillations. Asymptotically, the profile of component 1 in column IV is a train of rectangular waves of constant amplitude, between the two concentrations CJn+ ' and C"'*, the frequency of the waves tending towards infinity as the front position nears to A%,. These concentration plateaus have total asymptotic band widths that are respectively A%, - L"' and L"', with L"' = KBLC'/(KA(1 -KB)).
Simulated Moving Bed Chromatography
802
Figure 17.8 Concentration plateaus of the raffinate in column IV at the end of periods n and n + 1. Reproduced with permission from G. Zhong and G. Guiochon, Chem. Eng. Sci, 51 (1996) 4307 (Fig. 8).
For the extract, a similar result is obtained. Only if Acn = L will a constant concentration plateau be obtained at Cj • Otherwise, if nc is derived from the inequality A\_x < L < A\, we have a similar square-wave pattern on the concentration plateau in column II. The waves are characterized by two concentrations, Cj and Cj , respectively, as shown in Figure 17.8. Their respective total asymptotic lengths are At - Lh' and Lb>, with i / = Lb'/ (1 - Kc). So, in the general case, the concentration of the raffinate plateau varies between f C" * and C|" and the concentration of the extract plateau between Cj . We can easily identify that C\n+1)t* and C^ l)t* are equal to and C2,E in Eq. 17.28, respectively, and also that « ^(n-2)t* -2
1 4 - Ffli _ —
Fa2(l-
(17.30a) (17.30b)
If /3 = 1, these values are the same as Cirj? and ^E, respectively, so the two concentrations for each component reduce to one.
17.3.5 Comparison of Ideal, Linear SMB and TMB ChromatograA comparative analysis of the analytical solutions derived earlier in this section for TMB and for SMB chromatography permits a rapid identification of the equivalences and differences between these two operation systems. First, the steady
17.3 Analytical Solution of the Linear, Ideal Model of SMB
803
state concentrations of either the raffinate or the extract streams in a TMB are constant (see Eq. 17.14) while, in an equivalent SMB (i.e., same experimental parameters) the composition of the two streams may oscillate between two different concentrations. However, these two concentrations are not very different, especially if j6 is close to 1. They become equal when |6 = 1. The relative difference between the maximum concentrations of the product streams obtained by TMB (Eq. 17.14) and by SMB (Eq. 17.28) is
-1)
JT -2,E
=
(17.31b)
C™B fl2_ai/3 Thus, the maximum raffinate concentration obtained with a given SMB is smaller than that produced by an equivalent TMB while the opposite is true for the maximum extract concentration. However, the differences between these concentrations are small when /3 is close to 1. The concentrations of the product streams are equal in the two processes in the limit case when /3 = 1. Furthermore, it is possible to show that the time averaged concentrations of both the raffinate and the extract in SMB are equal to the constant concentrations of these streams obtained in TMB. This is required for mass conservation. Finally, the main difference between these two processes is that the steady state can be obtained only in an asymptotic sense in SMB, which means that it will take a large number of periods, hence a long time to reach it.
17.3.6 Conclusion on the Solution of the Ideal Linear Model There are simple algebraic solutions for the linear ideal model of chromatography for the two main counter-current continuous separation processes, Simulated Moving Bed (SMB) and True Moving Bed (TMB) chromatography. Explicit algebraic expressions are obtained for the concentration profiles of the raffinate and the extract in the columns and for their concentration histories in the two system effluents. The transition of the SMB process toward steady state can be studied in detail with these equations. A constant concentration pattern can be reached very early for both components in column III. In contrast, a periodic steady state can be reached only in an asymptotic sense in columns II and IV and in the effluents. The algebraic solution allows the exact calculation of these limits. This result can be used to estimate a measure of the distance from steady state under nonideal conditions. The analytical solution obtained for this model is not trivial as it is in the case of the elution and breakthrough problems. Accordingly, it is most useful for the investigation of separation carried out under linear or quasi-linear conditions. In such cases, it provides a useful starting point for further studies. Figure 17.9 shows a comparison between the experimental and the calculated concentration profiles along the column train of an SMB unit. It shows that the condition of ideality can be relaxed (see later, Section 17.4) and that the solution of the ideal
804
Simulated Moving Bed Chromatography
Figure 17.9 Comparison between the experimental and calculated concentration profiles along the columns of a SMB unit. T. Yun, G. Zhong and G. Guiochon, AIChE } . , 43 (1997) 935. (Fig. 4). Reproduced by permission ofthe American Institute ofChemical Engineers. ©1997 AIChE. All rights reserved.
model provides an excellent approximation of the actual profiles. Similarly, the condition of linearity can also be relaxed (see Section 17.7) and the solution of the linear model provides an acceptable approximation when the isotherm deviates slightly from linear behavior. Significant deviations, however, take place for moderate deviations because the velocity of the concentration fronts depends on the plateau concentrations. This explains the numerous failures of the process observed by those attempting an empirical optimization of an SMB unit. The front begins to move too fast and the safety condition is no longer fulfilled. Finally, knowing the analytical solution of the limit case provides a useful check of the numerical solutions obtained and of the accuracy of their algorithms. The excellent agreement between the experimental and the calculated concentration profiles in SMB (Figure 17.9) should not surprise. It is due to the fact that, although the system operates under linear conditions, the front of the raffinate concentration profile and the rear of the extract concentration profile are both shocks (see next section). Accordingly, in an actual SMB operated with columns that have usually a rather limited efficiency, the front of the raffinate and the rear of the extract concentration profiles are both shock layers. They move at the same velocity as that of the shocks in the ideal model and they are steep (see Chapter 14, Section 14.1.4).
17.3.7 The Zone Boundaries in Linear, Ideal SMB are True Shocks Under ideal conditions, i.e., when the column efficiency is infinite, there are no axial dispersion and no mass transfer resistances, a concentration shock is stable on the front of a profile if the velocity associated with a high concentration on a diffuse boundary is higher than the velocity associated with low concentrations (see Chapter 7, subsection 7.2.3). This is, for example the case in elution chromatography when the equilibrium isotherm of the compound considered follows Langmuir behavior. Conversely, if the isotherm follows anti-Langmuir behavior, the velocity associated with a high concentration on a diffuse boundary is lower than the velocity associated with high concentrations, the high concentrations tend to pile up at the back of the concentration profile, and a stable shock forms on the
17.3 Analytical Solution of the Linear, Ideal Model ofSMB
805
rear of the concentration profile. Under linear conditions, the velocity associated with a concentration is independent of the concentration, so shocks are neither stable nor unstable. In SMB chromatography, however, the cycle-average velocity of a given concentration on a diffuse profile depends on the concentration or, more exactly, on the position of this concentration at the beginning of a cycle. When a cycle begins, the concentration profile of the extract extends over columns I and II, with pure extract along the end part of column I, before the extract port. The concentration profile of the raffinate extends over columns II and III, with pure extract in the early part of column IV, downstream the raffinate port. During the cycle, the profiles migrate along the column train. Let us assume now that, at the beginning of a period, the initial rear front of the extract concentration profile along column I is diffuse. During the cycle, each concentration moves forward along this column at a constant velocity, V^E = MJ/(1 + k'E), where «/ is the liquid phase velocity in column I and k'E is the retention factor of the extract, i.e., the second component of the feed. Eventually, before the end of the cycle, this concentration moves past the extract port and enters into column 11. In this column, it migrates forward at the velocity VU,E = " n / (1 + k'E), where U\\ is the liquid phase velocity in column II. Assuming that the columns are all identical, the retention factor of the extract is the same in all columns. However, the integral mass balance at the extract node (Eq. 17.7c) states that Qi — QE = Qu, where the Qs are volumetric flow rates. Since all the columns have the same crosssection, this means that wj > u\\. In other words, the velocity associated with a concentration is higher in column I than in column II. Accordingly, the cycleaverage velocity of a concentration of the diffuse rear front of the extract will depend on the initial position of the concentration on that diffuse front. Obviously, the low concentrations are farther removed from the extract port than the high concentrations. Thus, the average velocity of low concentrations during a cycle is higher than that of high concentrations. The rear diffuse front of the extract will become steeper and steeper. The concentration discontinuity at the rear of the extract profile is stable, even though the SMB is operated under linear conditions. The same rationale applies to the front concentration profile of the raffinate. If we assume that, at the beginning of a period,, this front (located in column III) is diffuse, the velocity of a concentration on that diffuse front is VH^R = M////(l + k'R). When the concentration passes the raffinate port and enters column IV, its velocity becomes VIV,R — M/v/(l + k'R). The mass balance at the raffinate drawoff port is Qui — QR = Qiv, showing that Uui > Ujy and that the velocity of a concentration is higher in column III than in column IV. The later a concentration stays inside column III and passes the raffinate node, the higher its cycle-average velocity. Therefore, high concentrations on a diffuse front raffinate profile move faster than low concentrations. A concentration shock is stable and one will form sooner or later if the initial concentration boundary were to be initially diffuse.
806
Simulated Moving Bed Chromatography
17A Analytical Solution of the Linear, Nonideal Model of SMB It is simpler to analyze the behavior of a linear system in terms of its equivalent continuous counter-current flow model. The equivalent true counter-current moving bed (TMB) representation of the SMB process provides the detailed mathematical model of the process. This leads to simple analytical solutions for the concentration profiles and provides valuable insights into the main factors governing the behavior of the system. Assuming an axially dispersed plug flow of the fluid through the column, the mass balance equation for the fluid phase for a single adsorbable species is [15,27]: dC
+u
dC
dq
dq
Fu +F
ix Tz- 4 i
D
32C
,„„ „,
(17 32)
= ^
-
We assume a linear equilibrium isotherm (q = aC) and a lumped kinetic model with linear driving force mass transfer, we have at steady state: d
A
= k{q*
-q) + usfz= k(aC ~q) + usfz
(17.33)
Combining these two equations gives DL^-u-^-Fk(aC-q)=0
(17.34)
The adsorbed and fluid phase concentrations at the column outlet are related by an overall mass balance: Fus(qL-q0)=u(CL-C0)
(17.35)
with the appropriate Danckwerts boundary condition at the two column ends (Eq. 2.15). Rewriting these equations in dimensionless form, rearranging them using the integral mass balance between the column ends, z = 0 and z = L, gives
- (Pe + St)fx + PeSt(l - 7)«> = PeSt (l -
(17.36)
where 7 = aFu$ / (M) is the ratio of the solid to the liquid phases flux of the solute considered, x = z/L , qo and Q are the concentrations of the solute in the adsorbed and the mobile phases at the column inlet (z = 0), respectively, qL and Ci are the concentrations of the solute in the adsorbed and the mobile phases at the column outlet (z — L), respectively, Pe is the axial Peclet number (Pe — uL/DL), St is the Stanton number (St = kL/us),
dx
x=0
d(p ~dx
= 0 x=l
(17.37)
17A Analytical Solution of the Linear, Nonideal Model of SMB
807
The solution of this equation (i.e., the concentration profile) is l-qo/aCo m2x _
m^/,m2+m1x
- m2/Pe) - m2emi{l - m1/Pe)
(17.38)
where m\ and m2 are the roots of the following equations corresponding to the signs + and —, respectively: (17.39) At the column outlet, (z — 1), Equation 17.38 can be written as:
„ inr
,17.m
= A
(17.40)
1 — qo/aCo or: c
o
aC
7-1L
o
aC
o
where A is a function of Pe and St. win (-\
17A.I
/ n,.\
„
»/.
fi
/ Tt _ \
\
•r±±)
First Particular Case, Axial Dispersion is Negligible
In the plug flow limit, Pe —> oo, we have m\ = Pe 3> m.2 = (1 — j)St and A is equivalent to est(1~T). Therefore the concentration profile will be given by: ) _
_
<st{\-i)
or: Co
7-I
(1
This solution is valid for plug flow assuming a linear driving force model for mass transfer. The solution for a linear plug flow system in which the mass transfer rate is controlled by intraparticle diffusion rather than by the linear driving force model has been derived by Amundson and Kasten [29].
808
Simulated Moving Bed Chromatography
17.4.2 Second Particular Case, Mass Transfer Resistances are Small (Equilibrium Limit) In the equilibrium limit, the mass transfer resistances are small and axial dispersion originates essentially from axial mixing. Then St is nearly infinite, ni\ = St 3> wi2 = (1 — y)Pe and A reduces to (l/7)e Pe ( 1 ~'". The concentration ratio across any section of the bed is then given by 'e(L-V
(17 M)
i — I/O/«1M)
or:
° Co
=
' [(i_^_)^(i-T) + m-il (1 flC 7 - 1L o
(17.45)
For a multi-zone SMB, each zone of the TMB system can be described by the same differential mass balance and kinetic equations. However, these equations are coupled by the boundary conditions which incorporate the injection and withdrawal of fluid streams at the appropriate points. Analytical solution cannot be obtained, only numerical solution is available.
17.5 McCabe-Thiele Analysis Rather than using a continuous model and representing the system in terms of the dispersion model, one may consider each bed as equivalent to a number of theoretical equilibrium stages. A McCabe-Thiele diagram then provides a convenient representation of the system and shows clearly the effects of the process variables [10,30]. Equivalent Stages Model In the stage model, the bed is considered as equivalent to a certain number, n, of ideal equilibrium stages. The relationship between the continuous and the stage model may be derived by considering the Height Equivalent to a Theoretical Plate (HETP) as discussed by Ruthven [31]. The number of theoretical stages is given by
£ =f =
to
(1746)
-
And in the limiting case when the mass transfer resistances are small and St ^> Pe,
It is important to know that the HETP in SMB (H1) and in elution chromatography (H) are not identical, although both are related to the combined effects of axial dispersion and the mass transfer resistances. According to the above equation 17.46, H' depends on the flow rate ratio.
17.6 Optimization of the SMB Process
809
Ching et ol [30] compared the concentration profiles calculated from the continuous nonideal, linear model, with those calculated with the stage model and the McCabe-Thiele diagrams with the number of stages given by Eq. 17.46 and showed that they are almost identical, except in the region of z = 0. Although the continuous nonideal model is physically more realistic, the stage model combined with the McCabe-Thiele representation has the advantage that it provides a simple means of understanding how the concentration profile is affected by changes in the flow rate and the switching time and thus provides useful guidance for the selection of the optimal operating conditions. Like distillation, the McCabe-Thiele analysis is strictly valid only for a binary system. However, only two components are usually present at significant concentrations within each individual section of the column (and, besides, in practice, the SMB process is essentially used to separate binary mixtures). A preliminary analysis in which each section is considered as a pseudo binary McCabe-Thiele system can therefore provide useful guidance in the design of a multicomponent adsorption system.
17.6 Optimization of the SMB Process The optimization of SMB is easy when the unit operates under linear conditions. Several reliable methods are available. The problem becomes far more complex when the unit operates under nonlinear conditions. Then, the velocities of the concentration fronts are strong functions of the concentrations of the feed components and almost any inconsiderate adjustment of any parameter can lead the process to a costly crash.
17.6.1 Optimization of an SMB System with a Linear Isotherm, Using the Safety Margin Approach The successful operation of an SMB process relies on the successful fulfillment of a number of separation conditions. To achieve a good separation, the flow rates in all four sections must be chosen in such a way that, at the end of each cycle, the fronts and rears of the bands of the two components of the feed are located within specific ranges in the proper column (or section, in a more complex, multi-column unit). As we discussed earlier, an SMB separation can be studied as an equivalent TMB process, the solid-phase flow rate of which is related to the switching time of the SMB unit by Qs = (1 - e)Aus = (1 - e)AL/t*
(17.48)
where t* is the switching (or cycle) time, us the apparent velocity of the solid phase, A the cross section of the column, and L its length). The internal volumetric flow rate in the equivalent TMB unit, QjMB, can be calculated by using the relationship QTMB
=
QSMB _ Ql
(17.49)
810
Simulated Moving Bed Chromatography
where Q?MB is the volumetric flow rate in the SMB unit. The derivation of the separation conditions is based on the ideal or equilibrium model, i.e., on the assumption that axial dispersion and the mass transfer resistances are all negligible and that the column efficiency is practically infinite. In conventional studies of SMB, it is further assumed that the solid phase flow rate through each column and the void fraction of each column are the same. In the linear case, the ratio of the internal flow rate and the solid-phase flow rate can be combined with the slope of isotherm (a,-) by using a safety margin, j6y [25,27]: ^
(17.50a) TMB
^
(17.50b)
Qs
pin
Qs
piv
(17.50c) (17.50d)
The flow rate has to be larger in section III than in section II, since Qm = Qu + QF > Qn- Based on the safety margin this can be expressed by:
faifan < *
(17-51)
To achieve the complete separation of a binary mixture in the framework of the ideal model (i.e., with no axial dispersion and neglecting mass transfer resistance), the safety margins must be at least equal to one [24,27]. In practice /3 is chosen to be slightly larger than 1 (safety margin): /3y > 1
(17.52)
Equations 17.51 and 17.52 define a sufficient number of criteria to allow the correct choice of the operating conditions in an SMB operating under linear conditions. This set of conditions is equivalent to the one derived by Storti et al. [16] (see later, Subsection 17.6.5), the so called Triangle Method. However, both sets of conditions are based on the assumption that all columns have identical characteristics and an infinite efficiency. In practice, the different columns of an SMB separator cannot be identical. Their individual average porosity, permeability, retention factors, and efficiency are more or less different, however slightly. The influence of the possible differences between the columns of an SMB unit on its performance is discussed later (Subsection 17.7.1.5)
17.6.2 Standing Wave Design of SMB for Linear Isotherm Ma and Wang [32] proposed a new approach which they called the Standing Wave Design (SWD) for the optimization of the experimental conditions of the SMB separation of a binary mixture, using an SMB system for which they assume a linear
17.6 Optimization of the SMB Process
811
equilibrium isotherm and ignore the mass transfer resistances and axial dispersion (ideal system). The standing waves can be achieved if the velocity, uW/j of a specific wave in each zone is equal to the velocity of the port movement (v). Accordingly, they proposed the following definitions: • v = ulw2 is the velocity of the desorption wave of the more retained in zone 1. • v = M^J is the velocity of the desorption wave of the less retained in zone 2. • v = ul^\ is the velocity of the adsorption wave of the more retained in zone 3. • v = u^-y is the velocity of the adsorption wave of the less retained in zone 4.
component component component component
In the ideal case, with a linear isotherm, uwi=
"° , (17.53) 1 + Fa,-) By, substituting Eq.5 into Eqs. 1 to 4, they derived the following relationship between the interstitial velocity in each zone and the velocity of the port movement 1
u\
=
(1 + Ffl2)v
(17.54a)
M«
=
(1 + Ffli)v
(17.54b)
ul
u
=
(1 + Fa2)v
(17.54c)
ujy
=
(1 + Fflx)v
(17.54d)
With an ideal system, it is possible to achieve products exhibiting a 100% purity, with a 100% recovery yield, at any values of the feed flow rate, QF, provided that the standing wave conditions (Eq. 17.54) are met. Qf is related to the difference between u^11 and u^1 through
QF = eA{uin - ul1) = A(l - e){a2 - ax)v
(17.55)
v = -—
(17.56)
Or, ^
.
Therefore, it is possible to determine the zone flow rate {CAUQ) and the average port velocity, v, for a given feed flow rate. The switching time, ts, is derived from Eq. 17.56 (v = Lc/ts). These results are equivalent to those reported by Ruthven et al. [27] and by Zhong et al. [24] when the safety factor is set equal to to 1 (Eqs. 17.10a and 17.11a). If the operating parameters derived from Eq. 17.54 are applied to an SMB system operating under ideal behavior, the adsorption and desorption concentration waves are square waves. The adsorption waves of the less retained solute (solute 1) and of the more retained solute (solute 2) are standing in zone IV and III,
812
Simulated Moving Bed Chromatography
respectively, and the desorption waves of solutes 1 and 2 are standing in zones II and I, respectively. In this way, the components 1 and 2 are collected at the raffinate and the extract ports, respectively, both as pure products. Using the Standing Wave Design, algebraic equations were also derived for a nonideal SMB system with linear isotherms, by solving the equations for the four zone flow rates and for the port switching time. Operating with the corresponding parameters guaranties a high product purity and a high recovery yield [32,33]. SWD was thus applied to the optimization of the SMB process for the separation of phenylamine and tryptophan [34] and for the purification of paclitaxel [35]. 17.6.2.1 Standing Wave Design of SMB for Nonlinear Isotherm Mallmann et al. [36] tried to extend the SWD design to the case of nonlinear isotherm behavior of an SMB system. The specific waves must be identified and made standing, and their migration velocities must be determined in order to calculate the optimum operating conditions in nonlinear SMB. Nonlinear waves for TMB is analyzed by Rhee et al. [22,23] and by Storti et al. [16,20,28]. Mallmann et al. [36] tried to determine the wave migration velocities in SMB by using experimental results obtained in elution chromatography and the hodograph plot [22,23,37]. However, the concentrations Csi and CS2 of the plateaus in SMB are lower than those of the feed because the feed is diluted by the desorbent stream before entering zone III. Analytical solutions cannot be obtained for this plateau concentration. Therefore, a numerical iterative calculation procedure must be used to determine the plateau concentration and the optimum set of operating conditions for complete separation in nonlinear SMB [36].
17.6.3 Optimization of an SMB System with the Triangle Method In order to obtain an efficient separation, the SMB must be operated with flow rates that have to be chosen properly in every zone. In order to do so, the flow rate of the equivalent TMB can be determined using the equilibrium theory. The SMB flow rates are then determined from the equivalence between TMB and SMB fluid velocity over one switching period. Rhee et ah [22,23] derived an analytical solution of the ideal model in the case of the complete separation of a binary mixture following Langmuir isotherm behavior and for a single counter-current column operated under steady state conditions. Based on this theory, the triangle method was developed by Storti et al. [16,20,28], Mazzotti et al. [38^0], and Chiang et al. [41,42] to determine the working flow rates in every zone of the classical four zone TMB conditions under linear and nonlinear isotherm behavior. With the ideal model, a four-section TMB unit has four operational parameters. It is practical to define these four parameters as the ratios of the net fluid flow rate in each section to the solid flow rate, wij, vtiu, mm, and miy. These ratios are easily related to the liquid-phase flow-rates in each section of the TMB and to the normalized port switching time of the corresponding SMB process, through geometric and kinematics equivalencies [28].
17.6 Optimization of the SMB Process
813
The solutions derived by Rhee et al. [22,23], Storti et al. [16,20,28], and Mazzotti et al. [38-40] are valid only for a two-component mixture following Langmuir isotherm behavior, with a constant selectivity, i.e., for c\Si\ — qS/2 — CJS. The solution derived by Chiang et al. [42] was obtained for a binary mixture that follows a nonstoichiometric Gurvitsch isotherm behavior. This isotherm model has also a constant selectivity. Storti et al. [28] accounted for the influence of the different flow rates in the different zones of a counter-current separation unit on the fulfillment of the separation conditions set above, in the absence of mass transfer resistances and of axial dispersion. This led these authors to develop the triangle method, which was named for the triangular region bounding the solution space. They showed that the separation is complete if the operating point is inside a triangle, the sides of which can be derived from the flow constraints in each zone of the SMB unit [28,43]. This triangle method is used to discuss the operating conditions required to achieve complete separation and to optimize these conditions when the adsorption equilibrium isotherm is linear or follows Langmuir behavior. In the latter case, the triangle bounding the solution space becomes curvilinear. This method has been successfully applied to numerous SMB separations [20,21,43-45]. In this approach, the key parameters are the ratios rrij, with j = 1 to 4, between the net fluid flow rate and the solid phase flow rate in each section, j , of the unit. These ratios are given by the following equation [28] QTT M B
- -
Alternately, these equations can be written in terms of the operating parameters of the equivalent SMB unit by using the relationship Q™B = Q;SMB -Qs/F: Q SMB
m
i
=
or: m,
~^-p
(17 58a)
-
QSMBJ* _
=
where Q^MB is the volumetric flow rate inside section j of the SMB unit, t* is the switching time, e the total column porosity, and V the total or geometrical column volume. In the case of either a linear or a Langmuir adsorption equilibrium isotherm, the equilibrium theory demonstrates that a single, multi-section counter-current adsorption column, operated with constant boundary conditions, achieves eventually a steady-state. The concentration profiles and histories of this column depend on the values of the flow rate ratios, rrij [22]. Accordingly, the steady-state regime of a four-section TMB (j = 1, • • •, 4), as well as the corresponding cyclic steady-state regime of the equivalent SMB unit depend only on the feed composition and on the values of these four flow rate ratios, ttij,j — 1, • • • ,4. Therefore, for a given feed composition, the design issue of both the TMB and the SMB units
814
Simulated Moving Bed Chromatography
reduces to the development of suitable criteria for the selection of the values of the ntjS [28]. Only specific combinations of these values are compatible with a complete separation. The conditions that the flow rate ratios must satisfy in order to achieve such a separation were derived from the equilibrium theory [28,38]. These constraints imply that in sections II and III the net flow rate of components B and A must be negative and positive, respectively and that in sections I and IV the net flow rates of these two components must be positive and negative, respectively.
17.6.4 Optimization of the SMB Process with a Linear Isotherm Using the Triangle Method For a linear isotherm (Q,- = a,-Q,i = 1,2), following the procedure discussed above, it is easy to prove that the necessary and sufficient conditions for a complete separation are the following inequalities [43]:
^ l-ep
a2 < «i < fli <
mi < oo m2< a2 m3 < a2
(17.59a) (17.59b) (17.59c)
<
mA < «i
(17.59d)
These constraints define a region in the four-dimension space whose coordinates are the operating parameters m.\, m2, m^ and m4. The points of this region represent the operating conditions corresponding to a complete separation. Sections II and III of the TMB unit play a key role in performing the separation. A positive feed flow rate implies m^ > m2 and the constraints given by Eqs. 17.59b and 17.59c can be rewritten as : «i < m2 < 1113 < a2
(17.60)
These inequalities define the projection of the four-dimension region of complete separation onto the (m2lm^) plane, as shown in Figure 17.10. The regions in which the separation can be achieved are limited by the constraints m2 < a2 and m^ > a\. If either the former or the latter inequality is not fulfilled, the extract or the raffinate, respectively, is flooded with solvent and no separation is obtained. The region of the (m2, m^) plane where the separation can take place includes three other regions, besides the triangle-shaped completeseparation region. In the part where m^ > a2 and a.\ < m2 < «2 the constraint 17.59c is not fulfilled. Therefore, the extract is insufficiently retained, it is carried forward, pollutes the raffinate stream, and drives its purity below 100%, whereas constraint 17.59d being satisfied, the purity of the extract is still 100%. For similar reasons, the region where m2 < ci\ and a\ < m^ < a2 corresponds to operation conditions leading to a purity of the extract that is less than 100%, whereas the purity of the raffinate stream is 100%. The region where both m^ > a2 and m2 < a.\ corresponds to the operating conditions for which both components distribute
17.6 Optimization of the SMB Process
815
no pure outlet
pure Extract
/
w
Figure 17.10 The (ni2, m^) plane with the four different regions for a system characterized by a linear isotherm. Reproduced with permission from M. Mazzotti, G. Storti, M. Morbidelli, J. Chromatogr. A, 769 (1997) 3 (Fig. 3).
E+R pure
/ /
2 •
pure Raffinate / b
/ 2 m2
between the two product streams and both streams have a purity that is less than 100%. In summary, in the framework of the equilibrium theory, and provided that the flow rate ratios m\ and ra4 fulfill their relevant constraints (Eqs. 17.59a and 17.59d), the position of the operating point in the (m^, m$) plane in one of the four regions of separation allows a correct prediction of the separation performance of the unit [43]. The SMB design derived from the triangle method is typically represented by a two-dimensional plot of the quantities rn
17.6.5 Comparison of the Safety Margin and the Triangle Method for Linear Isotherms The two main methods used to select the experimental conditions under which there is complete separation in linear SMB are the safety margin app'roach [27,46] and the triangle method [16,28,43]. It is easy to show that both approaches are equivalent in the linear case [47]. Equations. 17.50a to 17.50c result from the Safety Margin approach. Rewriting them and using the classical relationship QjMB = QSMB _ Q s / f leads to the following equations for the SMB process: -iSMB
1
Q1MO/Qs - p
(17.61a)
1 "F 1
(17.61b) «2
(17.61c)
816
Simulated Moving Bed Chromatography
Qg--\
= f-
(17.61d)
Qs f piv The left hand sides of these four equations are identical to the corresponding ratios of the net fluid flow rates and the solid phase flow rate, rrij. These ratios are used as the basis for the separation conditions derived in the triangle method (see Eqs. 17.59a through 17.59d of the previous section). In these equations, mi must be larger than WI3 because the liquid phase flow rate must be larger in section III than in section II. Substituting mi and M3 in this condition (mi > m^) (Eq. 17.60) with the equivalent expressions mi = a\f>u and mo, = ail'$m, respectively, gives directly aifin > ailfim, i-e., Eq. 17.51 (pnpui < «)• Similarly, if m;- in Eqs. 17.59a to 17.59d is replaced by the corresponding expressions using Eqs. 17.61a through 17.61d, Eq. 17.52 (/3y > 1) is fulfilled. If we choose /3jy greater than one, m± = fli//5jy, which is positive, cannot be smaller than the lower limit value given by Eq. 17.59d (which has a negative value). The void fraction inside the particle, ep, can be neglected in this equation and the lower limit is in fact zero. In conclusion, the two sets of operating conditions given by the two methods are equivalent.
17.7 Nonlinear, Ideal Model of SMB The transient behavior of a continuous countercurrent multicomponent system was considered in detail by Rhee, Aris and Amundson [22,23] from the perspective of the equilibrium theory, i.e., assuming that axial dispersion and the mass transfer resistances are negligible and that equilibrium is established everywhere, at every time along the column. The final steady-state predicted by the equilibrium theory is simply a uniform concentration throughout the column, with a transition at one end or the other. Therefore, the equilibrium theory analysis is of lesser practical value for a countercurrent system, which normally operates under steady-state conditions, than for a fixed-bed (i.e., an SMB) system, which normally operates under transient conditions. The equilibrium theory analysis, however, reveals that, under different experimental conditions, several different steady-states are possible in a countercurrent system. It shows how the evolution of the concentration profiles may be predicted in order to determine which state is obtained in a particular case. In general, the feed injection in a countercurrent system takes place at one end of the column. In a long enough column, equilibrium is reached at a certain distance from this zone. The size of this zone depends only on the fluid and solid phase concentrations of the feed stream components and on the flow rate ratio, mj. It is independent of the initial state of the column [22,23].
17.7Nonlinear, Ideal Model of SMB
817
17.7.1 Optimization of the Operating Conditions for a Nonlinear Isotherm Using the Triangle Method In the development of a new SMB application, two design issues must be solved. The first one is the size of unit, namely the number of columns to be used in each section, their length, their diameter and the average particle size of the packing material to be used. Depending on the required production rate, these parameters will be chosen with the aim of optimizing the column efficiency while keeping the pressure drop below a given upper limit [48]. The second issue arises when an existing SMB unit is available and the objective is then to optimize the operation conditions in order to achieve the optimum separation (i.e., the production rate for a given purity of the product(s)). In this case, we assume that the SMB unit, i.e., its geometric dimensions and the nature of the packing materials are fixed and that the adsorption isotherm of the feed compounds is known. Therefore, the operation parameters to be optimized are the internal flow rate, Qj, the switching time, t* (corresponding to the solid flow rate, Q$, in the equivalent TMB unit, with V/t* = Qs/(1 — e)) and the feed composition. One approach for selecting the operation conditions consists in applying a McCabe-Thiele-like analysis to an ideal stage-by-stage model of the unit [13]. This approach can be applied to systems described by any kinds of isotherm but it is limited to binary separations. A second approach is the triangle method which was developed based on the equilibrium theory model which assumes that the adsorption equilibrium is established everywhere at any time in the column. The equivalent TMB configuration with a four-section unit will be considered here. The model equations consist in four sets of mass balance equations, one for each section j (j = 1, • • •, 4), with the relevant boundary conditions and the integral material balances at the column ends and at the nodes of the unit [16,28]. These equations were given earlier, in Section 17.2 (Eqs. 17.4 to 17.6). 17.7.1.1 Constant Selectivity and Competitive Langmuir Isotherm Let us consider the separation of a binary mixture whose adsorption equilibrium can be described by the classical competitive Langmuir isotherm:
^f
(= 12
'
(1762)
-
In the framework of the equilibrium theory, the mass balance equation can be solved analytically in the case of competitive Langmuir isotherm behavior for a TMB column with constant initial and boundary conditions [22]. Applying the method discussed earlier allows the derivation of the following necessary and sufficient conditions that the flow rate ratios, m,j, must satisfy for a complete separation to be achieved [28,38^0]: «2 = <
m1/min < rn\ < oo m2<m3< m3/max
(17.63a) (17.63b)
Simulated Moving Bed Chromatography
818
pure Extract
Figure 17.11 The (mj, m.3) plane with the four different regions for a system characterized by a nonlinear isotherm (see text). Reproduced with permission from M. Mazzotti, G. Storti, M. Morbidelli, J. Chromatogr. A, 769 (1997) 3 (Fig. 4).
-e-p 1-e-p
(17.63c)
m 4/max
The eluent must be fully regenerated at the exit of section IV which means that m4 has an upper limit. Mazzotti et al. [40] showed that this limit, m 4 m a x , is given by ?«4,max
f (m3 ~ tn2)-
=
m3
2
\
- m2)] - 4flim3 J
(17.64a)
where a\ and b\ are the first and the second parameter of the Langmuir isotherm for component 1, respectively, and C\ is the concentration of component 1 in the feed. The lower limit of m.\ and the upper limit of m^ are explicit. The lower limit of m\, m\iU^ = a2, does not depend on the flow rate ratios nor on the feed composition, whereas the upper limit of m4, m±irQSXI depends on the flow rate ratios m2 and m^ as well as on the feed composition. The two flow rate ratios, mi and j«3, correspond to the two sections of the SMB where the separation actually takes place. Their values do not depend on either m\ or m4. Hence, they define the complete separation region in the {m^, nio,) plane. This is the curvilinear triangular region that is shown in Figure 17.11. The explicit boundaries of this region were calculated by Mazzotti et al [40]. It is bound by the straight lines afb, bw, and wr, and by the curve ra. The points a, b, and / are on the bisector of the plane {m2lm3). The coordinates of these three points and of w and r are given by • Point a, {fl2, a2), where a2 is the slope of the isotherm of the more retained solute • Point b, (flj, ai), where a.\ is the slope of the isotherm of the less retained solute
17.7 Nonlinear, Ideal Model of SMB
819
• Point/, (*i,xi). • Point w,{ylry2). • Point r, (zi,Z2). The parameters x\ and x2 (x\ > x2 > 0) used in this derivation are the roots of the following second degree equation: (1 + b2C\ + feiCf )xz - [«2(1 + feiCf) + «i(l + b2Cl)]x + ata2 = 0
(17.64b)
The parameters, y\, y2, z\, z2 are given by the following equations: yi
=
^
(17.64c) a2 x1[x2(al-a1)+a1(a1-x2)] ax{a2-x2)
=
z1
=
x2 -^
(17.64e)
fl2 z
=
x\\x2(a2 - xi)(a 2 - fli) + gix x (g 2 - x2)\
2
Points / , w, and r depend on the feed composition through the parameters of X\ and x2. The equations of the boundaries of the complete separation region are • Straight line ab: m$ = m2, • Straight line wf: [a2 - *i(l + b2Cl)]m2 + b2C\xxmz = xx{a2 -
Xl)
(17.64g)
[a2 - «i(l + b2Ci)]m2 + b2Cla1m3 = ax(a2 - ax)
(17.64h)
• Straight line wb:
• Curve ra: (17.641) All these equations depend on the feed composition. The separation region shown in Figure 17.11 is similar to the one obtained in the linear case (see Figure 17.10). As in this simpler case, in the nonlinear case there are three different separation regions around the complete separation region. Proceeding clockwise from the lower left corner of the (m2,OT3)plane, there can be found the pure raffinate region, the no pure outlet region and the pure extract region (see Figure 17.11). There are important differences between the linear and nonlinear cases, however. In the nonlinear case, all the boundaries of the complete separation region are no longer linear and they depend on the feed composition. Also, the constraints on the flow rates are coupled and are no longer independent as they are in the linear case.
820
Simulated Moving Bed Chromatography
17.7.1.2 Nonlinear Multicomponent System Explicit criteria for the selection of the design and operating conditions can be obtained for systems in which the different components of the feed have Langmuir isotherm behavior with a constant selectivity, i.e., when the competitive isotherms are:
with a, — qsbj, where qs is the saturation capacity of the adsorbent which is assumed to be the same for all the components of the mixture [28,38,43]. 17.7.1.3 Optimum Operating Conditions We need now to determine what are the optimum operating conditions for producing the pure components of a feed having a given composition. The point representing the process must be located within the region of complete separation. The optimum operating conditions correspond to a minimum desorbent flow rate (and consumption) and to a maximum enrichment and production rate. All the performance parameters improve when the difference {mj, — m2) is increased, i.e., when the operating point is farther removed from the bisector of the axes, towards the vertex w of the complete separation region. It follows that point w represents the optimum operating conditions, as far as only the flow rate ratios m2 and WI3 are considered For Langmuir competitive isotherms, the optimum flow rate ratios are [43] m2,opt =
—
F
(17.66a)
«2 r
= *iai(a2-x2) At the optimum value of m2 and m^, the flow rate of desorbent that is necessary to operate the SMB decreases and the concentrations of the compounds 1 and 2 in the products increase with decreasing values of m\ and increasing values of m^. So, for Langmuir competitive isotherm behavior, we have Wl,opt = mi,min = «2
(17.67)
and ttld
nrit
=
TtlA mav
=
— I Ol "I" W^J + bl Ci (fffo — WT ) —
y [«i + m3 +fciCf(m3 - m 2 )] 2 - 4aim3 j
(17.68a)
While, for a linear isotherm, t h e o p t i m u m flow rate ratio will b e given b y : fl2 (17.69a) «i
(17.69b)
17.7 Nonlinear, Ideal Model ofSMB
821
The triangle method approach discussed earlier provides a useful tool for the rapid selection of satisfactory experimental conditions for operating an SMB under either linear or nonlinear conditions [45]. It can be used to determine safe and robust conditions that are not too far removed from the optimal conditions. Optimization using this approach is quick and efficient. Generally, an optimization procedure based on the triangle method includes the following four steps [45]. First, the adsorption isotherms are determined accurately, in a concentration range which must include the anticipated feed concentration. Second, the feed concentration and a value of the switching time (t*) are carefully chosen. The selection of the switching time is the result of a compromise: too short a switching time will give a high production rate and a short start-up time (the time required to approach a given distance from steady-state) but it will also lead to high flow rates and possibly to high axial dispersion (i.e., important deviations from ideal behavior). Therefore, there is a lower limit for t*, set according to the maximum pressure and/or to the maximum flow rate that can be allowed. Furthermore, the use of a short switching time would accelerate the wear and tear of the switching valves, hence would cause higher maintenance costs. Third, proper values are selected for m.\ and JM4, values that must fulfill the inequalities mentioned above (see Eqs. 17.63 and 17.64). Finally, the (m^m^) diagram is plotted and the values of mi and OT3 are selected so that the operating point lies inside the separation triangle. This point should be close to the triangle apex in order to achieve a high production rate, yet far enough to assure the robustness of operation and the purity of the collected streams, which can be achieved only at the expense of separation performance [43]. The safe distance of the operating point from the apex of the triangle increases with decreasing column efficiency because the triangle theory assumes this efficiency to be infinite. Due to the uncertainty of the parameters of the theoretical model, and to nonideal effect, the theoretical optimal operating condition obtained from the triangle method is not robust in actual application. 17.7.1.4 Robustness of the Operating Conditions Under the optimal operating conditions derived above, the performance of the unit is not robust [28]. It is very sensitive to various possible disturbances, such as perturbations in the operating conditions, inaccuracies in the physico-chemical parameters, and model uncertainties. Either perturbations in the operating conditions (e.g., in the flow rate or the switching time) or inaccuracies in the estimation of the geometrical characteristics of the columns, such as their inner volume or the void fraction of the bed, may modify the flow rate ratio j«y. Thus, the operating point may be moved from the expected location in the complete separation region to another position, which is outside of the complete separation region. On the other hand, perturbations in the feed composition or model uncertainties, such as the use of poor estimates of the isotherm parameters, modify the shape and the location of the complete separation region according to Eqs. 17.63a to 17.63c and Eqs. 17.64a to 17.64i, without having any impact on the values of flow rate ratios my. Also, when the mass transfer resistances are significant, the triangular region of complete resolution shrinks but it still defines an infinite set
822 Figure 17.12 Effect of an increase of the feed concentration on the shape of the complete separation zone in the (ni2, MJ3) plane. Isotherm parameters: qS)1 = 46.58 g/1, qSi2 = 49.75 g/1, h '= 0.0644 1/g, b2' = 0.0804 1/g; concentrations: Cy j = Cj 2 = CfM/2, with CfM =' 0.02 g/1 in g/1 in I, 2.5 g/1 in II; 5 g/1 in III; and 7.5 g/1 in IV. Reproduced with permission from S. Khattabi, D.E. Cherrak, K. Mihlbachler and G. Guiochon, ]. Chromatogr. A, 893 (2000) 307 (Fig. 2).
Simulated Moving Bed Chromatography
4.5
£3.5
3
2.5
X
Ex -
of combinations in the (m2,m3) plane that allow complete separation [49]. Based on these consideration, it is obvious that the optimum point w in the (jn-i, m^) plane is not robust since the slightest disturbance may drive the operating point out of the complete separation region, into one of the three separation regions where a complete separation is not achieved (see Figure 17.11). The robustness of the operation conditions can be improved only at the expense of the separation performance, i.e., by choosing an operating point within the complete separation region, far enough from the optimal point, w. Mihlbachler et al. [26] have shown that, if the adsorption isotherm behavior deviates profoundly from Langmuir behavior, it is still possible to operate quite satisfactorily an SMB unit and to produce highly pure products. However, in such cases, good actual separation conditions cannot be predicted with the triangle method (see Figure 17.12). The only method that allows an optimization of the experimental conditions consists in calculating the concentration profiles along the column train and the concentration histories at the draw-off ports and in searching for the set of parameters that optimize the objective function completed with the constraints required, e.g., the purities of the collected streams (see Section 17.9). 17.7.1.5 Influence of the Heterogeneity of the Column Set on SMB Performance Although all models of SMB and all theoretical discussions of this process assume that the columns of an SMB train are identical, it is impossible to achieve the preparation of eight to sixteen identical columns. The most reliable procedure consists in weighing as many samples of packing material as there are columns to pack, in making sure that these weights are identical within the limits of precision of the balance used, then, when packing the columns and closing them, in avoiding the formation of any void volume at their end. These steps are reasonably practical only if some form of axial compression is used. Even so, there will be differences in packing density, external porosity, and permeability between the columns obtained. Some data regarding the distribution of column character-
17.7 Nonlinear, Ideal Model of SMB
823
istics are available in the literature [45,50,51]. The reproducibility of commercial columns has been studied under analytical conditions for several different brands [52-56] and under nonlinear conditions for Kromasil columns (Eka-Nobel Bohus, Sweden) [57]. It was found to be extremely satisfactory, the fluctuations of the retention times of moderately polar compounds on columns from the same lot having a relative standard deviation (RSD) of the order of 1% [55,57]. However, it is unlikely that columns prepared in small series could achieve comparable RSDs. Thus, it is important in practice to assess the influence of fluctuations of the characteristics of the columns in an SMB train on the performance of the unit. This study was performed by Mihlbachler et al. [25,26]. The authors considered only the effect of fluctuations of the total porosity of the column. This parameter is important because it controls both the column permeability, hence the profiles of pressures and flow rates along the different sections of the SMB, and those of the phase ratio, F — (1 — e)/e. Accordingly, the column to column fluctuations of the phase ratio will be
Since e is usually between 0.65 and 0.80, the fluctuations of F will be several times larger than those of e. This has important consequences as it magnifies the effect of column to column porosities [26,45,50,51]. In the following we consider that each column has its own value of the phase ratio, F;. In the following discussion, we examine only the effect of differences in the column porosity. Let t\ be the porosity of column k. As discussed earlier Eqs. 17.51 and 17.52 define a sufficient number of criteria to allow the correct choice of the operating conditions in an SMB operating under linear conditions. This set of conditions is equivalent to the one derived by Storti et al. [16], the so called triangle method (see Subsection 17.10). However, both sets of conditions are based on the assumption that all columns have identical characteristics and an infinite efficiency. 17.7.1.6 Revised Set of Separation Conditions When the columns (k = 1, •••,«) of an SMB have different properties, the true period of the system, the time after which the system returns to its initial conditions, is no longer the switching time but is n times larger. We call this time, nt*, the superperiod of the SMB. In this subsection, we discuss the effects arising from the columns in an SMB train having different porosities, hence different retention factors for the components of the feed. The retention factor of component i in the physical column k (the column that will be successively part of the four different sections of the SMB) is proportional to the initial slope of its isotherm, «j, and to the phase ratio, Ffc. As discussed earlier, the phase ratio depends on the total porosity of the bed, ejfc. Both parameters are part of the equations used to calculate /Sy (Eqs. 17.49 to 17.50c). Thus, the exact values of these design parameters will vary from column to column. In some cases, it might happen that one of the j8^y is less than one
824
Simulated Moving Bed Chromatography
during one of the cycles constituting the superperiod while it remains larger than one during the other cycles (or at least during most of them). The separation conditions derived above have to be extended to cover such cases. The complexity of the situation arises from the consequence of this observation, that every column leads to a different set of separation conditions. Thus, the total number of these conditions equals the total number of columns in the SMB process considered. Obviously, the characteristics of the production (e.g., the amount of raffinate and extract produced per cycle and the purity of these two output streams) will vary during a superperiod. Our first aim becomes the derivation of a revised set of separation conditions that allows the choice of the flow rates through the four sections of an SMB process, taking into account the characteristics of the different columns used. Equation 17.58b can be written as Q SMB f *
_ y£
QSMBJ*
where t* is the switching time and V is the total column volume. Equation 17.71 can be rewritten for columns having different porosities ek:
In the relationship a\ < mi < m^ < a2 (Eq. 17.60), mi and mo, can be replaced with the above equation. Each therm of the inequality is multiplied by (1 — ek), and then ek is added. Q
ek< ^ L
QSMBX*
< M I L < (i _ £k)a2 + efc
{1773)
At this point the following new variables ry, gk and hk are introduced: rj
=
- ^ -
(17.74a)
{l-ek)ai+ek
(17.74b)
= (l-ek)a2 + ek
(17.74c)
gk =
hk
Introduction of these new variables into Eqs. 17.50a to 17.50d gives the revised set of separation conditions for SMB under linear conditions: %
<
r
i < °°
gk < rn < rin < hk 0 < rIV
(17.75a)
(17.75b) (17.75c)
The important advantage of this extension compared to a method using either the safety margins, /5y [27], or the triangle method [16,28], is that ry does not depend on the porosity of the columns. Thus, the operating conditions, which are described by the four internal flow rates, do not change depending on the subset of physical columns contained in the section considered during a given cycle.
825
17.7 Nonlinear, Ideal Model ofSMB Figure 17.13 Separation triangles based on the equilibrium theory for different porosities under linear conditions. Isotherm parameters: fli = 3.5 and «2 = 4.2. Solid line triangles - separation triangle for the porosities of e = 0.7 (L0.7) and e = 0.65 (L0.65) and for the average porosity of 8 columns of e = 0.6875 (L8av). Dashed line - horizontal scanning of the triangle (rui = 1-92). Reproduced with permission from K. Mihlbachler,}. Fricke, T. Yun, A. Seidel-Morgenstern, H. Schmidt-Traub and G. Guiochon, ]. Chromatogr. A, 908 (2001) 49 (Fig. 2).
L0.65
/
L8av L0.7
/
/ 1.95
rll
Only the boundaries of the inequalities that define the conditions to be satisfied by the ry's are functions of a,- and ek. These boundaries depend on the specific columns considered. As shown earlier [16,28], the constraints for the flow rates in the central sections of the SMB unit (Eq. 17.74c) can be represented by a simple graph: the separation triangle. To achieve complete separation based on the equilibrium theory [58], the operating conditions must be chosen in such a way that the corresponding point in the (rjj, rm) plane is located inside a triangle defined by the parameters of the separation (see Figures 17.12 and 17.13). Thus, when the columns are different, there is a different triangle in the (ru, rm) plane for each column. For complete separation, the point representing the experimental conditions selected must be inside all of these triangles. Since the production rate depends on the location of the point inside the triangle, variations between the column characteristics will cause a reduction in the productivity of the unit considered. A revised set of Eqs. 17.75a to 17.75c can be written to extend these separation conditions to nonlinear SMB operation [59]. In the case of a set of columns having different porosities, and on the basis of the triangle model, the separation area in the case of a binary mixtures the components of which follow competitive Langmuir isotherm behavior becomes defined by the set of equations a
2k
r
(17.76a) (17.76b)
l
+
[a*lk
hcfeed/1 (rUI - ru) (17.76c)
-\j[a *lk +
+ hcfeed,i{rui -
The lower boundary of r\y is set to zero due to the negligible value of e p . Mihlbachler et al. [26,59] discussed the triangle method for the optimization of
826
Simulated Moving Bed Chromatography
SMB using this revised set of conditions. They applied their results on the operation of SMB units fitted with trains of columns having slightly different porosities to the separation of the enantiomers of Troger's base on Chiralpak AD, a cellulose carbamate (see isotherm data in Figure 4.28), a case in which the competitive isotherms deviate strongly from Langmuirian behavior. Their results are discussed in Section 17.9.2.
17.8 Recent Improvements in SMB Performance with New Operating Modes In order to improve the performance of SMB units and to make them more efficient, more flexible, and more competitive, several new operation modes have been introduced. These includes the use of supercritical fluid SMB [60], the operation of conventional SMB units with a temperature gradient [61], a pH gradient [62], or a solvent gradient SMB [63], and the design of SMB processes that can deliver more than two streams of purified products [64,65]. The operation of SMB under gradient conditions improves the separation performance by allowing the separate optimization of the adsorption strength of each solute in the different sections of the unit by creating a gradient of pressure, temperature, or solvent composition along the unit. Finally, new processes have added flexibility and performance by no longer switching the different ports simultaneously or by changing the feed flow rate during each period.
17.8.1 Supercritical Fluid Simulated Moving Bed (SF-SMB) There are two main advantages in replacing the organic eluent traditionally used in SMB with a supercritical fluid, most often carbon dioxide. 1. The easy separation of the purified products and the desorbent and the full compatibility of the fluid phase with any product bound to be used by humans. This may be of particular importance for the production of pharmaceuticals of food additives. 2. The easy tunability of the properties of the mobile phase, such as its density, viscosity and solubility. From a technical viewpoint, the main performance improvements are due to the possibility of tuning in a rather wide range the elution strength of the mobile phase by adjusting the local pressure. This possibility is limited only by the impossibility of avoiding a significant pressure drop along a column, with the consequence that the isotherm coefficients depend on the position along the column. An SF-SMB unit can be operated in two different modes: 1. The isocratic mode, in which the pressure in the four section of the SMB unit is nearly uniform. 2. The pressure gradient mode, in which there are significantly different average pressure levels kept nearly constant along each section of the SMB by using back-pressure valves.
17.8 Recent Improvements in SMB Performance with New Operating Modes 827 Since the adsorption strength decreases with increasing density of the fluid phase, a pressure gradient decreasing along the column yields a correspondingly decreasing elution strength gradient. The fluid phase density, and therefore its elution strength, is kept at a maximum in section /, where the most retained component must be desorbed and eluted back into the extract stream to regenerate the column. The fluid phase density is kept at a minimum in section IV, where the least retained component must be adsorbed by the solid phase to avoid it escaping with the fluid phase and being recycled with it into section I [60,66]. Useful criteria to specify suitable operating conditions of SF-SMB were derived on the basis of the equilibrium theory [60,66]. The operating pressure and the density have a significant effect on the retention. So, by applying a suitable pressure gradient, with the pressure decreasing from section I to section IV, the productivity can be significantly improved compared to that achieved under constant pressure. This effect was shown theoretically [60,66] and demonstrated experimentally [67]. Denet el ol [67] reported the separation of the enantiomers of Tetralol by SF-SMB, using a set of Chiracel OD columns. The complete separation of the Tetralol enantiomers was achieved, whether the SMB was operated under isocratic or pressure gradient conditions. As theory predicts, the productivity was found to be three times higher in the second case, in which a pressure step of about 50 bar was imposed between section II and III of the SMB, compared with the case in which the pressure was kept constant.
17.8.2 Temperature Gradient SMB in Simulated Moving Bed The possibility of using a temperature gradient, such that sections I and II of an SMB unit would be operated at a higher temperature than sections III and IV has been investigated theoretically. This method would achieve with a solvent as the fluid phase the same goal as a pressure gradient does in SF-SMB. The advantages and limitations of this approach have been discussed in detail [61] but, so far, no experimental investigations have been carried out using a temperature gradient.
17.8.3 Operation of a Simulated Moving Bed Unit under a Solvent Gradient Gradient elution chromatography is a separation method that exploits the effect of the fluid phase composition on the retention behavior of the feed components. It is widely used, especially for analytical separations in the areas of the life sciences, in biochemistry, and in the biotechnologies (e.g., separation of complex mixtures of proteins or peptides). In its conventional implementations, SMB units are operated under isocratic conditions. The composition of the fluid phase, e.g., the organic modifier concentration, the pH, or the buffer concentration remain constant in all the sections of the SMB unit. However, it has recently been shown that SMB units can also be operated under solvent gradient mode (SG-SMB). Then, the feed and desorbent streams introduced have a different composition. The fluid phase composition is different in each section. It is chosen independently, in order to
828
Simulated Moving Bed Chromatography
Figure 17.14 Schematics representation of an equilibrium stage X. Re? . . , . produced with permission from D. An'
r
%,^SIM I J I_
J
tos and A. Seidel-Morgenstem,
*T"xlc,x " | .. v vp ' CiF or (Feed stage, F) i-,, " V D ,C l D =0 (Desorbent stage, D) x ~F.
x 1
>-.
J. Chro-
J
^ • csx*
matogr. A, 944 (2002) 77 (Figure 3).
*-> x+1
I
->vR,CiR
vE, c l 6
vTp
or (Raffinatestage, R)
(Extract stage, E)
w
reduce the adsorption and the retention times in sections I and II and to increase them in sections III and IV [63,68,69]. Stefanie et al. [68] studied a closed loop SMB unit in which two solvent mixtures of different compositions are used as the feed solvent and as the desorbent for a binary separation. For such SMB systems, these authors derived the region of separation and showed how the optimum operating conditions can be found, using the equilibrium theory, i.e., neglecting axial dispersion and the mass transfer resistances, and assuming linear equilibrium isotherms. They also assumed in their calculations that the separation performance of the SG-TMB unit is the same as that of the SG-SMB. They used the following relationship to account for the dependence of the affinity of the solutes for the solid phase in the presence of a fluid phase of variable composition {i.e., for the variation of the initial slope of the isotherm of the solute or its a parameter with the solvent composition)
ai
(i7 77)
=(i4^
-
where x is the volume fraction of the weak solvent, af is the initial slope of the isotherm of the strong solvent in the weak solvent {i.e., for x = 0), ki and «,• are positive, empirical numbers describing the effect of x on the retention of solute i. A mobile phase gradient is established by feeding two fluid phases of different compositions through the desorption and the feed ports. The volume fraction of the weak solvent in the desorbent and in the feed are Xp> and Xp, respectively. Within the SMB unit, the mobile phase composition varies between xp and Xp. Usually XD < Xp, the desorbent being a stronger solvent than the feed solvent in order to have a greater elution strength. This ensures that the retention time of the feed components will be smaller in sections I and II than in sections III and IV. The situation is illustrated in Figure 17.14 for an equivalent TMB and neglecting axial dispersion. Under steady-state, we have a step gradient, with a liquid phase of composition Xi in sections / and II and X3 in sections III and IV. These values depend on Xu and Xp and also on the external and internal flow rates. They can be calculated through the mass balances of the weak solvent at the desorbent and the feed nodes. =
x3Q™B
(17.78a)
=
B
(17.78b)
x2Q™
where Q™B is the volumetric flow rate in section i of the equivalent TMB. From these equations, the following inequalities can be derived: xD < x2 < x3 < xp
(17.79)
17.8 Recent Improvements in SMB Performance with New Operating Modes 829 In an SG-SMB, the actual profile of the concentration gradient is different from what it is in the ideal case, the one that is illustrated in Figure 17.14, due to the nonsteady state nature of the SMB behavior and to the axial dispersion that was neglected but cannot be avoided. 17.8.3.1
Operating Conditions of SG-SMB for Complete Separation
When the equilibrium isotherm is linear, the following conditions apply to the flow rate ratios. These conditions are necessary and sufficient to achieve complete separation. They were derived for isocratic conditions [16,27] and will hold in the gradient mode. a2{x2)
<
mi
(17.80a)
m2< a2(x2) m3
(17.80b) (17.80c) (17.80d)
Since a,- depends on the mobile phase composition through the parameters x2 and x3, the lower and upper bounds of the flow rate ratios, ntj, also depend on the mobile phase composition as well. The mass balance equations 17.77 and 17.78a can be rewritten in terms of the flow rate ratios as follow: m3(xF-x3)
= m2(xF-x2)
(17.81a)
m1(x2-xD)
= mA(x3-xD)
(17.81b)
Solving these last two equations for x2 and x3 in terms of the flow rate ratios and of XQ and xp and substituting the resulting expressions in Eqs. 17.80a yields a set of inequalities relating the four unknowns nij (j = 1, • • • 4). This system of equations is nonlinear due to the nonlinear dependences of the «,'s on x through Eq. 17.77. This system of inequalities defines a region in the four dimensional space spanned by the four flow rate ratios. In the framework of the equilibrium theory, the points belonging to this region correspond to the experimental conditions that allow the achievement of a complete separation of solutes A and B, one in the extract and the other in the raffinate. The boundaries of this region depend on the composition of the two streams pumped into the SMB unit, the desorbent, xp, and the feed, Xp, and on the retention times of the solutes, which is described by Eq. 17.77. Stefanie et al. [68] discussed the region of complete separation using the above approach in detail. They also used a numerical solution of the SMB process based on a kinetic model with a linear driving force model to describe the behavior of each individual chromatographic column [21]. The results of the numerical calculations carried out using this nonideal model confirm the validity of the theoretical analysis based on the equilibrium theory, particularly in identifying the typical separation regimes and in locating accurately the corresponding separation regions in the operating parameter space [21,49]. These authors compared also the solvent gradient mode with the isocratic mode in terms of productivity
830
Simulated Moving Bed Chromatography
and solvent consumption. They demonstrated that the solvent gradient mode reduces the solvent consumption and increases the concentrations of the products collected. Antos et ah [69] used an equilibrium stage model and assumed a linear isotherm. They derived numerical solutions by using the equilibrium-dispersive model. Various strategies for determining sets of suitable operating parameters were examined, based on extensive numerical calculations. Different strategies for designing the solvent gradient SMB process were discussed and compared with the goal of maximizing the productivities and of minimizing the desorbent consumption.
17.8.4 Improving SMB Performance by Operating Under Complex Dynamic Conditions Another direction taken to improve the performance of the SMB process, is based on the concept of operating the unit under more complex forced dynamic conditions. In the classical SMB mode of operations, the column configuration (i.e., the distribution of the columns into different sections, most often four), the liquid phase flow rates and the feed concentration are constant during a production campaign (except for control adjustments). Several new operation modes of SMB, the Varicol [70], the Powerfeed [71], and the Modicon [72] processes were recently developed. In these new operation modes of the SMB process, the column configuration, the liquid flow rates, and the feed concentration are varied during each successive cycle, respectively. Therefore, these processes can achieve better performance than the conventional, simple SMB process because more degrees of freedom are introduced in the conduct of the unit and the continuous chromatographic process can be operated under more complex dynamic conditions. 17.8.4.1
The Varicol Process
The principle of the Varicol process consists in a non-synchronous shift of the inlet and outlet ports of a multi-column system on a recycle loop [70]. The Varicol process is characterized by a even more efficient use of the stationary phase than the one achieved in the SMB process. Both TMB and SMB include four zones that are unchanged during operation. Both the SMB and the Varicol processes consist in a series of columns that are connected in series, with inlet/outlet lines connected between the columns. In the conventional SMB process all these inlet/outlet lines are shifted periodically and simultaneously. This switching insures that all the inlet (feed and desorbent) and outlet (extract and raffinate) ports are all switched in the same time in the direction of the liquid flow. Accordingly, the number of columns in each zone stays the same at each moment. In the Varicol process, there are also four column zones, but the inlet and outlet lines are no longer shifted simultaneously. In other words, the column distribution between the four zones varies during a cycle. Like the conventional SMB process, the Varicol process is periodic. It returns to the same status at the end
17.8 Recent Improvements in SMB Performance with New Operating Modes 831 of the period. However, the column distribution between the different zones does not stay the same during the period because the lines are shifted at different times. Thus, the column allocation in each zone changes accordingly. The Varicol process allows for a finer optimization of the distribution of the columns among the four zones, in a much more efficient way than what is feasible in SMB. The SMB restriction that there can be only an integer number - at least one - of columns in each zone is eliminated. It is possible to have, on the average, less than one column per zone. Thus, a 3-column Varicol can operate as a four-zone SMB. The fact that the column distribution can be optimized means that, in general, a Varicol process performs better than the SMB process that uses the same number of columns. This reduces separation costs, a direct consequence of the better distribution of the solid phase among the four zones. In the Varicol process, the two outlet lines between each column must be connected to the recycling line before the eluent and the feed lines are shifted in the direction of the fluid stream. This modification to the conventional design prevents the feed flux from polluting the extract or the raffinate streams when the number of columns in zone II or III, respectively, falls temporarily to zero. This also prevents the eluent flux from diluting the extract or the raffinate flux when the number of columns becomes equal to zero in zones I or IV. The additional flexibility of the Varicol process can be used to optimize the column distribution among the different zones of the separator and this explains the higher efficiency achieved in terms of amount of product separated per amount of packing material used, compared to what is achieved with the standard SMB process. This advantage was shown in two recent theoretical case studies made first on a mixture of sugars with linear adsorption isotherm behavior, and second on a mixture of amine isomers with nonlinear adsorption isotherm behavior [73]. Ludemann-Hombourger et dl. [74] performed the separation of a racemic mixture comparing the SMB and the Varicol processes and showed that, for a 6-column system, the Varicol process performs better than the standard SMB one, both in terms of increased specific productivity and in terms of reduced eluent consumption. They also showed that, by accurately distributing the columns among the four different zones, the Varicol process allows in the same time a reduction of the number of columns used in the separation unit and an improvement of the specific productivity. Toumi et cd. [75] discussed the optimization strategy to follow for a systematic comparison of the SMB and the Varicol processes and applied their method to the separation of a mixture of propanolol enantiomers on 20 pm Chiralpak AD particles. The eluent was a mixture of heptane, ethanol, and diethylamine (80:20:0.1) and the equilibrium isotherm was nonlinear. Experimental adsorption isotherm data were well accounted for by a modified Langmuir isotherm. The authors investigated the application of the triangle method based on the equilibrium theory to the Varicol process. As discussed earlier, according to the equilibrium theory, the flow rate conditions under which there is complete separation with the true moving bed (TMB) process are represented by the points located in a triangular-shaped region. SMB was shown to be equivalent to a TMB if the number of columns is sufficiently large. Toumi et ah, however, found that, unlike the
832
Simulated Moving Bed Chromatography
SMB process, the Varicol process is not equivalent to a TMB and that the triangle method completely fails to predict the optimum operating condition in the Varicol process. This is mainly due to the fact that, in the Varicol process, the discrete switching dynamics has a strong effect on the region of complete separation. A numerical solution of the general rate model was used [75], that accounts for all the important sources of nonideal behavior, i.e, axial dispersion, pore diffusion, and the mass transfer resistances between the liquid and the solid phases, and is based on the numerical method introduced by Gu [76], in which the finite element discretization of the mass balance in the bulk phase is combined with orthogonal collocation of the mass balance in the solid phase. This model was used systematically to optimize the operating parameters of both the SMB and the Varicol processes. The general goal of the program was to find the optimum values of the operating parameters allowing the achievement of a given separation using a given unit. For this comparison of the SMB and Varicol processes, the goal was to minimize specific separation costs on a given unit, meeting the required purity of the products after the process has reached its cyclic steady state. The zones II and III are longer in the Varicol process than in a standard SMB process, which enables the higher feed throughput that is observed with the Varicol process. In all cases, the Varicol process can handle a higher feed flow rate than the SMB process. A higher production rate can be achieved with the Varicol process than with a standard SMB. Due to a more flexible use of the solid phase, the Varicol process needs one fewer column than SMB in order to perform the same separation, e.g., five columns with Varicol versus six with SMB. Finally, the superiority of the Varicol process is most pronounced when the total number of columns is small. 17.8.4.2
The Powerf eed Process
Another possible approach to improve the separation performance of SMB units is to change the internal and external liquid flow rates during the switching period. The Powerfeed process, like the Varicol process, exhibits more degrees of freedom than the classical SMB process and, therefore, allows more room for optimization. This new process was first proposed in a patent [71], then described by Kloppenburg et al. [77] who calculated the optimum flow rates using a numerical optimization method and pointed out the high potential of this process to decrease significantly the solvent consumption. Zang et al. [78] also studied this process. Later, these authors [79] used a multi-objective optimization technique to investigate in detail the possibility offered by the Powerfeed process and to compare the optimum performance that can be achieved with the Powerfeed, the Varicol, and the classical SMB processes under the same experimental conditions. The same stage-in series model described by Zhang et al. [80] has been adapted to simulate the SMB, the Varicol and the Powerfeed processes, with a slight revision that enables the column flow rates to change in the same time. These authors considered four separation problems, involving either linear and nonlinear adsorption isotherms. For each of them, they optimized the performance of the three processes with the same constraints, which allows a comprehensive comparison of the SMB, the Varicol, and the Powerfeed operations and production rates.
17.8 Recent Improvements in SMB Performance with New Operating Modes 833 These results show that both the Powerfeed and the Varicol processes provide a performance that is significantly improved compared to that of the conventional SMB process, and that the extent of the improvement achieved is larger for more difficult separations. A rigorous comparison between the Varicol and the Powerfeed processes in general terms is not possible, but it seems fair to say that they are equivalent in terms of potential performance, although the implementation of the Varicol process may be simpler than that of the Powerfeed process. 17.8.4.3 The Modicon Process Another possible way to improve the separation performance of SMB units is to change the feed concentration during each successive period. This operation mode, called the Modicon process, was suggested by Schramm et al. [72]. These authors optimized the process using numerical solutions of the equilibrium-dispersive model and compared the performance of the Modicon process with that of conventional SMB. They concluded that a cyclic modulation of the feed concentration allows a significant improvement of the separation performance The productivity and the product concentration can be increased and the specific solvent consumption decreased compared to those achieved with conventional SMB.
17.8.5 Multicomponent Separations in SMB The major drawback of the classical four-zone SMB is its inability to handle a multicomponent mixture and deliver a purified product free from the lesser and the more retained components. It cannot separate a multicomponent mixture into three different fractions. It can only separate either pure component 1 or pure component 3 (the most or the least retained compound). It cannot separate component 2 pure in a single pass. Several concepts have been proposed to modify the classical SMB and achieve this goal. A first approach consists in using a four-zone SMB, but either using two different adsorbents [81] or having a variation of the working flow-rate with respect to time within a switching period [71]. Multicomponent separations using a single unit of SMB were first proposed by Szepesy et al. [82]. The idea, however, was not pursued until Hashimoto et al. [81] reported a SMB process which can resolve a three-component mixture into its individual components. The process employed a series of individual columns packed alternately with two types of adsorbents. To separate a feed containing components A, B and C, a pair of adsorbents was selected so that the component A is the most strongly adsorbed solute on one of these adsorbents, component B is the most strongly adsorbed solute on the other adsorbent and component C is the least strongly adsorbed solute on both adsorbents. Under properly selected operation conditions, it was demonstrated that component C can be recovered as pure component in the raffinate stream while components A and B are collected alternately as pure products in the extract stream. This approach may not find wide applications because a suitable combination of different adsorbents must be found for each multicomponent feed and the difficulty of changing adsorbent in the columns.
834
Simulated Moving Bed Chromatography
A second approach consists in adding a fifth zone and modifying the elution strength of the liquid phase within the fifth zone [83-85] or keeping the same eluent within the entire configuration [86]. A third approach consists in a cyclic process based on the method of column switching [64]. It should be noted that the cyclic process does not have the advantages associated with a countercurrent process. It is essentially a chromatographic elution technique, based on the method of column switching. Ching et al. [87] employed a column-switching procedure similar to that of Row et cd. [88], which allows the introduction of feed into the cyclic process on a continuous basis. It permits continuous introduction of the feed and is designed to produce higher product concentration compared to normal elution cyclic process. A last approach consists in having two SMB in series and operating them either separately or combined into a single unit [41,64,65]. Nicolaos et al. discussed the separation of a ternary mixture with either a linear [64] or a Langmuir competitive isotherm [65], using a procedure called Multifraction which they modeled with the equivalent TMB process. Consider a mixture of three compounds, 1, 2, and 3 and assume that compound 1 is the less retained and compound 3 the more retained component. If compound 1 (or 3) is the only target compound, the separation can be processed by a single four-zone SMB. However, in order to fractionate this ternary mixture into three fractions, four different configurations of two SMB in series were proposed [89]. In general, for a multicomponent mixture of M component, it is possible to define one key compound (K). All components between 1 and K exit from the SMB unit in the raffinate stream, the other ones, from K+l to M, in the extract. To apply this concept to a ternary mixture, we can define two cases. If the key component is 1, pure component 1 can be collected in the raffinate. If it is component 3, it can be collected as pure in the extract), the remaining binary mixture, made of 2 and 3 or of 1 and 2, will have to be separated with a second SMB. Nicolaos et al. [64,65] considered either a series of two independent TMB's units or a combination of them into a single system. In the first configuration, the ternary mixture is first separated into a binary mixture and a pure compound on the first classical fourzone SMB and this binary mixture is sent to the second classical four-zone SMB, in order to separate the two remaining compounds (see Figure 17.15) [64]. They used the equilibrium theory to determine the feed and the eluent flow rates for various configurations and compared their performance both under linear [64] and nonlinear (competitive Langmuir isotherm [65] conditions. They showed that neither a five-zone nor an eight-zone SMB can separate simultaneously a ternary mixture into three pure components. It is preferable to use a five- plus four-zone SMB configuration and to perform the most difficult separation with the second SMB. However, this conclusion may change, depending on the assumptions made and the objective function considered. The SWD approach was extended to ideal and nonideal linear multi-component systems in order to achieve any desired separation of three or more components in a Tandem SMB. This approach allows the development of optimal separation strategies for multi-component mixtures [90,91]. In tandem processes, two or more SMB units are linked in series. One of the desired products is directly pu-
17.8 Recent Improvements in SMB Performance with New Operating Modes 835
Solid Figure 17.15 Two four-zone SMBs in a row. Reproduced with permission from A. Nicolaos, L. Muhr, P. Gotteland, KM. Nicoud, M. Bailly, J. Chromatogr. A, 908 (2001) 71 (Fig. 1).
Liquid
IV
Feed 1,2'
Raffinate 1
III II
Extract 2
1 .
Eluent
rified by the first SMB unit while the second desired product and the unresolved impurities of the feed are sent to the second SMB unit which purifies the second compound. Such units may be used to separate multi-component mixtures. 17.8.5.1 Model Validation of SWD In order to validate the applicability of the mathematical model for predicting wave propagation in SMB, numerical calculations based on the application of the general rate model (the VERSE model implementation [92]) to SMB were carried out and their results were compared to those of SWD. The product purity and yield obtained from the calculations agree with the target values obtained from SWD. The VERSE model calculations confirm that the operating parameters derived from the SWD method for nonideal systems allows the achievement of high purity products with a high yield for the separation of three or more components. Berninger et al. [92] discussed various strategies for splitting three-component and N-component mixtures and applied SWD to mixtures with three and more components. They tried to identify the optimal splitting sequence in a tandem SMB process for the recovery of a component or a group of components with intermediate affinities. According to this study, the separation of a mixture with three or more components requires more solvent than does the separation of a binary mixture. When all three solutes in a ternary mixture need to be recovered at a high degree of purity through a tandem SMB, the solvent consumption depends not only on the adsorption strengths of the two nearest neighbors, but also on the retention strength of the most and the least retained solutes. For ideal systems, the solvent consumption is minimized and the product concentration maximized if the easier of the two separations is performed first. If only the intermediate component of a ternary mixture needs to be recovered, e.g., to purify a compound from less and
836
Simulated Moving Bed Chromatography
more strongly retained impurities, it is better to allow one of the two impurities to distribute between the two outlet ports of the first ring. This conclusion also holds for the recovery of a single component or a group of intermediate components from a mixture of more than three components. For nonideal system, SWD can be used to find the optimal splitting strategies for minimum desorbent consumption, which leads to the highest product concentration. The standing-wave equations can be solved to give the values of all the operating parameters that guarantee the desired purity and recovery yield in a tandem SMB.
17.9 Numerical Solutions for Nonlinear, Nonideal SMB Although the algebraic solution of SMB based on the equilibrium theory is a powerful tool for the modeling of the SMB process, the optimum conditions for the separation of a binary mixture derived from this model do not fully apply to the design of an actual SMB unit since the effects of axial dispersion and of the mass transfer resistances are frequently important. The useful range of the triangle shrinks when the mass transfer resistances and axial dispersion increase. Probably because the fronts of the raffinate and the extract bands are shock layers even under linear conditions (see earlier, Subsection 17.3.7) and because there are often several columns in each section of an SMB, a reasonable agreement is often observed between the experimental and calculated concentration profiles along the column train or the concentration histories at the product ports. This agreement is reached even though the columns used in SMB units have a rather low efficiency, rarely exceeding a few hundred theoretical plates. The assumption of an instantaneous equilibrium between the two phases is not realistic for optimization purposes in some cases, particularly when dealing with protein and large molecules [93]. When the effects of the mass transfer resistances and of axial dispersion are significant, the triangle containing the point representing the optimum solution shrinks [21]. Nevertheless, in many instances, engineers have applied the equilibrium theory and taken these effects into account by using a sufficient safety margin factor (see Section 17.6.5) in the results derived from the equilibrium models or they selected experimental conditions leading to an operating point located far from the optimum point predicted by the triangle method (point w in Figure 17.12). However, if the value of the safety margin is estimated arbitrarily, serious difficulties may arise from over- or underestimating the errors made. By contrast, when the mass transfer resistances and/or axial dispersion are considered, there is no analytical solution for an SMB operated under nonlinear isotherm conditions. A numerical solution of the applicable mathematical model must be used instead to calculate the performance of the SMB, to simulate the influence of the various design and operating parameters, and to search for the optimum flow rates and switching time that give the desired results. In this quest, the selection as a starting point of the optimum set of flow rates and switching time derived from the equilibrium theory permits a considerable reduction of the number of calculations. As discussed earlier by Ruthven and Ching [27], four
17.9 Numerical Solutions for Nonlinear, Nonideal SMB
837
different methods may be used to obtain these numerical solutions.
17.9.1 Numerical Solutions of the SMB Model Various numerical solution of the SMB process using a nonlinear isotherm have been discussed by several authors, for example by Hashimoto et al. [94] who calculated a numerical solution of this model by using the linear driving force model to account for the mass transfer kinetics. Lim and Ching [95] also used this model for numerical calculations. Zhong and Guiochon [96] used a finite difference method with a predictor-corrector scheme which is alternately upwind and downwind with a two-step period. However, orthogonal collocation on finite elements is more often used because of its higher accuracy. Such methods were used in many investigations of chromatography and, particularly, of SMB. Hassan et al. [97] calculated the separation of glucose-fructose and MEA-MOH. Mallmann et al. [36] used an algorithm based on the periodic port movement in a ring of fixed beds [32]. The method of Orthogonal Collocation on Finite Element (OCFE) [37] was used to solve the mass balance equation. The same algorithm can be used to calculate numerical solutions of both continuous SMB and TMB processes. Klatt et al. [89] and Dunnebier et al. [15] used a general rate model for the description of the behavior of chromatographic columns, the description of the periodic port switching, and used a node model [27] for mass balances around the inlet and outlet nodes for modeling SMB. The resulting system of coupled partial differential equations can be solved efficiently using the numerical technique introduced by Gu [76], where a finite element discretization of the bulk phase is combined with orthogonal collocation of the solid phase.
17.9.2 Numerical Solutions of the TMB Model Equivalent to an SMB In this approach, an equivalent countercurrent movement of solid is assumed instead of the SMB process. This equivalent true moving bed (TMB) neglects the dynamics associated with periodic switching and produces mean concentration profiles over a switching period. The model equations of the equivalent continuous TMB model under steadystate conditions simplify into a set of ordinary differential equations. Using any nonlinear isotherm model, this set of equations that can be solved easily, with one of several numerical methods, for example the fourth-order Runge-Kutta method, as was done by Gottschlich et al. [98] for a problem of purification of enzymes by biospecific affinity. Assuming axial dispersion, the linear driving force approximation for intraparticle mass transfer, and a Bi-langmuir multicomponent adsorption equilibrium, Pais et al [14,99,100] obtained a numerical solution for the true moving bed equivalent to an SMB unit, using a software package based on the method of orthogonal collocation in finite elements. The prediction of this model and that of an SMB model were compared in terms of steady-state performance, steady-state internal concentration profiles and transient behavior of the extracts and raffinate purities. Model and experimental results are compared and
838
Simulated Moving Bed Chromatography
found to be in good agreement for the separation of the enantiomers of 1, l'-bi2-naphthol, using an eight-column configuration. The authors concluded that, in spite of small differences that appear between these two strategies, the performance of an SMB operation, and therefore the optimization of its experimental conditions, can be done using the equivalent TMB approach. Assuming axial dispersion, the linear driving force model for mass transfer kinetics, and a modified competitive Langmuir isotherm, Lehoucq et al. [101] obtained a numerical solution, using the method of lines [102]. Good agreement between the model and the separation of two different racemic mixtures under various operating conditions was demonstrated [101]. Storti et al [103] also compared the steady-state TMB and the corresponding SMB, using the numerical solutions of suitable mathematical models. A finite rate of the mass transfer kinetics, axial dispersion and nonlinear multicomponent equilibrium isotherms are included in the model used. Intraparticle mass transfer is accounted for through the use of a lumped "pore-diffusion" model. Numerical solutions of the TMB are calculated using the orthogonal collocation method that reduces the system of partial differential equations to a larger system of nonlinear ordinary differential equations. The result of the equilibrium theory model {i.e., ideal model) is used as the first trial solution for optimization purpose. The SMB model also is solved numerically using the orthogonal collocation method. The resulting system of ordinary differential equations is solved through a fourth order Runge-Kutta method. The separation of a mixture of para- and meta-xylene in the vapor phase, using isopropyl benzene as the desorbent, was used as a practical application for the comparison of the performance of TMB and SMB. They concluded that, in general, a good agreement between the two models can be achieved even when the SMB has a relatively small number of columns. Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. For the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boundary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. The calculation programs used for the SMB and the TMB were validated with experimental data obtained over a wide range of concentrations and temperatures for an eight-column SMB plant, by comparing the calculated and the experimental results. Figure 17.16 compares the profiles calculated for the TMB and the SMB with the experimental results obtained in the middle of the first period of the tenth cycle of a campaign. Both models predict very well the actual experimental results. The results of the calculations for the TMB are also in good agreement with the experimental results. The profiles calculated for the TMB around the feed port are unrealistic but the more important concentration profiles around the two
17.9 Numerical Solutions for Nonlinear, Nonideal SMB Figure 17.16 Comparison between : experimental and calculated results for a highly concentrated SMB run; '• Cfed = 363 kgrrr 3 , C^eed = 322 | r kgm~3. Reproduced with permission \ from Y. A. Beste, M. Lisso, G. Wonzy, J 150. W. Arlt, J. Chromatogr. A, 868 f (2000) 169 (Fig. 5). I1
839 Fructose SMB Glucose SMB, BgF=0.000457 Fructose TMB Glucose TMB, B^O.000457 Glucose TMB,B,F=0.0 Fructose exp. Glucsoe exp.
product ports are well accounted for by the model. The results show that, instead of an SMB model, a TMB model could be used to calculate the concentrations profiles of an SMB, the advantage over the SMB model being a shorter calculation time. Because the solutions of the TMB model are calculated much more rapidly, the use of this model is particularly recommended to monitor repeated runs during the optimization procedure. Leao et al. [106] also used mathematical models based on the TMB description of the SMB process. They described four TMB models, two transient models (TM) and two steady-state models (SSM). These models assume axial dispersion in the liquid phase and plug flow for the solid phase. Intraparticle mass transfer was described in terms of a simple linear driving force approximation. Both linear and non-linear isotherms were used. The mathematical models for transient operation can be written either as a system of partial differential-algebraic equations (PDAEs), when the equilibrium isotherms are written as separate equations (model TM-1) or as a system of partial differential equations, when the algebraic equations representing the equilibrium isotherms are inserted into the mass balance/mass transfer equations (model TM-2). If we write that the terms containing the time differentials (i.e., dC/dt, dq/dt, dC/dT, dq/dt) in the TM models are equal to 0, we obtain SMM models. Therefore, the mathematical models that are needed to calculate the steady-state concentration profiles consist of systems of differential-algebraic equations, mixed systems of differential and algebraic equations (model SSM-1) or of ordinary differential equations (model SSM-2), depending on how the equilibrium isotherm equations are inserted into the mathematical model. The four mathematical models were solved numerically, using suitable public-domain solvers. The experimental results of the separation of mixtures of glucose/fructose in a 12-column SMB (three columns per section) were to validate these models. Two cases were considered: the first assumed linear adsorption isotherms for both components, the second a nonlinear isotherm for glucose and a linear isotherm for fructose. All four mathematical models predict well the steady-state concentration profiles and the performance parameters of the experimental SMB system. The two transient models predict the time evolution of the internal concentration profiles of each component and the number of cycles that is required to reach steady-state.
840
Simulated Moving Bed Chromatography
However when only the steady-state concentration profiles and the process performance of an SMB are required, the use of the model SSM-1 is recommended because it requires a much shorter computing time and allows the separate, easy adjustment of the adsorption isotherm when needed. 17.9.2.1 Separation Conditions for a Heterogeneous Column Train Mihlbachler et ol. [26,59] studied the separation by SMB of the enantiomers of Troger's base on Chiralpak AD, a cellulose carbamate CSP (see isotherm data in Figure 4.28). They compared their experimental results with the numerical solutions of an equivalent TMB process that was written within the framework of the equilibrium-dispersive model. The mass balance equation for each component i in each column k of each section / of the SMB is given by
°
(17 82)
"
In this equation, the provision of identifying each particular column k in any section j is included while most previous authors assumed that all the columns in the SMB train are identical and merely refer to a column in section ;. This refinement allows a more precise modeling of the operation by including the column characteristics into the chromatographic model [26]. Knowing the value of the column porosity, e^, the phase ratio Fj. = (1 — e^/e^ can be calculated for each column. Knowing the column efficiency, Nj^, permits the derivation of the apparent dispersion coefficient: Dap = ^
(17.83)
where the linear velocity of the mobile phase is Uj^. To solve this set of mass balance equations for the SMB process, the correct initial and boundary conditions must be defined [24,27,28]. The competitive adsorption behavior of the Troger's base enantiomers onto Chiralpack AD, using 2-propanol as the mobile phase, was found to be nearly unique [107]. In this case, the competitive isotherms deviate strongly from Langmuirian behavior. They can be described based on statistical thermodynamics describing the formation of multiple layers on the surface of stationary phase during the adsorption process [59,107]. The adsorption of the lesser retained enantiomer is cooperative with that of the more retained enantiomer while the adsorption of this more retained enantiomer is barely affected by the local concentration of the lesser retained enantiomer (see Figure 4.28). The FA isotherm data were best fitted to a three-layer isotherm [107]. The data were also fitted to two different Langmuir competitive models, one being the best-fit model and the other derived from the set of Henry coefficients d\ and a^ of the two enantiomers calculated from the analytical chromatograms obtained by injecting a very small amount of sample of the racemic mixture into the column.
17.9 Numerical Solutions for Nonlinear, Nonideal SMB
841
Figure 17.17 Evaluation of the separation area of the Troger's base enantiomers. Solid line: Analytical Solution with the Henry coefficients derived from the analytical chromatograms. Dashed line: Analytical Solution with the Best Fit Langmuir Isotherm Data. Symbols: +, calculated limits of the separation area based on the correct isotherm and the true porosity of each column; o calculated limits of the separation area based on the average column porosity; * experimental conditions of the experiments performed. K. Mihlbachler, A. SeidelMorgenstern, G. Guiochon, AIChE I., 50 (2004) 611 (Fig. 6). Reproduced by permission of the American Institute of Chemical Engineers. ©1997 AIChE. All rights reserved.
17.9.2.2 Determination of the Separation Area and the Operating Points The separation area of the SMB system in the rjj, rm plane must be defined first before finding the operating conditions required to perform a separation that is both optimum and robust. As discussed earlier an, algebraic solution can be derived from the "Triangle method" when the adsorption isotherm is correctly accounted for by the competitive Langmuir model (see Section 17.7.1). This solution cannot be extended for the complex isotherm model accounting for the competitive adsorption behavior of the Troger's base enantiomers [59,107]. In such a case, the SMB model must be solved numerically to determine the borders of the separation region. Since there is no algebraic solution, the set of the mass balance equations of the two components (see Eqs. 17.82 and 17.83) combined with the competitive adsorption isotherm of the two enantiomers was solved using the backward-forward finite difference method discussed in Chapter 10. The entire region was numerically scanned by calculating the steady-state concentration histories at the raffinate and extract ports for a dense network of points representing the possible combinations of experimental conditions in the separation area and by determining the locations in the plane defining an area within which the experimental conditions lead to the production of either enantiomer at a purity better than 98%. The shape of this area is related to the competitive isotherm model used and to the column characteristics [59]. The borderlines defining the separation areas obtained with three different isotherm models, the three-layer isotherm model and the two Langmuir isotherm models are shown in Figure 17.17. The five operating points labeled * along a straight line parallel to the diagonal of the r\i, rjjj plane represent the experimental conditions corresponding to different SMB operation conditions the results of which were discussed in detail [59]. The separation area in Figure 17.17 that was calculated using the three-layer adsorption isotherm model was determined in two different ways. The data points
842
Simulated Moving Bed Chromatography
labeled + were obtained when the calculations were made with the exact porosity values of each column used. The data labeled o were obtained when all the columns were assumed to have the same porosity, a porosity equal to the average porosity of all the columns. The separation area bordered by the + symbols is the true separation area since it was calculated with the true porosities of each column. It is slightly narrower than the area derived from the average porosity of the column set. The solid and the dashed lines in Figure 17.17 represent the analytical solution of the "Triangle method" for the two Langmuir competitive isotherms, the one obtained with the Henry coefficients derived from an analytical chromatogram and the one obtained with the best-fit competitive Langmuir isotherm model. The shapes of these areas are quite different, confirming that the shape of the separation areas is a strong function of the isotherm model. Note that the true separation area obtained when using the porosity of each column is narrower than the separation area obtained using the average porosity. The true separation area is also smaller than the one derived from the "best" competitive Langmuir isotherm (a model that does not fit correctly the data [107]). On the other hand, the separation area available is much larger than the one predicted by the triangle theory when using the best Langmuir competitive isotherm derived with using the Henry coefficients measured from an analytical chromatograms. Only the use of accurate isotherm models and of the exact values of the column characteristics may allow the achievement of a high production rate and of a guaranteed product purity. Among the five different operating points chosen along the straight line (*) to acquire experimental data (see Figure 17.17), points # 3 gave purities exceeding those required and points # 2 and 4 gave purities close to 98%. Finally, it was found that the production rate of the raffinate measured for the combinations of flow rates corresponding to these data increases with increasing values of rjj and rui while that of the extract decreases. This result also shows how the threelayer isotherm model accounts properly for the unusual competitive adsorption behavior of this pair of enantiomers and permits a satisfactory description of the performance of the 8-column SMB unit and a correct prediction of the experimental purity values. Figure 17.18 shows the concentration histories during one cycle, after steady state has been reached, at two different ports of an 8-column SMB, when it is operated under the experimental conditions corresponding to the operating point labeled 1 in Figure 17.17. The combination of the signals of a polarimetric detector, Figure 17.18a, (which has response factors equals in absolute value but opposed in sign for the two enantiomers) and a UV-detector, 17.18b, (which has the same response factor for both enantiomers) permit the correct determination of the concentration histories of both compounds at the port where these two online detectors are located. Figure 17.18 shows that the actual raffinate stream is pure, in agreement with the prediction of the calculation. The difference between the experimental and calculated purities of the extract stream is 1.35%. This deviation may be attributed to the accumulation of the feed impurities in the SMB process (see Figure 17.18b). These two figures also show a reasonably accurate prediction of the oscillations of the concentration histories [59].
17.9 Numerical Solutions for Nonlinear, Nonideal SMB 4.0-
843
b
Con centraition C [g
3.S3.0-
N
2.52.01.5-
I\
1.00.5-
Rafflnate
o
2 Extract
, 'v. Feed
Raffinate
Figure 17.18 Internal Concentration Profiles of the Troger's base enantiomers measured with (a) a polarimeter and (b) a UV detector. Concentrations of each enantiomer. Solid line: experimental data at port 6. Dotted line: experimental data at port 5, (*) Calculated profiles using the 3-layer isotherm model. K. Mihlbachler, A. Seidel-Morgenstern, G. Guiochon, AIChE J., 50 (2004) 611 (Figs. 8 and 9). Reproduced by permission of the American Institute of Chemical Engineers. ©1997 AIChE. All rights reserved.
17.9.2.3
Practical Importance of the Heterogeneity of a Column Train
Simple elution chromatograms calculated for a binary mixture exhibiting competitive Langmuir isotherm behavior (see Figure 17.19a) show a significant variation in retention times and band widths with changes in the total column porosity. Due to the longer retention, the profile obtained for a given sample size are broader and eluted later with the lower porosity column than for the higher porosity one. The separation areas calculated with the same isotherm parameters for an 8-column SMB are shown for two homogeneous trains and for the train having an average porosity of 0.69375 and made with one column with a 0.65 porosity and seven with a 0.70 porosity. The separation area calculated for the real heterogeneous column train (see Figure 17.19b) differs somewhat from the areas obtained for the two ideal, homogeneous trains. Note that there is only a small common area for the triangles corresponding to the different columns in the heterogeneous train (see Figure 17.19b). These results suggest that care should be taken to avoid using columns having too different characteristics in the operation of an SMB train, to avoid a significant loss in the production rate and/or the product purity. The closer the experimental conditions to those corresponding to the optimum production rate, the more sensitive the purity of the products and the production rate to column-to-column fluctuations of the physico-chemical characteristics. This sensitivity increases also rapidly with decreasing value of the separation factor of the binary mixture. Finally, the robustness of the separation increases with increasing number of columns in a section. There is little difference between the separations achieved if the differing columns are alternated in the train rather than adjacent to each other [26].
Simulated Moving Bed Chromatography
844
Figure 17.19 Influence of the porosity in HPLC. (a) Elution profiles of a binary mixture with competitive Langmuir isotherm behavior, calculated for two columns differing only by their permeabilities. Solid lines: eT = 0.70; Dotted lines: eT = 0.65. (b) Separation triangles calculated for different column porosities. Dashed line triangles: Porosity of all columns, 0.65. Dotted line triangles: Porosity of all columns, 0.70. Solid line triangles One column has a porosity of 0.65, the other seven a porosity of 0.70 (average porosity, 0.69375). The separation area was scanned along the solid straight line parallel to the diagonal. The symbols * and + indicate two operating points. Reproduced from K. Mihlbachler, ]. Fricke, T. Yun, A. Seidel-Morgenstern, H. Schmidt-Tmub, G. Guiochon, J. Chromatogr. A, 908 (2001) 49 (Figs 5 and 6).
17.9.3 Numerical Solutions of the Equivalent Mixing Cell TMB Model The equivalent TMB model can also be considered as a series of well-mixing tanks [108-113]. The bed volume, V, is considered as equivalent to a certain number of theoretical stages (IV), each stage being considered as an ideal mixing cell of volume V/N distributed between the fluid and the stationary phases, in accordance with the bed phase ratio, with N
(17.84a) (17.84b)
A differential mass balance for each stage is given by: (17.85) This approach, the stage model, is simple because the equations are first-order ordinary differential ones that can be easily solved by Gear's algorithm. However, it is less accurate than the continuous model, especially when dealing with columns
REFERENCES
845
having a low efficiency or with feed components having very different retention times, hence a large value of cc. Ching and Ruthven, compared the solutions of the mixing cell model and the continuous model for an SMB system [18,30].
17.9.4 Numerical Solutions of the Equivalent Mixing Cell SMB Model A numerical solution of the SMB process in the nonlinear case, using the mixing cell method and applying the McCabe-Thiele operation diagram, was described by Ching etal. [114,115].
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Chapter 18 Optimization of the Experimental Conditions in Preparative Chromatography Contents 18.1 Definitions 18.1.1 18.1.2 18.1.3 18.1.4 18.1.5 18.1.6 18.1.7 18.1.8
851
Throughput Sample Size and Loading Factor Cycle Time Production Rate Cut Points Recovery Yield Purity of a Component Specific Production
851 851 852 853 854 856 857 857
18.2 The Economics of Chromatographic Separations 18.2.1 18.2.2 18.2.3 18.2.4
The Components of the Production Cost The Different Objective Functions Identification of the Experimental Parameters An Economic Analysis
18.3 Optimization Based on Theoretical Considerations 18.3.1 Simultaneous Optimization of the Production Rate and Recovery Yield using the Ideal Model 18.3.2 The Knox and Pyper Approach 18.3.3 Optimization for Touching Bands Using the Ideal Model 18.3.4 Optimization for Overlapping Bands with No Yield Constraint 18.3.5 Optimization for Overlapping Bands with Yield Constraint
18.4 Optimization Using Numerical Solutions 18.4.1 18.4.2 18.4.3 18.4.4
Maximum Production Rate in Elution Simultaneous Optimization of the Production Rate and Recovery Yield Minimum Solvent Consumption in Elution Compromises between Maximum Production Rate and Minimum Solvent Consumption 18.4.5 Maximum Production Rate in Gradient Elution Chromatography 18.4.6 Maximum Production Rate in Displacement Chromatography 18.4.7 Comparison between Elution and Displacement Chromatography 18.4.8 Comparison Between Isocratic Elution, Gradient Elution and Displacement Chromatography 18.5 Recycling Procedures 18.6 Practical Rules 18.7 Optimization of the SMB Process 18.7.1 Comparison of Batch and SMB Chromatography References
849
857 859 861 862 863
867 867 869 871 878 882
883 884 893 895 897 898 903 907 912 915 920 924 934 935
850
Optimization of the Experimental Conditions
Introduction As with any industrial process, preparative chromatography needs to be optimized. Such an optimization has to be based on a thorough understanding of the process variables and its economics. It requires a thoughtful decision regarding the choice of the objective function. As a matter of fact, the optimization of the experimental conditions for economic production is the best justification for the detailed study of the fundamentals of nonlinear chromatography. It is difficult to optimize a separation without a clear understanding of how the thermodynamics of competitive phase equilibria, the finite rate of mass transfer, and dispersion phenomena combine to affect the individual components' band profiles. In the end, it is these profiles that determine the cut points and therefore the purity, the amount produced, and the cycle time. These factors determine the yield, the production rate, and ultimately the cost of the separation. Depending on the economics of the separation, the need and the direction of the optimization may lead to widely different approaches and results. In the first two sections, we define the terms used in this chapter and we discuss the different optimization approaches which stem from different economic situations. It is well known that there are no technical optima in industry, only economic optima. For reasons that are discussed in this section, several objective functions of the optimization can be defined, corresponding to maximum production rate, or to maximum amount of purified component prepared per unit volume of mobile phase, or to an almost complete recovery, or to any intermediate combination of these extremes. There are three possible approaches to the optimization of a separation: (1) a theoretical approach based on the solutions of the models discussed, i.e., essentially on the analytical solution of the ideal model, with appropriate correction for apparent dispersion; (2) a numerical approach which uses classical computational methods; and (3) an empirical approach. While in the end, any process optimization needs some fine-tuning which can proceed only through series of experiments, an understanding of the fundamentals of chromatography aids in carrying out this goal faster and with less risk. It also permits the formulation of simple rules which may guide toward the rapid development of laboratoryscale and low-volume process-scale separations, for which the time and resources required for a complete optimization are often lacking. Whatever the approach selected, in order to optimize a nonlinear, preparative chromatographic system, one needs to consider four different factors, the objective function selected, the experimental parameters that can be optimized, the decision variable, and the constraints that must be satisfied [1]. The objective functions are discussed later, in Section 18.2.2). They include the production rate, its cost, the recovery yield, the specific solvent consumption, or some combination of the above. Many experimental parameters of a preparative chromatography separation cannot be changed during the optimization process. Typically, such parameters are the nature of the feed components, their relative compositions, and the nature or even the brand of packing material. Sometimes, the column diameter is also fixed by prior investments. The decisions variables are those parameters that can be changed during the optimization process, in order to maximize the ob-
18.1 Definitions
851
jective function. These variables usually include the feed load, the mobile phase flow rate, and its composition. The constraints are either physical limits, such as the solubility of some feed components or mobile phase additive, or some arbitrary constraints, such as the recovery yield and the product purity. A successful optimization procedure requires that clear statements be made about these four factors at the very beginning. In practice, prior to beginning the actual process of optimization of the experimental conditions of a preparative separation, extraction, or purification, it is necessary to perform the choice of the chromatographic system to be used. This requires the measurement of the most important characteristics of the performance of several combinations of stationary and mobile phase combinations. It is imperative to maximize the selectivity of the chromatographic system while making sure that its capacity is important. The feed solubility in the mobile phase must be high and the saturation capacity of the stationary phase important. The actual focus of this chapter is the optimization process following the selection of the chromatographic system (mobile and stationary phases).
18.1 Definitions For the sake of clarity, we want to state here the definitions of the main parameters employed to characterize the procedures used [2].
18.1.1 Throughput The throughput is the amount of the mixture to be treated (i.e., amount of feed, amount loaded, or sample size) that is introduced in the column per unit time. The throughput is equal to the product of the sample size, or amount, and the frequency of the sampling. This frequency is the reverse of the cycle time. Throughput and production rate are related through the recovery yield.
18.1.2 Sample Size and Loading Factor The sample size or amount of feed injected at the beginning of each cycle is the product of the concentration of the feed, Cf and the volume of feed injected, V^, or the product of the feed concentration, the duration of the injection, tp, and the mobile phase flow rate, Fv m = C°Vmj = C%FV
(18.1)
In many cases, and especially in theoretical discussions, it is convenient to use a dimensionless unit in order to report the sample size. Since adsorbents usually have a finite saturation capacity that corresponds to the formation of a dense monolayer of adsorbate, a convenient reference to express the sample size in dimensionless units is the column saturation capacity, or amount of sample needed to cover the adsorbent with a monolayer. Similarly, there is a saturation capacity for other retention mechanisms, e.g., in ion-exchange equilibria, the amount of an
852
Optimization of the Experimental Conditions
ionized compound needed to exchange all the ions of an ion-exchange resin. The sample size is thus expressed as the loading factor I U f' (l-e)SLqs>i (l-e)SLqs4 ^ where S is the column cross-sectional area, L the column length, e the total porosity of the bed, qsj the column saturation capacity, Cf the feed concentration, and Vjnj the sample volume. L {
18.1.3 Cycle Time The cycle time, Atc, is the time that separates two successive injections. The cycle time is somewhat arbitrarily defined. Several definitions and approximations can be used, depending on the complexity of the separation: • The cycle time is the analysis time of the sample, tA, i.e., the time between injection and the complete elution of the last component band. This includes the time necessary for the concentration of this last compound to return to an arbitrarily small value and is typically equal to three or four standard deviations (an) of the last component band (under linear conditions), beyond the retention time of the last component (f^ n). So, &c = tA = t%n + s(Tn
(18.3a)
If the sample does not contain late running impurities requiring column regeneration, a second sample can be injected when a time shorter than t& has passed after the injection of the first sample. Obviously, nothing is eluted between the injection and the elution of the nonretained compound. This leads to the next definition. • The cycle time can be the corrected analysis time, or the analysis time minus the holdup time Atc = tA-t0
(18.3b)
This definition is particularly easy to implement in theoretical work where all the time values characterizing the different features of the elution profile of the band of a binary mixture contain to [2]. Also, because bands rarely extend much beyond the analytical retention time, t^o, when the isotherm is convex upward (e.g., Langmuirian), t°Rn — to = k'Onto is often taken as definition of the cycle time [2], with t\ n retention of the last component under linear conditions. • We can also assume that the cycle time is equal to the time between the moment when the concentration of the first component exceeds a certain threshold, i.e., 1 x 10~9 M, or 1 x 10~6 mg/mL, and until the concentration of the second component becomes smaller than the same threshold [3-7]. This definition gives smaller values of the cycle time (i.e., the corresponding production rate is larger), which depend on the resolution between the two bands, i.e., on most
18.1 Definitions
853
of the experimental parameters. This definition can be used for simple experiments, or in numerical determination of the optimum experimental conditions. In practice, simple definitions of the cycle time, such as those just given, can be used to determine the optimum operating conditions, leaving the other contributions for later adjustment. Since these contributions tend to be constant, there is a simple linear relationship between this approximate optimum and the true optimum production rate. • In actual practice, the cycle time is often more complex to estimate. In addition to the elution time, it must include the time required for column regeneration, the time required for stabilization after regeneration, and the dead time between runs. The time of each of these steps depends on the flow rate selected for the corresponding operation. The results obtained in optimization studies may differ slightly when different definitions are used [4].
18.1.4 Production Rate The production rate is the amount of feed turned into product, i.e., purified at the required degree of purity per unit time.1 In this chapter, the production rate refers to one of the components of the mixture; that is, we assume that there is only one desired component in the feed. If there are more than one, a weighted combination must be used. The production rate is equal to the product of the feed volume, the concentration of the corresponding component in the feed, and the recovery yield, divided by the cycle time. Thus, we have
^ P
(18.4a)
where Vr\ is the production rate of component i, V\nj the sample volume, C° the concentration of component i in the sample, and Yj the recovery yield. The loading factor is the ratio of the sample size to the column saturation capacity, so n,- = VinjC? — (1 — e)SLqsLtj. It we take the definition of the cycle time given in Eq. 18.3b, with IA = £gn, where t\ n is the retention time of the last component at infinite dilution, we obtain _ (1 - e)qSiiSLLf/iYi K
r
0,fi 0
_ (1 - e)qS/iSLf/iYiu K O ,H
_ eSLf/iYiU k'Qi °i
K
0,n
where Lfi is the loading factor for the component i and Y,- is the recovery yield. This equation shows that the production rate is proportional to the column crosssectional area, to the loading factor, to the recovery yield, and to the mobile phase x We have always assumed that the production rate is proportional to the column cross-sectional area. In practice, this assumes that columns can be packed with the same material, at the same efficiency, irrespective of their diameter, and that the feed can be distributed homogeneously over the entire cross section. Although meeting these requirements may not be always easy, this seems to be achieved routinely in many instances.
Optimization of the Experimental Conditions
854 C mg/mL
b
a
B1 A1 A2
B2
0
0 5
tR,2
tc,1
5
tR2,o
tc,2
C mg/mL
c
16
d
20%
12
10%
5% 10% 20% 5%
8
6
0
0 5
8
11
5
8
11 Time, min
Figure 18.1 Illustration of the cutpoints. (a, b) Definition of the areas A\, A2, B\, and B2 used to calculate the production rate and the recovery yield. See text and Eqs. 18.5 to 18.8. Column length, 10 cm; Fv = l mL/min; k[ = 6, k'2 = 7.2, b\ = 0.03, b2 = 0.036; tp = 0.46 min; N = 2000. (a, b) nx = 23.3, n2 = 7.7 mg; Lfil = 9%, Lfi2 = 3%. (c, d) Effect of purity on the position of the cutpoints. (c) same parameters as for (a) and (b). (d) n\ = 7.7, n2 = 23.3 mg. velocity and that it is proportional to the column saturation capacity, qs;. However, Y; depends on the resolution between bands and is inversely correlated to Lfj and u.
18.1.5 Cut Points Cut points specify the beginning and end of the collection of a fraction. Cut points can be based on time, although in practice time is not the most stable parameter, due to possible fluctuations in the flow rate. They can be based on threshold values of the UV-detector signal or on volume of eluent through a mass flowmeter. Cut point locations can also be a combination of a characteristic feature on the chromatogram and volume. Cut point locations for valve switching may also be based on the response of less traditional detection instrumentation such as the on- line measurement of density, conductivity, temperature, specific ion detection, near infrared, or refractive index. In this chapter, we define two cut times, tCi\ and tC/2, and three fractions for a binary mixture (however, four cut points could also be considered, as shown in the Figure, at the front of the first peak and end of the second one). The first fraction contains the purified first component, a mixed fraction follows, and the third fraction contains the purified second component. The production rate for
18.1 Definitions
855
component i is defined as follows (see Figure 18.1 and definitions of A, and Bz below, Eqs. 18.6a to 18.6d): P,= ^
(18.5)
where n, is the amount of component i injected {i.e., contained in the sample size). Thus, the yield is Y; = («,- — A,)/n;. The amounts of the two components contained in the three different fractions are calculated as follows [2]. • For i = 2, A2 is the amount of component 2 that is eluted before the second cut point, and, accordingly, is lost for the production of this component (see Figure 18.1): A 2 = FV
rtc,2
C2 dt
(18.6a)
•ltd
Fv is the mobile phase volume flow rate, which is assumed to be constant. For i = 1, A\ is the amount of component 1 that is eluted after the first cut point and is lost (if the intermediate fraction is wasted) or which has to be recycled: AX = = FFVV II °'°'2 Q dt Jt J c,i
(18.6b)
The production rate Pr^ is the amount of the corresponding compound collected per unit time. Thus, there is a certain degree of uncertainty, as the total amount collected for a fraction is «; — A; + £>,-, but the production rate of i is assessed at its actual content in the corresponding fraction containing compound i and is given by Eq. 18.4b. The amounts of the components of the binary mixture which end in the wrong fractions are Bj, calculated as follows. For i - 2, B2 is the amount of the first component that is eluted after the second cut point and, accordingly, coelutes with the second component, dilutes it and contributes to increase its production and to decrease its purity (see Figure 18.1): B2 = FV f2
Q dt
(18.6c)
4,2 where t2 is a time when the concentration of the first component in the eluent has fallen below the detectable level. For i = 1, B\ is the amount of component 2 that is eluted before the first cut point and coelutes with the first component C2 dt
(18.6d)
where t\ is a time when the concentration of the second component in the eluent has not yet risen above the detectable level.
856
Optimization of the Experimental Conditions
The concentrations Q and C2 and the times t\ and li can easily be derived from the numerical solution of the equilibrium-dispersive model as discussed in Chapters 11 and 12 and from the purity requirements. They can also be calculated from the analytical solutions of the ideal model (Chapter 8), when this model is used for optimization studies. In practice, understanding the individual component band profiles by collecting a large number of fractions and performing their quantitative analysis is critical. Then, the purity may be calculated from these profiles. This may be a first step to providing a starting and ending point for cut point location determination [8]. For short campaigns, when small columns are used, or when complex mixtures are separated, this may be the best strategy for determining the cut point locations. However, a more economical approach for routine manufacturing or large production campaigns consists in using the knowledge of the individual band profiles to create a cut point strategy based on eluent volume, time, or the response of a detector or other instrument outputs. In this latter approach, fewer fractions need to be collected and fewer analysis are required. The purity requirements determine the location of the cut points and, therefore, the yield, the production rate and the economics of the process. This is illustrated in Figures 18.1c and d for a 1:3 and a 3:1 mixture, respectively. As the purity decreases, the yield increases. Note that the amount of each feed component injected into the column is split into three fractions. The first fraction is the purified product, the second is contained in the mixed zone fraction, to be recycled or sent to waste, and the third is impurities of other purified products, if any.
18.1.6 Recovery Yield The recovery yield is the ratio between the amount of the desired component collected at the required purity in the purified fraction and the amount injected in the column with the feed. The recovery yield is a function of the purity at which the products must be prepared. The recovery yield for component i is given by (see Figure 18.1) Y{ = ^—^
(18.7)
ti
As defined here, the yield is the fraction of the component that is recovered in the purified fraction, as the final product. The fraction that is not recovered is not necessarily lost, however. A small fraction may be truly lost, but a large portion or even all the mixed zone of the chromatogram may be and often is recycled. The recycled fraction may be reprocessed under the optimum conditions for the mixture that has the composition generated under steady-state operation by adding the intermediate, recycled fraction to the feed (see later, Section 18.5). Alternatively, the same equipment may be employed under different experimental conditions, optimized for the recycled fraction, or a second dedicated chromatographic process may be used. Complex strategies may be designed, involving two or more steps and/or recycling [9-11], which give a higher production rate and a higher recovery yield, lower production cost compared to a single purification step, and permit the achievement of a global 90 to 95% yield (see Section 18.5 below). This is done at the cost of a higher process complexity.
18.2 The Economics of Chromatogmphic Separations
857
18.1.7 Purity of a Component The purity of a component, i, is the concentration of this component in the collected fraction (solvent excluded). It is given by
Usually the purity is a constraint of the separation. The typical value considered below is 99%. The purity requirements determine the location of the cutpoints and therefore the yield, the production rate, and the economics of the process. This effect is illustrated in Figures 18.1c and d for a 3:1 and a 1:3 mixture, respectively. As the required purity decreases, the yield increases.
18.1.8 Specific Production As we discuss in the next section, the amount of solvent consumed per unit amount of purified product prepared is an important contribution to the total cost of production in many cases. The amount of solvent used during a cycle is the product of the cycle time and the flow rate. The amount produced per cycle is the product of the amount injected and the recovery yield. Thus the solvent consumption is given by CSi
_ AtF ~
u
Alternatively, we can consider the amount of purified component produced per unit volume of solvent used. This amount is the specific production and it is equal to
sp
<=ck = it
(18 9b)
-
The specific production, the production rate, and the recovery yield are used for the definition of objective functions.
18.2 The Economics of Chromatographic Separations As explained in Chapter 1, we can distinguish two main areas in preparative chromatography, laboratory and industrial scale separations. Laboratory scale preparative chromatography corresponds to the need to prepare a certain amount of a purified compound as an intermediate step in the collection of further information (e.g., on its physical, chemical, biochemical, pharmacological, or toxicological properties). Industrial scale preparative chromatography is concerned with the production of a purified compound for sale as a final product or as a critical component of a product. In the case of laboratory scale separations, information is highly valuable, and, except for the time spent, the operating costs are insignificant, due to the short duration of the project. Hence, optimization must be performed rapidly in this case. However, the initial capital investment can be large,
858
Optimization of the Experimental Conditions
depending on whether large, semipreparative laboratory equipment or small, industrial scale equipment is chosen. Furthermore, the composition of the feedstock is often incompletely or even poorly known and the time spent in developing the methodology and in running the separation is the essential component of the final cost. In such cases, there is little incentive in acquiring data regarding the equilibrium isotherms and in calculating optimum conditions. Simple rules, based on the fundamentals of nonlinear chromatography but of easy application, are needed to permit a rapid selection of experimental conditions not too remote from the optimum ones. These rules are discussed in Section 18.6. When a purified compound is required as an intermediate or a final product, the situation is entirely different. The nature of the feedstock is better defined. The production costs become of primary importance and must be reduced by a proper optimization of the design and operating conditions. The 2004 "strategic initiative" of the Food and Drug Administration recommends that manufacturers of Pharmaceuticals show that the control of their processes is based on a profound understanding of the physico-chemical principles of the operations involved. It will become relatively easy for those who master a process to proceed to later adjustments of its parameters. Since there is no better way to demonstrate such a knowledge than to show the ability to control and optimize the process, these new regulations will encourage systematic investigations of the chromatographic process and its optimization in the early stage of the project development. The possibility of using several sets of operating parameters to permit easy adjustment of the process to changing economic conditions must be considered at the stage of the process design. Also, the variation of the parameters of the feedstock, caused by a change in raw materials or supplier, by adjustments of the upstream part of the production process, or by unavoidable statistical fluctuations must be taken into account. Thus, the optimization of industrial scale preparative chromatography requires a systematic approach. Ideally, this approach should rely on the determination of the operating conditions and the calculation of the production rate, the recovery yield, and the elements of the production cost. Moreover, an economic analysis aimed at minimizing the product cost per unit amount of product must be performed. We need first to discuss the components of the production costs and to define the objective functions of the possible optimization strategies. Finally, we want to point out that considerable savings can often be achieved by modifying slightly the composition of the feed to simplify the chromatographic separation, provided this is compatible with the other operations involved in the manufacturing of the products. Different authors have previously discussed the economics of preparative chromatographic separations [12-14]. The main difficulty of the exercise is in the separation of the technical and the economic parameters. The influences of the various technical parameters of the process on the characteristics of the production performance are easy to figure out. They can be discussed in detail relatively independently of the specific details of a given separation problem. By contrast, the economic parameters are usually very specific to a product and even to the company that runs the separation and they are never constant over the long term.
18.2 The Economics of Chromatographic Separations
859
Table 18.1 Contributions to Production Costs"
Cost Contributions Capital Labor Total Fixed Costs Solvent a - Recycling b - Losses Energy Packing Material Maintenance Total Operating Costs
Case I (Peptides)
Case II (Fatty Acids)
Case III (Steroids)
24 58 82
27 20 47
23 15 38
4 6 5 1 2
20 20 4 7 2
29 9 4 18
18
53
62
"In %, after H. Colin, in Preparative and Production Scale Chromatography, G. Ganetsos and RE. Barker Eds., M. Dekker, New York, NY, 1993,p. 35.
18.2.1 The Components of the Production Cost The total cost of a purification carried out by chromatography, or production cost (PC), can be divided in three different components, the fixed costs, FiC, the operating or proportional costs, OC, and the cost of the lost feed, FeC PC = FiC + OC + FeC
(18.10)
The distribution of the different cost contributions between these three components (Table 18.1) leaves only a few uncertainties [12]. A correct estimate of the actual cost of these contributions is more difficult, especially because each company has different accounting procedures, compounded by administrative differences and the arbitrary distribution of the overhead contributions. 18.2.1.1 Fixed Costs (FiC) These costs include first the amortization of the capital investments made in the preparative chromatograph, its installation, the building where it is located, and the associated equipment (pumps, tanks, tubings and valvings, control equipment). It is uncertain whether the cost of the packing material should be capitalized. It seems more reasonable to include it as an operating cost, because packings have a finite lifetime which rarely exceeds two years. In opposition to Colin [12], we consider that labor costs (including workers' health care and insurances, and the corresponding overhead charges) are a component of fixed costs. For a given unit, labor costs depend essentially on safety regulations, labor contracts, and the other ongoing operations in the facility but little on the actual production rate,
860
Optimization of the Experimental Conditions
whether it is adjusted for maximum production rate or maximum specific production [13]. For example, doubling the production rate of an industrial chromatograph will not double the number of operators required. A number of other costs, which would seem at first to belong to operating costs, are in fact fixed costs. This includes regulatory costs, auditing costs, recurring periodic maintenance and calibration required for GMP, and the heating/cooling of the building. This should also include provisions for workers' retraining after personnel changes and upgrading of the equipment by replacing the components that become obsolete before being worn-out. Once a given piece of equipment has been bought and made available for a certain separation, the corresponding capital and labor costs are almost independent of the production rate. Their contribution to the total product cost will be minimized by maximizing the production rate of that equipment. 18.2.1.2
Operating Costs (OC)
These costs include the products and services needed to operate the chromatograph, the solvents (including the cost of their regeneration and waste disposal), the packing material and the solvent used for packing the column, utilities, and the energy consumed. The inclusion of the maintenance costs [12] as operating costs is arguable. Part of the maintenance (e.g., pumps) belongs to the operating costs, while it is more logical to include the recurring periodic maintenance within the fixed costs. This is a minor point, however, given the size of the contribution (around 2%). The operating costs are proportional to the throughput, i.e., to the production rate and to the inverse of the recovery yield. The most important contributions are the solvent and the energy costs [12]. They can be lumped together because most of the energy needed is spent on concentrating the collected fractions and regenerate the solvent for reuse in the mobile phase. The solvent and energy costs are proportional to the production rate. Their contribution to the total product costs will be minimized by maximizing the specific production, i.e., by minimizing the amount of solvent needed to produce a unit amount of final product. The operating costs are proportional to the cost of the solvent (SC), the production rate, and the solvent consumption (CS,). Hence, it is inversely proportional to the specific production, and OC = Prt CS{ SC = f^-SC = wSC
(18.11)
Trends in legislation suggest that the costs associated with solvent storage, recycling, reprocessing, and disposal will increase relative to the other costs. 18.2.1.3
Feed Cost (FeC)
In order to produce a certain amount of purified compound, we must introduce a larger amount as feed. The recovery yield or final amount of product recovered divided by the amount of product in the crude cannot be 1. This loss causes an additional cost [12]. Poor yields may add significantly to the cost when the chromatographic purification is undergone after completion of a complex series of
18.2 The Economics of Chromatographic Separations
861
reactions and operates on a high-value-added mixture or on an expensive crude. The feed cost is proportional to the production rate, to 1 — Y;, where Y,- is the yield and to the unit cost of the feed, FeCw, at this stage of the production FeC = FeQ Pr( (1 - Y{)
(18.12)
Estimates of the cost of the crude can be made based on final product cost minus the cost of the chromatographic step or on the sum of the costs of the steps to produce the crude. Estimates of the amount of crude lost are based on the yield and the possibility of recovering part of the crude contained in the mixed zones of the chromatogram. In conventional chromatographic procedures, a mixed zone appears both before and after the component of interest. When the production rate is high and the yield moderate, the mixed zone contains an important amount of product which could be recycled off-line or on-line. This amount could often be repurified and recovered as valuable product, using one of several possible approaches [9-11]. For example, the recycled product could be added to the feed and processed with it, or it could be stored and reprocessed under optimized conditions (see later, Section 18.5).
18.2.2 The Different Objective Functions The essential goal of an optimization procedure is the determination of the values of the experimental parameters that maximize or minimize an objective function, while satisfying a number of constraints. The selection of the most suitable objective function is an important step. Because it is difficult to make any reasonable estimates of the costs of all the different contributions to the production cost, we can consider in a general discussion only some simple scenarios and their possible combinations. Since the feed cost depends essentially on the value of the recovery yield, it can be minimized by either considering optimization procedures with a constraint of minimum recovery yield or maximizing the product of the production rate and the recovery yield. Another possibility is to minimize the operating costs, or the fixed costs. Most practical cases are intermediate between these two extremes and can be taken care of by a proper mixing of the two objective functions (see Section 18.4.4). There are several cases where it is necessary to determine the optimum conditions for maximum production rate: (1) when the demand for a product exceeds the production capacity, and (2) when the capital cost of the purification is significant compared to the operating costs and to the cost of the unrecovered crude. As a matter of fact, sales, hence production, can rarely be kept for a long period at any predetermined level. There must be alternative strategies to adjust the production rate and the recovery yield, while minimizing costs. The absolute maximum production rate can be looked for, or the maximum production rate with a recovery yield constraint. Other combinations will permit the minimization of the production costs. Alternatively, since we know from the literature [12] that the costs associated with the loss, the processing, and the regeneration of the solvent used often account for nearly 40% of the total production costs, we can choose as
862
Optimization of the Experimental Conditions
the objective function the amount of solvent needed to produce a unit amount of purified component. This function can then be minimized [13]. Thus, there are two main objective functions: (1) the production rate and (2) the specific production, or amount of product purified per unit amount of solvent used. By using these two functions, we can take into account all the major cost contributions identified by Colin [12], capital, labor, solvent, and energy, in our analysis, without compromising its conclusions seriously. Obviously, when numerical methods of optimization are used, there are no serious conceptual difficulties in itemizing and including the various costs contributions. We may also calculate the unit price of the purification. It is the ratio of the purification cost and the production rate. Combining Eqs. 18.10 to 18.12, we obtain PnCe
PC
=
* =
FiC + OC + FeC
¥r
Provided the various parameters are known or can be estimated, Eq. 18.13 defines the most important of the objective functions. It is difficult, however, to discuss on a general basis the results of an equation that contains so many parameters that are specific to any given separation problems.
18.2.3 Identification of the Experimental Parameters It is interesting to reconsider at this stage the mass balance equation (Chapter 2) (18.14) Ul
VL
<J&
OL~
where C and q are the concentrations in the mobile and stationary phase, respectively, t is the time, z the distance along the column, F is the phase ratio, u the mobile phase velocity, and Dav the apparent dispersion coefficient which accounts for the finite column efficiency, and is given by D.
P
=^
(18.15)
where L is the column length and IV the column efficiency under linear conditions. The initial and boundary conditions have been discussed in Chapters 2, 7, and 10. We can rewrite Eq. 18.14 in dimensionless variables, with
X = I
(18.16a)
T =
(18.16b)
-
where to = L/u is the holdup time of the column. With these variables, Eq. 18.14 becomes dC
da
dC _
1 32C (
]
18.2 The Economics of Chromatographic Separations
863
This equation is the same as Eq. 6.27, but written slightly differently for convenience. In practice, F cannot be adjusted. The phase ratio is related to the particle porosity and is a characteristic of the stationary phase used. Therefore, for given eluite isotherms, the band profiles depend exclusively on the column efficiency and the loading factor (which appears in the boundary conditions). If we can assume that the column efficiency is the same for the two components, then the production rate and the recovery yield at a certain degree of purity depend on these band profiles and on their degree of interference. This assumes that the column porosity and its efficiency do not change over time. In practice, depending on the column technology, this approximation may not be entirely valid [15]. We can rewrite the objective functions for the specific solvent consumption and the production rate in dimensionless coordinates ^ Pr.
(18.18a) £
=
JihIru=™=euSP.
(18.18b)
where Axc is the dimensionless cycle time and Xp the dimensionless injection time. The recovery yield depends on the degree of overlap of the product and its impurities in reduced coordinates, hence it depends only on the reduced sample size and the column efficiency. For a given value of the loading factor, Y; increases with increasing column efficiency. Hence, the solvent consumption is minimum when the column efficiency is maximum. The solvent consumption per unit amount of product, or specific production, is independent of the actual value of the mobile phase flow rate, while the production rate is proportional to the flow rate. Given a plate height equation, there is an infinite number of combinations of column lengths, particle sizes, and mobile phase velocities which give a certain column efficiency. The maximum specific production will be achieved by operating the column at the maximum pressure drop available and by choosing the length and particle size in such a way that the column efficiency is maximum at the corresponding flow rate. The optimization for maximum production rate requires more complex calculations. We discuss successively the use of theoretical considerations combining the ideal model with a correction for band dispersion and the use of numerical methods.
18.2.4 An Economic Analysis A conventional method used in chemical engineering to evaluate the economies that are due to the scaling up of different unit operations and of their equipments (e.g., reactors, blenders, heat exchangers, filtration systems) consists in plotting the logarithm of the cost of the equipment versus the logarithm of the surface area that it occupies. In most cases, a linear plot is obtained. For example, for a glassed lined reactor, the straight line has a slope of 0.54 (see Figure 18.2); for a doublecone rotary blender, it has a value of 0.49. In the frequent advent when limited information is available on the cost effects of scaling up, which still is the common case in preparative HPLC, it is conventionally assumed that a slope value
864
Optimization of the Experimental Conditions log ($ Capital)
6.56.5
Figure 18.2 Cost of a chromatographic column skid versus the cross-sectional surface area of the column. Reproduced with permission from A. Katti, in Handbook of Analytical Separations, Vol. 1, "Separation Methods in Drug Synthesis and Purification," K. Valko, Ed., Elsevier, Amsterdam, The Netherlands, 2000, p. 213 (Fig. 7.7).
6.0
Slope=0.53 5.5
5.0 2.0
2.5
3.0
3.5 log (Surace Area, cm2)
4.0
of 0.6 is a reasonable estimate. The value of the cost of the process equipment required for the scaling up of the process is estimated from the cost of the equipment used at a lower production rate, this log-log relationship/ and a slope of 0.6. This assumption implies that preparative chromatography, as a unit operation, is cost effective upon scale-up, which is consistent with the conventional wisdom in the field. Accordingly, it is important to use a column and the associated instruments required to produce the amount desired, rather than a battery of smaller units. The design and the selected operating conditions must be optimized to minimize the cost. Katti [16] performed an economic analysis of a binary separation, in order to study the dependence of the various cost contributions on a few critical parameters. This study affords a new perspective on the optimization of the chromatographic process. This analysis was based on the systematic use of the numerical solution of the set of partial differential equations for chromatography, with the cost per gram of purified compound produced as the objective function. Assumptions were made regarding the required product purity, the retention and the separation factors, the feed mixture composition, the column capacity, the axial diffusivity of the main component, the mobile phase viscosity, the parameters of the Knox plate height, the selected operating pressure, the average particle size, the equipment costs, its depreciation, the solvent costs, the packing life time, the installation costs, the system availability and/or the downtime, and the manpower costs. The loading amount, the flow rate, the column length were simultaneously optimized. This analysis showed that, when the separation factor is increased from a value (a = 1.05) corresponding to a difficult separation to a value (a. = 1.4) corresponding to an easier separation while all the other parameters remain constant, all the cost contributions decrease strongly in absolute values (see Figures 18.3a and b). However, the relative contribution of the solvent costs increase considerable while that of the packing costs decreases markedly. This is because, when the separation becomes easier, the optimum values of the mobile phase flow rate and of the loading factor increase while that of the column length decreases (see Figures 18.4a to d). This result occurs in a large part because the larger the separation factor, the lower the column efficiency needed.
18.2 The Economics of Chromatographic Separations a
Separation Factor Factor Separation
Separation Factor 1.4
b
1.05 5.2 Solvent 24%
7.9 Packing 36%
1.7 Lost Crude 8% 1.8 System 8%
0.05 Labor 19%
5.2 Labor 24%
0.05 System 7%
0.07 Packing 10%
0.02 Lost Crude 7%
Crude Cost Cost $0.1/g $0.1/g
d
0.34 Packing 23%
0.14 Solvent 58%
Total Cost/g 0.24 Crude Cost $100/g $100/g 0.48 Solvent 34%
0.12 System 8%
0.13 Labor 20%
0.03 Lost Crude 5%
e
0.02 Packing 9.0%
0.02 System 7%
Total Cost/g 21.9
c
865
0.12 Packing 11%
0.38 Solvent 58% Total Cost/g 0.65
ire 10 Pressure
0.41 Labor 35%
Cost/g 1.45 1.45 Total Cost/g
Bar Bar
1.19 Total Cost/g 1.19
0.13 Packing 20% 0.03 System 5%
f
0.34 Solvent 28%
0.14 System 12%
0.18 Lost Lost Crude Crude 12% 12%
0.33 Labor 23%
0.17 Lost Crude 14%
0.09 Labor 14%
Pressure 200 Pressure
Bar Bar
0.32 Solvent 47% 0.09 Lost Crude 14%
Cost/g 0.66 0.66 Total Cost/g
Figure 18.3 Relative Cost Contributions for different values of the separation factor, the crude cost, and the operating pressure selected, a, b: a = 1.05 and 1.4; c, d: crude cost, 0.1 and 100 $/g; e, f: pressure, 10 and 200 bar. Reproduced with -permission from A. Katti, in Handbook of Analytical Separations, Vol. 1, "Separation Methods in Drug Synthesis and Purification," K. Valko, Ed., Elsevier, Amsterdam, The Netherlands, 2000, p. 213 (Figs. 7.9, 7.13, and 7.15).
The cost of the crude has a significant influence on the optimum experimental conditions. Figures 18.3c and d show that, when the crude cost increases by three orders of magnitude, all the cost contributions increase significantly but their relative contributions change because the optimization process must lead to a higher value of the recovery yield. Thus, the relative contribution of the solvent cost decreases, that of the packing cost increases, and, although the amount of lost crude
Optimization of the Experimental Conditions
866 Fv L/min
Fv, L/min 12
N
20
12000
10
6000
N 10000
N
Fv, L/min 21
.--•i
6000
14 8
6000
3000 7
0
0
Loading, product/g packing Loading, mg product/g packing
100
a
100
50
0
0
80
120
45
60
0
$/g
c
b
Separation Factor
25
c 1000 1000 kg/yr
$/g
8.0
1
15
1.4
30
c
1
4.0
2
1.2
84 b Cycle Time 35
1.5
1
1
89
0
10
4
0.1
22
40
1000 kg/yr 2
$/g 6
10 Total
94
15
0 1.6
0 0.1
d
a
Yield
29
L, cm 80
50
10
0
0 Loading, mg product/g Loading, mg product/g packing packing
Cycle Time 90
40
30
100
70
L, cm 70
1000 kg/yr
a
85
10
b
Cycle Time
10
2000 Yield 100
17
50
L, cm
4 Loading, mg product/g packing 24
Yield
0.5
1
10
Crude Cost $/g
100
0 1000
0 0
d
50
100 Pressure, Pressure, bar bar
150
0.0 200
d
Figure 18.4 Optimum operating parameters as functions of the separation factor, the crude cost, and the inlet pressure, a, b, c, d: Effects of the separation factor; e, f, g, h: Effects of the crude costs; i, j , k, 1: Effects of the inlet pressure, a, e, i Variations of the flow rate and the column efficiency; b, f, j : loading/g packing and recovery yield; c, g, k: Column length and cycle time; d, h, 1 cost per grams produced and production rate. Reproduced with permission from A. Katti, in Handbook of Analytical Separations, Vol. 1, "Separation Methods in Drug Synthesis and Purification," K. Valko, Ed., Elsevier, Amsterdam, The Netherlands, 2000, p. 213 (Figs. 7.8, 7.14, and 7.18).
decreases, the fractional cost of that lost crude increases. This is because, as the crude cost increases, the optimum value of the flow rate decreases, that of the column length increases, and the loading factor of the column decreases in order to increase the yield and reduce the total costs (see Figures 18.4e to 18.3h). Finally, Figures 18.3e and 18.3f show that, as the operating pressure increases, the total cost of a unite production decreases by a factor two, the relative contributions of the different costs origin shifting again considerably. The fractional costs of the solvent and the packing material increase (their total costs are hardly changed). The cost of lost crude remains constant but the corresponding fractional cost is doubled. However, and most importantly, the labor and system costs decrease considerably, nearly five-fold. This can be understood from Figures 18.4a to d. As the maximum allowable operating pressure increases, the optimum values of the flow rate and of the column length increase while the loading factor remains constant in order to minimize costs. The feed amount injected per cycle increases with the column length and the cycle time decreases, explaining the larger production rate of the unit.
18.3 Optimization Based on Theoretical Considerations
867
18.3 Optimization Based on Theoretical Considerations It would be very attractive to derive analytical expressions for the optimum experimental conditions from the solution of a realistic model of chromatography, i.e., the equilibrium-dispersive model, or one of the lumped kinetic models. Approaches using analytical solutions have the major advantage of providing general conclusions. Accordingly, the use of such solutions requires a minimum number of experimental investigations, first to validate them, then to acquire the data needed for their application to the solution of practical problems. Unfortunately, as we have shown in the previous chapters, these models have no analytical solutions. The systematic use of these numerical solutions in the optimization of preparative separations will be discussed in the next section. The ideal model is the only model for which analytical solutions are available for the prediction of band profiles, provided that the competitive equilibrium isotherms of the two components could be accounted for by the Langmuir model (Chapter 8). However, the use of analytical solutions of the ideal model for optimization purposes suffers from two serious drawbacks. First, the competitive Langmuir model accounts only approximately for the competitive adsorption behavior of most binary mixtures (Chapter 4). Second, the ideal model assumes the column efficiency to be infinite, which severely limits its usefulness. All columns have a finite efficiency, with the result that the mixed band between the end of the collection of the purified first component and the beginning of the collection of the purified second component is wider than predicted by the ideal model. Accordingly, the recovery yield and the production rate are lower. Finally, the solution of the ideal model alone cannot say anything useful regarding the optimum values of the parameters that control the column efficiency, i.e., the mobile phase flow velocity and the particle size, unless some suitable correction is designed and used. It can inform on the optimization of other parameters and particularly of the loading factor of the column, as we discuss in the next subsection. A combination of the solution of the ideal model with a simple model of band broadening, following the approach initially suggested by Knox and Pyper [17], permits one to account for the influence of the column efficiency on the production rate of the second component. Some useful conclusions have been reached, on which we report here.
18.3.1 Simultaneous Optimization of the Production Rate and Recovery Yield using the Ideal Model Felinger and Guiochon [18] maximized the product of the production rate and the recovery yield using the ideal model. The recovery yield for the more retained component is [2]: 1 —/
if Lf I--,
i_i
, otherwise
868
Optimization of the Experimental Conditions
where r\ is the positive root of Eq. 8.12 and x is given by Eq. 18.52. For the definition of the cycle time, they considered the time required for regeneration and re-equilibration of the column after each run. Assuming that j column volumes of solvent are needed to regenerate the column, the cycle time will be defined as the analytical retention time of the more retained component plus / times the void time. With this definition, the production rate can be expressed as: Pr2
= { w+j+y
<•
M0K9
(a—lj/a
—
'
i
u
—'
^
L - ~ J
(1820)
.1
otherwise
The maximum production rate is reached at the limit of touching bands, and it remains constant when the loading factor is increased further. For the less retained component, assuming that 100% purity of the collected fraction is required, the recovery yield of the less retained component is [2]: 1+
^
(18.21)
otherwise The production rate of the less retained component is (18 22)
^ = fc^+y + i)
'
The combination of Eqs. 18.19 to 18.22 defines the combined objective function, which is Pr2xY2={ —
-
u0k'2
Xlm _ l 4 % I_X
*-i)/gl 2 _J_ [(*-!)/ if L•f,2 f, < ^ ^ rL L XT ^%^JJ
/ ^ otherwise
-
(18.23)
for the more retained component, and b2{V2+j+l)
x Y1 =
1 - LM(i+1) h(k'2+j+i) [*•
(^-^iy^)2]
(!8-24)
otherwise
for the less retained component. Note that Eqs. 18.19 to 18.24 are valid only if qsi = qS2 which is the condition of validity of the competitive Langmuir isotherm. Figure 18.5 shows the results calculated with the ideal model for the combined objective function of production rate and recovery yield. When the separation of the less retained component is optimized, the ideal model fails to identify an optimum value of the loading factor for maximum production rate. The production rate increases monotonously with increasing loading factor while the recovery
18.3 Optimization Based on Theoretical Considerations
869
Figure 18.5 Plot of the production rate, recovery yield, and their product for the (a) less retained and the (b) more retained component against the loading factor, based on the ideal model, a = 1.2; k'2 = 2, Cj = 100 mg/mL, Cf = 300 mg/mL, qs = 260 mg/mL Reproduced with permission from A. Felinger and G. Guiochon, ]. Chromatogr. A, 752 (1996) 31 (Figs. 1 and 2).
yield decreases with increasing value of L^j beyond the value at which touching bands is reached. With the Pr x Y objective function we are able to find an optimum value of the loading factor, although the maximum is very flat in most instances. When the purification of the more retained component is optimized, the recovery yield is 100% until the bands of the two components touch each other. The production rate increases linearly with increasing value of the loading factor. When the two bands begin to overlap, the recovery yield begins to decrease with increasing value of Lf/2, while the production rate remains constant. Choosing the product of these two quantities Pr x Y as the objective function, we observe a sharp maximum at the loading factor where band overlap begins. The yield achieved is 100% and the production rate is maximum, an ideal situation.
18.3.2 The Knox and Pyper Approach Knox and Pyper [17] made the first systematic study of the optimization of the experimental conditions in preparative liquid chromatography based on the use of a simple chromatographic model. They considered touching band separation {i.e., in the ideal model, £R/2 — ^E (Figure 8.6), or approximately a unit resolution between the two bands) with the following assumptions: 1. There is no competition between the two components. Accordingly, the elution band profiles of the two components are the same, whether a given amount of each component is injected pure or as a mixture. They recognized that this assumption is inexact when the resolution between the bands is moderate {i.e., lower than 2). Due to the large influences of the displacement and the tag along effects on the band profiles, this assumption is unrealistic for bands having a resolution of the order of 1. 2. The elution band profile of each component would be a right triangle if the
870
Optimization of the Experimental Conditions
column efficiency is infinite. We know that this is true only for the isotherm q = fl(l — bC)C, the parabolic isotherm. It is incorrect for a Langmuir isotherm (Chapter 7). 3. The apparent column HETP, Htot, defined as the ratio of the second centered moment of the elution band to the column length, can be separated into two additive contributions, Htot = H^n + H^, where Hjyn and H^ are the kinetic and the thermodynamic contributions, respectively. The term H^n is the same for a finite or a very small sample and is equal to the column HETP measured under analytical conditions. The term Hth is the contribution to band broadening due to the nonlinear behavior of the isotherm. This is only a first order approximation, as we have shown in Chapter 10 (Section 10.2.4). However, this approximation is reasonable [19]. The main conclusions of their work were the following [17]: a. The volume of sample solution injected has no effect on the width of the elution band, provided that it is smaller than half the volume bandwidth of an analytical size sample {i.e., 2
18.3 Optimization Based on Theoretical Considerations
871
value of c£,/L, it is independent of the separate values of the particle size and the column length. This optimum ratio is given by the relationship Dm
n
V 3 N* C AP k0 opt
(18.26)
'
When a complete plate height equation is used (e.g., h = b/v + Av033 + Cv), the dependence of the throughput on dp and L at constant value of di/L is small. The most serious limitation to the validity of the conclusions of this work is the assumption that the competitive interactions between the components of the mixture can be neglected. Even in the case of touching bands, when the elution bands of the two components have a resolution unity, this resolution is achieved only at the column exit, which means that the two bands were overlapping during their entire migration along the column. If the two components are simultaneously present in the mobile phase at finite concentrations, they compete for access to the stationary phase. As a result of this interaction, the separation is better than predicted by the model when the concentration of the second component exceeds that of the first one (displacement effect) and worse in the opposite case (tag-along effect), as shown in Chapter 8 and in the next section.
18.3.3 Optimization for Touching Bands Using the Ideal Model Golshan-Shirazi and Guiochon have investigated the optimization of the experimental conditions using the analytical solution of the ideal model [20-24]. In the case of touching bands, the recovery yield is practically total (~ 100%). Therefore, the same experimental conditions assure the maximum production rate for both components. Their assumptions are limited to the following two: 1. The apparent column HETP can be approximated by the sum of kinetic and thermodynamic contributions, as suggested by Knox and Pyper [17]. As noted in Chapter 10, this approximation seems reasonable at a high degree of column overloading, because then the band profiles are essentially controlled by the thermodynamics of nonlinear phase equilibria and the kinetic contribution becomes a small correction. The results of Golshan-Shirazi et ol. [20-24] and those of Lucy and Carr [19] confirm the quality of this approximation. 2. The competitive interactions of the components of a binary mixture are accounted for by the competitive Langmuir isotherm model. Using this model, several optimization problems have been discussed—the touching band problem (this section) and the overlapping band problems, with or without a recovery yield constraint (next two sections). In the case of touching bands [20], this approach differs from the one of Knox and Pyper [17] essentially by the fact that the competition between the mixture components is now taken into account. Also, since the Langmuir isotherm model
Optimization of the Experimental Conditions
872
*
K
[\ i \ s
\
|
S| s
! s &'
\
\
s ci'
s Tkrats)
Figure 18.6 Comparison of the true band profiles at the optimum sample size predicted by the noncompetitive model of Knox and Pyper (dotted lines) and by the ideal model with competitive Langmuir isotherms (solid lines) in the case of touching bands. k'01 = 6. a. = 1.2. Isotherm coefficients: b\ = 2.4; \>i = o(b\. Column length: 25 cm. Phase ratio: F = 0.25. Mobile phase velocity: 0.6 cm/s. (a) Feed composition 1:9. (b) Feed composition 3.6:1. (c) Feed composition 9:1. Reproduced with -permission from S. Golshan-Shirazi and G. Guiochon, J. Chromatogr., 517 (1990) 229 (Figs. 3 to 5).
is used, the band profile is not triangular, but this difference is relatively minor (Figure 18.6). Different results are obtained, depending on the intensity of the tagalong effect in the band interactions. In turn, this intensity depends on the degree of band interference (i.e., on the separation factor and on the sample size) and on the relative composition of the feed. Figure 18.6 illustrates the differences between the band profiles calculated with the ideal model, in the cases of the Knox-Pyper and the Golshan-Shirazi-Guiochon optimization models. The results show that the simplified model of Knox and Pyper, because it neglects competition, predicts sample sizes and production rates that are too low when the second component is the more concentrated and too high when it is the less concentrated. This is because the separation is enhanced by the displacement effect in the former case, depressed in the second case by the tag-along effect. As explained in Chapter 8, in the discussion of the solutions of the ideal model, touching bands take place whenever the loading factor of the second component is equal to
0*-I) 2 b2 )
cc2{l
(18.27a)
with
x=
(18.27b) a, qS/1
where a = a
18.3 Optimization Based on Theoretical Considerations
873
of the problem.2 The degree of band overlapping in the chromatogram depends on the relative value of the loading factor compared to L/,2,j- Eor a column of infinite efficiency, the bands overlap if Ly/2 > ^f,2,t- The bands are touching when L^2 = ^f,2,t a n d the plateau on the top of the band profile of the second component caused by the interaction between the two components is intact. If Lfi2 = ^f,2,p'
the plateau caused by the tag-along effect has completely disappeared and the second component profile is the same as if this component were pure. When Lf/2 is intermediate between Lft2t. and Lf/2p, the plateau subsists in part. Thus, two different equations should be used to calculate the band profile of the second component (see Chapter 8, Tables 8.4 and 8.5). For real columns, the efficiency is finite and the loading factor required to achieve touching band separation is smaller than for an ideal column. The actual value depends on the column efficiency, and increases with increasing efficiency. It could be smaller than Lf/2iP, or between Lf,2,p
an(
18.3.3.1
^ Lfi2,t.
Touching Bands with Lf/2 < Lf/Zp (No Tag-Along Effect for the Second Component)
In this case (L = zj,; see Chapter 8, limit between cases 4 and 5, Figure 8.8a and Table 8.5), the loading factor of a column with finite efficiency is the solution of the following equation [20]: 16
,,z . K
,„
. . -,,. ,
(lg29)
0,2
Solving this equation for Lf/2 gives the optimum loading factor for touching band separation. Note that this solution is valid only as long as it is smaller than or equal to Lf2/p (Eq. 18.28). Otherwise, the equation given in the next section (b) must be used. Putting this value in the equation for the production rate (Eq. 18.4b) gives the production rate of the second component. This result is complicated, as the equation has no analytical solution, although it is easily solved numerically. Several simplifications are possible, however. i. First Simplification We can assume that the loading factor obtained is relatively small and neglect Lf2 compared to i/^/,2 m Eq. 18.29. In doing so, we assume that the band profile of the second component is a right triangle, as did Knox and Pyper [17], but we still conserve the competitive interaction between the two components, which is introduced by the use of the competitive Langmuir 2
See Chapter 8, Eq. 8.10a, ocb\C2r2 - (u - 1 + a&iA - b2C2)r - b2Cj = 0.
Optimization of the Experimental Conditions
874
isotherms. With this assumption, the modified Eq. 18.29 is easily solved, and we have: , « - l , 16 a No
^,2
(18.30)
Inserting this value of Lr2 into Eq. 18.4b gives the production rate: P?2
qs,2U{& — I ) 2
1
2
(l-e)S
cc-1
16
(18.31)
ii. Second Simplification As a further approximation, we may neglect the competitive interactions between the two components of the mixture, thus ignoring the term L / 2 / ( « - 1) in the RHS of Eq. 18.29. Then, we obtain: -) 2
-l
(18.32)
No
The combination of Eqs. 18.4b and 18.32 gives the production rate: p
r 2
(l-e)S
-
-
'
• -
• •'
• -
• z |
(18.33)
This equation is identical to the equation given by Knox and Pyper [17] (combination of their Eqs. 40, 47, 63, 67), in the case when the cycle time is equal to tR0,2 — to= ^0,2^0-
iii. Optimization Using a Simple Plate Height Equation This requires the use of a plate height equation to relate the column efficiency and the mobile phase flow rate. There are two possible approaches, depending whether a classical plate height equation is used (e.g., the Knox equation [25]) or a simplified equation is preferred since the column is operated at a high flow velocity [3,4,6,7]. The latter method gives approximate equations, but permits a detailed discussion of the effect of each parameter; the former requires a numerical solution [4,20,22]. We present first its conclusions [20]. The use of the Knox plate height equation is discussed at the end of this section. The linear mobile phase velocity, u, is related to the column design and operating parameters by the following equation (Chapter 5, Section 5.3): k0APd2p u=
(18.34)
wherefcois the column specific permeability (of the order of 1 x 10 3 ), AP, the pressure drop between the column inlet and outlet, dp, the average particle size of
18.3 Optimization Based on Theoretical Considerations
875
the packing material used, t], the mobile phase viscosity, and L, the column length. Since the column is operated at a high mobile phase velocity, we can reasonably assume that the height equivalent to a theoretical plate (HETP) of the column is given by the simplified equation [17]: h = Cv (18.35) where h is the reduced plate height (h = H/ dp, with H equal to the HETP) and v is the reduced velocity (v = udp/Dm). The column efficiency under linear conditions (i.e., at infinite sample dilution) is No = L/H^ or (1836) ^ As shown by Rnox and Pyper [17], both the linear velocity of the mobile phase and No depend only on the ratio d2 / L, not on the separate values of the column length and the particle size. Thus, the production rate also depends only on this ratio. As long as Eq. 18.35 is valid, long columns packed with coarse particles give the same production rate as short columns packed with fine particles, provided that they have the same ratio dl/L. Thus, if the column length can be adjusted easily, a variety of packing materials of different sizes can be used (within reason). The experimental demonstration of an optimum particle size at constant column length has been given recently [26], in agreement with the results of a previous study [27]. Note that this conclusion is general, and is valid in the cases of touching bands as well as overlapping bands. The combination of Eqs. 18.4b, 18.29,18.34, and 18.36 gives the production rate of the second component. Differentiation of this equation with respect to d\/h permits the calculation of the optimum value of this ratio corresponding to the maximum value of the pressure drop allowed. The equations cannot be solved in closed form, but the numerical calculation is easy. Some possible simplifications are interesting and will be discussed below. Assuming that the band profiles are right triangles, a reasonable assumption when the relative retention is close to 1 and the loading factor has to be small, we can use Eqs. 18.30 and 18.31 instead of Eqs. 18.29 and 18.4b. Then, the equation giving the optimum value of the ratio di/L can be solved to give
a
- i J(a - l/2)Voc2-a
-
(a2-cc-1/12) (18.37)
Combination of Eqs. 18.30,18.36, and 18.37 gives the optimum value of the loading factor for touching bands: (18 38) ^ ^ ^ " The value of the optimum production rate is obtained by combining Eqs. 18.31, 18.34,18.37, and 18.38:
-i)3
Optimization of the Experimental Conditions
876
ot- 1/2) y V - a + 1/36 - (a2 - a - 1/12)
(18.39)
Finally, if we adopt the model of Knox and Pyper [17], neglect the competitive interactions between the two components of the mixture, assume that the band profiles are right triangles, and use the value of the loading factor given by Eq. 18.32, we obtain (18.40)
The optimum value of the loading factor becomes _ 1 fa-l /' oPt ~ 6 V oi
(18.41)
2
Inserting Eq. 18.40 into Eq. 18.35 gives the optimum column efficiency: (18.42)
T>— and the maximum production rate is
(l-e)S
ec-\ 24(l + fc(,2) V cc
kpAPDm
(18.43)
These equations are equivalent to those derived by Knox and Pyper [17]. 18.3.3.2
Touching Band with Lf/2 > Lf/2rP (Residual Tag-Along Effect for the Second Component)
When the ratio of the two loading factors, Lf^/Lf^ increases, i.e., when the first component concentration increases, the tag-along effect becomes dominant, and the optimum loading factor increases toward L/,2,p- This is the case 4 in Chapter 8 (see Figures 8.6a, c, and d). When the optimum value of Lf^ becomes larger than Ly 2,p Eq. 18.29 and the equations derived in the previous section are no longer valid. In practice, this happens when X (Eq. 18.27b) is larger than 0.4 [20]. In this case, instead of Eq. 18.29, the loading factor is given by the following equation [20]: (18.44) 16
*
\
Nofe
18.3 Optimization Based on Theoretical Considerations
877
Equations 18.34,18.35, and 18.36 of the previous subsection still permit the derivation of the optimum mobile phase flow velocity, column plate height, and plate number at infinite sample dilution, respectively. The production rate is obtained by combining Eqs. 18.4b, 18.34,18.36, and 18.45. Writing that the differential of the production rate by respect to d2 / L is 0 gives the optimum value of this ratio [20]: a+ll ^I[X 2 + 3X+^± opt
1__2 \
2
1
T + [X 2 +3X+^±1] 2 (18.45) \ 4
fX:
where X is given by Eq. 18.27b. When X becomes large, the square root term in the RHS of Eq. 18.45 is practically equal to unity and the equation simplifies considerably to (18.46) opt
In this case, the optimum value of the ratio C&/L is independent of the feed composition. Inserting the optimum value of dl/L given by Eq. 18.45 into Eq. 18.36 gives the optimum column efficiency (at negligible sample size). Combination of these equations with Eq. 18.43 gives the optimum loading factor
4X(1 + X) 2
1+8
3-
(18.47)
\
If X is large, the optimum loading factor becomes ,2
1 V.opt - 2X-
(18.48)
CL
It is equal to the value predicted by the noncompetitive model of Knox and Pyper [17] for X = 3, shorter when X is larger than 3, and longer when X is smaller than 3. In this case, the column efficiency is No,0pt = 6 4
oc-1
(18.49) ^0,2
This efficiency corresponds to a resolution of 2 between the analytical peaks. Finally, combination of Eqs. 18.4b, 18.36,18.45, and 18.47 gives the maximum production rate that can be achieved with a certain value of the inlet pressure.
878
Optimization of the Experimental Conditions
18.3.3.3
Optimization Using the Knox Plate Height Equation
More accurate results can be obtained using the classical Knox plate height equation [25]. No analytical solutions can be derived in this case, but the optimum conditions can be derived by following a simple procedure. A specific value of the particle size is selected, and the value of the mobile phase flow velocity achieved with the maximum pressure drop available is calculated for a series of increasing values of the column length (Eq. 18.34). The column efficiency (No = L/hdp) is derived using the Knox equation h = ^- = -+ Avl/3 + Cv dp v
(18.50)
Equations 18.29 and 18.43 remain valid (depending on the value of Lf2), independently of the plate height equation selected. Introducing the column efficiency in the proper equation (Eqs. 18.29 or 18.43, depending whether Lf^/ctLf^ is smaller or larger than 0.4) gives the loading factor. Finally, inserting the loading factor and the velocity in Eq. 18.4b gives the production rate for each column length. This permits the derivation of the optimum column length for maximum production rate (if using particles of a given size) [20]. Alternatively, the particle size could be varied at constant column length, to derive the optimum particle size for a given column length.
18.3.4 Optimization for Overlapping Bands with No Yield Constraint In the case of overlapping bands, the optimum experimental conditions are different whether maximum production rate of the first or the second component is looked for. The calculations of the optimum experimental conditions using the ideal model can be carried out for the second component, because a correction can be derived to take into account the dispersion of the two bands in the mixed zone [22]. For the first component, this is not possible and this method cannot be used to derive the optimum experimental conditions for maximum production rate of the first component. Production rate and recovery yield of this compound depend on the thickness of the shock layer, which cannot be calculated in overloaded elution. Golshan-Shirazi and Guiochon [21,23] have derived from the solution of the ideal model [22] the equations giving the optimum loading factor, L',2, for maximum production rate of the second component and the corresponding sample size, in the case of overlapping bands (see ref. 1, Eqs. 19 and 20)
L 2
/< - U r ^ y J T-^E-TT^f-
n tot
=
L}i2(l + ^)(l-e)SLqSi2
i+x
(18 51a)
-
(18.51b)
18.3 Optimization Based on Theoretical Considerations
879
where X is given by Eq. 18.27b, and x=
1 - P « 2 _,
/C°(1-PM2)
aC?PM2
(18.52)
where Pu2 is the required purity of the second component.3 Using the same relationship as Knox and Pyper [17] to relate the apparent column plate height and the efficiency under linear conditions, and to take the finite column efficiency into account, these authors showed also [21] that, under these conditions, the production rate is given by Pr2
(18.53a)
(l-e)S
(18.53b) w iith
4
^2,*
=
W
=
2JLf2/P
(18.53c)
2,th
— Lf2/P
(18.53d)
and the recovery yield is given by (18.54) When there are no yield constraints, the yield calculated at the maximum production rate is usually around 60% [21,23]. Figure 18.7 shows a plot of the production rate of the second component versus the sample size. In this figure, the solid lines are given by Eq. 18.53a, while the symbols are derived from the calculation of numerical solutions of the equilibrium-dispersive model [21]. Equation 18.51a predicts correctly the optimum sample size for maximum production rate of the second component, although it has been derived from the ideal model [2]. Use of a Simplified Plate Height Equation If the plate height is assumed to be given by the simplified Eq. 18.35 (h = Cv), Eq. 18.36 remains valid, and d^/L is given by tjDn k0 AP C No
(18.55)
3 rj is the positive root of Eq. 8.12, and is practically equal to . «is the separation factor. «,-, b\ are the coefficients of the Langmuir isotherm. i = (b2 + fl*iri)/(fc2 + hn) (see Eq. 8.24), and LfAp is given by Eq. 18.28.
Optimization of the Experimental Conditions
880
Figure 18.7 Influence of the sample size on the production rate of a preparative chromatographic column. Plot of the production rate per unit mobile phase flow rate, Pr/Fv, versus the loading factor, Lfz- Solid lines derived from Eq. 18.53a. Symbols from numerical solution of the equilibrium-dispersive model. Experimental conditions: Column length: 25 cm; particle size: 10 }im; efficiency: 5000 theoretical plates. Isotherm coefficients: k® = 6.0; a = 1.2; b\ = 2.5; b2 = 3.0. Sample composition: C° = 0.5 M; C\ = 4.5 M (1:9 mixture), tp = 1 s. Purity requirements: curve 1: 99.5%; curve 2: 99%; curve 3: 98%. Reproduced with permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 1368 (Fig. 2). ©1989, American Chemical Society.
0.0
C.I
Elimination of di/L between Eqs. 18.34 and 18.55 gives AP Dn
u=
(18.56)
Combination of Eqs. 18.53a and 18.56 gives Pr2
AP Dn
(l-e)S I-ax
1\
V . w2/th .
1+
16 w
lth
k> 0,2
(18.57)
This equation shows that there is an optimum column efficiency which permits the achievement of the maximum production rate for a given separation. This efficiency is obtained by solving the algebraic equation obtained by writing that the differential of Eq. 18.57 is equal to 0. It is easier to derive from Eq. 18.57 the production rate for several values of the column efficiency and to find the maximum value by interpolation. The optimum values oidl/L and u are obtained from Eqs. 18.55 and 18.56, respectively. Equation 18.57 shows also that the production rate is proportional to \fA~P and increases always with increasing pressure P at which the equipment can be operated. Figure 18.8 illustrates this influence, hi Figure 18.8a, the production rate is plotted versus di/L for different values of AP, showing that there is an optimum
18.3 Optimization Based on Theoretical Considerations
881
Figure 18.8 Influence of the column inlet pressure on the production rate, (a) Plot of the production rate per unit column cross-sectional area, Pr-ileS, versus the ratio d^/L. Feed composition: 1:9. Separation factor oc = 1.10. Curve 1, AP = 200 atm. Curve 2, AP = 500 atm. Curve 3, AP = 1,000 atm. (b) Plot of the maximum production rate versus the inlet pressure. Values of the ratio dj/L (in 10" 8 cm): 1, 21.1 (P=20); 2,13.1 (P=50); 3,9.3 (P=100); 4, 7.65 (P=150); 5, 6.65 (P=200); 6, 5.4 (P=300); 7, 4.75 (P=400); 8, 4.25 (P=500). Reproduced from permission from S. Golshan-Shimzi and G. Guiochon, Anal. Chem., 61 (1989) 1368 (Figs. 6b and 6c). ©1989, American Chemical Society.
value of the geometrical parameter. This optimum decreases with increasing pressure. In Figure 18.8b, the maximum production rate of the optimum column is plotted versus the pressure. The column should always be operated under the maximum pressure possible, and the column length and particle size optimized for maximum production rate. Finally, the dependence of the production rate on the separation factor is complex since No, 7, x, and X depend on oc [21]. x varies rapidly with a when the relative concentration of the second component is large, and the displacement effect is dominant [2]. Nevertheless, it can be shown that at constant pressure AP, the maximum production rate obtained with an optimized column is approximately proportional to [(a — l)/cc]3 [28]. For a given (i.e., nonoptimized) column, the production rate is proportional to [(«. — l)/a]- v , with y between 2 and 3, depending on the importance of the difference between the given and the optimum columns [28]. More accurate results could be obtained with the Rnox plate height equation (Eq. 18.50), using a simple numerical procedure [21].
Optimization of the Experimental Conditions
882
Table 18.2 Trade-offs between the Production Rate and the Recovery Yield"
Yield
No
Rhs
(/o)
60c 80d 90d
95 d 99d
99.9d Touching Band 2
L (m)
V
Loading Factor (%)
550 650 825 1050 1800 3000
0.76 0.83 0.94 1.05 1.38 1.79
0.084 0.097 0.111 0.127 0.173 0.234
113 103 90 78.7 57.7 42.9
4.82 3.60 3.20 3.04 2.92 2.9
3230
1.85
0.244
41
1.24
Production Rate
Pr2/S(l-e) (mole m~2 sec"1) 0.46 0.413 0.361 0.316 0.232 0.172 0.0705
"Optimum experimental conditions for maximum production rate at various specified values of the recovery yield. Experimental conditions: oc - 1.2, k'o 1 = 3; column saturation capacity: qS/i = qSf2 = 5; phase ratio, F = 0.25; maximum available pressure, AP = 100 bar; solute molecular diffusion coefficients, Dm = 1 x 10~9 m 2 /s; mobile phase viscosity, r\ = 1 cP; packing particle average diameter, dp = 10 ^m; degree of purity of the collected fractions, 99%, cycle time, Atc = £R(Q2 —fy;plate height equation, h = 2/v + v0-35 + O.li/; composition of the binary mixture, 10% of the first component. h Rs is the resolution observed between the two component bands at very low value of the loading factor, i.e., under linear conditions. c Maximum possible production rate calculated without yield constraints, using the method described in [21] (Section 3). Values calculated using the procedure described in [23]. ^Values calculated using the procedure described in [20] (Section 2.b).
18.3.5 Optimization for Overlapping Bands with Yield Constraint Golshan-Shirazi and Guiochon have also derived the optimum experimental conditions for production rate with a yield constraint in the case of overlapping bands [23]. They have shown that the optimum loading factor is given by 0.6
(18.58)
where L% 2 is given by Eq. 18.51a, and Y2 is the required recovery yield. However, no analytical expressions can be derived for the optimum column efficiency in this case. The optimum efficiency has to be derived from the numerical solution of the equilibrium-dispersive model (Chapters 10 and 11). Table 18.2 summarizes the optimum experimental conditions for a given separation using: (1) overlapping bands without recovery yield constraint; (2) overlapping bands with different yield constraints; and (3) the touching band method. In the first case, the optimum conditions have been derived using Eq. 18.53a, and the procedure described in Section 18.3.4. The recovery yield is around 60%. hi
18.4 Optimization Using Numerical Solutions
883
the third case, the optimum conditions have been derived as explained in Section 18.3.3. In the second case, the yield constraint is handled using the following procedure [23]. Equation 18.4b is rewritten Pr2
_
Lf>2Y2qs,2u (1859)
Since the recovery yield achieved under the experimental conditions giving the maximum production rate is of the order of 60%, a first-order approximation of the loading factor giving a recovery yield Y is Lf = Q.6L*r/Y. Then the column efficiency is adjusted to achieve the required recovery yield. This method has been adopted because the results of numerical optimization suggest that there is a quasi-linear relationship between the recovery yield and both the mobile phase velocity and the loading factor [7]. The values of the production rate predicted by this optimization strategy for different values of the required recovery yield, from 60% (which is the yield obtained in practically all cases when optimization is carried out for maximum possible production rate, without yield constraints) to nearly 100% (the yield that corresponds to the touching-band case) are compared in Table 18.2. The production rate decreases slowly at first with increasing required recovery yield, the loss of production rate being 10% for an 80% yield and 22% for a required yield of 90%. Beyond 90%, the production rate decreases rapidly with increasing required yield, and this drop becomes precipitous above 99%. For the touching-band condition, the production rate is 15% of the maximum production rate possible and still only 22% of the production rate possible with a recovery yield of 95%. Allowing a recovery yield of 99.9%, which is equivalent to total recovery for all practical purposes, still gives a 2.5 times larger production rate than the touching-band case. However, the reason for this strong increase in the production rate is that we have accepted to produce 99% pure fractions, not that we have accepted a yield loss of 0.1%. Touching bands permit, at least in theory, both a total recovery yield and the production of 100% pure products. When the concentration of the less retained component in the feed is much less than the concentration of the more retained one, allowing a small amount of impurity in the product permits a large increase in the production rate [13]. This is no longer true at higher concentrations of the less retained component. The data in Table 18.2 also show that the required efficiency, as well as the column length, increases rapidly with increasing required recovery yield, while the reduced velocity and the loading factor decrease steadily. Although the throughput decreases, allowing important savings on the solvent costs, the column required is longer and the amount of packing material needed per amount of feed purified increases.
18.4 Optimization Using Numerical Solutions As we have seen, there are two simple objective functions for a numerical optimization: the production rate and the specific production. More complex schemes
884
Optimization of the Experimental Conditions
can also be studied, either by including a yield constraint in the maximization of the production rate, or by using as objective function linear combinations of the production rate and the specific production. Other strategies are also possible, using one of several possible recycling methods. These methods are not discussed here; they are summarized in the next section. We investigate the maximization of the production rate, with or without yield constraints, the maximization of the specific production, and the combination of these two optimization methods. Finally, the optimization of the experimental conditions in gradient elution and in displacement chromatography will be discussed and the relative performances of isocratic elution, gradient elution, and displacement chromatography compared.
18.4.1 Maximum Production Rate in Elution Snyder and Dolan [29] attempted to optimize numerically the sample size and the efficiency of a given column for maximum production rate, under specific purity requirements and recovery yield constraints for a given value of the separation factor. They used a numerical solution of the Craig model which was designed to reduce computing time. The optimum conditions were obtained by mapping the production rate as a function of the two parameters. For different values of the loading factor, the production rate and the recovery yield were calculated and plotted versus the column efficiency. When the production rate and the recovery yield have been mapped for a given value of a, the production rate and the corresponding value of the column efficiency for a given value of the recovery yield can be plotted versus the loading factor. From these plots, the optimum loading factor and column efficiency for maximum production rate at the required recovery yield are derived for the required purity and the chosen value of the separation factor. The method has been extended to the optimization of the same parameters in gradient elution [30]. This approach suffers mainly from the limitation of the number of parameters (only two) that can be optimized and from the fact that these two parameters are optimized separately, which does not generally afford the true optimum conditions. The maximization of the production rate by numerical methods was first studied by Ghodbane and Guiochon [7], who used the equilibrium-dispersive model to calculate the band profiles under a specific set of experimental conditions, the Rnox [25] equation to relate the mobile phase flow velocity and the column efficiency, and a Simplex algorithm for the optimization. They discussed only the simultaneous optimization of two parameters, the mobile phase velocity and the sample size, or the particle size and the column length. Their main conclusions were that 1. The optimum experimental conditions for maximum production rate of the first and the second component are quite different. 2. When the production rate obtained with a given column is optimized, the maximum production rate is achieved at high mobile phase velocities. The production rate depends heavily on the column efficiency under analytical conditions; most of this efficiency is traded off for a short cycle time.
18.4 Optimization Using Numerical Solutions
885
Figure 18.9 Plot of the production rate versus the loading factor. 1:3 mixture; a = 1.09. Sample volume in standard deviations, s, of the Gaussian profile obtained for the first component at infinite dilution. From top to bottom, Is, 3s, 5s, 10s. (a) Production rate for component 1. (b) Production rate
for component 2. Reproduced from permission from A. Katti and G. Guiochon, Anal. Chem., 61 (1989) 981 (Fig. 1). ©1989, American Chemical Society.. 3. The maximum production rate is achieved with a column that separates the two components with a resolution close to unity or lower, under linear conditions. 4. The production rate increases steadily with increasing allowed inlet pressure. 5. The recovery yield, typically about 60% at maximum production rate, can be increased to ~ 80% by accepting a moderate reduction in the production rate. Higher yields require a much larger reduction of the production rate. This agrees well with the results presented in Section 18.3.5 above. Katti and Guiochon [31] studied numerically the influence of the sample size on the production rate, and the separate optimization of the sample volume and amount, assuming 99% purity. Figure 18.9 demonstrates that there is an optimum sample size. This optimum amount is greater for the first component (Figure 18.9a) than for the second (Figure 18.9b), and it is less sharply defined, especially at small sample volumes (concentrated feed). The effect of the sample volume and size on the production rate is much less important when the separation factor is large. For the production rate of the second component, the maximum is sharp because the relative importance of the tail of the first component, behind the second shock layer, increases rapidly with increasing sample size, causing a rapid loss of recovery yield and a decrease of the production rate.
Optimization of the Experimental Conditions
886
ta
1 \1 \
Figure 18.10 Calculated chromatograms corresponding to the maximum production rate of one of the components of a binary feed. All optimum sample volumes, 1 cr of the analytical peak, (a) First component, 1:3 mixture; a = 1.09; sample size, Lf = 4.25%. (b) Second component. Same mixture as in Figure 18.10a, except Lf = 0.90%. (c) Second component, 3:1 mixture; u = 1.7; Lf = 29.1%. (d) Second component, 9:1 mixture, a = 1.7; Lf = 33.2%. The vertical tick marks show the position of the cut points. Reproducedfrompermission from A. Katti and G. Guiochon, Anal. Chem., 61 (1989) 981 (Figs. 5a,b, 6d, 7d). ©1989, American Chemical Society.
The chromatograms corresponding to the maximum production rate of each component for a = 1.09 and a 99% purity are given in Figure 18.10a and b, for this simple, unidimensional optimization problem. The loading factor corresponding to maximum production rate is always greater for the first component than for the second. The displacement effect can be used to enhance the production rate of the first component [8]. By contrast, as noted above, the tail of the first component behind the front of the second component increases rapidly with increasing sample size beyond a critical value not much above touching band conditions. Figure 18.10c and d also shows the chromatograms for maximum production rate of the second component of a 3:1 and a 9:1 mixtures with a separation factor a = 1.7. In these chromatograms the cut points for 99% purity are shown by vertical lines. Due to the strong tag-along effect that takes place for the second component in these last two figures, the production rate is not very important and the loading factors of these chromatograms are barely larger than those corresponding to touching bands. These conclusions were confirmed by later studies which also led to more detailed results [3,4,32,33]. Katti et al. [3] used a similar method to optimize the production rate of a given chromatographic column operated in the overloaded elution and the displacement modes, and compared these performances (see Section 18.4.7). The results obtained were in agreement with those reported above. The main limit of this work, as of the previous one, was that no more than two parameters were optimized simultaneously. Later, Felinger and Guiochon [4] discussed the four-dimensional simplex optimization of a separation by overloaded elution. This permits the simultaneous optimization of the column design parameters (column length and average particle size), and the operating conditions (mobile phase flow velocity and sample size). Systematic calculations were made to study the influence of the retention factor, which is not usually considered as an optimizable factor but has a profound
18.4 Optimization Using Numerical Solutions
887
Figure 18.11 Plot of the maximum production rate in elution versus the retention factor of the less retained component of a binary mixture. Separation factor 1.2. Each data point gives the maximum production rate after optimization of the mobile phase velocity, the sample size, the particle size, and the column length. Reproduced with permission from A. Felinger and G. Guiochon, ]. Chromatogr., 591 (1992) 31 (Fig. 12).
influence on the maximum production rate. As did the previous authors, they assumed in their calculations: (1) competitive Langmuir isotherm behavior; (2) the Knox plate height equation; and (3) the validity of the equilibrium-dispersive model of chromatography. They used the supermodified simplex algorithm of Morgan et al. [34], as described by Dose [35]. Their conclusions can be summarized as follows: 1. The optimum mobile phase velocity for maximum production rate is always higher than the velocity at which the column efficiency is maximum, between 10 (a = 1.1) and 30 (a = 1.8) times higher. Extremely high values of the optimum velocity may be obtained when the production rate of a given column is optimized, if the length of the selected column is too long for the desired separation, and the maximum column efficiency is much higher than needed. This is because the column efficiency required under optimum conditions is usually quite low, comparable to the efficiency giving a resolution of 0.9 between the two analytical bands. The excess efficiency is traded for an increased production rate by using a high flow velocity. A still higher production rate would be obtained with a shorter column. 2. Although there are separate optima for the column length and the particle size, the production rate varies slowly with either L or dp when the ratio &\IL is kept constant. The plot of the production rate versus L and dp has a ridge which is nearly parallel to the direction dS/L = constant (see Figure 18.12). This confirms the theoretical results of Knox and Pyper [17] and of Golshan-Shirazi and Guiochon [20,21]. 3. The optimum retention factor of the first component of interest is extremely low, often below 0.5, as previously suggested by Golshan-Shirazi and Guio-
Optimization of the Experimental Conditions
10.0
Figure 18.12 Contour plots of the production rate of the first component in elution as a function of the column length and the particle size (solid lines), dp in fim, L in cm. Mixture composition 3:1;« = 1.5; k\ = 6.0. Contour plots of d^/L = constant (numbered lines, value of the ratio in A). Reproduced with permission from A. Felinger and G. Guiochon,}. Chromatogr., 591 (1992) 31 (Figs. 8 and 10).
chon [36]. This result is illustrated in Figure 18.11, which shows a plot of the production rate versus the retention factor of the first component. This result is quite different from the conclusions of investigations made on the optimization of the experimental conditions for rapid separations in analytical chromatography [37]. The use of such low retention factors raises obvious technical problems, all the more because the band maximum is quite sharp, and accordingly the cut points must be recognized accurately. Valves have to be operated at much higher frequency, hence more precise control is required. Control of the chromatograph has to be tight. Finally, and most importantly, the calculations are carried out using narrow, rectangular injection profiles. In practice injection profiles are wide, and the rear of the elution profile of the unretained material may interfere with the front of the band of interest if k[0 is too small. Nevertheless, the advantages of low retention factors are obvious in terms of the production rate (Figure 18.11) and of the concentration of the collected fractions (see later, Figure 18.32). 4. The optimum column design parameters depend strongly on the retention factor of the first component (see Figure 18.13 and Table 18.3). The purification of the second component requires a longer column than that of the first one. 5. The optimum sample size for the more retained solute is close to the value
18.4 Optimization Using Numerical Solutions
889
Figure 18.13 Plot of the optimum value of the ratio d^/L in elution versus the retention factor of the first component of a 1:3 mixture. Separation factor: curve 1: 1.1; curve 2,1.2; curve 3,1.5; curve 4,1.8. Reproduced from permission from A. Felinger and G. Guiochon, f. Chromatogr., 591 (1992) 31 (Fig. 17).
Figure 18.14 Plot of the production rate in elution versus (a — l ) / a . k' = 4. 1, First component of a 3:1 mixture; 2, second component of a 1:3 mixture; 3, first component of a 1:3 mixture; A, second component of a 3:1 mixture. Optimum parameters in Table 18.3.
predicted by the ideal model (Eq. 18.51b) [2]. 6. The production rate increases very rapidly with cc — 1. It is proportional to [(a — l)/a]3 at low values of a (approximately below 1.03), but it increases somewhat more slowly with increasing cc at higher values. These results are illustrated in Figure 18.14, which shows the variation with a of the maximum
Optimization of the Experimental Conditions
890
Table 18.3 Optimum Conditions for Maximum Production Rate of the Components of a Binary Mixture. Influence of the Separation Factor+
3:1 mixtures
1:3mixtures
a 1.02 1.03 1.04 1.06 1.08 1.10
22470 0.72 14468 1.45 9600 2.06 5488 3.21 3551 4.17 2542 5.28
6.11 10" 3 1.8110"2 3.46 10" 2 7.89 10~ 2 1.35 H T 1 2.00 H T 1
m N 103.5 42395 134.3 24596 126.6 18513 112.7 10333 94.8 6695 85.9 4796
1.02 1.03 1.04 1.06 1.08 1.10
39392 22224 16507 9503 5966 4369
3.49 10"4 1.10 H T 3 2.64 10"3 7.47 10~3 1.46 10"2 2.37 10"2
10.1 23017 12.8 15779 16.9 10949 21.9 6058 23.7 4100 26.3 3304
N
Pr
0.04 0.09 0.16 0.36 0.62 0.94
1.0110"3 3.57 10" 3 8.43 10" 3 2.18 10~ 2 3.92 10~ 2 5.98 10~ 2
m 57.0 121.2 168.2 177.9 155.5 148.3
0.06 2.09 10" 3 0.11 7.19 10~ 3 0.19 1.52 10" 2 0.40 3.94 10~ 2 0.68 7.35 10" 2 0.98 1.16 10"1
8.8 11.8 13.3 15.5 17.8 20.7
Pr 0.21 0.77 1.42 2.69 3.63 4.83
+
Top part, optimization for maximum production rate of the first component. Bottom part, optimization for maximum production rate of the second component. k' = 4. N, optimum plate number; he, optimum loading factor; m, optimum reduced or apparent sample size (Eq. 10.15c); Pr, maximum production rate.
Figure 18.15 Chromatograms of binary mixtures corresponding to maximum production rate (a = 1.20). (a) First component of a 3:1 mixture, (b) Second component of a 1:3 mixture, (c) First component of a 1:3 mixture, (d) Second component of a 3:1 mixture. The position of the vertical line indicates the cut point for production of the corresponding component at 95% purity.
20 a
b 4
15
10 2 5
0
0 60
80
100
120
50
15
60
70
80
90
10 c
d 8
10 6 4 5 2 0 100
0 120
140
160
180
200
80
100
120
production rate of both components of a 1:3 and a 3:1 binary mixture. Figures 18.15b to 18.15e show chromatograms giving the maximum production rate of one or the other component under different sets of experimental conditions. 7. If a compound must be purified, a higher production rate will be obtained if the impurities are eluted first. If a compound must be extracted from a mixture where it is not a main component, the production rate will be higher if it is
18.4 Optimization Using Numerical Solutions
891
Table 18.4 Comparison of Calculated and Experimental Optimum Production Rates for Rand S-2-phenylbutyric Acid
Ratio 1/1 1/1 1/10 10/1
Isomer Sc Se Re Re Re Re Sc
se
1-PrOH 8.7 8.9 7.8
10.0
Time (min) 5.4 5.2 5.7 5.5 5.3 5.0 4.9 4.8
Y
Pr
^g/(cm2min) 90 85 90 84 90 86 90 90
7.0 6.6 5.9 5.5 14.0 13.4 25.6 25.6
Sc, Rc, calculated data for the maximum production rate of the enantiomer. Se, Re, experimental data corresponding to the same cut times. eluted first. These numerical procedures have been applied to the solution of practical problems of optimization of the experimental conditions of separations by overloaded elution. For example, they made possible the calculation of the maximum production rate under yield constraints of the components of several racemic mixtures. The results of this numerical optimization procedure were compared with experimental data [32,33]. Very good agreement between the two sets of results was reported in the two cases investigated. In the former case [32], the production rate of 99% pure enantiomers from the racemic mixture of R- and S-2-phenylbutyric acid was maximized as a function of the sample size and the mobile phase composition. The calculations were based on the column performance and the equilibrium isotherms of the two components (bi-Langmuir isotherms, Chapter 3). The separation was performed on immobilized bovine serum albumin, a chiral stationary phase, using water-methanol solution as the mobile phase. The retention times decrease with increasing methanol content, but so does the separation factor. For this reason, the optimum retention factor is around 3. Calculated production rates agree well with those measured (Table 18.4). The recovery yield is lower than predicted. In the second case [33], the production rate of D- or L-alanine from the racemic mixture was optimized as a function of the sample size and the mobile phase velocity, using the same chromatographic system, and applying the same calculation procedures. Again, there is very good agreement between calculated and measured production rates (Table 18.5). The band profiles calculated and those measured by fraction collection and analysis are compared in Figure 18.16. Figures 18.16a and b show the profiles optimized for maximum production rate of either the L-isomer (first eluted) and the second D-isomer. The agreement between calculated and measured profiles is excellent, although the symbols are system-
Optimization of the Experimental Conditions
892
Table 18.5 Comparison of Calculated and Experimental Optimum Production Rates for Dand L-alanine
Isomer Lc L e ,i Le,2
Dc D e ,i
De,2
Reduced Velocity 13.6 13.6 13.6 13.6 13.6 13.6
Purity (%) 99 99 96 99 99 92
Cut Time (min) 1.05 0.95 1.05 1.38 1.55 1.40
Y (%) 82 70 76 96 74 90
Pr fig/(cm 2 min) 35.0 29.9 32.5 29.3 22.7 27.6
Lc, Dc, Calculated data for the maximum production rate of the enantiomer. Le, De, experimental data corresponding to the same cut times. atically above the solid line for the part of the first component profile which is in the mixed zone, showing a small but significant error, which explains the difference between predicted and measured production rates. Note that, although the
Figure 18.16 Comparison of experimental and calculated production rate. Preparation of 99% pure enantiomers of N-benzoyl-D,L-alanine from the racemic mixture. Experimental (symbols) and calculated (solid lines) band profiles for the optimum sample size for maximum production rate at v = 13.6 of (a) the L-isomer (sample size, 43 pg), and (b) the Disomer (sample size 30 fig). BSA immobilized on a quaternary ammonium anion-exchange resin, mobile phase, 50 mM phosphate buffer with 1% acetonitrile. Reproduced with permission from S. Jacobson, A. Felinger and G. Guiochon, Biotechnol. Progr., 8 (1992) 533 (Figs. 4 and 5). ©1992, American Chemical Society.
18.4 Optimization Using Numerical Solutions
893
9 L Y=43.0%,Pr=0.0994 Pr=0.0994 =12.9%, Y=43.0%, Lff=12.9%,
14
7 6 C (mg/mL)
10 C (mg/mL)
Lf-7.2%, Lf=7.2%, Y=26.7%, Pr=0.0510
8
12
i 55
8 \L Y=73.4%, Pr=0.0805 Pr=0.0805 Lff=6.5%, =6.5%, Y=73.4%,
6
4
4 2 2 0
< ^ e " . * \>03 D 1 > Pr=0.0422 Pr-0.0422 Lf=2.6%, Y=90.3%,
h
3
\
•
11
40
60
80 80
100 100
time (s)
120 120
140 140
160 160
18( 180
0
50
'
\
X\ 100 100
150 150
200 200
250 250
300 300
350 350
400 400
. 450 450
500 500
time (s)
Figure 18.17 Optimum separations calculated by the equilibrium-dispersive model for isocratic overloaded elution for the purification of the less retained (left) and the more retained (right) component. The production rate (left graph in each figure) and the product of the production rate and recovery yield (right graph in each figure) was maximized, respectively, a = 1.2,fcj= 2, Cj = 100 mg/mL, Cj? = 100 mg/mL. Reproduced with permission from A. Felinger and G. Guiochon, J. Chromatogr. A, 752 (1996) 31 (Figs. 4 and 5).
two enantiomers have very similar properties, the production rate is 25% larger for the first eluted isomer than for the second one, due to the enhancement of the separation of the two isomers caused by the displacement effect.
18.4.2 Simultaneous Optimization of the Production Rate and Recovery Yield Felinger and Guiochon [18] used the Pr x Y objective function to simultaneously optimize the production rate and recovery yield. They found that the use of Pr x Y instead of Pr as an objective function has an enhanced advantage for difficult separations. When merely the production rate is maximized, the recovery yield is often poor due to an important overlapping mixed zone, which is unacceptable if the feed is expensive. The chromatograms on the right-hand side of Figures 18.17a and b were obtained for the optimum separation when Pr x Y is maximized instead of Pr. The optimum loading factor is smaller for both components in the case in which PrxY is maximized and the recovery yield is significantly higher, whereas the production rate is only slightly smaller than in the case in which Pr is maximized. Figure 18.18 illustrates the shift of the position of the optimum experimental conditions when Pr is replaced by the Pr x Y objective function for the optimization. The maximum production rate is at point A, while Pr x Y reaches its maximum at point B. The contour lines clearly show that the production rate is hardly lower at the new optimum. On the other hand, the recovery yield is improved when the experimental conditions are shifted from point A to point B. The surface determined by Pr x Y exhibits a well defined maximum, which makes the numerical path toward optimization stable. In order to improve the recovery yield with a small sacrifice in the production rate, the loading factor should be reduced and the efficiency of the column should
Optimization of the Experimental Conditions
894
0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
Pr1 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
0
B 2
4
6 Lf
8
10
12
14 0
1 0.8 0.6 0.4 0.2 0
0
B 2
4
6 Lf
8
10
12
14 0
0.06 0.05 0.04 0.03 0.02 0.01 0
0
B 2
4
6 Lf
8
10
12
14 0
B 0
6000 5000 4000 3000 A 2000 N 1000
2
A
4
6
8
Lf
10
12
14 0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 B 0
2
A
4
6
8
Lf
10
12
14 0
6000 5000 4000 3000 N 2000 1000
0.035 0.03 0.025 0.02 0.015 0.01
Pr2 Y2 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 B 0
6000 5000 4000 3000 N 2000 1000
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Y2
6000 5000 4000 3000 A 2000 N 1000
0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01
Pr1 Y1
0.06 0.05 0.04 0.03 0.02 0.01 0
6000 5000 4000 3000 A 2000 N 1000
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Y1
0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01
Pr2
2
A
4
6 Lf
8
10
12
14 0
6000 5000 4000 3000 N 2000 1000
Figure 18.18 Left hand side: Plot of the production rate of the less retained component against the loading factor and plate number (a = 1.2, k[ =2,C\ = 100 mg/mL, C\ = 100 mg/mL.), plot of the recovery yield, and plot of PT\ X Y\. Right hand side: same plots for the more retained component. Reproduced with permission from A. Felinger and G. Guiochon, J. Chromatogr. A, 752 (1996) 31 (Figs. 6 to 11).
be increased. Nagrath et al. used the general rate model of chromatography coupled with the kinetic form of the steric mass action isotherm to optimize the ion exchange chromatography of proteins [38]. Figure 18.19 illustrates that the production rate, the yield, and their product change with the loading factor in a similar manner to the trends plotted in Figure 18.18, where the equilibrium-dispersive model and the Langmuir isotherm model was employed.
18.4 Optimization Using Numerical Solutions
20
|
• » »
895
9 1 % purity 95% purity 99% purity
15
s «
5
5
10 Feed Load (DCV)
Figure 18.19 Optimization results for the later eluting component in a tertiary mixture at three different purities on a 90 jim. FF Sepharose stationary phase: 91%, 95%, and 99% purity constraints. Column conditions: diameter 1.6 cm; length 10.5 cm. Feed conditions: ribonuclease A, a chymotrypsinogen A, and the artificial component at 0.5 mM each. All optimal results are presented as a function of column loadings (dimensionless column volume), (a) Optimal production rate times yield (mmol/min/mL). (b) Optimal production rate (mmol/min/mL). (c) Optimal yield. Reproduced with •permission from D. Nagmth et ah, Biotechnol. Prog., 20 (2004) 163 (Fig. 8).
18.4.3 Minimum Solvent Consumption in Elution As we have shown (Section 18.2.1), the costs associated with the purchase, purification, recycling, recovery, loss, and waste of solvents are an important contribution to the global cost of the purification [12]. Accordingly, it is interesting to study the optimization of the production rate for minimum amount of solvent consumed per unit amount of purified products. This question has been discussed by Felinger and Guiochon [13]. A detailed discussion of the use of the specific production (amount of purified product per unit volume of mobile phase used) is given in Section 18.2. The specific production depends only on the loading factor and the column efficiency. If we assume that the production rate depends only on the ratio d^/L, not on the separate values of the column length and the particle size, which is approxi-
896
Optimization of the Experimental Conditions
mately true, the production rate depends on a single additional parameter. Thus, a very simple and straightforward scheme emerges for the practical optimization of separations for maximum production rate at the maximum possible specific production. We want to operate the column at the maximum possible flow rate (highest production rate), with the maximum possible efficiency (highest specific production). So we should design a column that operates at the efficiency optimum velocity when the maximum pressure drop that we can use is applied.5 The reduced velocity is related to the pressure drop by the equation (Eq. 5.23)
Combination with the definition of the plate height (H = L/N) and the Rnox [25] plate height equation (h — H/dp — B/v + Av1/3 + Cv) gives N= BDmyL APMkody
(APMk0\1/3 P\DmVLj
2
^
r
APMkod4p ^ DmrjL
(18.61)
Knowing dp, it is easy to derive the length of the column for which N is maximum. As there are usually several size grades of the selected stationary phase, it is possible to choose the particle size that gives the best possible efficiency. The column design is then optimized, and the optimum velocity is the one achieved when the column is operated under the maximum pressure available. The optimum sample size is then easily derived by a simple series of experiments. Calculations show that the specific production increases indefinitely with increasing column efficiency, but that this increase becomes very slow past a certain efficiency (Figure 18.20). Obviously, for each efficiency, there is an optimum loading factor for maximum specific production, and the recovery yield increases approximately as the specific production. Again, as in the maximization of the production rate, there is an optimum value of the retention factor. This value is also extremely low, below 0.5. This is in part due to the fact that, in this case, the columns are operated at higher efficiencies than in the simple maximization of the production rate. For practical reasons discussed already, however, it is probably not advisable to operate under conditions where k' is lower than 1. It is noteworthy that, if the chromatographic system can be adjusted to perform the required separation with a good value of a. at low retention factors, there is little difference between the parameters corresponding to maximum production rate at any specific production, or at the highest possible specific production [13]. On the contrary, at high values of the retention factor, a painful choice must be made, to reduce production rate and save on the amount of solvent required, or to consume large amounts of solvent to maximize the production rate, which increases the operating cost markedly (Table 18.6). 5
The selection of the maximum pressure at which a certain separation can be conducted has to take into account the pressure ratings of the pump and the column, as well as the stability of the packing material used. One should be satisfied that the column life at the selected pressure, hence flow rate, is sufficiently long.
897
18.4 Optimization Using Numerical Solutions
500
1000
1500
2000
2500
Figure 18.20 Plot of the specific production in elution versus the column efficiency, N. a = 1.2 andfc'a= 6. Less (•) and more (o) retained components of a 3:1 mixture; less (A) and more (+) retained components of a 1:3 mixture. Reproduced from A. Felinger and G. Guiochon, AIChE } . , 40 (1994) 594. Fig. 3a). Reproduced by permission of the American Institute of Chemical Engineers. ©1994 AIChE. All rights reserved.
18.4.4 Compromises between Maximum Production Rate and Minimum Solvent Consumption We have discussed in Section 18.2 the components of the unit price of the purification (Eq. 18.13). This price can be used as an objective function of optimization. It is difficult, however, to carry out extensive studies of the results obtained in this area on a general basis. Specific values have to be used for the capital costs, the unit solvent cost, and the unit feed cost, as well as for the characteristics of the separation itself. Felinger and Guiochon [13] have investigated a simpler problem. Neglecting the unit feed cost, they studied the maximization of the objective function Pr* defined as 1
xv (18.62) Pr SP The maximization of the parameter Pr* is equivalent to the minimization of the cost in Eq. 18.13, with FeCK = 0, and w = SC/ (FiC + SC), where FiC is the capital cost (interest plus amortization). Obviously, the capital cost and the production rate have to refer to the same time period. Depending on the proportion of capital cost and operating cost, the optimization of experimental conditions gives different results. Figure 18.21 illustrates the dependence of the optimum experimental conditions for minimum production cost (i.e., for maximum value of Pr* in Eq. 18.62) on the proportion of the solvent costs in the total costs, w. There are two situ-
P~r*
Optimization of the Experimental Conditions Table 18.6 Comparison of the Optimum Conditions for Maximum Production Rate and for Maximum Specific Production*
q/q 3:1 3:1 3:1 3:1 1:3 1:3 1:3 1:3 3:1 3:1 3:1 3:1 1:3 1:3 1:3 1:3
Compound 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
K
i
N
dv jim
2 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6
1396 5000 1966 5000 2550 5000 1238 5000 859 5000 1117 5000 1628 5000 605 5000
12.6 8.32 11.3 8.32 10.4 8.32 12.9 8.32 14.4 8.32 13.3 8.32 11.9 8.32 15.9 8.32
L
f %
10.0 14.8 3.3 3.1 9.3 15.5 3.3 3.7 10.7 17.0 3.5 3.3 10.0 19.7 4.0 3.8
Pn
mg/(cm2s) 0.973 0.706 0.153 0.123 0.300 0.260 0.672 0.495 0.440 0.263 0.078 0.046 0.137 0.098 0.295 0.183
SP mg/mL 1.231 2.042 0.241 0.354 0.556 0.753 0.803 1.430 0.424 0.760 0.069 0.134 0.193 0.284 0.234 0.529
Yi % 47.9 51.9 65.6 87.0 61.4 53.8 76.7 96.6 46.4 52.1 59.4 90.0 59.5 49.7 58.5 98.3
A = 1.2.
ations, depending on the value of w, and a rather sharp transition range in between. When the solvent costs are larger than the capital costs, the optimum column efficiency is indefinitely large, as in the optimization for minimum solvent consumption. When the production costs are lower than the capital costs, the optimum column efficiency remains low, close to the value that gives the maximum production rate. In the intermediate region, the optimization problem is ill-posed, and the optimum conditions depend only slightly on the column efficiency.
18.4.5 Maximum Production Rate in Gradient Elution Chromatography The optimization of gradient elution preparative chromatography has been sparsely studied. Snyder and Dolan [30] determined the experimental conditions that maximize the production rate of a purified product in overloaded gradient elution chromatography by using a computer simulation of the Craig distribution model. They showed that there is a close similarity between the results afforded by separations made in gradient elution and those obtained with the "corresponding" isocratic run, this run being defined as the one that gives an isocratic capacity factor k' equal to the apparent capacity factor obtained in gradient elution, under the experimental conditions considered. This correspondence condition implies that
18.4 Optimization Using Numerical Solutions
899 = 0.2
,00
Figure 18.213-D plot of the hybrid objective function representing the total production cost in elution (defined in Eq. 18.62). Plot for the less retained component of a 3:1 mixture at a separation factor oc = 1.5 and a retention factor k[ = 6. A. Felinger and G. Guiochon, AIChE ]., 40 (1994) 594 (Fig. 8). Reproduced by permission of the American Institute of Chemical Engineers. ©1993 AIChE. All rights reserved.
two requirement are satisfied. First, all the experimental conditions (column, flow rate, temperature, • • •) are the same in both modes, except the composition of the mobile phase, which is constant in the isocratic run but variable during the gradient run. The second requirement deals with the value of the gradient steepness (see Chapter 15, discussion below Eq.15.33): G =
(18.63)
Optimization of the Experimental Conditions
900
6
produced with permission from A. Felinger and G. Guiochon, J. Chromatogr. A, 752 (1996) 31 (Fig. 12).
Pr=0.143 Lff=6.5%; Y=31.7%, Pr=0.143
5 Lp2.8%; Y=89.3%, Pr=0.121 Pr=0.121 L f=2.8%; Y=89.3%,
4 C (mg/mL)
Figure 18.22 Optimum gradient elution separations for the purification of the more retained component. The production rate (left) and Pr x Y (right) were maximized, respectively, oc = 1.2, k'Q1 = 10, G = 0.5, C° = 100 mg/mL, C°2 = 300 mg/mL. Re-
j"
S | 33
\
J
2
11
0fi
1
50
\
\ l\ {1 \ \ ,V. 1 1 60
70
80
90
\
\
100
110
120
time (s)
where cp is the volume fraction of the modifier, tG is the gradient time, and S is the solvent strength parameter (the slope of the In k' vs. q> plot, see Chapter 15). The gradient steepness, G, controls the retention time in analytical gradient elution chromatography [39]:
t0 + £
k'(cpo)G) =
k'G
(18.64)
where tp is the injection time and k'((po) is the retention factor at the initial mobile phase composition. k'G = G^1 ln(l + fc'(^o)G) is an apparent gradient retention factor that controls the retention time in analytical gradient elution. Accordingly, it also gives the elution time of the diffuse rear end of the band profile in the case of convex upward isotherms. Therefore, k'G can be used to define the cycle time in gradient elution chromatography as tc = tp + t0 (; + 1 + k'G2)
(18.65)
assuming that ; column volumes of solvent are required for the regeneration of the stationary phase after each run. The second requirement for the gradient run and the corresponding isocratic run is that the steepness of the gradient is related to the capacity factor, k', of the corresponding isocratic separation by G = 1/1.15k'. Then, Snyder and Dolan [30] showed that the production rates for either component of a binary mixture in isocratic and in gradient elution are essentially the same. In both modes, the production rate increases steeply with increasing separation factor of the two solutes. The optimum sample size and the optimum plate number for maximum production rate are the same in gradient elution and for the corresponding isocratic separation. Felinger and Guiochon used the equilibrium-dispersive model to study the effect of experimental conditions on the maximum production rate and Pr x Y [40,41]. We have seen in Section 18.2.3 that in isocratic elution chromatography, the loading factor and the column efficiency are the critical parameters to be optimized for maximum production rate. In gradient elution there is an additional parameter, the gradient steepness The concentration profiles relative to each other do not depend on the gradient steepness [42]. If the gradient steepness is increased, the band profiles are
18.4 Optimization Using Numerical Solutions
901
Parelo frontiers of tertiary m iitureatSHt purity 1.1
0.3
.33"
'0
5
10 15 rn 25 30 35 40
Figure 18.23 Pareto frontiers and outlet concentration profiles for a tertiary mixture (a chymotrypsinogen A, ribonuclease A, and artificial component) on a 90 f*m FF Sepharose stationary phase. The 1st component is the early eluting ribonuclease A, the 2nd component is the middle eluting a-chymotrypsinogen A, and the 3rd component is the later eluting artificial component. The triangles, stars, and squares are the Pareto solutions for the optimization of the first, second, and later eluting components, respectively. Outlet concentration profiles for the tertiary mixture for the six cases indicated in (a) are shown in (b), (c), and (d) when the component of interest is the first, second, and the third component, respectively. Column diameter, 1.6 cm; length, 10.5 cm. Feed concentrations 0.5 mM of each component. The average separation factors are 1.55 between ribonuclease A and a-chymotrypsinogen A is 1.55, and 1.35 between a-chymotrypsinogen A and an artificial eluting component. Reproduced from D. Nagrath et ah, AIChE }., 51 (2005) 511 (Fig. 7). Reproduced by permission of the American Institute of Chemical Engineers. ©2005 AIChE. All rights reserved.
compressed but the relative concentrations, and accordingly the recovery yield, remain constant [30]. The Pr x Y objective function can successfully be applied to the optimization of overloaded gradient elution chromatography [40,43]. Figure 18.22 compares the chromatograms obtained under the optimum conditions given by the Pr and
902
Optimization of the Experimental Conditions
Pr x Y objective functions for the purification of the more retained component of a binary mixture using linear gradient elution. The optimum gradient steepness is different for the less and the more retained component. It varies with the separation factor. For the less retained component, the production rate usually reaches a plateau without a maximum and, in most cases, there is no production rate gain above G = 0.4-0.6. Although the cycle time should decrease when the gradient steepness is increased, the optimum column efficiency increases with increasing gradient steepness and these two effects compensate each other, resulting in a nearly constant cycle time. For convergent solutes, when the separation factor decreases with the increase of the fraction of the organic modifier in the mobile phase, a flat gradient profile should be used, flat enough to still provide a separation factor not too close to unity and avoid reversal of the elution order of the two components at the end of the elution. Schramm et al. [43] used a simplified classic cell model to describe gradient elution chromatography. This model was used to optimize the duration of a linear gradient and the volume of the injected feed, using a simplex algorithm, in the case of the chromatographic separation of a ternary mixture, using two different objective functions (Pr x Y and the specific solvent consumption, CS{ = Pvx/m[, where Fv is the volumetric flow rate of the mobile phase, x is the period between two successive injections and m, is the mass flow of component i). Jandera et al. studied the effect of the gradient profile on the production rate during the reversed phase gradient elution of the binary mixture of o-cresol and phenol [44]. They concluded that the initial concentration of the organic solvent in the mobile phase has a stronger effect on the production rate than the gradient steepness. An optimum initial mobile phase composition can be determined for maximum production rate. Separations carried out at steep continuous gradients outperform the production rate of isocratic elution even if the cycle time in gradient elution is much longer due to column regeneration [45]. Gallant et al. [46] discussed the optimization of the purification of proteins by preparative ion-exchange chromatography, using linear gradient separations. They employed the Steric Mass Action (SMA) model to account for the multicomponent competitive adsorption of proteins in ion-exchange chromatography, over a wide range of salt concentrations [47,48]. The SMA models protein adsorption in ion-exchange chromatography as a stoichiometric exchange [49,50]. Using this adsorption model in conjunction with the equilibrium-dispersive model, Gallant et al. obtained numerical solutions for the band profiles of proteins in preparative ion-exchange chromatography, using linear concentration gradients (see Chapter 15). The model was able to predict the separation of a-chymotrypinogen A, cytochrome C and lysozyme under overloaded conditions. The determination of estimates of the parameters of the SMA model for the compounds involved in this separation was discussed by Gallant et al. [46]. The optimization of preparative ion-exchange elution chromatography was discussed under the constraints of baseline resolution and of induced sample displacement (in other words, in the3 case of bands exhibiting a moderate degree of interference, leading to a significant displacement of the first component by the second while preserving a sufficient
18.4 Optimization Using Numerical Solutions
903
resolution. The effect of the adsorption properties of the feed, the feed volume, the gradient slope, the purity required, and the product concentrations were considered, using numerical solutions of the problem. The results obtained indicate that, under appropriate conditions, sample displacement can be employed dramatically to improve the production rate while experiencing only minor losses in the product recovery yield or their purity. All these specific results are consistent with the theoretical results that were discussed earlier in this section. Gallant et al. [1] determined the optimum feed volume and gradient slope and showed that gradient elution dramatically increases the production rate while the recovery yield and the purity are only slightly affected. They could increase the concentration of the collected fractions by one order of magnitude over their concentrations in the feed. This indicates that protein concentration should be carefully monitored during the design of a gradient separation so that aggregation or precipitation is avoided. Nagrath et al. used multiobjective optimization, an optimization strategy that overcomes the limits of a single objective function, to optimize preparative gradient elution chromatography [51]. In the physical programming method of multiobjective optimization, one can specify desirable, tolerable, or undesirable, ranges for each design parameter. They obtained optimum experimental conditions using bi-objective (production rate and recovery yield) and tri-objective (production rate, recovery yield, and product pool concentration) optimization. The multiobjective optimization is a promising tool for the optimization. In Figure 18.23a the Pareto frontiers6 at 95% purity level and the concentration profiles of a ternary mixture are plotted using the bi-objective optimization. The Pareto frontier for the purification of the second component is below those for the other two components because it is surrounded with two impurities. In Figures 18.23b to d the optimum chromatograms for different scenarios are plotted for the three target compounds, respectively: reasonable recovery yield (Case 1) or reasonable production rate (Case 2) are the objectives. Cases IIP (high yield) and 12P (large production rate) are the optima for the production of the first component. The results of the bi-objective optimization correlate well with the results obtained with the Pr x Y objective function.
18.4.6 Maximum Production Rate in Displacement Chromatography The optimization of the experimental conditions in displacement chromatography for maximum production rate has been studied less than the optimization of overloaded elution, reflecting the lesser importance of this method in industrial practice. Frenz et al. [52] performed an experimental study of the dependence of the throughput on the operational parameters in reversed-phase displacement systems. They demonstrated that both the nature and the concentration of the displacer must be appropriately selected to optimize the throughput. Jen and Pinto [53] have used the ideal model, and the /i-transform approach to maximize 6 A Pareto frontier is a limit for the solution along which the improvement in one objective can be achieved only at the cost of another objective.
904
Optimization of the Experimental Conditions
Figure 18.24 Plot of the maximum production rate of the two components in displacement chromatography versus the retention factor of the first eluted one. Separation factor, a = 1.2; feed composition 3:1. Curve 1, first component, 3:1 mixture; curve 2, second component, 1:3 mixture; curve 3, first component, 1:3 mixture; curve 4, second component, 3:1 mixture. Reproduced with permission from A. Felinger and G. Guiochon, }. Chromatogr., 609 (1992) 35 (Fig. 9).
the production rate of a given column (i.e., L and dp are fixed). They have shown that the column should be operated at the resolution point, when the isotachic train is just formed, and that the optimum sample size is independent of the column efficiency, provided it is not very low. Katti et al. [3] have studied the optimization of the performance of a given column in the displacement mode, using the competitive Langmuir isotherm model, the equilibrium-dispersive model to calculate band profiles, and the Rnox equation [25] to relate the apparent dispersion coefficient of the model and the mobile phase velocity. They have shown that the maximum production rate is achieved under experimental conditions such that the isotachic train is not completely formed. Accordingly, the recovery yield is far below unity, a result which comes as a surprise only to those addicted to considering the ideal model of chromatography as a realistic model of displacement. The optimum mobile phase velocity is rather high. The optimum retention factor is low, much lower than typically used in conventional applications. The collected fractions are highly concentrated, which seems to be the major advantage of this method. Felinger and Guiochon [54] carried out a systematic investigation of the optimization of the experimental conditions for maximum production rate, using the same model as for their similar study on the optimization of elution [4] (competitive Langmuir isotherm, equilibrium-dispersive model [24], Knox equation [25], and super-modified simplex algorithm [34]). Their main conclusions are the following. 1. There is an optimum value of the retention factor of the first component of interest. This optimum is typically between 1.2 and 2.0, i.e., always much lower than the factors used in conventional practice. The optimum retention factor
18.4 Optimization Using Numerical Solutions
2.
3.
4. 5.
905
depends slightly on the feed composition and is different for production of the first or second component of a binary mixture. When k'xp increases/ the required column efficiency increases, thus the cycle time increases too. Figure 18.24 illustrates this result in the case of a binary mixture with a separation factor of 1.2. There is no optimum displacer concentration. The production rate increases always with increasing displacer concentration. The increase becomes very slow at high concentrations, however. So, in practice, there will always be a maximum, either at the displacer solubility or when one of the assumptions of the model breaks down {e.g., non-Langmuir isotherm, or viscous fingering; see Chapter 5). There is no well-defined optimum column length. The production rate increases constantly with increasing column length, although it does so very slowly beyond a certain length (ca. 50 cm, in most practical cases). There is a practical optimum value for the ratio <&/L, indicating that there is a weak optimum particle size for columns of a given length. The production rate is rather sensitive to the parameters of the column design, i.e., to the ratio di/L. It can easily drop several fold if the proper column is not selected. This is especially important when the production rate of the more retained component of a binary mixture is optimized, and when this second component is the less concentrated (e.g., 3:1 mixture).
Zhu and Guiochon [55] have studied the optimization of the production rate in displacement chromatography under isotachic train conditions. This problem is different from the previous one. One does not obtain the maximum production rate possible in displacement chromatography by operating under isotachic train conditions. However, this procedure is attractive as it gives the maximum possible recovery yield with a given chromatographic system, and probably a higher specific production. The use of the shock layer theory permits the derivation of an analytical expression giving the production rate as a function of the experimental conditions in the case of competitive Langmuir isotherm behavior of the feed components [55]. Thus, the optimization of the experimental parameters by differentiation is easy. The conclusions of these authors are in agreement with previous experimental results from Horvath et al. [52]. The production rate decreases rapidly with decreasing value of cc — 1 (see Figs. 12.7 and 12.8). There are optimum values of the displacer concentration, of the displacer retention factor, and of the mobile phase velocity. Figure 18.25 illustrates these results by showing a plot of the production rate versus the normalized breakthrough time of the displacer, IN = {tR,d ~ *o)/^o = Kd/ where £j>^ is the breakthrough time of the displacer front [52]. These optimum values are different from those found by Felinger and Guiochon, which is expected since the optimization problem is different, but the trends observed are similar. Especially noteworthy is the fact that the optimum retention factor of the displacer is not very large (rarely more than a few units). Natarajan et al. [56] discussed the optimization of ion-exchange displacement separations of proteins by using numerical solutions of the solid-film linear driving force model. The equilibrium isotherms of the protein-salt multicomponent
Optimization of the Experimental Conditions
906
„
0.16
U
0.1Z-
Product A 100*
1
4
r
Product E 94%
0.03 -
S 2
0.04 " o n Pi
0 1 2 3 Normalized Breakthrough Time of the Displacer
Figure 18.25 Comparison of experimental and theoretical results. Optimization of the displacer concentration for maximum production rate under isotachic train conditions, (a) Experimental results. Plot of the maximum production rate versus the normalized breakthrough time of the displacer. (b) Optimum calculated with the shock layer theory. F = 0.416; DL = 0.000023 cm 2 /s; Ifcy = 0.33 s" 1 . Langmuir isotherm coefficients (and k' values): k\ = 1.5, k'2 = 2.5, k'3 = 4.0, k'x = 6.0; k'd = 9.0; bx = 0.04; b2 = 0.07; b3 = 0.12; bx = 0.18; bd = 0.6. Reproduced from (a) } . Frenz et al, ]. Chromatogr., 330 (1985) 1 (Fig. 8), and (b) } . Zhu, G. Guiochon, AIChE }., 41 (1995) 45 (Fig. 5.7a). Reproduced by permission of the American Institute of Chemical Engineers. ©1995 AIChE. All rights reserved.
mixture were measured on the ion-exchange system and modeled using the Steric Mass Action (SMA) model. This model has been shown accurately to predict the chromatographic behavior of single components and of multicomponent mixtures in ion-exchange, under isocratic [48], gradient elution [46,57], and displacement conditions [58]. Algorithms based on the finite difference method were employed for the calculation of numerical solutions. Since the SMA isotherm is implicit, a Newton-Raphson technique was used at each step of the calculation to derive the equilibrium concentrations from the SMA isotherms. These authors presented an iterative optimization scheme whereby one can identify the optimum operating conditions for displacement separations. Results indicate that the use of this optimization scheme leads to significantly better performance than the conventional old rule of thumb. As objective function, they used only the production rate. The feed components, the concentrations of these compounds in the feed, the stationary phase, and the pH of the mobile phase were selected as the fixed parameters in this problem. The decision variables included the feed load, the mobile phase flow rate, the salt concentration, and the displacer partition ratio (A = q^^/Cdisp, with Cj^p and Q!Sf,, the stationary and the mobile phase concentrations of the displacer). Finally, there were three constraints, the purity (product purity better than 95%), the recovery yield (recovery yield better than 80%), and the main component solubility (maximum protein concentration,
18.4 Optimization Using Numerical Solutions Input: SMA parameters, transport parameters, level of loading. Initial guesses for the flow rate (F) and A should be at their lower bounds.
Optimize the salt concentration
Increase A till the yield constraint is satisfied
Stop
.Yes
907
No
Is I Fnew - Fo,dl < tol
Is the yield constraint satisfied'?
Yes No
, = Fl)U1 + 0.25»(Fl,-Fl>M)
'*
Is I F, TO - F oU I < tol w=
Foid-0.25*(F,,-Fi)
Yes Fold = F m
F. = F r a
No
Has the production rate increased?
,,No Increase A till the yield and solubility constraints are satisfied
Yes
Figure 18.26 Iterative scheme for optimizing displacement separations. Reproduced with permission from V. Natamjan, B. W. Bequette, S. M. Cramers, ]. Chromatogr. A, 876 (2000) 51 (Fig. 1).
4mM). An iterative optimization scheme was developed and used to carry out the optimization of several displacement systems (see Figure 18.26). Using this scheme, one can identify the optimum operating conditions for displacement separations at a given level of feed loading, on a given resin packing material. The iterative optimization scheme was employed to carry out the optimization of displacement systems on three different resin materials (two polymethyl methacrylate sulfopropyl-bonded resins, one 8, the other 40 y.rs\ from Waters and a 34 }im HP Sepharose), for three different classes of separation problems (converging, parallel, and diverging affinity lines) [56,57]. The results suggest that the characteristics of the resin play a very important role in the production rate that can be achieved for a given separation problem. A high-capacity resin may be particularly effective for displacement separations. The higher capacity allows the use of a higher load and may help to improve the mass transport characteristics by allowing a higher surface diffusion flux. It was demonstrated that a material made of relatively large particles (e.g., 34 }im Sepharose) can perform better than material made of small particles (e.g., 8 }ira Waters PMMA) [56].
18.4.7 Comparison between Elution and Displacement Chromatography Few comparative studies have been published so far. Some work based on experimental data has been presented by Liao et al. [59], by Viscomi et al. [60], and
908
Optimization of the Experimental Conditions
Figure 18.27 Comparison of the performance of displacement and elution chromatography. Maximum production rate in the separation of a certain mixture on a given column: 25-cm- long column packed with 20 |im particles. 1:3 mixture, a = 1.20; k'2 = 9.78. Top: plot of maximum production rate of second component versus reduced velocity of the mobile phase. Figures by the symbols on the lines (upper line, elution, lower line, displacement) are the concentration of the collected fractions. Bottom: recovery yield of the second component. Reproduced with permission from Katti et al., J. Chromatogr., 540 (1991) 1 (Fig. 10).
a
_ . -~-^~^
1.8
\
^-^2.2
/3.1
/3.B I BO
200
HO
280
Reduced Velocity
by Lee et al. [61]. In these studies, the same column was used to carry out the same separations, first by displacement chromatography, then by elution. The optimization of the experimental conditions for maximum production rate by displacement was carefully carried out. The development work devoted to the overloaded elution experiment was less complete. There is no reason that the same column design, the same mobile phase composition (hence, retention factors), and the same mobile phase velocity would be optimum for both methods. In spite of these limitations, we observe that in the former paper [59], the production rate achieved in the elution mode was more than half that in displacement, and the regeneration time which should be included in the cycle time was not figured in, as usual in the literature on displacement chromatography. Lee et al. [61] calculated, based on the ideal model, the production rate, the yield, and the enrichment factor achieved when producing a 99.99% pure product. The production rates of both modes were comparable. The fractions collected in displacement were approximately 10 times more concentrated in displacement than in elution, and the yield of the former method was better. However, the calculations were done in a single case. Furthermore, the ideal model does not permit the optimization of the mobile phase velocity, and does not take into account the effect of the finite column efficiency. A more systematic investigation was done by Katti et al. [3]. Using the competitive Langmuir isotherm model, the equilibrium-dispersive model [24], and the Knox equation [25], these authors optimized the operating parameters of given columns for maximum production rate of either the first or the second component of binary mixtures with various separation factors (1.2 < cc < 1.7) and composition. Constraints of purity (98%) and maximum inlet pressure (125 atm) were included, and also, in some cases, a recovery yield (60 or 90%) constraint. The maximum production rates achieved with the two modes are comparable when there is no yield constraint. However, the recovery yield is lower in displacement than in elution, because the maximum production rate is achieved under non-
18.4 Optimization Using Numerical Solutions
909
Figure 18.28 Comparison of the perfor, mance of displacement and elution chromatography. Maximum production rate in the separation of a certain mixture on a | given column: 25-cm- long column packed I with 20 y,m particles. 1:3 mixture, a = 1.70; | k'2 = 9.78. (a): plot of maximum production * rate of second component versus reduced velocity of the mobile phase. Figures by the symbols on lines are the loading factor « and, in the case of displacement, the disI placer concentration, (b): concentration of | the collected fractions (mg/mL). Reproduced ° with permission from A.M. Katti et at,}. Ch.romatogr., 540 (1991) 1 (Fig. 9).
5 Reduced Velocity
isotachic conditions. The same conclusions were reached by Frenz et al. [52]. So, with a minimum yield constraint, displacement has to be carried out under experimental conditions closer to those corresponding to the isotachic train. Then, the production rate achieved is much higher in the elution mode at low values of cc — 1 (Figure 18.27), comparable at high values (Figure 18.28). Figure 18.27 illustrates the dependence of the maximum production rate optimized for loading factor in elution, and for loading factor and displacer concentration in displacement, as a function of the mobile phase velocity. The yield and production rate advantages of elution are obvious. On the other hand, the concentration of the collected fractions is one to two orders of magnitude higher in displacement than in the elution mode, as shown in Figure 18.28. Although providing some useful terms of comparison, this study has the drawback of comparing the performance obtained with the same column, optimized successively for the two modes. As noted above, there are no reasons why the same column design would be optimum for both modes. The extremely high optimum value calculated for the mobile phase velocity is an indication that the column selected is too long for the elution mode. Its efficiency is too high, and it is traded for a shorter cycle time. A higher production rate could be obtained with a shorter column. This was confirmed by later studies [5]. A more general comparison has been made by Felinger and Guiochon [5]. Using the same model as Katti et al. [3] and similar programs, they optimized simultaneously the design and operating conditions in the displacement and the elution modes. They also studied the influence of the retention factor. Their conclusions can be summarized as follows. 1. In both modes, there is an optimum value of the retention factor. This optimum value is very low in the elution mode, around 0.5, and the optimum is quite sharp (Figure 18.29). The optimum value is larger in displacement, around 2, and the optimum is flat, which is more usual in chromatography. This result is valid whether or not the column design has been optimized. It is also valid
Optimization of the Experimental Conditions
6.0
910
a —
1.5
5.0
1:3 6
\
at
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\
• * -
6.
odu
c o JP o o PJ"
'Q
a.
i
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^
'.
2
" • &
1.0
x
'"••&••
*
-
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—
—
o
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.0
1.0
2.0
3.0
4.0
8.0
b2
°1 6.0
k'
Figure 18.29 Plot of the maximum production rate of the two components versus the retention factor of the first eluted one. Separation factor, a = 1.5. All symbols correspond to the production rate of the optimum column, operated at the optimum velocity and sample size. Curve 1, first component, elution. Curve 2, second component, elution. Curve 3, first component, displacement. Curve 4, second component, displacement, (a) Feed composition: 3:1. (b) Feed composition: 1:3. Reproduced with permission ofWiley-Liss Inc., a subsidiary of John Wiley & Sons, Inc. from A. Felinger and G. Guktchon, Biotechnol. Bioeng., 41 (1993) 134 (Fig. 4). ©1993, John Wiley & Sons.
when a recovery yield constraint is applied (Figure 18.30). 2. The production rate is much higher in elution than in displacement chromatography if the retention factor has been optimized. If a large value of the retention factor is used, as is conventionally done, the production rates achieved with both modes are comparable. 3. The production rate advantage of the elution mode decreases with increasing separation factor. It is larger for the second component than for the first one (Figure 18.31). 4. If the optimized columns are used, the recovery yields achieved at the maximum production rate with both modes are comparable, and around 60%. 5. Although in both modes, there are separate optimum values of the column length and the average particle size, these optima are flat. The production rate varies only slowly if both L and dp are changed while keeping the ratio di/L constant and equal to its optimum value. 6. The concentration of collected fractions is much higher in displacement chromatography than in elution (Figure 18.32). However, this concentration in-
6,0
18.4 Optimization Using Numerical Solutions
a
=
911
1.5
4.0 /
O
3.0
oduction rate
5.0
3:1
••-ii
V \
2.0
Q.
p 0
.."^Z
0.0
"*1*1* -"-o.,,",
0.0
1.0
2.0
1
1
3.0
4.0
1
5.0
1
6.0
0.0
6,0
k'
Figure 18.30 Plot of the maximum production rate of the two components versus the retention factor of the first eluted one. Same as Figure 18.29, but recovery yield constraint, 90%. Reproduced with permission ofWiley-Liss Inc., a subsidiary of John Wiley & Sons, Inc. from A. Felinger and G. Guiochon, Biotechnol. Bioeng., 41 (1993) 134 (Fig. 7). ©1993, John Wiley & Sons.
creases with increasing retention factor in displacement, while in elution it increases with decreasing retention factor. At the (low) value of the optimum retention factor in elution, the collected fractions are only one order of magnitude less concentrated in elution than under optimum displacement conditions. 7. Any comparison between the production rates achieved with the same chromatographic system (but not the same column, nor necessarily the same mobile phase composition) in the elution and displacement modes is biased by the lack of clear estimates of the cycle time in the displacement mode. The time required for regeneration of the column must be factored in. This time cannot be less than a few times to, and can be as high as t°R d, the displacer retention time at infinite dilution. Experimental comparisons are missing at this stage. Nevertheless, it can be concluded that, for the large-scale production of chemicals whose adsorption behavior is described by a Langmuir or related isotherm, the preferred mode of implementation of preparative chromatography should be elution. Unless the higher concentration of the collected fractions is a critical advantage or one has a special need to remove impurities squeezed between the bands of major components (see Chapter 12, Section 12.1.5, Figure 12.9), the method does not seem to offer signif-
912
Optimization of the Experimental Conditions
Figure 18.31 Ratio of the maximum possible production rates in elution and displacement versus the separation factor. From top to bottom: first and second components of a 1:3 mixture, second and first components of a 3:1 mixture, respectively. Reproduced with permission of Wiley-Liss Inc., a subsidiary of John Wiley & Sons, Inc. from A. Felinger and G. Guiochon, Biotechnol. Bioeng., 41 (1993) 134 (Fig. 5). ©1993, John Wiley & Sons.
icant advantages. Displacement is more complicated than elution. It requires the use of a displacer which will appear in some of the fractions. One must be careful to avoid the introduction of new impurities in otherwise purified fractions when the displacer is selected. It also requires that the column be regenerated, a step that has not been factored out in the cycle time in any of the comparisons discussed above. Some caution is in order, however, because feedstocks are rarely made of binary mixtures (even in the case of the preparation of pure enantiomers from binary mixtures). Displacement is more effective than elution at sweeping and concentrating the late eluted impurities and may be a valuable alternative when serious problems arise from the presence of such compounds. Finally, the effect of a change in feedstock composition, which is a normal production event and usually causes changes in impurity concentrations, on the control of a displacement separation is an area that has yet to be studied.
18.4.8 Comparison Between Isocratic Elution, Gradient Elution and Displacement Chromatography The comparison of the optimum performance of isocratic and gradient elution chromatography indicates that [40,41]: 1. The recovery yield achieved under optimum conditions is the same in gradient and in isocratic elution. 2. The optimum loading factor is higher in gradient elution than in isocratic elution, because the band compression diminishes the tag-along effect of the more retained component. Accordingly, the average concentration of the collected
18.4 Optimization Using Numerical Solutions
913
*
*
o
8-
a. •••"
Q
.•••*"'
o_
a = 1.2 o CO
j
CJ
/
o q 1,0 O-
"" 0—
6 1
0.0
1.0
2.0
3.0
4.0
5.0
6.0
k' Figure 18.32 Concentration of the collected fractions versus the retention factor of the less retained component. Curve labels: 1, first component; 2, second component; O, overloaded elution; D, displacement, (a) Separation factor, a = 1.2. (b) Separation factor, cc - 1.8. Reproduced with permission of Wiley-hiss Inc., a subsidiary of John Wiley & Sons, Inc. from A. Felinger and G. Guiochon, Biotechnol. Bioeng., 41 (1993) 134 (Fig. 8). ©1993, John Wiley & Sons.
fractions and the production rate are higher in gradient than in isocratic elution. 3. There is no need for a high column efficiency in gradient elution because the average retention factor is higher than the optimum value during the separation. 4. The optimum gradient steepness depends mostly on which is the most important feed component, the more or the less retained. It is higher for the purification of the less retained component than for that of the more retained one. Felinger and Guiochon studied the performance of isocratic and gradient elution as well as displacement chromatography with the Pr x Y objective function at different retention factors [41]. In isocratic elution and in displacement chromatography, the retention factor remains constant during the separation, as opposed to gradient elution where it continuously changes. We can, however, use the apparent gradient retention factor k'G (see Eq.18.64) to include gradient elution in the comparison. Figure 18.33 summarizes the results of a comparison of the maximum performance of the three modes of chromatography for a separation factor oc — 1.5, for different mixture compositions, and for the production of either the first or the
Optimization of the Experimental Conditions
914 oc=1.5; 1:3; 1st component
a=1.5; 1:3; 2nd component
0.35 Isocratic — Displacement •---
—
0.3
Isocratic Displacement Gradient
b.
0.8
Gradient —
0.75 0.7 -
0.25
r" ~ * " ~ > " t ' « . .
0.65 -
0.6 - , ' .
0.2
f/
0 3 5 - 14 0.5 - •
0.15
0.45 0.1
04 2
4
6
8
->
A
10
10
ce=1.5; 3:1; 1st component
<x=1.5; 3:1; 2nd component
1.2 1.1
*
/
0.2 Isocratic — Displacement Gradient —
0.18
1 0.9
Isocratic Displacement Gradient
d.
0.16
0.8
1
0.14
0.7
*
\
4
\
" • • • - * . . . .
I/
I•
0.6 0.12 03 -i 0.4
0.1 10
Figure 18.33 Comparison of the performance {Pr x Y) of isocratic elution, gradient elution, and displacement chromatography for different mixture compositions and elution order at cc = 1.5 Reproduced with permission from A. Felinger and G. Guiochon, J. Chromatogr. A, 796 (1998) 59 (Fig. 6).
second feed component. The dependence of the separation performance on the retention factor is most significant in gradient elution. Displacement chromatography is the mode that is the least sensitive to changes of the retention factor. While the arbitrary selection of a particular experimental set-up (mixture composition, elution order, separation factor, etc.) may vastly influence the separation performance, we can conclude that either gradient elution or isocratic elution generally outperforms displacement chromatography in all cases in which this last method requires extensive column regeneration (with five column volumes or more). On the other hand, when the displacement separation does not require column washing, all three modes give similar results. In the present case, a gradient steepness that results in a gradient retention factor no larger than k'G = 5 is preferred. The maximum performance of gradient elution was found for values of k'G slightly below 3 for the purification of the less retained component and slightly above 3 for the purification of the more retained component. The performance of gradient elution chromatography is very sensitive to the value of the gradient retention factor Displacement chromatography appears to be the method of choice if the retention factor (or the gradient retention factor) is larger than 5. Displacement
18.5 Recycling Procedures
915
chromatography is more attractive than isocratic elution for retention factors in excess of 2 when important washing is not needed for column regeneration. If a cleaner feed can be used, elution becomes more attractive in a much wider range of retention factors. The same phenomenon is also observed for more difficult separations. In all three modes of chromatography, the optima of the Pr xY objective function are found for such combinations of column design and operating parameters that the recovery yield is—depending on the separation factor—between 70 and 90%, while the maximum production rate is often associated with values of the recovery yield lower than 50%. Usually both gradient elution and displacement chromatography offer higher optimum loading factors than isocratic elution, due to the band compression effect of the former two modes. Gradient elution chromatography requires less column efficiency than isocratic elution, i.e. shorter columns packed with larger particles can be used which will result in shorter cycle times to the increased flow rate. Natarajan et al. compared the optimum performance of gradient elution and displacement ion exchange chromatography of proteins [62]. They used the solidfilm linear-driving force model of chromatography combined with the steric mass action isotherm. They showed that the initial salt concentration is much more important in displacement mode than in gradient elution. They proposed that in gradient elution, the salt concentration should be kept as low as possible and it should not be considered as an optimization parameter. The optimum gradient slope is the maximum allowable one unless the sample volume is rather large. They found that for difficult separations, the production rate of displacement chromatography outperforms that of gradient elution.
18.5 Recycling Procedures In overloaded elution, the production rate of the first component, especially when its concentration is lower than that of the second component, can always be increased by increasing the loading factor considerably beyond touching-band conditions. This is done at the expense of a loss in recovery yield. An intermediate fraction is collected, which contains the purified component at a concentration that is too low. This fraction can be collected and reprocessed. There are several methods of performing this operation. Some off-line methods, collecting the fraction and storing it, are simple and obvious, but they are cumbersome as they require the use of different experimental conditions, the mixed fraction having a composition quite different from that of the feed. Some on-line procedures of recycling are more attractive. Bailly and Tondeur [9] have discussed on a theoretical basis different implementations of recycling. Using the ideal model and assuming competitive Langmuir behavior, they have shown that recycling saves solvent and gives more concentrated fractions compared with the classical elution process. The subsequent reinjection of insufficiently purified fractions has been studied by Crary et al. [63]. Later, Seidel-Morgenstern and Guiochon [10] and Charton et al. [11] car-
Optimization of the Experimental Conditions
916
2
3 Time (min)
4
2
3 Time (min)
4
2
3 Time (min)
4
Figure 18.34 Band profiles after one to ten cycles (progressive build up of the steady state profiles). Left: calculated profiles. Right: Experimental profiles recorded. Top: First to fifth cycles. Bottom: Cycles 6 to 10. Circles and thin solid line, cycle 1; squares and dotted line, cycle 2; diamonds and dash-dotted line, cycle 3; triangles and dashed line, cycle 4; stars and thick solid line, cycle 5. (b) Experimental profiles for cycles 6-10. Circles and thin solid line, cycle 6; squares and dotted line, cycle 7; diamonds and dash-dotted line, cycle 8; triangles and dashed line, cycle 9; stars and thick solid line, cycle 10. Reproduced with permission from I. Quinones, C. M. Grill, L. Miller, G. Guiochon. ]. Chromatogr., 867 (2000) 1 (Figs. 22 and 23).
ried out detailed investigations of recycling procedures based on the use of the equilibrium-dispersive model and compared the results of their calculations with experimental data. In "recycling with shaving," a large band of a binary mixture is injected, and the fractions on both sides of the unresolved bands that are within the specified purity are collected at the end of each cycle, the mixed band being recycled. The sample load decreases during the processing of one batch. The production rate is almost always lower than in regular elution [10]. In "recycling with mixing," the mixed zone is collected at the end of each cycle, mixed with a certain amount of fresh feed, and reinjected in the column. Thus, a few cycles are necessary at the beginning of each production campaign to reach steady state. Charton et al. [11] have shown theoretically, and confirmed by systematic experiments, that this procedure reduces significantly the consumption of solvent without great loss of production. Accordingly, these procedures are economically attractive and should be given more attention. Quinones et al. [64] investigated the performance of a closed-loop steady-state recycling process (SSR) in which a band train is recycled through a chromato-
18.5 Recycling Procedures
917
graphic column, the leading and the tailing edges of the train are collected at each cycle, and an equivalent amount of feed injected at the heart of the train. The process was applied to the separation of two enantiomers on Chirapak AS, a cellulose tris-methylbenzylcarbamate, using acetonitrile as the mobile phase. The competitive adsorption isotherms were determined by FA and accounted for by a Langmuir isotherm. The concentration profiles of the two enantiomers in the band train were calculated using the equilibrium dispersive model and accounting for the contributions of the extra-column volumes to the axial dispersion. The progressive formation of a steady-state profile and the production rate of the SSR process were accurately accounted for. Figure 18.34 compares the experimental profiles recorded when the train passes through the detector between ten successive cycles and the profiles calculated. It illustrates the progressive trend toward a steady state profile. The difficulty of reaching general conclusions is most serious, however. Comparisons are possible only between carefully optimized procedures, which may take a significant amount of time, as it requires the acquisition of a large amount of accurate thermodynamic data. Band profiles depend so much on the exact features of the isotherms that a high degree of accuracy, first in the measurement and then in the modeling of the equilibrium data, is required. A systematic investigation of the optimization of the experimental conditions for the purification of a compound from a binary mixture, using a recycling process, was done by Teoh et al. [65]. These authors compared the experimental band profiles obtained with their instrument and those calculated with a computer programmed for the equilibrium-dispersive (ED) model, to verify that this model applied to their system. Good agreement between the calculated and experimental elution profiles confirmed that the ED model could be employed for the optimization of their chromatographic process. Then, these authors used the ED model for the optimization of a preparative HPLC process using closed-loop recycling operation, in order to investigate what improvements recycling can bring to the performance and the yield of a simple HPLC process. Binary mixtures of 1,3,5-tri-tert-butyl-benzene (TTB) and 1,3,5-tri-hydroxybenzene (PHL) were used as the feed. The separation was carried out on a 0.46 x 25 cm column packed with microcrystalline cellulose triacetate, with an average particle size of 5 fim. Two different volumes (10 and 50 fiL) of mixtures of different compositions (1:1, 1:3, and 3:1) were injected into the column. All the experiments were performed under isocratic conditions, with pure methanol as the mobile phase, and low concentration samples. Teoh et al. [65] ignored the competitive adsorption between the two feed components on the grounds that their column was efficient, the band resolution rather high, and the samples used rather dilute. So, they assumed that each compound follows single-component Langmuir adsorption isotherm behavior, with
Considering that recycling processes in preparative HPLC separations are designed to purify high value chemicals, Teoh et al. [65] did not consider the ab-
918
Optimization of the Experimental Conditions
solute production rate to be a major concern but rather optimized the amount of purified products collected,ft,-Y,-(f)(with ft,- amount of i injected and Y,-, recovery yield), subject to some minimum purity and yield constraints. The optimization problem can be defined as consisting of finding the values of the switching time between recycling and product collection, tj, of the final time, tj, and of the number of cycles that maximize a function of n, and Y,-, subject to Pwf™1 ymin
< <
PUJ < 1 Yj
(18.67) (18.68)
The amounts of the purified compounds (TTB or PHL) produced were optimized separately, both subject to the minimum purity and recovery yield constraints. A minimum purity of 99% was chosen. The purity of component i, PUJ, was defined as the concentration of component i in the first collected fraction (excluding the solvent) Hj — Apj
u
i ~ zncz. i 5~
Where ti{ is the amount of component i injected into the column, Api is the amount of component i that is lost because it elutes in another collected fraction, and B; is the amount of the impurity that co-elutes with the component z, diluting it and thereby decreasing the product purity (but increasing its production). The yield Y( was defined as the ratio of the amount of the desired component that is collected at the required purity in the purified fraction to the amount that was injected in the feed. Yt =
Ui
~ Api n
(18.70)
A control vector parameterisation approach [66,67] implemented with the gPROMS process modeling tool was employed to solve this dynamic optimization process [68]. The optimum values of the switching time, tj, and of the final time, tr, were determined. The optimal operating conditions were found for different numbers of operating cycles for either TTB or PHL as the main product. The optimum number of cycles, and hence the effective column length, is thereby determined. Optimization of the production of TTB (first feed component) Figures 18.35 left and right show the variations of the recovery yield of TTB with the number of cycles for the injection of 10 and 50 fiL samples of different compositions. Figure 18.35 shows that the yield in TTB (for a 1:3 mixture) increases from 30% (without recycling) to nearly 80% with only two cycles, in which case the effective column length is increased by a factor two. Figures 18.35 left and right show also that there is an optimum number of cycles in this recycling process. The width of the bands increases with increasing extent of recycling, the peaks become flatter and broader. Further recycling beyond the optimum number of cycles deteriorates the separation and causes a reduction of the resolution achieved.
18.5 Recycling Procedures
919
Figure 18.35 Yield of TTB versus the number of cycles for samples of different compositions, •, 1:3; o, 1:1; x, 3:1 mixtures. Left 10 pL samples; Right 50 jiL samples. Reproduced with permission from H. K. Teoh, M. Turner, N. Titchener-Hooker, E. Sorensen., Comp. Chem. Eng., 25 (2001) 893 (Figures 7 and 8).
Figure 18.36 Yield of PHL versus the number of cycles for samples of different compositions, •, 1:3; o, 1:1; x, 3:1 mixtures. Left 10 }iL samples; Right 50 }iL samples. Reproduced with permission from H. K. Teoh, M. Turner, N. Titchener-Hooker, E. Sorensen., Comp. Chem. Eng., 25 (2001) 893 (Figures 9 and 10).
Optimization of the production of PHL (second feed component) Figures 18.36 left and right show the variations of the recovery yield of PHL versus the number of cycles for injections of 10 and 50 fiL samples of different compositions. The yield increases also with increasing number of cycles, except for the second cycle. Actually, the purification of PHL becomes worse in this particular case. This might be due to diffusive tailing effect of the first component or, quite probably, for the neglect of the influence of the competitive interactions of the two compounds with the stationary phase. The yield in PHL does not increase as much as in the case of TTB. In addition, the purity is slightly lower than that achieved without recycling, especially when the concentration of PHL is low and it is the minor component of the feed (e.g., for the 3:1 mixture with 50 jiL samples). This result indicates that recycling does not always guarantee an increase in the amount of purified product obtained. The separation failed to achieve a minimum purity constraint at the fifth cycle. The opposite influences of the number of cycles on the product purity and the recovery yield determine the number of cycles required. Therefore, careful consideration of the parameters controlling these two dependencies is required to choose the optimum operating conditions for the column, especially when the main objective is to maximize the production of the second component.
920
Optimization of the Experimental Conditions
18.6 Practical Rules Numerous publications have discussed the issue of optimization of the experimental conditions in preparative chromatography. Most papers in this area are based on empirical observations. They report on the conclusions derived by people who have acquired long-term experience and familiarity with the method. Each author has dealt with a variety of problems, but the scope and range of these problems vary considerably from author to author. Without a solid theoretical background to sift through this experience and place it into perspective, the validity of the conclusions and, more importantly, the range in which they are valid are still much in doubt. Laboratory and industrial scale preparative chromatography have different problems, issues, and requirements. The former needs the rapid development of separation schemes that are easy to implement rapidly, but it is rarely demanding regarding the production cost. The latter allows more time and greater means to develop the process but requires severe control of the cost. It seems difficult to optimize industrial separations otherwise than by a systematic approach, involving the following 1. A careful investigation of the available stationary phases and solvent combinations to develop a chromatographic system offering a large separation factor, relatively small values of the retention factors, good solubility of the sample components in the mobile phase,7 and a large column saturation capacity. 2. The accurate determination of the competitive equilibrium isotherms of the feed components of importance in the chromatographic system selected, and the measurement of the other parameters of importance (column efficiency as a function of the mobile phase velocity, viscosity of feed solutions in the mobile phase). 3. The computation of the optimum experimental conditions for the required production, taking into account the important cost components, the required purity, and the various constraints (yield, inlet pressure). 4. The validation of the results of the calculations by an appropriate series of experiments, preferably at the pilot scale. For the development of laboratory scale separations, there is usually no time available to acquire the parameters on the thermodynamic and dynamic behavior of the feed components in a series of chromatographic systems. The potential cost savings would not justify such an approach. The following set of practical rules appears as a good compromise between the opposite requirements of a fundamental and a practical approach. 7
The purification of Buckminster fullerene (Cgo) is a typical example of a production rate controlled by the capacity of the mobile phase. In a solvent that gives a retention factor near unity, the solubility is only 800 ppm [69]. In 1-methyl naphthalene, the solubility is close to 5%. To use this solvent, an adsorbent that retains strongly the fullerenes is needed [70]. A silica-bonded tetraphenyl-porphyrin gave satisfactory results [70,71].
18.6 Practical Rules
921
1. Using conventional analytical columns no longer than the preparative column that is intended for use (e.g., 15-40 cm long columns), search for a chromatographic system which gives the maximum possible resolution between the critical feed components, i.e., the component of interest and its nearest neighbors, a retention factor for the first component not exceeding 2, and which uses a good solvent of the feed. This last condition is critical, but is not always possible to satisfy (e.g., fullerenes [69,70]). It is important that the search for the chromatographic system be limited to phases that can be used in preparative applications. The rest of the development work should be done with a sample of the phase which has been selected. 2. The next step is the optimization of the column length and the particle size for maximum production rate. In practice, there is an optimum for the ratio di/L but no separate optima for L and dp. A satisfactory approximation of the optimum ratio d%/L can be obtained using the following equation, which is derived assuming a simple plate height equation and neglecting the effect of competition
opt
1 + k'o 2 V 3k0CAP
where AP is the maximum pressure at which the equipment available can be operated safely on a routine basis and C is the numerical coefficient of the third term in the Knox plate height equation (Eq. 18.50). Thus, the most convenient average particle size can be chosen among the different grades available for the selected stationary phase. Then, the column of optimum length with this particular phase is prepared or a commercial column having a value of d\/h close to the optimum is selected. 3. After the column has been packed and equilibrated with the mobile phase, the flow rate is optimized by injecting small samples and increasing the flow rate until the resolution obtained is approximately equal to 1.7-2 if the separation must be carried out under touching-band condition, or a resolution near unity if the separation is done under overlapping band condition. 4. An estimate of the optimum sample size is given by Eqs. 18.37 and 18.48 in the case of a touching bands strategy and by Eqs. 18.51a and 18.58 in the case of overlapping bands strategy. Experiments carried out with an analytical column having the optimum geometry permit an empirical determination of the optimum sample size for either approach. 5. The injection profiles at column inlet should be close to rectangular. The effect of the injection function on the band profile is illustrated in Figure 18.37 for a 1:4 and a 4:1 mixture. In the inset, the injection function is employed. The diffuse injection was made by utilization of a Gaussian distribution at the end of a short rectangular pulse. This injection function was used because it has been found to give a good representation of experimentally determined injection profiles, as illustrated in Figure 2.3b. The amounts of the main component and impurity in the two figures are the same. Also the amounts injected with the
Optimization of the Experimental Conditions
922 C mg/mL
a
32 o
C 1:C 2=4:1 24
b
20
R
R
15
D
0
Lf,1=4.5 n1=30.3 mg
Co1:Co2=1:4
R
o
L=10 cm N=2000 R
15
16
10 D
8
5
R D
D 0
0 12
22
D
32
42
12
22
R
32
t min
42
Figure 18.37 Effect of the injection profile on band profiles. Heavy dotted line, profile of the impurity for a rectangular injection, thick solid line, profile of the main component for a rectangular injection, thin shaded line, profile of the impurity for a diffuse injection, thick shaded line, profile of the main component for a diffuse injection. Porosity = 0.8, k'01 = 15, k'o 2 = 20, &i = 0.03, £>2 = 0.036, tp = 6.06 min (rectangular injection), (a) 4:1 mixture, Q),i = 5 mg/mL, CQ^ = 20 mg/mL. Inset: rectangular and diffuse injection profiles of same area, (b) 1:4 mixture, Qy = 20 mg/mL, CQ^ = 5 mg/mL. two injection functions are the same. The rear boundary associated with the diffuse injection becomes less than 0.1 mg/mL at 12 minutes, twice the value of tp for the rectangular injection. The figure shows how the injection function Recovery Total Yield, Combined a 99% Purity 70.9% (4.3 o) I 16
33 min
3% Ethyl Acetate/Hexane
Figure 18.38 Effect of the mobile phase composition on the production rate. Elution profile of a 6 g sample of a 25:75 mixture of benzosuberone and a-tetralone. Chromatograms, recovery and production for different concentrations of ethyl acetate in nhexane (2%, 3%, and 3.85%). Flow rate, 50 mL/min. Reproduced with permission from } . Newburger and G. Guiochon, J. Chromatogr., 523 (1990) 63 (Fig. 11).
86.5% (5.2 g)
13
77.0% (4.6 o)
25 min
6fl 3.85% Ethyl Acetate/Hexane 90.0% (5.4 g) I 11.5 m > 99% Purity
80.0% (4.8 g)
21 min E3+E3 z 95% Purity
l < 95% Purity
18.6 Practical Rules
923
ho ml /mfn
44
(0.5cm/mln)
84
24 (0.5cm/min)
44
14
(1cm/min)
27
Recovery Total Yield, Combined ^ 95% Purify
86% (3.4 g)
9 1 % (3.6 g)
a 99% Purity
49% (2.0 g)
44% (1.7 g) m a 99% Purity
90% (3.6 g) 49% (2.0 g) E3 + E l > 95% Purity
• < 95% Purity
Figure 18.39 Effect of mobile phase flow rate on the production rate. Elution profile of a 4 g sample of a 25:75 mixture of diethyl phthalate and /3-tetralone. Chromatograms, recovery yield and production rate obtained at different flow rates. Reproduced with permission from J. Newburger and G. Guiochon,}. Chromatogr., 523 (1990) 63 (Fig. 13).
can strongly affect the rear boundary of the main component and its impurities. Thus, extracolumn band broadening can incur additional yield losses or affect the purity of the product. The location of the cut points is also changed. A well-designed system can avoid such effects and permits a more economical process. This procedure usually gives excellent results. The production rate may often be increased, if needed, by increasing the sample size, reducing the retention factor, and increasing the mobile phase velocity. This requires further experiments. Figures 18.38 and 18.39 show how the mobile phase composition and flow rate can be optimized experimentally to increase the production rate of a purified component at a given purity [72]. Increasing the concentration of the strong solvent in the mobile phase permits a decrease of the retention time (at constant a in the particular case), and in the same time a significant increase in the recovery yield at constant sample size, as illustrated in Figure 18.37, in agreement with the theoretical results reported earlier (see Figure 18.11). Figure 18.38 shows that increasing the volume flow rate threefold permits a threefold increase in the production rate at constant feed load and recovery yield. An example of application is shown in Figure 18.40. An analytical column was used to isolate less than 20 mg each of the enantiomers of a benzodiazepinone derivative by repetitive injection [73]. Figure 18.40a illustrates the overloaded elution preparative chromatogram. This chromatogram was obtained after adjusting the flow rate, mobile phase composition, and temperature to increase the resolution of the analytical separation. Then the sample size was increased to raise the production rate of each component. Due to the tailing of the early eluting enantiomer, a higher sample size was not chosen in order to obtain greater than 95% purity of the second component. The wavelength of detection was adjusted to 295 ran to prevent saturation of the detector. Fraction B was collected at greater
Optimization of the Experimental Conditions
924
Time (min)
Figure 18.40 Example of purification by preparative chromatography. Purification of the enantiomers of a benzodiazepinone on 4.6 x 250 mm column of cellulose tribenzoate coated on 10 }im silica. Mobile phase, n-hexane-2-propanol (40:60), 0.25 mL/min, at 49oC. (a) Preparative chromatogram, 0.75 mg. (b) Analysis of fractions B (dashed line) and D (solid line). Reproduced with permission from A. Katti, P. Erlandsson and R. Dap-pen, } . Chromatogr., 590 (1992) 127(Figs. 3 and 5).
than 99% enantiomeric purity. Analysis of fractions B and D are provided in Figure 18.40b. Fraction D contains 7% of the early eluting enantiomer and fraction E 5%. Fraction C was recycled. A second chromatographic pass allows purification of the second component to 98% purity.
18.7 Optimization of the SMB Process In Chapter 17, we discussed the optimization of the flow rate ratios in the four zones of the SMB process and that of the switching time. The triangle theory allows the determination of the optimum conditions for maximum production rate and minimum eluent consumption. Due to the complexity of the simulated moving bed process, most current studies limit studies on the optimization of an SMB unit operation to investigating the influence of these parameters. Few data are available on the optimization of many other experimental parameters (e.g., pressure drop, product purity) and column design conditions (e.g., column length, particle size, efficiency) or on that of the column configuration (optimum number of columns in the individual zones). In general, the selection of the optimum operating conditions of SMB units is rather difficult because these units have a complex behavior that is not yet completely understood due to the intensity of the nonlinear, competitive equilibrium effects. As a consequence, there are systematic tendencies among users to operate SMB units under rather dilute conditions, in order to minimize the consequences of these nonlinear effects, or empirically to search for the best operating parameters that provide for optimal nonlinear operating conditions [74]. Storti et al. [75] developed a simple and accurate procedure for the selection
18.7 Optimization of the SMB Process
925
of the optimum operating conditions of SMB separation units. This procedure is based on the equilibrium theory of chromatography. It permits in the same time the minimization of the amounts of adsorbent and desorbent required and the maximization of the amounts of extract and raffinate produced. These authors found that suitable modifications of the ratios of the flow rates in the various sections of the operation unit led to the use of shorter columns and resulted in a higher degree of process robustness with respect to small changes in the operating conditions. Lehoucq et ah [76] analyzed several issues related to the selection of the optimum operating conditions in chiral separations, using both an empirical approach and the results of numerical calculations. They reached the following conclusions: 1. Due to the difficulties encountered in the selection of robust operating conditions at high feed concentrations and to the strong influence of fluctuations of the feed concentration on the zone of complete separation, the maximum feed concentration should be the smallest concentration that leads to the saturation of the selective sites of the CSP chosen. 2. The highest productivity is achieved with the smallest number of columns and the shortest switching period. The switching period is limited by the maximum pressure drop that the SMB unit can stand and by the number of columns. 3. Depending of the component of interest, it might be appealing to consider operating conditions that are outside the region of complete separation and that lead to the production of the pure product of interest only. This allows a higher productivity. Mazzotti et ah [77] presented a general theory and derived explicit criteria for the choice of the operating conditions of SMB units that would permit the achievement of the desired separation of binary mixtures. Klatt et ah [78] developed a model-based optimization and process control of SMB with a two-layer control architecture. The optimum operating conditions are calculated off-line by a detailed model of the chromatographic process, such as the general rate model. A control algorithm is then used to keep the process operating within the optimum conditions against disturbances. The cost function can be reduced to the required desorbent inflow, provided that the feed inflow is pre-specified [78]. Therefore, the optimization problem consists in minimizing the desorbent inflow for a given purity requirement of both the extract and the raffinate. The temporal changes of the model parameters due to aging can be estimated on-line [79]. Dunnebier et ah [79] used a staged sequential optimization algorithm, which require an excellent initial guess of the optimal solution, to maximize a generalized cost function with constraints on the product quality of both the extract and the raffinate. Multiobjective optimization of the SMB and Varicol processes by a non-dominated sorting genetic algorithm (NSGA) which does not require any initial guess of the optimum solution was carried out by Zhang et ah [80] who used in that process an objective function that maximizes the feed flow rate (maximum throughput). J = f(Q2,F,ts,x)
(18.72)
Optimization of the Experimental Conditions
926 (a)
Current 1st subinterval (0-V4)
1'01 1
>
1 —
TA
subinterval
\
„/ "s 1 '1
subinterval
Is*
)1 i
E,.
~-
1
subinterval
Neit Switching 1" subinterval (0-V4)
f
O1
V
!O
1™
Figure 18.41 (a) SMB system with 6 columns, (b) Principle of operation of SMB and 4subinterval Varied systems (port switching schedule). Reproduced from Z. Zhang et ah, AIChE } . , 48 (2002) 2800 (Fig. 1). Reproduced by permission of the American Institute of Chemical Engineers. ©2002 AIChE. All rights reserved.
where the four optimization parameters are: Q2 the fluid flow rate in zone II, F the feed flow rate, ts the switching time, and x the column configuration (the distribution of the number of columns along the four zones). In order to keep the purity of both the extract (Pur E) and the raffinate (Pur R) at the specified 95%, the following modified objective function can be minimized, instead of maximizing Eq. 18.72 J =
w
1 - Pur E 0.95
+w
1 - Pur R 095
(18.73)
A large value of w (5 x 104) is needed to keep the purities within ±0.2% of the desired value. In the SMB processes column configuration is decided at the design stage a priori and remains unchanged during the separation process. In the Varicol process, however, the inlet and outlet ports are no longer shifted equally and synchronously. This is illustrated in Figure 18.41 for one switching period. Figure 18.41 .b shows the four-subinterval Varicol process. The number of columns in each zone varies during a global switching cycle. Therefore the Varicol process can have several column configurations which adds flexibility to optimize this parameter {i.e., x)- The optimization of the four parameters of Eq. 18.73 showed that the throughput of the Varicol process is superior to that of the SMB. The determination of the optimum distribution of the number of columns over the four zones of the process results in a complex mixed integer nonlinear program. The multiobjective optimization allows the simultaneous maximization of the
18.7 Optimization of the SMB Process
927
0.98
0.96 -- » - *
0.94
0.92
0.92
0.94
0.96
0.98
PurR
Figure 18.42 Pareto optimal solution for the multiobjective optimization of the SMB and Varicol systems. Reproduced with permission from Z. Zhang et ah, AIChE ]., 48 (2002) 2800 (Fig. 2).
purity of the extract and the raffinate as well as the minimization of the solvent consumption at a constant throughput. In Figure 18.42 the Pareto line for the purity of the extract is plotted as a function of the purity of the raffinate for a 5-column and a 6-column SMB, and for a 5-column Varicol process. The optimization of the 5-column SMB yields line DBC, along which the purity of one of the two streams can only increase at the cost of decreasing the purity of the other stream. The discontinuity of line DBC at point B corresponds to a change in optimum column configuration. Lines ABC and DBE correspond to fixed column configurations. A comparison of line DBC with lines ABC and DBE confirms that column configuration is an important optimization parameter in SMB separations. With a 6-column SMB, in a similar manner, Pareto line NOS is obtained when the column configuration is also optimized, whereas for fixed column configurations lines NOP or QOS are the optimal. For the 4-subinterval 5-column Varicol process, the optimum is found along line FGLM. The multiobjective optimization effectively shows the variability of the decision variables and operation conditions of the continuous chromatographic separation. The 5-column Varicol process offers higher purities than the equivalent 5-column conventional SMB, using the same amount of stationary phase. The multiobjective optimization can also be used simultaneously to maximize the throughput and minimize the eluent consumption [80]. The effect of the particle size was also considered with the multiobjective optimization, using a genetic algorithm [81]. The particle size has a dual effect on the operation of an SMB unit. Larger particles allow a lower back pressure but lead to a poorer column efficiency. The effect of the particle size in the case of a 5-column SMB process is illustrated in Figure 18.43. The largest particle size considered in the study (dp — 40^m) leads to the poorest performance. It requires the highest flow rate and the bad column efficiency limits the productivity. The use of an in-
Optimization of the Experimental Conditions
928
(b) ._ . _ . .
50
:
• 40
A
J•
£
§30 A
A
cT 20
A
•
•
• dp-40 10
;•
iflp=30 • dp-20
0 0
20
40
60
80
100
0
120
20
40
60
80
100
120
Prod, g/(l d)
Prod, g/(l d)
I _
(o) 150
c
; •
i •
A
•
120
* • dp=40
<• 90
Adp=30 • dp=20
J 60
i !
i 140
i 1 •
1
• ! :
30
• dp=40
•
A
A dp=30
•
• dp=20
0 0
20
40
60 Prod, g/(l d)
80
100
120
20
40
60
80
100
120
Prod,g/(l d)
Figure 18.43 Pareto optimal solution for the multiobjective optimization (maximum purity and production rate) of the SMB unit for various particle sizes. The (a) purity of the extract, (b) flow rate, (c) number of theoretical stages per column, and (d) pressure drop are plotted as the function of production rate. Reproduced with permission from Z. Zhang et al, J. Chromatogr., 989 (2003) 95 (Fig. 4).
termediate particle size (dp = 30^m) causes a flow rate decrease and a pressure increase, as expected. The same production rate can be achieved and the product purities are higher than in the case of 40 jvm particles. When the particle size is further reduced to dp = 20^m, however, the production rate will be smaller than with 30 jinx particles, because the pressure drop limits the maximum allowable flow rate. The Varicol process was found to be superior to the corresponding SMB when 40 ,wm or 30 pm particles were used. For the same production rate, Varicol can give a higher extract purity. In the case of 20 ftm particles, however, the Varicol could not outperform the SMB process. The performance difference of the two processes diminishes as the particle size decreases and it disappears at 20 fim in that particular instance [81]. The performance of the units increases asymptotically when the number of columns is increased. At 6-8 columns the difference between the performance of Varicol and SMB vanishes and additional columns do not offer any significant gain in productivity. Jupke et al. studied the effects of the column length in the design of an optimum SMB process for the chiral separation of a racemic mixture of a precursor
18.7 Optimization of the SMB Process
4.5
929
5 5.3 column length [cm]
6
6.5
Figure 18.44 Influence of the column length on the productivity for two different flow rates. Reproduced with permission from A. Jupke et ah, ]. Chromatogr., 944 (2002) 93 (Fig. 13).
-O-specific productivity ™*"specific eluent consumption -•-specific separation costs (1000 kg/a) -•-specific separation costs (5000 kg/a)
Figure 18.45 Influence of the plate number on the specific productivity, specific eluent consumption and specific separation costs. Reproduced with permission from A. Jupke et at, ]. Chromatogr., 944 (2002) 93 (Fig. 17).
of a drug (EMD53986) [82]. Figure 18.44 shows that, for each flow rate in zone I, there is an optimum column length at which the maximum production rate is found. The two curves in Figure 18.44 run parallel; the higher the flow rate, the longer should the column be. The practical consequences of this conclusion are limited by the constraint of a maximum allowed pressure drop. When the particle diameter and the column length are not optimized simultaneously while taking the maximum pressure constraint into account, one may find that the maximum production rate is found at flow rates that would be larger than those allowed by the largest allowable pressure drop.
Optimization of the Experimental Conditions
930
SMB
BATCH 8.6%
8.3%
b)
S.4%
production 1000 kg/a separation costs 2.18 $/g 4.2%
production 1000 kg/a separation costs 2 11 $/g
4.2%
10.3 %
production 5000 kg/a separation costs 1.94 $/g I
I eluent :::j
lost cnida
2.1%
9.2%
production 5000 kg/a separation costs 1 72 $/g I
I capital \ adsorbent
|
operation (labour and maintenance)
Figure 18.46 Contribution of the different cost types to the total specific separation costs in optimized batch and SMB separations for different production rates. Reproduced with permission from A. Jupke et al, J. Chromatogr., 944 (2002) 93 (Fig. 18).
Jupke et al. also studied the economics of the SMB separation. They determined the specific productivity, the specific eluent consumption and the specific separation costs as functions of the column plate numbers. The specific costs were determined for two different production rates 1000 and 5000 kg/year. They could calculate the optimum plate number for maximum value of the specific productivity. The eluent consumption and the specific separation costs continuously decrease with increasing column plate number. Both are relatively high at the maximum specific productivity. These results are similar to those derived in the case of batch chromatography. The economic optimum is a compromise between the effects of conflicting experimental parameters. A comprehensive optimization study of a simulated moving bed reactor (SMBR) was reported by Subramani et al. [83]. It deals with the direct synthesis of methyl tert-butyl ether (MTBE) from tertiary butyl alcohol (TBA) and methanol. The applicability of the Varicol process for a reactive system was explored. The optimization problem involves a relatively large number of decision variables, some continuous variables such as the flow rates in the various sections of the unit and the length of the columns and some discrete variables, such as the number of columns and the column configuration. Multi-objective (involving two or three different objective functions) optimization was performed for an existing system (optimization of the operating conditions) and for the design of new SMBR and Varicol systems. The efficiencies of these different systems are compared. A non-dominating sorting genetic algorithm (NSGA) optimization technique allowed the handling of the complex optimization problem involved in this study. A genetic algorithm (GA) is a non-traditional search and optimization method that has recently become very popular in engineering optimization. It mimics the
18.7 Optimization of the SMB Process
931
principles of genetics and of the Darwinian principle of natural selection. When a single objective function optimization method is used, one attempts to find the best design that is usually also the global optimum. With multiple-objective optimization problems, there could exist an entire set of optimum solutions that are equally good. These solutions are known as Pareto-optimal (or non-dominated) solutions. A Pareto-set is described by a set of points that are such that, when one moves from one point to any other, at least one objective function improves while the other worsens. Since none of the non-dominated solutions in the Pareto set is superior to any other, any one of them is an acceptable solution. Optimization of an existing SMBR system Maximizing the purity of a fraction and the yield of a compound and minimizing the solvent consumption are chosen as the three objective functions. Six decision variables were used in this optimization study, the switching time (ts), the number of columns in sections II, III, and IV, the amount of raffinate produced, and the eluent consumed. Since the optimization of an existing system is considered, the number of columns, their lengths and their diameters were kept fixed, but the sensitivity of the results to the number of columns on the Pareto shift was studied. The flow rate in section II and the temperature of the columns were also kept constant in order to allow a comparison of the optimum results at constant operation cost. Of the two throughput parameters, the raffinate flow rate (fi) was selected as a decision variable, in order to determine the optimum raffinate flow rate for a constant feed flow rate. Figure 18.47a,b show the Pareto optimal solutions for the optimization of an SMBR and compare the performance of the SMBR system equipped with different numbers of columns. The results show that there is no significant improvement in the purity (PMTBE) n o r m the yield (Y^BTE) of MBTE when the column number is increased beyond six. The purity of the MTBE obtained could be as high as 99% and the yield obtained 90%, although these two values are not achieved at the same time. The yield of MTBE showed a sharp decline at very high PMTBE value. The plot of the eluent flow rate (7) versus PMTBE and of the raffinate flow rate (j3) versus YMTBE are shown in Figure 18.47b. The raffinate flow rate, which is directly related to Y^BTE/ exhibits a similar trend, indicating that a higher yield requires a greater raffinate flow rate. The optimum switching time obtained was around 440 seconds. The optimum column distribution in each section of the SMBR unit for a given Ncoi is given in Table 18.7 The optimum column distribution in sections IV and I was found to be always the same (one column), for Ncoi =6,7 or 8. It follows that the number of columns in sections III and II has a greater influence on the extent of the reaction and the separation. Optimization of SMBR Systems at the design stage The optimization of any process at the design stage provides far more freedom than the optimization of an available unit, when one is constrained merely to optimize its operating parameters and maximize the performance of the existing system. At the design stage, several additional decision variables become available for optimization. Subramani et al. [83] optimized an SMBR system at its design stage, attempting to maximize the purity of the desired product while minimizing the fixed costs
Optimization of the Experimental Conditions
932
o.s
Figure 18.47 Pareto optimal solutions and corresponding decision variables in the optimization of an existing SMBR unit. Reproduced with permission from H. J. Submmani, K. Hidajat, A. K. Ray, Comput. Chem. Eng., 27 (2003) 1883 (Fig. 3).
Table 18.7 Optimization of an Existing SMBR system
NCol
6 7 8
Section III, IV, I, II III, IV, I, II III, IV, I, II
Optimum column configuration {%) 2/1/1/2 3/1/1/2 or 2/1/1/3 4/1/1/2
Reproduced with permission from H. J. Subramani, K. Hidajat, A. K. Ray, Comput. Chem. Eng., 27 (2003) 1883 (Table 3).
{i.e., minimizing the total amount of adsorbent and catalyst immobilized in the unit) and the operational costs {i.e., minimizing the volume of eluent circulated). Since the packing material is expensive, the length of each column was optimized. The authors allowed the number of columns to vary between 5 and 6, since it was found in the optimization of an existing SMBR system that there is no significant improvement when the number of columns is increased beyond 6. In addition, based on the results of the optimization of an existing SMBR system, the number of columns in sections IV and I was kept fixed at 1. They found optimum lengths
18.7 Optimization of the SMB Process
933
OSCol-SMBB • IS Col-SMBR
A AAAAO*
.
A A Afe •
O» A
A
O*
A
/<• & ° !i A A A A
Figure 18.48 Pareto optimal solutions and corresponding decision variables in the optimization of a Varicol and a conventional SMBR. Reproduced with permission from H. J. Submmani, K. Hidajat, A. K. Ray, Comput. Chem. Eng., 27 (2003) 1883 (Figure 6).
of about 15 cm for the six-column SMBR and of about 20 cm for the 5-column SMBR. The optimum column configuration for the five- and the six-column SMBR for sections III, IV, I, and II were 2/1/1/1 and 2/1/1/2, respectively. Optimization of Varicol Systems and Comparison of Their Performance to Those of a Conventional SMBR Subramani el al. [83] discussed the optimization process for an existing Varicol system and compared its best performance with the performance of optimized five- and six-column SMBR. In a Varicol process, there are, in principle, an infinite number of column configurations. In order to restrict this variety, these authors considered only four subintervals. The length of the columns was assumed to be 20 cm.
Optimization of the Experimental Conditions
934
Table 18.8 Comparison of Objective Functions and Decision Variables for a few Chromosomes in the Pareto Sets shown in Figure 18.48
Pt#in Fig. 18.48 1 2 3 4 5 6
System
5-SMBR 5-SMBR 5-Varicol 5-Varicol 6-SMBR 6-SMBR
PMTBE
YMTBE
(%)
(%)
70.07 98.18 70.40 98.05 70.81 98.16
72.64 53.58 81.51 58..00 86.08 75.88
T(-)
Ms)
H-)
1.300 2.439 0.996 2.440 1.114 2.462
599 588 591 553 581 555
0.701 0.192 0.493 0.178 0.595 0.295
A A D-A-A-A D-A-A-A H H
Reproduced with permission from H. J. Subramani, K. Hidajat, A. K. Ray, Comput. Chem. Eng., 27 (2003) 1883 (Table 4).
A detailed comparison of the optimum solutions obtained for the different SMBR and Varicol systems are shown in Figure 18.48a,b, that show plots of the yield, YMTBE> a n d of the eluent flow rate (7) versus the purity (PJVTTBE)- The figure shows that the performance of a five-column Varicol is far superior to that of a five-column SMBR. For the same value of PMTBE/ the five column Varicol results in a higher value of Y^TBE than that of the five-column SMBR. The sixcolumn SMBR performs slightly better, giving a slightly higher Y^BTE/ than the five-column Varicol, but it requires a larger amount of adsorbent/catalyst has has a higher investment cost. The performance of the five-column Varicol system is even more impressive when we look at the objective function and the decision variable for some selected points from Figure 18.48a in Table 18.8 [83]. For example, when point 3 (see Figure 18.48a for the five-column Varicol) is compared with point 1 (Figure 18.48a for the five-column SMBR), the five column Varicol results in a higher yield and uses less eluent (the eluent flow rate, 7, is smaller) and gives almost the same purity of MTBE (see Table 18.8). When point 3 (five-column Varicol) is compared with point 5 (six-column SMBR), a higher yield is achieved with the six-column SMBR, but only when the eluent flow rate (7) is increased (see Table 18.48). The optimum column configuration (x) are A (2/1/1/1) for the five-column SMBR, H (2/1/1/2) for the six-column SMBR and D-A-A-A ( 1 / 1 / 1 / 2 - 2/1/1/1 - 2/1/1/1 - 2/1/1/1) for the five-column Varicol system.
18.7.1 Comparison of Batch and SMB Chromatography At the time of writing, no general comparison is available yet regarding the performance of batch chromatography relative to continuous separation processes. Some case studies have compared the performance of the two modes for specific separations. The specific results depend much on the adsorption isotherms of the components to be separated, on the relative importance of the different cost con-
REFERENCES
935
tributions, and on the relative degrees of skills and effort in the investigation of the optimizing of the different processes compared. Results confirm that the SMB process can offer much higher productivity and lower eluent consumption than batch chromatography. Strube et al. compared the performance of optimized batch and SMB chromatography [84], For large scale fructose/glucose separation they found that the optimum batch productivity is 1.7 times higher than the optimized SMB productivity. On the other hand, the optimized solvent requirement is twice as high, which is important because the most important cost factor of this process is the price of the eluent. Thus, the specific separation cost for the fructose/glucose separation is much lower for the SMB than for the batch chromatography process. Jupke et al. studied the separation of a racemic mixture of a precursor of a drug (EMD53986) [82] by batch and SMB chromatography. That chiral separation was performed at a smaller scale than the aforementioned sugar separation. The cost contribution of various factors is summarized in Figure 18.46. In that instance the separation costs of batch chromatography were only marginally higher (3%) than those of SMB at the production rate of 1000 kg/year but the cost gap increases to 13% at a production rate of 5000 kg/year. The very small difference found at the low production rate is due to the high capital cost of the SMB unit. The adsorbent costs are higher for batch chromatography than for SMB. The dominant cost contribution comes from the eluent consumption (49 to 78%); the eluent costs are lower in batch chromatography. Seidel-Morgenstern compared SMB to conventional batch chromatography, to a recycling chromatography process, and to annular chromatography for the separation of some steroid isomers [85]. When the experimental conditions (feed concentration and flow rate) were optimized to maximize the production rate, SMB offered an approximately 2.5-fold gain over the other three separation modes investigated. The best eluent consumption was achieved with recycling, but the SMB process gave a rather similar value. When the solvent consumption was minimized, the optimum was found at much smaller production rates but the corresponding solvent consumptions were similar to those obtained for optimization at maximum production rate. Again, SMB gave the highest production rate; however, it also gave the largest solvent consumption. These examples confirm that the comparison of batch and continuous chromatography is not straightforward. The special benefits of each mode largely depend on the scale of production as well as on the adsorption isotherms and on the specific cost factors.
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1970. D. Nagrath, B. W. Bequette, S. M. Cramer, A. Messac, AIChE J. 51 (2005) 511. J. Frenz, P. Van der Schrieck, Cs. Horvath, J. Chromatogr. 330 (1985) 1. S. C. D. Jen, N. G. Pinto, J. Chromatogr. 590 (1992) 3. A. Felinger, G. Guiochon, J. Chromatogr. 609 (1992) 35. J. Zhu, G. Guiochon, AIChE J. 41 (1995) 45. V. Natarajan, S. M. Cramer, J. Chromatogr. A 876 (2000) 63. V. Natarajan, B. W. Bequette, S. M. Cramer, J. Chromatogr. A 876 (2000) 63. S. D. Gadam, S. R. Gallant, S. M. Cramer, AIChE J. 41 (1995) 1676. A. W. Liao, Z. El Rassi, D. M. LeMaster, Cs. Horvath, Chromatographia 24 (1987) 881. G. Viscomi, S. Lande, Cs. Horvath, J. Chromatogr. 440 (1988) 157. A. L. Lee, A. Velayudhan, Cs. Horvath, in: G. Durand, L. Bobichon, J. Florent (Eds.), 8th Intern. Biochem. Symp., Societe Franchise de Microbiologie, Paris, 1989, p. 593. [62] V. Natarajan, S. Ghose, S. M. Cramer, Biotechnol. Bioeng. 78 (2002) 365. [63] J. R. Crary, K. Cain-Janicki, R. Wijayaratne, J. Chromatogr. 462 (1989) 85. [64] I. Quinones, C. M. Grill, L. Miller, G. Guiochon, J. Chromatogr. A 867 (2000) 1. [65] H. K. Teoh, M. Turner, N. Tichener-Hooker, E. Sorensen, Comp. Chem. Eng. 25 (2001) 893. [66] V. S. Vassiliadis, R. W. H. Sargent, C. C. Pantelides, Ind. Eng. Chem. Res. 33 (1994) 2111. [67] V. S. Vassiliadis, R. W. H. Sargent, C. C. Pantelides, Ind. Eng. Chem. Res. 33 (1994) 2123. [68] Process System Enterprise Ltd., UK, gProms Advanced User Guide: Release 1.7 (1999). [69] M. Diack, R. N. Compton, G. Guiochon, J. Chromatogr. 639 (1993) 129. [70] M. Kele, R. N. Compton, G. Guiochon, J. Chromatogr. A 786 (1997) 31. [71] E Gritti, G. Guiochon, J. Chromatogr. A 1053 (2004) 59. [72] J. Newburger, G. Guiochon, J. Chromatogr. 639 (1990) 129. [73] A. Katti, P. Erlandsson, R. Dappen, J. Chromatogr. 590 (1992) 127. [74] E. Kusters, G. Gerber, F. D. Antia, Chromatographia 40 (1995) 387. [75] G. Storti, M. Masi, S. Carra, M. Morbidelli, Chem. Eng. Sci. 44 (1989) 1329. [76] S. Lehoucq, D. Verheve, A. Vande Wouwer, E. Cavoy, AIChE J. 46 (2000) 247. [77] M. Mazzotti, G. Storti, M. Morbidelli, J. Chromatogr. A 769 (1997) 3. [78] K. U. Klatt, G. Dunnebier, S. Engell, F. Hanisch, Comput. Chem. Eng. 24 (2000) 1119. [79] G. Dunnebier, S. Engell, A. Epping, F. Hanisch, A. Jupke, K.-U. Klatt, H. SchmidtTraub, AIChE J. 47 (2001) 2493. [80] Z. Zhang, K. Hidajat, A. K. Ray, M. Morbidelli, AIChE J. 48 (2002) 2800. [81] Z. Zhang, M. Mazzotti, M. Morbidelli, J. Chromatogr. A 1006 (2003) 87. [82] A. Jupke, A. Epping, H. Schmidt-Traub, J. Chromatogr. A 944 (2002) 93. [83] H. J. Subramani, K. Hidajat, A. K. Ray, Comp. Chem. Eng. 27 (2003) 1883. [84] J. Strube, S. Haumreisser, H. Schmidt-Traub, M. Schulte, R. Ditz, Org. Proc. Res. Devel. 2 (1998) 305. [85] A. Seidel-Morgenstern, Analusis 26 (1998) M46. [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61]
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Glossary of Symbols8 A A Ai
B
surface area of the adsorbent, 3 coefficient in the Rnox and the Van Deemter equations, 2 amount of component i not collected in the corresponding purified fraction and lost, 18 area of a rectangular injected pulse, Ap = n/Fv, 6 activity of a component, 3 Courant number, 10 molecular radius, 5 external surface area of the particles per unit volume, 6/dp for spherical particles, 5 initial slope of the equilibrium isotherm, 3; slope of a linear isotherm, k'Q = Fa, 2; distribution coefficient in linear chromatography, ratio q/C,6 first coefficient of the Langmuir isotherm, 3; coefficient of the Freundlich isotherm, 3 coefficient in the Knox and the Van Deemter equations, 2
B
c o l u m n permeability, B — k$dp, 5
B{
Biot n u m b e r Bi =
Bj
amount of component i collected in the purified fraction of its neighbor, ;', 18 numerical coefficients in isotherm equations, 3, particularly, second coefficient of the Langmuir isotherm, 3 final concentration of a breakthrough curve, 14 initial concentration of a breakthrough curve, 14 critical displacer concentration or watershed point, below which the zone of component i does not belong to the isocratic point, 9 constant concentration of additive in the mobile phase contained in the column, 2 dimensionless concentration, Eq. 6.27, 6 history of the band profiles along the column, 8 mobile phase concentration (of component i), 3 feed or initial concentration, 2 mobile phase concentration during the i step in frontal analysis, 3 concentration of component i in the column ahead of the first front (in binary frontal analysis), 4
Ap a a a a a
a, a,
b, bj Cf C! C'n crit C° Q Q (z, t) Q Cf Q Cifl
kfR/Dp,2
8 Symbols which are used only in one single section of a chapter are not included in this Glossary. The number at the end of each line indicates the chapter in which the symbol is introduced
939
940
Glossary of Symbols
Cijj
concentration of component i in the column behind the second front (in binary frontal analysis), 4 concentration of component i in the intermediate plateau (in binary frontal analysis), 4 plateau concentration in the isotachic train, 9 concentration of component i in the mobile p h a s e , 2 maximum concentration of a band, 3 concentration of the solute in the mobile phase, 6 displacer concentration, 9 concentration at the point z = nh, t = jx of the integration grid, 10 concentration of the solute inside the pores, Eq. 6.62, 6 concentration of the solute in the stationary phase, 6 concentration of component i in the stationary phase, 2 intersection of the hodograph transform and the C\ axis. Concentration of the front plateau of the first component in frontal analysis, Eq. 8.13a, 8
Qm Cf Cmj CM Cm Cn Cn Cp Cs CS/i Cf
Cf Cj
C^1 C^1 C^1 C* dC/dz CSi d D DA/B D°A B DA,B,0
D| A Dfl Dji Dip i ^ j
concentration of the first component at the front of the second component front, 8 intersection of the hodograph transform and the Ci axis. Concentration of the rear plateau of the second component in overloaded elution, Eq. 8.13b, 8 concentration of the first component at the rear of the second component front, 8 concentration of the second component at the rear of the second component front, 8 maximum concentration of the first component at the top of the first component front, 8 solute mobile phase concentration in equilibrium with the solid phase concentration Cs, 2 concentration gradient along the column, 5 solvent consumption, amount of solvent needed per unit amount of purified component produced, 18 column inner diameter, 2 Damkohler number, D = kadp/u, 6 diffusivity or diffusion concentration of solute A in solvent B, 5 diffusivity of A at infinite dilution in B, 5 reference diffusion coefficient, 5 diffusivity of B at infinite dilution in A, 5 apparent axial dispersion coefficient, 2 main diffusion coefficient, 5 cross-coefficient of diffusion, 5
Glossary of Symbols
941
Di, Diri
axial dispersion coefficient of the component i in the mobile phase, 2 Dm molecular diffusion coefficient, 2, or molecular diffusivity in the mobile phase, 6 Dp diffusion coefficient inside the particles (in the pores), 2 dp average particle size, 2 T Faraday constant, 5 F phase ratio, F = (1 - e) /e, 2 Fv volume flow rate of the mobile phase, 3 / friction factor, 5 / (0' / (r) distribution of the solute in the plate models, 6 f(t) elution profile in the plate models, Eq. 6.7,6 fi{Cm,j) functional dependence of the competitive equilibrium isotherm on the mobile phase concentrations of all components, 2 FeC feed cost (cost of product lost), 18 FiC fixed costs (amortization and labor), 18 G characteristic of a gradient, G = S/3£o/ H g(C) function related to the isotherm, g(C) = (C + Fq) /u, Eq. 7.47, 7 g'(0), g"(0) first and second initial derivatives of the isotherm, Eq. 7.46, 7 Gfl(|s Gibbs free energy of the adsorbate, 3 Gij interaction parameter in the calculation of the viscosity of mixtures, 5 gi(Cm,j>Cs,k) functional dependence of the mass transfer kinetics on the mobile and stationary phase concentrations of all components, 2 H height equivalent to a theoretical plate, 2 Hap apparent HETP, or experimental value with a linear isotherm, 10 Hkin/ H HETP under linear conditions, or column HETP, 10 Htf, HETP contribution due to a nonlinear isotherm, 10 h size of the integration mesh in the length direction, 10 h reduced plate height, h = H/dp, 2 Xv T? 1\ / / K Ka k'
intensity of the displacement effect, 8 intensity of the tag-along effect, 8 first-order Bessel function, Eq. 6.49a, 6 longitudinal mass flux of a component, 2 diffusional flux, 5 equilibrium constant, 3, also partition coefficient in nonlinear chromatography, K = k'0/(l + bC0), 14 adsorption equilibrium constant, 6 retention factor, k' = (tR - to)/to, 3
fc0
retention factor at infinite dilution, 3
942
Glossary of Symbols
kQ i
retention factor of a compound i under linear conditions or at infinite dilution, aF, 2 retention factor of component i in presence of a modifier, 13
k0 i km
apparent (lumped) mass transfer coefficient in the liquid film linear driving force model, 2
ks k^ ka, kaj kd> kd,i kf kf
apparent retention factor of the modifier, 13 adsorption rate constant, Eq. 6.63, 6 first-order rate constant of adsorption for the component i, 4 rate constant of desorption for the component /, 4 boundary or film mass transfer coefficient, Eq. 6.57b, 6 lumped mass transfer coefficient in the solid film linear driving force model, 2 apparent mass transfer coefficient, Eq. 6.43 and 6.44,6 specific permeability, 5 column length, 2 loading factor, Ly = n/([l — e]SLqs), 3 loading factor of component i, 8 minimum loading factor of the second component needed to achieve a wide band elution profile, 8
km, k'm fco L Lf Lfi Ll 2 L'r 2 L%2 Lr 2
m i n i m u m loading factor of the second component needed for t h e elution profile to retain a part of the injection plateau, 8 m i n i m u m loading factor of the second component needed for the elution profile to include a mixed zone, 8 m i n i m u m loading factor of t h e second component needed for its elution profile to include a plateau at C\, 8
Lr
apparent loading factor of a binary mixture, 8
I M.\ MA MB Mp
rank of a stage in t h e plate models, Eq. 6.1, 6 n t h m o m e n t of a distribution, 6 molecular weight of the solute, g, 5 molecular weight of the solvent, g, 5 mass flux of solute from the bulk solution to the external surface of the particle, Eq. 6.60,6 sample size, 3, also apparent sample size, m = N[k'0/(l + k'0)]2Lf, Eq. 10.115,10 Avogadro number, 5 column efficiency, 3; number of theoretical plates of the column, 5 total number of stages in the Martin and Synge plate model, Eq. 6.6a, 6 column plate number or efficiency, Eq. 6.56, 6 total number of stages in the Craig plate model, Eq. 6.14,6
m jV N N Nap = N Nc
Glossary of Symbols Nuisp Nf N{iZ Nftn, No Nif Nm Np Nrea Nrea Nsf Nth n na nai}s OC P PC Pe Pep Pez Pxr Pri Pu{ Pz p p , pl'i Qi q )
943
number of dispersion units in the column, Eq. 6.50, 6 number of mass transfer units corresponding to diffusion across the film around the particle, Eq. 6.68, 6 flux of component i entering a column slice at z, 2 column efficiency under linear conditions, 10 reduced length in the liquid film linear driving force model, 14 number of mass transfer units, Nm = k'okfL/u, 6,14 number of mass transfer units due to the diffusion kinetics in the pores, Eq. 6.69, 6 number of transfer units, N rea = k^to, 14 number of mass transfer units corresponding to the rate of adsorptiondesorption, Nrea = (FkaijsL)/u, Eq. 6.86, 6 reduced length in the solid film linear driving force model, 14 contribution to column efficiency due to a nonlinear isotherm, 10 number of moles; also amount of component injected (mole), 3, also 1/n is the exponent in the Freundlich isotherm, 3 number of moles of adsorbent, 3 number of moles of adsorbate, 3 operating costs (solvent, energy), 18 probability density function, Eq. 6.113, 6 production cost, 18 Peclet number, 2 particle Peclet number, Pep = (udp)/Dp, 2 axial Peclet number, Pez = (UL)/DL, 2 probability of finding a molecule in the stage number I after r operations have been completed with the Craig model, Eq. 6.11,6 production rate of component i, 18 purity of a collected fraction, 18 local value of the pressure in a column (i.e., P(z)), 5 pressure, 3 coefficient in the competitive Freundlich-Langmuir isotherm, 4 amount of compound adsorbed by the column packing after the ith step in frontal analysis, 3 adsorbed phase concentration averaged over a particle, 5 stationary phase concentration in equilibrium with the maximum concentration of a band, Eq. 7.8, 7 concentration in the stationary phase in equilibrium with the mobile phase, 2 equilibrium concentration of component i in the stationary phase. Stationary phase concentration in the ideal and equilibrium-dispersive models, 2 column saturation capacity, 3
944
Glossary of Symbols
qSti 1Z lZif
column saturation capacity (same units as C s ), 2 ideal gas constant, 3 ratio of the loading factor of the t w o components, 8
R
fraction of solute in the mobile phase, R = {vmCm)f (vmCm + vsCs) = 1/(1 + Fa) = 1/(1 + k'Q), Eq. 6.13, 6; also particle radius, 2 dimensionless loading parameter, Req — 1 / (1 + bQ)), 14 particle radius, Eq. 6.61, 6 Reynolds number, Re = (upe^dp) /r\, 5 number of operations performed during the operation of a Craig machine, Eq. 6.13,6 auxiliary variable, r = dC\ I dCi, 8 positive root of Eq. 8.10a, 8 negative root of Eq. 8.10a, 8 cross-sectional area of the column, 2 fraction of the adsorbent surface covered by the molecules of component A, 4 fraction of the adsorbent surface covered by the molecules of component B, 4 entropy of the adsorbate, 3 slope of the plot of In k'o versus the modifier concentration, 11 fraction of the adsorbent surface covered b y the molecules of solvent, 4 Schmidt n u m b e r (Sc = t]/pDm), 5 Sherwood number (Sh = (kfdp)/Dm), 5 specific production, or amount produced per unit amount of solvent pumped through the column, 18 Stanton number (St = kfL/u), 14 absolute temperature of the system, 3 critical temperature, 5 dimensionless parameter in the study of the kinetic models (T^ =
Req Rp Re r r r\ Ti S SA SB Sfl(js Si So Sc Sh SPj St T Tc T,|
time, 2 First moment of a peak, Eq. 6.31,6 analysis time, 18 Cycle time for periodic injections, Eq. 6.67a, 6 fth cut point, 18 dimensionless time, Eq. 6.24, 6 retention time of the end of the diffuse boundary of the first component, 8 time at which the elution of a band ends, Eq. 7.16, 7
Glossary of Symbols tpt\ tf tj fjvt tp tji tRfi fR i tjir2 t°R 2 to t(C) Ua
Vs vs VQ
945
retention time of the rear of the injection plateau of a wide injection band, 8 retention of a breakthrough curve, 3 retention time of the end of the second component plateau at Cf, 8 retention time of the band maximum, 10 width of a rectangular injection pulse, 3 retention time of the band maximum, 3 retention time at infinite dilution, i.e., under analytical (3) or linear conditions, Eq. 7.16, 7 retention time of the front shock of the first component, 8 retention time of the front shock of the second component, 8 retention time of a zero concentration of the second component, 8 holdup time of the column (to = L/u = Vm/Fv), 2, 6 retention time of a concentration C, Eq. 7.4, 7 internal energy of the adsorbate, 3 velocity of a concentration shock, 8 velocity of the front shock of a band having a maximum concentration CM, Eq. 7.6,7 local mobile phase velocity, 2; mobile phase linear flow-velocity, or average interstitial velocity (w — L/to), 6 optimum mobile phase velocity for minimum shock layer thickness, 14 velocity associated with a concentration, 3, Eq. 7.3, 7 velocity associated with a concentration C = 0,10 volume of mobile phase passing through the column, 2 volume occupied by the adsorbate, 3 volume of adsorbent in the column, 4 molar volume (cm 3 g mole^ 1 ) of the liquid solute at its normal boiling point, 5 volume of adsorbent in the column, 3 retention volume of the inflection point of the ith breakthrough curve in frontal analysis, 3 volume of mobile phase in the column, 2 volume of mobile phase in a stage of the plate models, Eq. 6.4, 6 retention volume of a sample pulse, 3, also total volume of mobile phase passed through the last stage of a Craig model during the entire process (Vg = rvm, Eq. 6.15,6 volume of stationary phase in the column, 2 volume of stationary phase in a stage of the plate models, Eq. 6.4,6 column void volume or holdup volume, 3
946 V\, V2 Wi 2w X x x x Yj Zf z Zrf Z\ Zj ZK ZL
Glossary of Symbols elution volumes of the two breakthrough fronts (in binary frontal analysis)/ 4 baseline bandwidth, 10 interaction energy between molecules in a pair of neighbors, 3 reduced variable, Eq. 6.113,6 dimensionless distance (x = z/L), 14 mole fraction of a component, 3 reduced volume, Eq. 6.3,6 recovery yield of component i, 18 charge of the ion, 5 abscissa or distance along the column length, 2 dimensionless distance (z^ = z/L), 6 maximum column length permitting the achievement of a wide band profile, 8 maximum column length allowing a part of the feed concentration plateau to be included in the elution profile, 8 maximum column length permitting a mixed zone in the elution profile, 8 maximum column length for which there is still a plateau at Cf in the elution profile of the second component, 8
Greek Letters oc /5 7 7
separation factor of t w o components (a = Uj/Ui), 4, also characterizes the pressure dependence of the viscosity, 5 rate of variation of the modifier concentration in gradient elution, 11 activity coefficient, 3; also tortuosity of t h e packing, 6 auxiliary variable (7 = « / ( l + feCf) = (*^irl + ^2)/(^l r i + ^2) =
F
dimensionless concentration, in the case of a Langmuir isotherm
A{j AC AHj AP APM Aq
corrective terms in the LeVan-Vermeulen equation, 4 amplitude of a concentration shock in the mobile phase, Eq. 7.5, 7 heat of adsorption, 2 difference between the inlet and outlet column pressures, 5 maximum pressure at which the column can be operated, 18 amplitude of a concentration shock in the stationary phase, Eq. 7.5, 7 cycle time, 18 duration of the sample injection, 2 width of the second component plateau at Cf, 8 width of the residual injection plateau, 8
Atc Ati Ati At\
Glossary of Symbols
Arjt At]z 6 8{i) S(tj) e €{, €p ij tj 7/B,O 7/B ©i 6 A A? }In \i }iads Hi y.Q Hi fin Us v TZ p a a2 crj of of o% T T T
947
width of the shock layer in dimensionless units (see 6), 14 width of the shock layer in time units, 14 width of the shock layer in length units, 14 thickness of the Nernst diffusion layer, 6 Dirac pulse injection, 6 dimensionless Dirac pulse injection, 6 total c o l u m n porosity or void v o l u m e fraction, 2 interstitial, or external porosity, or interparticle void fraction 5 internal or intraparticle porosity of the packing particles, 2 reduced retention coordinate (tj = x — kt), 14 mobile p h a s e viscosity, 5 reference viscosity (under atmospheric pressure), 5 viscosity of the solvent (cP), 5 surface coverage (©, = qi/qs),4 p a r a m e t e r characterizing t h e w i d t h of t h e shock layer, 14 dimensionless velocity of the shock layer (A = Usu), 14 conductance of t h e i o n of charge sign i, at infinite dilution, 5 n t h centered m o m e n t , Eq. 6.78, 6 chemical potential, 5 chemical potential of t h e adsorbate, 3 chemical potential of c o m p o n e n t i, 3 zeroth m o m e n t of a distribution (peak area), 6 first m o m e n t of a distribution (retention time of t h e m a s s center), 6 nth absolute or normalized m o m e n t , Eq. 6.77, 6 m e a n location of t h e a m o u n t of solute, Eq. 6.6a, 6 reduced mobile p h a s e velocity (v = (udp)/Dm), or particle Peclet number, 2 spreading pressure, 3 density of the mobile phase, 5 standard deviation of a Gaussian peak, 2 variance of a distribution, 6 variance of the chromatographic peak in dimensionless form, Eq. 6.28a, 6 variance of the distribution of the solute, Eq. 6.5, 6 variance of a chromatographic distribution in time, 6 variance of a chromatographic distribution in volume, Eq. 6.18, 6 retention time per plate, Eq. 6.4,6 size of the integration mesh in the time direction, 10 dimensionless time {r = ut/L), 14 change in internal energy per unit mole of adsorbent due to the spreading of the adsorbate over the surface of the adsorbent, 3
948 <2> (pa (t) (ps(t) TpB X X XA Xi Co,
Glossary of Symbols fraction of modifier in the mobile phase, 11 gradient profile in gradient elution, 2 injection profile, 2 association constant for the solvent, 5 empirical interaction energy parameter in the Fowler isotherm, 3 dimensionless concentration (x = C/CQ), 14 solute compressibility, 5 coefficient in the competitive Fowler isotherm, 4 characteristic parameter in the displacement theory, 9
Glossary of Terms Accuracy: Characterizes the difference between the result of a measurement and the true value of the measured data. It is difficult to estimate properly because the true value is most generally unknown. Computer experiments permit an easy estimate of the errors due to the use of incorrect models, but are unable to account for instrumental bias. Additive: In various modes of chromatography, the mobile phase used is a mixture containing a solvent and one or several additives, used, e.g., to adjust the retention factors of the feed components, or to permit their detection by a detector which does not respond to changes of their concentrations. If the compounds added to the mobile phase can partake of the retention mechanism, they will compete with the feed components. The base solvent used is called weak solvent, and the compounds dissolved in it are called additives or strong solvent. Adsorption: A physicochemical phenomenon causing the equilibrium concentration of one or several compounds to be different in a bulk fluid and at the interface between this fluid and either a solid surface or another fluid, immiscible in the first one. Depending on the nature of the interface, gas-solid, gas-liquid, liquid-solid, or liquid-liquid adsorption can take place. In liquid-solid adsorption, the effect is controlled by the interaction energy between the molecules of solvent and solutes, and the solid surface. Depending on the nature of the solid and the solvent used, adsorption can be controlled by the solute polarity (silica/nonpolar solvent, as in normal phase chromatography) or by dispersive or Van der Waals forces, hydrophobic interactions and solute molecular weight (graphitized carbon black or C18 chemically bonded silica, as in reversed phase chromatography). In the dynamic sense, adsorption indicates the process by which a molecule moves from the bulk solution to the interface. Analytes: Name given by analysts to the components of samples which are analyzed. Anastomosis: The interconnection in a network of channels (e.g., arteries, streets, pores in a porous particle) by which circulation is maintained if one of the channels is obstructed. Anion exchange: See ion-exchange, in cases when the resin contains immobilized cations (e.g., quaternary ammonium ions) or easily ionizable groups (e.g., DEAE or diethylaminoethyl groups, PEI or polyethyleneimine chains) and may exchange anions (i.e., negatively charged ions) with the solution. Anti-Langmuir isotherm: An isotherm which is convex downward or, by extension, an adsorption behavior which is opposite to the one observed with a Langmuir isotherm. With the Langmuir isotherm (see Chapter 3), the solute concentration at equilibrium in the adsorbent, q, increases less rapidly than the solution concentration, C. With an anti-Langmuir isotherm, q increases more rapidly 949
950
Glossary of Terms
thanC. Apparent dispersion coefficient, Dap\ The apparent dispersion coefficient lumps all the contributions to axial dispersion arising from axial molecular diffusion, tortuosity, eddy diffusion, and from a finite rate of mass transfer, adsorptiondesorption, or other phenomena, such as reactions, in which the eluites may be involved. It is used in the equilibrium-dispersive model of chromatography to account for the finite efficiency of the column (Eq. 2.53 and 10.11). See equilibriumdispersive model. Asymptotic solution: The solution of a differential or partial differential equation after an infinitely long time and an infinitely long distance along the space coordinate axis (see Chapters 7 and 14). Although such solutions may seem unrealistic, they are in fact quite relevant. It is often impossible to distinguish the asymptotic solution and the true solution after a short period of time (e.g., the isotachic train in displacement chromatography). Average mobile phase velocity, u: Column length divided by the breakthrough or holdup time (u = L/IQ). Axial dispersion, D^: When a band migrates along a column packed with nonporous particles, it spreads axially because of the combination effects of axial diffusion and the inhomogeneity of the pattern of flow velocity in a packed bed. This combination of effects is accounted for by a single term, proportional to the axial dispersion coefficient. It is independent of the mass transfer resistance and of the other contributions of kinetic origin to band broadening. Backflush: A procedure consisting of reversing the direction of the mobile phase stream in order to eliminate components (usually impurities) with high retention factors, by eluting them through the column inlet. Band: Volume of mobile phase containing a component. The band profile is the plot of the concentration of this component versus the volume of mobile phase passed, or the length occupied by the mobile phase in the column, or the time. Usually, a band is a wide, unsymmetrical peak, such as those obtained under nonlinear conditions. In displacement chromatography, a band is the box-car shape of an individual profile in an isotachic train. Band broadening: The result of a combination of processes, of thermodynamic and kinetic origins, which makes the bands of feed components exiting the column wider than the injection pulse. Band height: Maximum concentration of the band (or maximum detector signal above the baseline). Bandwidth: The width of a concentration profile at a certain fraction of its height (e.g., its half-height) or length of the base-line between the intersection with the front and rear inflection tangents. Baseline: The detector signal recorded when there are no components in the column eluent. The measurements of the intensity of the detector signal during a separation are done by reference to the baseline.
Glossary of Terms
951
Bi-Langmuir isotherm: An isotherm model which is the sum of two Langmuir terms (Eq. 3.53). Binary mixture: Mixture containing only two components. Binomial distribution: A distribution based on the coefficients of a binomial expansion, (« + b)n, with a + b = 1. When n increases indefinitely it tends toward a Gaussian distribution. Biot number: The Biot number characterizes the rate of the film mass transfer kinetic in the general rate model, Bi = (kfdi)/(2epDp), with kt the film mass transfer coefficient, dp, the particle size, and Dp the diffusion coefficient inside the particles. Boundary condition: Mathematical term for the conditions which the solution of a differential or partial differential equation must satisfy at its boundary, hi chromatography, translation into mathematical terms of the conditions imposed by the experiment to the composition of the mobile phase at the column inlet and outlet (e.g., pulse injection). Breakthrough curve: The concentration profile observed at the column exit as a response of the column to an abrupt change in the composition of the mobile phase stream pumped at the inlet. Capacity factor or column capacity factor, k': A dimensionless retention time or volume, more generally called the retention factor (see this term). The original name is a source of ambiguity in nonlinear chromatography because it is related to the slope of the tangent to the isotherm, not to its saturation capacity. The retention factor is the product of the initial slope of the isotherm and the phase ratio. Carrier: A fluid which acts only as a transport agent, without being otherwise involved in the separation process, e.g., the gaseous mobile phase in gas chromatography. hi liquid chromatography, the term is mainly used for the mobile phase percolating the column in displacement chromatography, before the displacer solution starts being pumped into the column. Cation exchange: See ion exchange. A cation exchanger is a resin containing immobilized anion groups (e.g., -SO^", -CO^~, carboxymethyl groups) bound to the resin that can exchange cations with a solution. Characteristic lines: The existence of these lines is an essential feature of the theory of quasilinear partial differential equations of the type of the mass balance equation in the ideal model (dC/dt + u / ( l + Fdq/dC)dC/dz = 0, Eq. 7.2), and a key element of their solutions. The lines (dx/dt)c = u/ (1 + Fdq/dC) in the (x, t) plane are called characteristic lines or characteristics. C is constant along each of these lines. See Chapter 7, Section 7.2.2. Chemical potential, ft: The differential of the Gibbs free energy with respect to the concentration of a component. The chemical potential indicates by how much the free energy of the system would change if the concentration of the corresponding component were to vary slightly. If a system contains several different phases (e.g., solutions, adsorbed layers), a chemical potential can be defined for each
952
Glossary of Terms
component in each phase. Since at equilibrium the free energy of the system is at minimum/ the chemical potential of each component must be the same in all the phases of the system. Chiral: Molecules which contain a carbon atom carrying four different substituents (e.g., CH3-C*H(OH)-CO2H) are found to exist under two different structures or isomers, which are mirror images. An exception is when the molecule has a center of symmetry. (NB: There are some other sources of chirality too complex to be discussed here. See an Organic Chemistry textbook). The corresponding isomers are called enantiomers or optical isomers. They have different properties when reacting with another pure enantiomer, and behave differently when placed in a system containing only one of two possible enantiomers. This is the case of drugs in the body, since amino acids, carbohydrates and many fats exist there as nearly pure enantiomers. Chiral relates to these property differences and to the corresponding environment. Chromatogram: The plot of the detector signal versus time. In this book, we usually consider that a nonselective detector is used, i.e., the chromatogram is considered also as a plot of the total concentration of the components in the mobile phase versus time. Chromatographic velocity: The chromatographic velocity is the ratio of the mobile phase flow rate to the total volume in the column that is available to the mobile phase. See Chapter 2, Section 2.3.3. Coelution: A situation where two different components exit the column at the same time, or at times which are so close that no visual changes of the band profile occur when their relative concentration is altered. Coherence: Coherence is a state in which the variations of all the dependent variables move in a synchronized fashion, i.e., without shift of their profiles relative to each other. In the language of the theory of wave propagation and the method of characteristics, such a behavior is called a simple wave. Thus, simple waves are coherent by definition, and the coherence principle expresses that an arbitrary perturbation will eventually be resolved into a set of simple waves. See EG. Helfferich, Chem. Eng. Comm., 44 (1986) 275. Column efficiency, N: See efficiency. Competitive frontal analysis: Frontal analysis carried out with multicomponent mixtures, for the determination of competitive isotherms or the enrichment of certain feed components. Competitive isotherms: When the solution concentration is not negligibly small, the different components compete for access to the adsorbent surface (in adsorption) or more generally for participation in the retention mechanism. As a result of this competition, the adsorption isotherm of one component depends not only on its concentration but also on the concentrations of all the other retained compounds, whether feed components or mobile phase additives (see Chapter 4). Compressibility: The volume of a given mass of fluid decreases with increasing pressure. The compressibility is the value of dV/(VdP), usually at atmospheric
Glossary of Terms
953
pressure. For an ideal gas, it is equal to —1/P, but for a liquid it is of the order of 1 x 10~4atm~1. The compressibility is most often negligible in liquid chromatography for inlet pressures below 200 atm, around 2800 psi. Concentration, C: 1. Amount of a compound per unit volume. The conventional units are in mass of compound per unit volume (g/L, mg/L, mg/mL) and in number of moles per unit volume, or molarity (1M = 1 mole/L). 2. A separation process which increases the absolute concentration of a component in the final product with respect to the feed concentration (see Chapter 1, Section 1.2.3). Concentration shock: See shock. Constant pattern: The asymptotic solution in frontal analysis or displacement chromatography. Each point of the concentration profile moves at the same velocity, so the profile migrates but its shape remains constant. Constriction: Ratio of the extreme diameters along a flow streamlet in a porous medium. Constriction contributes to axial dispersion. Counter-ion: An ion used to ensure the electrical neutrality of a solution, but which is not otherwise involved in an electrochemical phenomenon (e.g., in ionpair or in ion-exchange chromatography). Cl~ and Na + are typical counter-ions. Courant number, a: A number characterizing the risk of instability in the numerical integration of partial differential equations (see Eqs. 10.86 to 10.88). Craig machine: A separation device based on liquid-liquid extraction, whose principle is analogous to chromatography. A large number of interconnected tubes each contain the same amount of a "stationary" liquid. After the sample is introduced in the first tube, the operation involves two successive operations repeated a large number of times. The tubes are shaken to quicken mass transfer and achieve partition equilibrium between the two liquids. Then the "mobile" liquid in the last tube is collected, the mobile liquid in each other tube is moved from one tube to the next, and fresh mobile liquid is added to the first tube. Eventually, the sample components are all extracted and separated. Cut points: Times, mobile phase volumes, or threshold values of the concentration when fraction collection begins or ends in preparative chromatography. Desorbent: The recycled mobile phase in a simulated mobile bed. Desorption: The process through which a molecule moves from the surface on which it was adsorbed to the bulk fluid phase. Detector calibration: An intermediate step in concentration measurements during which the relationship between the detector signal and the composition of the mobile phase flowing through it is determined. If the relationship is linear, the result of the calibration is a response factor. Otherwise, it is a calibration curve. Diastereoisomers: When a molecule contains several asymmetrical atoms of carbon (see chiral), two types of isomers, enantiomers and diastereoisomers, exist. The former are mirror images. The latter contain some carbon atoms which have an identical structure, others which are enantiomeric. For example, if we have a molecule with two asymmetrical carbons, the first carbon atom can have
954
Glossary of Terms
one of two possible geometrical structures, LI or Dl, which are mirror images and, similarly, the second atom can have the structures L2 or D2. Therefore, the molecule can have four geometrical structures: L1-L2, L1-D2, D1-L2, D1-D2. There are two pairs of enantiomers, (1) L1-L2 and D1-D2; (2) L1-D2 and L2-D1. The pairs L1-L2 and L1-D2 are not enantiomers but are called diastereoisomers. The physicochemical properties of enantiomers are the same (except as related to polarized light or to interactions with other pure enantiomers); the properties of diastereoisomers are usually different, making their chromatographic separation possible on conventional achiral phases. Diffuse boundary: Part of a band profile along which the concentrations migrate at a velocity close to the velocity which is associated with them in the ideal model (the difference decreases with increasing column efficiency). Concentrations change slowly as a function of time or distance along diffuse boundaries. The associated velocity (Eq. 7.3) is a function only of this concentration and of the isotherm equation, and is proportional to the mobile phase velocity. The other part of the profile, where concentration changes rapidly, is a shock layer. Diffusion: A physicochemical process through which concentration gradients are relaxed and disappear progressively (see Chapter 5). Diffusion coefficient, Dm: Coefficient characterizing the intensity of the diffusive flux or the rate of diffusion along a concentration gradient. In principle, the diffusion coefficient is a function of the concentration, and the diffusive flux of one component depends on the concentrations of all the components of the system. These effects are neglected here, since only dilute solutions are used in chromatography. For the same reason, diffusion coefficients at infinite dilution are used. Dirac injection, 5: A mathematical function used to describe an injection profile with an infinitely narrow width, an infinitely high concentration, but a finite amount injected (usually small). This profile provides a convenient mathematical boundary condition which, for most practical purposes, is reasonably matched by the actual injection profiles achieved under analytical conditions (see Figure 2.3). Dimensionless plot: Plots in which dimensionless parameters are used. For example, if we replace a concentration versus time plot of a chromatogram of a component with Langmuir adsorption behavior by a plot of bC versus (t — to)/ (£RO ~~ to), we obtain a plot which depends only on a dimensionless sample size, the loading factor (see Figure 7.3). The chromatograms of all pure components following a Langmuir isotherm would overlay if the injected size corresponds to the same loading factor. Discontinuity: See shock. Dispersion coefficient, D^: See apparent dispersion coefficient. Displacement chromatography: A chromatographic process in which, after the sample is injected, the mobile phase is replaced by a solution of a compound more strongly adsorbed than any feed component, the displacer. Eventually, the feed components are separated into a series of successive zones, forming an isotachic train which propagates without further change. If the column efficiency, the sepa-
Glossary of Terms
955
ration factor, and the component concentrations are high enough, each component zone contains primarily the pure component, with minimum interference between its two neighbors. Displacement effect: When the local concentrations of the mobile phase components are not negligible compared to the capacity of the stationary phase, they compete for interaction with this phase (e.g., adsorption in normal or reversed phase liquid chromatography). When the bands of two components of the feed interfere, the more retained one tends to force the desorption of the lesser retained. The apparent retention of the less retained component decreases as it is displaced by the more retained one. Displacer: A compound more strongly retained than all the components of the feed. This solution is used to displace the feed components. The displacer is usually dissolved in the carrier. Distance Time diagram: In displacement chromatography, diagram indicating the trajectory of the concentration shocks and the regions where diffuse boundaries appear. Elution by characteristic points (ECP): A method for the measurement of isotherms using the relationship derived in the ideal model between the mobile phase concentration and its retention time on the diffuse boundary of a band. The method is accurate only if the column efficiency exceeds several thousand theoretical plates (preferably 5000). Eddy diffusion: The flow in a packed column is uneven because the size distribution of the packing particles has a finite width and because the local packing density fluctuates. The channels available to the mobile phase are irregularly shaped and anastomosed. At the points where channels diverge or merge, minute eddies form. The molecules which travel along different paths have different residence times. These effects of hydrodynamic origin combine into a source of axial dispersion commonly referred to as eddy diffusion. This is one of the major contributions to the axial dispersion of chromatographic bands. Efficiency, N: The column efficiency characterizes the combined effects of the sources of band broadening due to axial dispersion and mass transfer resistance. It is derived from the width of the elution peak observed as the response to the injection of a small, narrow pulse of a dilute solution of a compound. It is difficult to correct for the contribution of the extracolumn sources of band broadening which have to be kept small. In preparative and nonlinear chromatography, there is a correlation between the column efficiency and both the steepness of the shock layer and the duration of the band beyond the retention time IRQ. However, the column efficiency is essentially a concept of linear chromatography, and it is difficult to extend to and use in nonlinear chromatography, except through the shock layer thickness concept. Eluate: The mobile phase at the column exit. May contain some feed components, the eluites. Also called the effluent. Eluent: Synonymous with the pure mobile phase, at column inlet.
956
Glossary of Terms
Eluite: The components of the feed, dissolved in the mobile phase, at the column exit. Eluotropic strength: A parameter characterizing the ability of the mobile phase to elute compounds on a certain stationary phase, with a low or moderate value of the retention factor. The eluotropic strength provides some measure of the strength of a solvent or, more generally, a mobile phase. Elution: A chromatographic process in which a pulse of feed is injected in the stream of mobile phase at the column inlet and carried to the column outlet. If the phases are properly chosen, each component exits as a separate band. Enantiomers: See chiral. Enrichment: A separation process which increases the product purity moderately (see Chapter 1, Section 1.2.3). Equilibrium-dispersive model: Model of chromatography assuming near equilibrium between the stationary and the mobile phases. Specifically, it assumes that the concentrations in these two phases are related by the isotherm equation, and that the effect of the finite rate of mass transfer can be lumped together with the axial dispersion coefficient. This model is valid when the column efficiency is larger than a few hundred plates. Excess isotherm: The conventional isotherm is a plot of the stationary phase concentration versus the mobile phase concentration. The excess isotherm is a plot of the difference between the concentration at the interface and in the bulk versus the bulk (e.g., mobile phase) concentration (Chapter 3, Section 3.1.7). External porosity, ej,: See interstitial porosity. Extract: A simulated moving bed unit produces two fractions. The extract is the one that contains the more retained components of the feed. Extraction: A separation process which collects from the feed a certain component of that feed, or most of it (see Chapter 1, Section 1.2.3). Most commonly associated with liquid-liquid extraction. Feed: Name given by chemical engineers to the sample introduced into the column. Feed cost (FeC): The part of the separation or purification costs corresponding to the amount of crude feed lost during the operation because of a recovery yield, Y, below 100%. The feed costs are proportional to 1 — Y. See Chapter 18, Section 18.2.1. Fick's laws: Classical laws of diffusion relating the flux and the concentration gradient (first law, Eq 5.4), and the rate of variation of the local concentration to the gradient of the concentration gradient (second law, Eq. 5.5). Fingering: A viscous instability. When a viscous band is pushed by a less viscous fluid, the interface is unstable. When the less viscous fluid penetrates through channels inside the viscous zone, the hydraulic resistance to forward migration along these channels is decreased, causing a preferential migration of the fluid along them. Viscous fingering may appear when a less viscous fluid pushes a
Glossary of Terms
957
more viscous one and is detrimental to a preparative separation. Finite difference methods: Methods used for the calculation of numerical solutions of systems of partial differential equations. The differential elements in the differential equations are replaced by corresponding finite differences, giving difference equations. Stability and accuracy conditions must be satisfied (Chapter 10, Section 10.3). Finite element methods: Methods used for the calculation of numerical solutions of systems of partial differential equations (Chapter 10, Section 10.3.6). Fixed costs (FiC): Part of the separation or purification costs corresponding to the amortization of capital. See Chapter 18, Section 18.2.1. Flux, /: Amount of a component transported per unit time through a surface of unit cross-section area, in the direction perpendicular to this surface. Fowler isotherm: See Eq. 3.59 and Chapter 3, Section 3.2.3.1. Freundlich isotherm: See Eq. 3.56 and Chapter 3, Section 3.2.2.4. Frontal analysis: A chromatographic process in which a feed solution is abruptly substituted for the mobile phase and pumped through the column. Each component has its own breakthrough curve, but only the least retained one gives a pure zone. This method is used by chemical engineers for bulk purification requirements. In chromatography, it is an accurate method of isotherm measurement because the retention time of the fronts is related to the amounts adsorbed and is independent of the column efficiency. Also called breakthrough analysis. Gaussian profile: "Bell-shaped" profile following the Gauss equation (Eq. 6.5). General rate model: Model of chromatography taking into account separately the contributions of each of the various sources of mass transfer resistances. It includes a bulk and a pore mass balance equations, and the relevant kinetic equations. Golay equation: A correlation between the column height equivalent to a theoretical plate and the mobile phase velocity. This equation is valid for an open tubular column (Eqs. 6.105a and 6.105c). Gradient elution: A procedure used in analytical and preparative chromatography to separate mixtures of components with a wide range of distribution coefficients. During the separation, the mobile phase composition is changed continuously (or by steps). Initially, the mobile phase is rich in weak solvent, and the weakly adsorbed components are still retained and separated. At the end of the separation, the mobile phase is richer in strong solvent, and the strongly adsorbed components can be eluted out of the column. This procedure is expensive, as solvent consumption is high and solvent regeneration costly. Guard column: Auxiliary, short, relatively inexpensive column placed upstream of the main separation column. Its main role is the extraction of strongly retained impurities, thus preventing fouling of the main column. It may also achieve a simple preseparation of the feed. Heat balance: A relationship expressing the conservation of energy in a closed
958
Glossary of Terms
system (e.g., in a chromatographic column). HETP, H: The height equivalent to a theoretical plate is equal to the column length divided by the column efficiency. Hodograph: Plot of the concentration of one component versus the concentration of the other one during the elution of a mixed binary band. Can be extended simply to multicomponent mixtures. Holdup time or volume, to or VQ: Retention time (or volume) of a nonretained component. ft-Transform: See Chapter 9, Section 9.2.1, Eq. 9.16. Hydrophobic interaction chromatography: Mode of chromatography using as a retention mechanism the molecular interactions between the hydrophobic or nonpolar regions of the external surface of protein molecules and chains bonded to a silica support. Although C18 alkyl chains would qualify, the name is reserved to polyether or polyamide chains which cause little degradation of the purified proteins. Ideal adsorbed solution (IAS) theory: A model of adsorption based on the Gibbs adsorption isotherm and on a general thermodynamic criterion for interfacial systems. It permits the prediction of multicomponent, competitive isotherms from single component isotherms alone, whether the two single-component isotherms are accounted for by the same model or not. The LeVan-Vermeulen isotherm model is obtained by applying the IAS theory to the particular case in which the single-component isotherms follow the Langmuir model (see Chapter 4, Section 4.1.5, and Chapter 11, Figures 11.23 and 11.28). Ideal model of chromatography: A model of chromatography assuming no axial dispersion and no mass transfer resistance, i.e., that the column efficiency is infinite (H = 0). This model is accurate for high-efficiency, strongly overloaded columns. It permits an easy study of the influence of the thermodynamics of phase equilibrium (i.e., of the isotherm) on the band profiles and the separation. See Chapters 7 to 9. Indirect detection: A method which permits the quantitation of components with a detector which does not respond to their concentration changes (e.g., analysis of carbohydrates with a UV detector). A component to which the detector responds and which is retained on the column is added to the mobile phase in constant concentration. The system peak associated to each component peak can be detected and recorded (see Chapter 13, Section 13.1.3). Infinite dilution: Conditions under which the concentrations of the solutions used are so small that the equilibrium isotherms can be considered as linear and noncompetitive. Initial condition: Mathematical term for the condition(s) the solution of a differential or partial differential equation must satisfy at the time origin (t = 0). In chromatography, translation into mathematical terms of the conditions imposed by the experiment to the composition of the mobile phase inside the column at the beginning of the experiment (e.g., column equilibrated with pure mobile phase,
Glossary of Terms
959
with a solution of additive or component of given concentration, etc.). Injection: Process transferring the desired amount of feed or sample into the mobile phase stream, just upstream of the column inlet. In analytical chromatography, it is carried out using syringes. In preparative chromatography, it is done by valve switching or by replacing the mobile phase by the feed at the pump. Internal or intraparticle porosity, ep: Fraction of the column volume contained inside the particles. Also called the fractional pore volume. Fraction occupied by the stagnant mobile phase. Interparticle or interstitial porosity, ey. Fraction of the column volume outside the pores, occupied by the bulk mobile phase and available for the mobile phase flow through the column. Typically around 40-41% in analytical columns, this fraction is somewhat less in compressed beds. Intersticial velocity: The ratio of the mobile phase flow rate to the geometrical volume of the column tube. See Chapter 2, Section 2.3.3. Ion exchange: Physicochemical phenomenon used as a retention mechanism in chromatography, through which a solution exchanges ions with the surface of a porous solid or the bulk of a permeable solid, usually a cross-linked polymer or copolymer, sometimes a zeolite. Typical ion-exchangers, called resins when appropriate, contain a large density of ionizable functions (e.g., carboxylic or amino groups) which can exchange ions with an appropriate solution. A resin containing bound CO2H carboxylic groups or SO3H sulfonic groups can exchange H + ions for Na + ions or R + organic molecules (or proteins); it is called a cation exchanger. A resin containing bound NR^ quaternary ammonium groups could exchange Cl~ for RCO^ groups (again, R can be a protein); it is called an anion exchanger. Irreversible adsorption: Particular case of adsorption where the rate of desorption is zero or negligible within the time frame of the experiment. Isotachic train: Asymptotic solution of the displacement problem. Each component zone pushes in front of it the zones of the less strongly adsorbed components and is pushed by those of the more strongly adsorbed ones (see Chapters 9 and 12). A constant pattern is reached and all the zones move at a constant velocity. Isotherm: The relationship between the concentrations of a component in the stationary and the mobile phases at equilibrium. The isotherm gives preferably the stationary phase concentration of the component as an explicit function of its mobile phase concentration and the mobile phase concentration of the other components of the chromatographic system. Isotherms provide the thermodynamic information required to design a chromatographic separation. Single-component isotherms are discussed in Chapter 3, multicomponent ones in Chapter 4. Jovanovic Isotherm: See Eq. 3.52 and Section 3.2.1.2. Kinetic models: See Lumped kinetic models. Knox equation: Correlation between column efficiency and mobile phase velocity (Eq. 6.105c). Langmuir isotherm: The simplest nonlinear isotherm model, q = (aC)/(l + bC),
960
Glossary of Terms
where q and C are the stationary and mobile phases concentrations, respectively, and a and b are numerical coefficients. Langmuir kinetics: A simple competitive kinetic model of adsorption-desorption. The rate of adsorption is proportional to the free surface area, the rate of desorption proportional to the surface area occupied by the adsorbate. Langmuir Freundlich isotherm: See Eq. 3.57. LeVan-Vermeulen isotherm: Isotherm model which often applies well when the single-component isotherms both follow the Langmuir model, but with different values of the column saturation capacity. Linear addition rule: Rule stating that the contributions of axial dispersion and all the sources of mass transfer resistance to the band broadening are additive. This rule is valid in linear chromatography, but has limited applicability in nonlinear chromatography. Linear chromatography: A model of chromatography assuming that the equilibrium isotherm is linear. This model accounts for the various sources of band broadening in the column, and permits their study independently of the properties of the isotherm. Synonymous with analytical chromatography. Linear isotherm: Equilibrium isotherm in which the stationary phase concentration is proportional to the mobile phase concentration. Although this isotherm occurs only rarely, the model is valid at low concentrations, where the true isotherm can be replaced by its initial tangent. It is the thermodynamic condition of analytical chromatography. Liquid film linear driving force model: Simple model of mass transfer in chromatography, where it is assumed that the rate of variation of the stationary phase concentration is proportional to the difference between the local concentration of the component in the mobile phase and the mobile phase concentration which would be in equilibrium with the local stationary phase concentration (Eq. 2.43). Loading factor: Ratio of the sample size to the column saturation capacity. Dimensionless parameter characterizing column overloading (Eq. 7.26). Lumped kinetic models: Simple kinetic models in which near-equilibrium is assumed, the mass balance equation in the pores is omitted, and the rate of variation of the stationary phase concentration is related simply to the deviation of the local concentrations from equilibrium (see Section 2.2.3 and Chapters 14 to 16). Lumped pore model: Simplification of the general rate model assuming that the rate of the adsorption/desorption kinetics is infinitely fast and using a linear driving force model to account for the mass transfer kinetics between the mobile phase percolating through the bed and the mobile phase stagnant inside the particles of the packing material. Mass balance: A relationship expressing the conservation of the mass of a certain compound. Mass transfer: All the phenomena of nonthermodynamic origin taking place during the migration of a concentration band from injection to elution out of the col-
Glossary of Terms
961
umn. This includes diffusion, dispersion, and adsorption-desorption. Mass transfer kinetics: The rate at which components move between the two phases of the chromatographic system. Mass transfer resistance: Since the mass transfer kinetics is not infinitely fast, some phenomena slow it down. They are the source of mass transfer resistance. Microbore columns: Columns used in analytical chromatography which have inner diameters less than about 2 mm. These columns are convenient for thermodynamic and kinetic studies using expensive or toxic chemicals. Migration velocity, uz: The velocity of a concentration in the ideal model (Eq. 7.3), or of a chromatographic band in linear chromatography. Mixed zone: In elution chromatography, when successive bands are not fully resolved, the intermediate zone. For binary separations, the zone containing the two components (Figures 8.3 and 18.1). Mobile phase modifier: See additive. Models of chromatography: Besides linear chromatography (Chapter 6), which assumes a linear equilibrium isotherm, there are four main models, differing in their treatment of the mass transfer kinetics. In the ideal model (Chapters 7 to 9), the column is assumed to have an infinite efficiency; there is no axial dispersion and the mass transfer kinetics is infinitely fast. In the equilibrium-dispersive model (Chapters 10 to 13), the rate of mass transfer is assumed to be very fast and is treated as a contribution to axial dispersion, independent of the concentration. In the lumped kinetic models (Chapters 14 to 16), the rate of mass transfer is still high, but their dependence on the concentration is accounted for. The general rate model (Chapters 14 and 16) takes into account all the possible sources of deviation from equilibrium. Moreau isotherm: See Eq. 3.64.
Nonselective detector: A detector which gives a response (i.e., a change in current or voltage) upon a concentration change for any component. Ideally, the detector should be linear (i.e., its response should be proportional to the component concentration), and this proportionality coefficient, or response factor, should be the same for all components. There are no such detectors. The differential refraction index detector comes closest to being nonselective. Normal phase chromatography: A mode of adsorption chromatography in which the mobile phase is nonpolar and the stationary phase is a polar adsorbent (e.g., n-hexane or methylene chloride on silica). Objective function: A function of the experimental conditions for which an extremum is looked in optimization studies. See Chapter 18, Section 18.2.2 Operating costs (OC): Part of the separation or purification costs corresponding to the products and services needed to operate the chromatograph (e.g., solvents, energy). See Chapter 18, Section 18.2.1. Operating line: Straight line between the origin and the point of the displacer isotherm at the displacer concentration used. Its intersection with the component
962
Glossary of Terms
isotherms gives the plateau concentrations of their bands in the isotachic train. Optical isomers: See Chiral. Optimization: Search of the experimental conditions allowing the achievement of the best possible separation, e.g., maximum production rate, minimum production cost. See Chapter 18. Orthogonal collocation: A finite element method for the determination of numerical solutions of partial differential equations. See Chapter 10, Section 10.3.6. Overdisplacement: A phenomenon which takes place in displacement chromatography when the displacer concentration is too high, and, although the isotachic train forms, the band plateaus are very narrow or nonexistent, the retention times of the bands are close to the holdup time and/or the shock layers are too thick. Little or no separation of the bands take place. Overlapping bands: The chromatogram obtained when the bands of closely eluted components are incompletely separated, with a mixed zone of significant size taking place between them. In preparative chromatography, the optimum experimental conditions correspond often to overlapping or unresolved bands. Overloaded elution chromatography: Name given to elution chromatography when a large sample is used, so the column is operated under nonlinear isocratic conditions. This distinguishes the preparative applications of elution from its analytical applications, which use small samples. Particle size, dp'. Average diameter of a single particle. Packing materials used in chromatography are usually heterogeneous. Many are made of irregular shaped particles for which a dimension, let alone an average size, is hard to define. The procedure used to measure the size is an averaging process which defines what it measures (e.g., Coulter counter, optical devices, permeability). To be meaningful a particle size should include (i) the method used, (ii) the average size, and (iii) the variance of the distribution. The distribution itself should be preferred. Peak: Volume of mobile phase containing a component. The peak profile is the plot of the concentration of this component versus the volume of mobile phase passed, or versus the length of the volume occupied by the mobile phase in the column. Usually, a peak is narrow and symmetrical, as the bands obtained under linear (i.e., analytical) conditions. Peclet number, Pe: A dimensionless number used frequently by chemical engineers. Pe = (ux)/Dm, where x is a characteristic length, u the mobile phase velocity, and Dm the molecular diffusivity. In chromatography, there are two Peclet numbers, defined by respect to the column length and to the particle size. The former must be large (i.e., 100 or more) for the simple models of chromatography to be valid. The second is known by chromatographers as the reduced mobile phase velocity. Permeability: A numerical coefficient characterizing the hydrodynamic resistance of a column (see Chapter 5, Section 5.3.4). Phase ratio, F: The ratio between the volume fractions of the columns which are
Glossary of Terms
963
occupied by the stationary and the mobile phases, F = (1 — e) /e, with e total porosity of the column. This definition is of engineering origin, and is not what chromatographers usually mean. Chromatographers define the phase ratio as the ratio of the volumes occupied by stationary and mobile phases in the column. This definition is difficult to apply in adsorption chromatography. Plateau concentrations: Several plateaus are considered in chromatography theory, besides the trivial plateau concentration corresponding to the elution of the top of a wide rectangular band. These are (i) the plateaus of the zones in the isotachic train in displacement chromatography, (ii) the plateaus of the less retained components displaced by the more retained ones in frontal analysis (concentration higher than the feed concentrations), and (iii) the rear plateau of the more retained component whose band tags along the band of a less retained component (concentration lower than the feed concentration). Plate height, H: See HETP. Plate number (apparent): Derived from the ratio of the retention time and the width of a band, using equations valid in linear chromatography. The apparent plate number has no sense in nonlinear chromatography beside indicating the degree of band broadening due to thermodynamics. Plate model: A model of chromatography which assumes that the column is a series of stages in each of which equilibrium between the stationary and the mobile phase is achieved. Poisson distribution: Distribution following the Poisson equation (Eq. 6.3). Pore diffusion: Diffusion inside the pores of the stationary phase particles proceeds more slowly than in the bulk mobile phase and sometimes following a different mechanism. This mass transfer process may cause a critical contribution to mass transfer resistance and limit the column efficiency. See Chapter 5, Section 5.2.6. Pore volume: Total volume of the pores inside the particles of the packing material. See internal porosity. Porosity, e: Fraction of the column volume which is void of mobile phase. See internal, interparticle, and total porosity. Powerfeed: An implementation of the simulated moving bed process in which the feed flow rate varies during the cycle. Precision: Characterizes the distribution of the results of measurements around their average value. The repeatability characterizes the ability to repeat the same value under the same conditions. The reproducibility, the ability to do so under somewhat different experimental conditions, e.g., by a different operator or using a different instrument. Pressure drop, AP: The difference between inlet and outlet pressure. Production rate, Pr: Amount of purified product obtained per unit time. Pulse injection: Introduction of the feed or sample over a short but well-defined duration.
964
Glossary of Terms
Purification: A separation process which increases the product purity by a large proportion (see Chapter 1, Section 1.2.3). Quadratic isotherm: See Eq. 3.61 and Chapter 3, Section 3.1.7. Racemic mixture: A mixture of two enantiomers in equal concentrations. Raffinate: A simulated moving bed unit produces two fractions. The raffinate is the one that contains the less retained components of the feed. Recovery yield, Y: Proportion of the product contained in the feed which is eventually recovered in the final product. Recycling: A chromatographic process in which a fraction of the mixed zone is reinjected on-line, without intermediate collection and pooling. See Chapter 18, Section 18.5. Recently, recycling has been generalized to include various procedures of rechromatography of collected fractions. Reduced HETP, h: The ratio H/dp of the column HETP to the average particle size. Reduced velocity of the mobile phase, v Name given by chromatographers to the particle Peclet number. Regenerant: Product or solution used to regenerate the column. A requirement in displacement chromatography, less frequently used in overloaded gradient or isocratic separations. Resolution: In linear chromatography, the resolution is defined as the ratio of the difference between the retention times of the maxima of two successive bands and their average width. The definition is valid in nonlinear chromatography as well, but has limited interest. Response factor: When the response of a chromatographic detector is linear, the ratio of the component concentration to the detector signal (e.g., absorbance with a UV detector). Retention factor, k'\ Dimensionless retention time or volume. Ratio of the times spent by the component in the stationary and the mobile phases, k! = {1%$ — to) I to under linear conditions, k' is proportional to the initial slope of the isotherm and to the phase ratio, k1 = Fa. Number of column volumes needed to elute a compound. Retention time, t%: Time elapsed between injection and a certain event, e.g., the elution of the band maximum, of a shock, or of a certain concentration, C. Reversed phase chromatography: A mode of adsorption chromatography in which the mobile phase is a polar solvent and the stationary phase is a nonpolar adsorbent (e.g., acetonitrile-water on an ODS-silica packing). Rotating annular column: An implementation of preparative chromatography using a cylindrical, annular column rotating around its axis. A continuous feed stream is injected at a fixed position just above the packing. Each component band follows a helical trajectory and is collected as a continuous stream along a fixed sector of the outlet annulus.
Glossary of Terms
965
Sample: Name given by chromatographers to their feed. Saturation capacity of the stationary phase, qs: Amount of a component needed to saturate the stationary phase in the column. In adsorption, amount needed to make a monolayer on the adsorbent surface. In ion exchange, amount required to exchange all the ion sites on the resin. Scatchard plot: Plot of q/C versus q, where q and C are the equilibrium concentration of a compound in the stationary and the mobile phases, respectively. It is one form of linearization of the Langmuir isotherm (see Eqs. 3.14 and 3.47). Separation factor, cc: The ratio of the initial slopes of the isotherms of two components, or ratio of the retention factors. As far as possible, a, > 1. Selective detector: A detector which gives widely different responses for different components. The response factors may differ by orders of magnitude. Typical examples are electrochemical detectors and spectrofluorescence detectors. Self-sharpening: A self-sharpening boundary is a shock layer, a boundary which sharpens when the injection is not a sharp front, and retains a constant steepness during its migration. The band fronts of compounds with Langmuirian isotherms are self-sharpening. Shave-recycling: A mode of recycling in which a large sample is injected and incompletely resolved bands are eluted. The zones containing the pure components in the front and rear of the mixed band are collected separately and the incompletely resolved mixture is recycled through the same column, until the sample is exhausted. Shift-Variant (-Invariant) Convolution: Convolutions in which the same spreading function is applied to every time element or data. This is the case in linear chromatography where the spreading caused by each plate on the passing distribution is the same. In nonlinear chromatography, the effect of each plate on the profile depends on the concentration in that plate. The convolution is said to be shift-variant. As a consequence the rules of variance addition do not apply. Shock: Concentration discontinuity arising at the front of a chromatographic band when the isotherm is convex upward, at its rear when it is convex downward, if the column efficiency is infinite (ideal model, Chapter 7). The discontinuity is stable and forms because in this case a velocity is associated with each concentration, and this velocity increases with increasing concentration for a convex upward isotherm. Points on the front profile at high concentrations move faster than points at low concentrations, and pile up at the front of the band. However, the area of the band is proportional to the sample size and is finite. So, a discontinuity must form. Shock Layer: Because the efficiency of actual columns is finite, concentration shocks are not stable. They are eroded by axial dispersion and the finite rate of mass transfer. A steep concentration gradient is formed instead. The steepness of the profile depends on the axial dispersion and the mass transfer resistance. In frontal analysis, a constant pattern, or steady-state profile, forms after an infinite period of time and an infinite migration length. In this case, the shock layer profile
966
Glossary of Terms
is constant. It just migrates unchanged. Shock Layer Thickness, SLT: Distance (in time, space or dimensionless coordinates) between the two points of the constant-pattern breakthrough profile where the concentration is equal to an arbitrary fraction of the concentration step. Simple Wave: Particular boundary conditions for which the solution of the nonideal chromatographic models is mathematically simple. In the case of a breakthrough curve, in frontal analysis or in the injection of a wide rectangular pulse, the concentration of each component varies between two constant values. The solution is said to be a simple wave solution, by reference to the theory of wave propagation which is governed by a similar equation. Simplex Algorithm: An optimization algorithm which is robust and simple, and has become popular. The value of the objective function is calculated for n + 1 different sets of experimental conditions, n being equal to the number of parameters to optimize. The values obtained are compared, the less favorable set is eliminated and replaced by a third set derived from the previous three by following simple rules. The algorithm converges if the objective function is well behaved. Simulated Moving Bed: An implementation of preparative chromatography in which a series of identical columns is used. The continuous injection of a feed stream and of the mobile phase stream, the continuous collection of two fraction streams are made in positions which are periodically moved by one column length. The result is a semisteady concentration profile for each component, which oscillate slowly, and permits the collection of two streams of constant composition. The process is equivalent to moving the stationary phase down the column while the mobile phase flows upward, the faster moving component eluting at the top, with the mobile phase, the slower moving one at the bottom, with the stationary phase. Size Exclusion Chromatography: A mode of chromatography using porous particles having pores of a size comparable to that of the molecules to be separated. Large molecules do not access the pores and elute at the external porous volume. Small molecules access all pores and elute at the total pore volume. Intermediate size molecules access part of the pores. This method applies to polymers, and especially to proteins. The solvent and the porous solid must be selected to avoid adsorption of the feed components. Solid Film Linear Driving Force Model: Simple model of mass transfer in chromatography, where it is assumed that the rate of variation of the stationary phase concentration is proportional to the difference between the local concentration of the component in the stationary phase and the stationary phase concentration which would be in equilibrium with the local liquid phase concentration (Eq. 2.42). Solvent Consumption, SC: Amount of solvent required to purify a unit amount of product. Solvent Strength: See Eluotropic Strength. Solute: Compound in solution, usually somewhat dilute, in the mobile phase.
Glossary of Terms
967
Sorption Effect: A nonlinear effect due to the difference in the partial molar volumes of the component in solution in the mobile phase, and adsorbed on the stationary phase. In liquid chromatography this effect is negligible. It is important in gas chromatography. Specific Production: Amount of purified product obtained per unit amount of solvent passed through the column. Split Peak: When the mass transfer resistance is high, part of the sample may not see the stationary phase and percolate through the column unretained. It elutes as a sharp peak, at the hold-up time. The rest of the sample is retained and elutes as a wide band. Step Gradient: A method of gradient elution in which the composition of the mobile phase is changed by incremental steps. Strong Solvent: See additive. Superficial velocity: The ratio of the mobile phase flow rate to the external or extra-particle column volume. See Chapter 2, Section 2.3.3. System peaks: System peaks take place when the mobile phase contains one or several compounds, or additives, which equilibrate between the two phases of the chromatographic system. Bands of additives, most of them coeluting with the feed components, arise from perturbation by the feed components of the additive equilibrium. See Chapter 13. Tag-Along Effect: An effect resulting from the competitive nature of the interactions between molecules of the feed components and the stationary phase. When the bands of two components interfere, and if the first eluted one is in excess, its molecules crowd those of the second component out of the stationary phase and into the mobile phase. Accordingly, the front of the second band is eluted earlier than when the same amount of the pure component is injected. Tailing: A type of band asymmetry in which the band front is steep and the rear returns only slowly to the base-line. It is often due to a convex upward isotherm. However, a slow kinetics of mass transfer, or an exponential injection profile can also account for this type of profiles. Tortuosity, 7: Ratio of the column length and the average path length of the channels available to the mobile phase stream and along which diffusion takes place. Total porosity, e: Fraction of the column volume occupied by the mobile phase. T6th isotherm: See Eq. 3.54 and Section 3.2.2.2. Touching Bands: Degree of separation between two bands. Touching bands separation is achieved when, under ideal model conditions (i.e., with an infinitely efficient column), the retention time of the front shock of the second component is equal to the retention time of a zero concentration of the first component. Unless the column efficiency is poor, this corresponds to a 100% recovery yield of each component. Unilan isotherm: See Eq. 3.55.
968
Glossary of Terms
Vacancy Chromatography: A chromatographic process in which a sample solution is used as the mobile phase and a sample of pure solvent is injected periodically. A vacancy of each component migrates along the column, instead of a positive peak in conventional elution. Van Deemter equation: A correlation between column efficiency and mobile phase flow velocity. Varicol: An implementation of the simulated moving bed process in which all the ports of the unit are not switched at the same time. It simulates the use of a fractional number of columns in each Section of the unit. Velocity: There are three different definitions of the mobile phase velocity, the chromatographic, the interfacial, and the superficial velocity. They are all defined as
the ratio of the mobile phase flow rate to an estimate of the column volume and differ by the estimate of the column volume used. The chromatographic velocity uses the total volume in the column that is available to the mobile phase; the superficial velocity, the geometrical column tube volume; and the interstitial velocity, the external or extra-particle volume. See Chapter 2, Section 2.3.3. Viscosity, rj: Numerical coefficient characterizing the distribution of shear in a flowing liquid, or the velocity of a liquid in a pressure gradient inside a given channel. Void volume, VQ: Hold-up volume of the column. Volume of mobile phase contained in the column. It includes the interstitial and the intraparticle volumes. Watershed Point: Concentration of the displacer at which the displacement of a component becomes impossible because the velocity of the displacer front is lower than the velocity of a zero concentration of that component. Zone: Somewhat synonymous of band, with an implication of a complex profile. However, the bands in displacement chromatography are called zones, even in the isotachic train.
Index Martire isotherm, 107 accuracy and precision of isotherm data measurement, 138 Moreau isotherm, 103 adsorption quadratic isotherm, 100 fundamentals, 70 S-shaped isotherm, 100 thermodynamics, 71 Toth isotherm, 93 adsorption energy distribution, 109 Unilan isotherm, 96 adsorption isotherms Van der Waals isotherm, 75 competitive isotherms, 151-219 Virial isotherm, 75 bi-Langmuir isotherm, 160 Volmer isotherm, 75 adsorption of ions, 106 determination, 191-214 adsorption of organic modifier Fowler isotherm, 180 effect on band profile, 709 Freundlich-Langmuir isotherm, 180 affinity energy distribution, 109 Langmuir isotherm, 154,165 API project, 6 LeVan-Vermeulen isotherm, 169 Martire isotherm, 184 apparent particle density, 60 statistical isotherms, 179 axial dispersion steric mass action model, 189 Gunn equation, 245 in porous media, 240 stoichiometric displacement model, 186 backward-forward differences, 499 Toth isotherm, 184 band asymmetry, 335 determination bandwidth data processing, 135 influence of sample size, 482 elution by characteristic point, 126 bi-Langmuir isotherm, 89,160 frontal analysis, 123,191,196 frontal analysis by characteristic point, bi-Moreau isotherm, 109 bonded alkyl chains density, 60 125 boundary conditions, 29, 38 h-mot method, 201 Danckwerts, 33 hodograph method, 210 displacement chromatography, 33 inverse method, 131, 212 frontal analysis, 33 nonlinear frequency response, 133 gradient elution, 33 pulse methods, 127, 202 breakthrough curve retention time method, 130 under constant pattern condition, 653 simple wave method, 210 analytical solution, 654 static method, 134 effect of axial dispersion, 657 single-component isotherms, 68-149 numerical solution, 657 bi-Langmuir isotherm, 89 shock layer theory, 658 bi-Moreau isotherm, 109 determination, 122-135 characteristic function, 329 excess isotherm, 78 characteristics extended liquid-solid BET, 104 theory of, 450 Fowler isotherm, 98 chromatographic mobile phase velocity, 61 Freundlich isotherm, 96 closed-closed boundary condition, 294 Jovanovic isotherm, 88 coherence theory, 196, 461 Langmuir isotherm, 74, 81 applied to frontal analysis, 198 Langmuir-Freundlich isotherm, 98 comparison of batch and SMB chromatogralinear isotherm, 73 phy, 934 liquid-solid equilibria, 80 969
970 competition for adsorption, 153 competitive surface diffusion, 256 concentration process definition, 14 concentration pulse method, 127, 204 coupling of molecular and eddy diffusion, 315 Craig plate model, 286 cut points definition, 854 cycle time definition, 852
ESIDEX influence of separation factor, 578 isotachic train, 439, 570 critical column length, 461 operating line, 442 plateau concentration, 457 steady state, 439 trace components, 580 watershed point, 445 displacement effect ideal model of chromatography, 416
economics of chromatographic separations, 857 Danckwerts boundary conditions, 33 eluotropic strength, 700 density of packing material, 60 elution by characteristic point, 126 diffusion, 222 EMG, 336 in intraparticle pores, 250 enantiomers in micropores, 238 competitive isotherms of, 163 in porous media, 236 enrichment two-mode pore diffusion, 252 definition, 14 Maxwell-Stefan approach, 232 equilibrium constants binary mixtures, 233 definition, 61 surface, 238, 254 equilibrium isotherms diffusion coefficients, 224 influence of mobile phase composition, correlations for macromolecules, 227 121 correlations for proteins, 227 influence of pressure, 117 Hayduk-Laudie equation, 226 influence of temperature, 119 influence of concentration, 229 equilibrium-dispersive model, 47 influence of pressure, 230 approximate analytical solutions, 476 influence of temperature, 231 Haarhoff-Van der Linde solution, 478 King equation, 226 Houghton solution, 477 Lusis-Ratcliff equation, 226 range of validity, 480 measurement, 232 displacement chromatography, 570 Reddy-Doraiswamy equation, 226 fundamentals of the model, 473 Scheibel equation, 226 in linear chromatography, 290 Wilke-Chang equation, 225 multicomponent systems, 531-567 diffusivity, 224 applications, 542 dispersion numerical solution, 492,532 in short coated tubes, 325 comparison of calculation methods, displacement chromatography 513 applications, 587 comparison with experimental results, bandwidth, 457 518-527 critical displacer concentration, 457 direct calculation, 496 equilibrium-dispersive model, 570 estimation of numerical errors, 495 impurities in displacer solution, 583 finite difference methods, 494, 533 influence of column length, 575 finite element method, 505, 539 influence of displacer concentration, 444, replacement of axial dispersion by nu572 merical dispersion, 497, 500 influence of HETP, 571 range of validity, 668 influence of sample size, 572 relaxation model, 490
971
INDEX single components/ 471-529 system peaks, 605-649 excess isotherm, 78 expectation-maximization, 112 exponentially modified Gaussian, 336 extended liquid-solid BET isotherm, 104 external film mass transfer resistance, 249 Kataoka equation, 249 penetration equation, 250 Wilson-Geankoplis equation, 249 extra-column tailing, 336 extraction definition, 14 Fick's first law of diffusion, 223 Fick's second law of diffusion, 223 finite difference methods, 494 finite element method, 505, 539 flow instability, 269 forward-backward differences, 499 forward-backwardn+i differences, 500 Fowler isotherm, 98,180 Freundlich isotherm, 96 Freundlich-Langmuir isotherm, 180 frontal analysis competitive, 191 reverse ^-transform, 196 single-component, 123 frontal analysis by characteristic point, 125
VERSE model, 724 volume-overload analytical solution, 703 Gunn equation, 245 fe-root method, 201 h-transform, 196 Haarhoff-Van der Linde solution of the equilibrium-dispersive model, 478 Hayduk-Laudie equation, 226 heterogeneous kinetics, 332,339 hodograph method, 210 hodograph transform, 544 hold-up time, 136 homogenous surface diffusion model, 763 Houghton solution of the equilibrium-dispersive model, 477
ideal adsorbed solution, 166 model for dilute liquid solution, 175 multicomponent adsorption of gas mixtures, 167 ideal chromatography definition, 13 ideal model, 46 ideal model of chromatography displacement chromatography, 437-469 for multicomponent systems, 387—437 dimensionless plot, 414 displacement effect, 416 elution of a narrow band, 401 general rate model, 51,301 elution of a wide band, 395 gradient elution chromatography, 721 gas chromatography, 421 homogenous surface diffusion model, intermediate plateau, 400 763 method of calculation, 407 Maxwell-Stefan surface diffusion, 764 position of concentration shocks, 396 numerical solution, 753 rear diffuse profiles, 398 VERSE model, 769 tag-along effect, 419 with pore diffusion, 754 indirect detection Gibbs isotherm, 71 Golay equation, 324 using system peaks, 618 gradient elution chromatography initial conditions, 29, 38 internal porosity analytical chromatography, 701 general rate model, 721 definition, 40 ion-exchange chromatography, 726 interstitial mobile phase velocity, 61 isotherm representation, 708 interstitial porosity numerical solution, 711 definition, 58 of peptides, 719, 721 intraparticle pore diffusion, 250 solid film linear driving force model, inverse method of isotherm determination competitive, 212 719 single-component, 131 under nonlinear conditions, 699-733
972 ion-exchange chromatography, 726 ions, adsorption of, 106 isotachic train, 439,570 critical column length, 461 selectivity reversal, 446,585 isotherm measurement competitive isotherms, 191—214 single-component isotherms, 122-149
INDEX transport-dispersive model, 296 lumped pore diffusion model, 54 numerical solution, 758 Lusis-Ratcliff equation, 226
Manhattan project, 5 Martin and Synge plate model, 284 Martire isotherm, 107,184 mass balance Jovanovic isotherm, 88 in the bulk mobile phase, 40 in the pores, 42 Kataoka equation, 249 near-isothermal system, 35 kinetic model nonisothermal system, 35 analytical and numerical solutions, 669 radially heterogeneous column, 36 comparison of kinetic models, 680 mass balance equation Langmuir, 49 derivation, 21 linear, 50 solution in linear chromatography, 290 linear driving force model, 747 two space variables, 36 lumped, 49 mass transfer resistance multicomponent problems, 735-777 external film, 249 numerical solutions Kataoka equation, 249 breakthrough curve, 671 penetration equation, 250 pulse injection, 674 Wilson-Geankoplis equation, 249 reaction-kinetic model, 670 in porous media, 240 analytical solution, 671 influence of bed heterogeneity, 246 single components, 651-697 kinetics in porous media, 247 King equation, 226 Maxwell-Stefan diffusion, 232 Knox-Pyper approach for optimization, 869 binary mixtures, 233 Maxwell-Stefan surface diffusion, 764 Langmuir isotherm, 74,81,154,165 McCabe-Thiele analysis, 808 in gas-solid equilibrium, 74 measurement of isotherms in liquid-solid equilibrium, 81 competitive isotherms, 191-214 Langmuir kinetic model, 49 single-component isotherms, 122-149 Langmuir-Freundlich isotherm, 98 mobile phase velocity Lax-Wendroff scheme, 497 chromatographic, 61 Lax-Wendroff two-step scheme, 498 definition, 40 LeVan-Vermeulen isotherm, 169 interstitial, 61 linear chromatography, 281-345 linear, 61 definition, 13 superficial, 61 extension to nonlinear, 341 Modicon process, 833 statistical model, 328 moment analysis, 310 stochastic model, 328 moments, 310 linear driving force model, 747 Monte Carlo model linear isotherm, 73 of nonlinear chromatography, 693 linear kinetic model, 50 Moreau isotherm, 103 linear mobile phase velocity, 61 multiobjective optimization, 925 linear solvent strength (LSS) theory, 701 loading factor nonideal chromatography definition, 851 definition, 14 lumped kinetic model, 49,295 nonlinear chromatography reaction-dispersive model, 296
INDEX
973
in nonlinear chromatography, 482 in perfusion chromatography, 320 objective function, 861 Van Deemter equation, 299 open-closed boundary condition, 293 plate model, 283 open-open boundary condition, 291 Craig model, 286 operating line Martin and Synge model, 284 in displacement chromatography 442 porosity optimization, 850 definition, 58 comparison of batch and SMB chromatoginternal, 59 raphy 934 interstitial, 59 comparison of elution and displacement, total, 58 907 Powerfeed process, 832 comparison of isocratic, gradient elu- production cost tion and displacement, 912 components, 859 for overlapping bands feed cost, 860 no yield constraint, 878 fixed cost, 859 yield constraint, 882 operating cost, 860 ideal model, 867 production rate touching bands, 871 definition, 853 Knox-Pyper approach, 869 purification maximum production rate, 884 definition, 14 displacement chromatography, 903 purity gradient elution, 898 definition, 857 maximum production rate and recovery yield, 893 quadratic isotherm, 100 minimum solvent consumption, 895 reaction-dispersive model, 296 objective function, 861, 867 production rate and recovery yield, reaction-kinetic model Thomas model, 670 867 real adsorbed solution, 177 practical rules, 920 recovery yield recycling procedures, 915 definition, 856 simulated moving bed process, 809,924 recycling procedures, 915 linear isotherm, 809 Reddy-Doraiswamy equation, 226 nonlinear isotherm, 817 retention of organic modifier, 705 safety margin approach, 809 triangle method, 812, 814, 817 S-shaped isotherm, 100 theoretical considerations, 867 sample size using numerical solutions, 883 definition, 851 using the Knox equation, 878 optimization of the experimental conditions, Scheibel equation, 226 Shirazi number, 510 850-937 shock layer case of Langmuir isotherm, 738 partition shock layer theory, 658 in short coated tubes, 325 case of Langmuir isotherm, 661 peak asymmetry, 335 for binary mixture, 737 peak tailing, 335 shock layer thickness penetration equation, 250 binary frontal analysis, 740 plate height equation, 310 controlled by axial dispersion, 738 general rate model, 313 controlled by mass transfer resistance, Golay equation, 324 739 definition, 13
974
INDEX
dependence on displacer parameters, 745 surface heterogeneity, 338 dependence on mobile phase velocity, system of mass balance equations 745 solution, 42 dependence on separation factor, 746 system peaks, 605-649 in displacement chromatography, 742 high concentration, 626 in frontal analysis, 662 peak profiles, 627 influence of concentration step, 665 theory, 609 influence of mobile phase velocity, 663 tag-along effect staircase frontal analysis, 666 ideal model of chromatography, 419 simple wave method, 210 theory of characteristics, 450 simulated moving bed process, 780-848 in displacement chromatography, 453 analytical solution of linear, ideal model, wave interactions, 454 785 Thomas model, 670 equivalent mixing cell model, 845 influence of the heterogeneity of column throughput definition, 851 set, 822 total porosity linear, ideal model, 785 definition, 58 linear, nonideal model, 806 T6th isotherm, 93,184 McCabe-Thiele analysis, 808 tracer pulse method, 128, 204 modeling of, 783 transport parameters Modicon process, 833 evaluation from peaks, 326 multicomponent separations, 833 transport-dispersive model, 296 nonlinear, ideal model, 816 triangle method, 812 nonlinear, nonideal model, 836 linear isotherm, 814 Powerfeed process, 832 nonlinear isotherm, 817 solvent gradient, 827 true moving bed chromatography, 780 standing wave design, 810, 812 equivalent mixing cell model, 844 temperature gradient, 827 numerical solution Varicol process, 830 equivalent to an SMB, 837 skeleton density, 60 Tswett, invention of chromatography, 3 solid density, 60 specific production Unilan isotherm, 96 definition, 857 staircase frontal analysis, 666 vacancy chromatography, 623 standing wave design Van Deemter equation linear isotherm, 810 origin of, 299 nonlinear isotherm, 812 Van der Waals isotherm, 75 statistical isotherms, 179 Varicol process, 830 statistical model, 328 versatile reaction-separation model, 769 statistical thermodynamics of adsorption, 76 VERSE model, 724, 769-775 steric mass action model, 189, 729 Virial isotherm, 75 stochastic model viscosity, 257,259 of linear chromatography, 328 of liquids, 257 of molecular migration, 333 viscosity of mobile phase, 259 of nonlinear chromatography, 693 calculation of inlet pressure, 266 stoichiometric displacement model, 186 importance of, 264 supercritical fluid simulated moving bed, 826 influence of composition, 262 superficial velocity, 61 influence of pressure, 261 surface diffusion, 238,254 influence of temperature, 259 competitive, 256 viscous fingering, 269
INDEX Volmer isotherm, 75 watershed point, 445 wave interactions, 454 Wilke-Chang equation, 225 Wilson-Geankoplis equation, 249
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