Termodynamics and Kinetics of Complex Systems

ADVANCES IN

CHEMICAL ENGINEERING Editor-in-Chief GUY B. MARIN Department of Chemical Engineering

Ghent University

Ghent, Belgium

Editorial Board DAVID H. WEST Research and Development

The Dow Chemical Company

Freeport, Texas, U.S.A.

JINGHAI LI Institute of Process Engineering

Chinese Academy of Sciences

Beijing, P.R. China

SHANKAR NARASIMHAN Department of Chemical Engineering

Indian Institute of Technology

Chennai, India

Advances in CHEMICAL ENGINEERING THERMODYNAMICS AND KINETICS OF COMPLEX SYSTEMS

VOLUME

39 Edited by DAVID H. WEST GREGORY YABLONSKY

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CONTRIBUTORS

Walter G. Chapman, Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005, USA Valeriy V. Ginzburg, The Dow Chemical Company, Midland, MI 48674, USA Miroslav Grmela, Ecole Polytechnique de Montreal, Montreal, QC H3C 3A7, Canada Shekhar Jain, Shell Technology India Pvt. Ltd., Bangalore 560048, India Prasanna K. Jog, The Dow Chemical Company, Midland, MI 48674, USA Boris M. Kaganovich, Energy Systems Institute, Russian Academy of Sciences, Siberian Branch, 130 Lermontov Street, Irkutsk 664033, Russia Alexander V. Keiko, Energy Systems Institute, Russian Academy of Sciences,

Siberian Branch, 130 Lermontov Street, Irkutsk 664033, Russia

Semion Kuchanov, Physics Department, Lomonosov Moscow State University,

Moscow 119991, Russia; A.N. Nesmeyanov Institute of Organoelement

Compounds, Russian Academy of Sciences, Moscow 119991, Russia

Vitaliy A. Shamansky, Energy Systems Institute, Russian Academy of

Sciences, Siberian Branch, 130 Lermontov Street, Irkutsk 664033, Russia

Rakesh Srivastava, The Dow Chemical Company, Midland, MI 48674, USA Jeffrey D. Weinhold, The Dow Chemical Company, Freeport, TX 77515, USA

vii

PREFACE

Thermodynamics and kinetics can surely be counted—along with transport phenomena, chemistry, unit operations, and advanced mathematics—as subjects that form the foundation of Chemical Engineering education and practice. Thermodynamics is of course a very old subject. For example, it was the same Rudolf Clausius, who in 1865 coined two immortal sen­ tences (1) “The energy of the universe is constant” and (2) “The entropy of the universe tends to a maximum,” that developed the famous Clausius– Clapeyron equation, one of the most basic physico-chemical relationships. Classical thermodynamics was largely complete in the 19th century, before even the basic structure of the atom was understood. The early 20th century saw the development of a statistical description of thermodynamics which successfully explained the macroscopically observable thermodynamic relations in terms of the statistical behavior of ensembles of atoms or molecules. By the late 1920s the statistical treat­ ment of molecular thermodynamics was starting to connect chemical thermodynamics to chemical kinetics through the development of transi­ tion state theory. Consequently, in 1930–1950 the pioneers of chemical engineering science had most of what they needed to begin putting che­ mical engineering on a firm mathematical foundation. That was indeed the focus of theoretical research in chemical engineering for the next 30 years or more. Since about 1970, chemical engineering has diversified and bifurcated into practically every adjacent area of science including polymer chemis­ try, materials science, molecular biology, microelectronics, nanotechnol­ ogy, and biomedical engineering. What has enabled chemical engineers to infiltrate these adjacent fields so successfully? We believe part of the answer is that chemical engineers are well equipped by their education to analyze systems in terms of the governing equations of thermody­ namics and kinetics. Whether the system is composed of fluids, molecules, unit operations, cells, repeat units, micro-electronic devices, or organisms, we wish to know the possible states of the system, their stability, the relationships between them, and the rate of change from one state to another. Whether at the basic scientific level or the practical technological level, such questions are essentially thermodynamic and kinetic questions.

ix

x

Preface

From the very beginning Gibbs formulation of thermodynamics was highly mathematical, in particular geometrical. In accordance with this tradition, several scientific schools have applied modern geometrical methods to continue progress in chemical thermodynamics and chemical kinetics. Three schools have to mentioned: the Belgian school of nonequilibrium thermodynamics (de Donder, De Groot, Mazur, Defay, and the Nobel laureate Prigogine), the American Minnesota School (Amund­ son, Aris, Horn, Feinberg), and the Russian Siberian School (Gorban et al.). In this book we present two new examples of this chemo-geometrical activity. First, Miroslav Grmela from the Montreal Ecole Polytechnique (Mon­ treal, Canada), in the paper “Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering,” develops a unifying thermo­ dynamic framework for multiscale investigations of complex macroscopic systems. In this new theoretical paradigm, the key conceptual role is played by the Legendre transformation, and the time evolution representing the approach to a more macroscopic level of description is introduced as a continuous sequence of Legendre transformations. This method is devel­ oped for chemically reacting multilevel systems. It can be considered as one of the cornerstones of the emerging multiscale engineering, which seeks to combine nano-, micro-, and macroscales. In the paper “Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems” by Boris Kaganovich, Alexander Keiko, and Vitaliy Shamansky (Melentiev Energy Institute, Irkutsk, Russia), the authors pre­ sent updated information regarding the model of extreme intermediate states. The authors build up this approach which started in the mid-1980s based on the chemo-geometrical ideas of Russian and American collea­ gues (Russian Siberian Team—Gorban, Yablonsky, and Bykov and US schools—Horn, Feinberg and Shinnar, respectively). However, during more than two decades they developed this model significantly, trans­ forming it into the original and powerful tool for analysis of many pro­ cesses of chemical engineering (flows in hydraulic circuits, coal combustion, isomerization, etc.). The phase equilibrium of materials is an inherently multiscale phenom­ enon which spans from the functional group (or atomic) scale through the morphological-structure scale to the macroscopic scale. Two texts pre­ sented in this volume are devoted to this problem. The contribution “Application of Meso-Scale Field-based Models to Predict Stability of Particle Dispersions in Polymer Melts” by Prasanna Jog, Valeriy Ginzburg, Rakesh Srivastava, Jeffrey Weinhold, Shekhar Jain, and Walter Chapman examines and compares Self Consistent Field The­ ory and interfacial Statistical Associating Fluid Theory for use in predict­ ing the thermodynamic phase behavior of dispersions in polymer melts. Such dispersions are of quite some technological importance in the

Preface

xi

plastics, materials, and electronics industries. This contribution highlights a recent advance by some of the authors: the inclusion of compressibility in the thermodynamics of nanoparticle dispersion. This is an important advance in mesoscale simulation which correctly expands the region of intercalated microstructures. The chapter demonstrates that mesoscale field-based simulation is finding practical applications for industrial nano-material development. In the final chapter “Principles of Statistical Chemistry as Applied to Kinetic Modeling of Polymer Obtaining Processes” by Semion Kuchanov (Lomonosov Moscow State University, Moscow, Russia), the contempor­ ary problems of bridging models of micro- and macrostructure are dis­ cussed. The hierarchical analysis of chemical correlation functions (so-called chemical correlators) is a subject of the author’s special interest. These problems are presented conceptually stressing that the problem of crucial importance is revealing the relation between the process mode and the chemical structure of polymer products obtained. We conclude with a famous quotation attributed to the great theore­ tical physicist, Arnold Sommerfeld Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more. This quotation is referred to often, maybe too often. Nevertheless, its lasting truth suggests that it makes sense to check from time to time the progress of chemical thermodynamics and kinetics, despite how old are the subjects, in particular as it concerns areas adjacent to chemical engineering. David West The Dow Chemical Company Gregory Yablonsky Parks College, Department of Chemistry Saint Louis University May 27, 2010

CHAPTER

1 Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems B.M. Kaganovich, A.V. Keiko
, and V.A. Shamansky

Contents

1. Introduction 2. Substantiation of the Equilibrium Thermodynamics

Capabilities for Describing Irreversible Processes 2.1 The experience of classics 2.2 New interpretations of equilibrium and reversibility 2.3 Equilibrium interpretations of the basics of

nonequilibrium thermodynamics 2.4 Equilibrium approximations 3. Models of Extreme Intermediate States 3.1 MEIS with variable parameters 3.2 MEIS with variable flows 3.3 MEIS of spatially inhomogeneous systems 3.4 Variants of kinetic constraints formalization 3.5 Geometrical interpretations 4. Comparison of MEIS with the Models of Nonequilibrium

Thermodynamics 4.1 Introductory notes 4.2 On the areas of effective applications of equilibrium

and nonequilibrium thermodynamics 4.3 Comparison of computational efficiency of

equilibrium and nonequilibrium approaches 5. Examples of MEIS Application 5.1 Introductory notes

2

5

5

8

12

16

17

17

20

26

29

33

38

38

39

46

50

50

Melentiev Energy Systems Institute SB RAS, Irkutsk, Russia Translated from Russian by M.V. Ozerova and V.P. Ermakova


Corresponding author. E-mail address: [email protected]

Advances in Chemical Engineering, Volume 39 ISSN: 0065-2377, DOI 10.1016/S0065-2377(10)39001-6


2010 Elsevier Inc. All rights reserved.

1

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B.M. Kaganovich et al.

5.2 Isomerization 5.3 Formation of nitrogen oxides during coal combustion 5.4 Stationary flow distribution in hydraulic circuits 6. Conclusion: What we have and what will be? 7. Problems of Equilibrium Thermodynamic Descriptions of Macroscopic Nonconservative Systems 7.1 Substantiation of the capabilities of equilibrium descriptions and reduction of the models of irreversible motion to the models of rest 7.2 Analysis and development of MEISs 7.3 Analysis of computational problems in MEIS application and MEIS-based devising of methods, algorithms, and computing system 7.4 Solution of specific theoretical and applied problems on MEIS Acknowledgments References

Abstract

50 54 64 66 69

69 70

70 71 72 72

The possibility is substantiated to model open and closed systems as well as reversible and irreversible processes on the basis of classic equilibrium thermodynamics statements. The consideration is given to new modifications of the model of extreme intermediate states (MEIS) built originally in the mid-1980s. They include constraints on irreversible macroscopic kinetics presented in a thermodynamic form, i.e., without the time variable. MEIS is compared with models of chemical kinetics and irreversible thermodynamics from two viewpoints: (1) the range and versatility of application areas, and (2) the simplicity and self-descriptiveness of computational experi­ ments. The potential of equilibrium modeling is explained on the examples of analysis of chemical systems and hydraulic circuits.

1. INTRODUCTION The subject of the studies to be discussed is modeling of macroscopic dissipative systems on the basis of the classical equilibrium thermody­ namics principles. The modeling tool is the model of extreme intermediate states (MEIS) suggested in the mid-1980s at the Melentiev Energy Systems Institute of Siberian Branch of Russian Academy of Sciences (Antsiferov et al., 1987; Gorban et al., 2001, 2006; Kaganovich et al., 1989, 1993, 1995) and which, unlike the traditional thermodynamic models intended for search of the final equilibrium point, allows one to study the entire attainability region from a given initial state of the studied system and find a point of partial or complete equilibrium that corresponds to the extreme value of a property the researcher is interested in (for example, concentration of target or harmful products of the processes that may

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

3

occur in the system). Various modifications of MEIS have been created over the past years. They include the constraints on irreversible macro­ scopic kinetics that are written in thermodynamic form (without time variable) (Gorban et al., 2006; Kaganovich, 2002; Kaganovich et al., 2004, 2005a, 2005b, 2006a, 2006b, 2006c). The equilibrium modeling of reversible and irreversible processes has been a traditional approach for studies in the natural science (and since the mid-19th century in the social and economic sciences as well). The theore­ tical foundations of this approach were laid by Galileo whose principles of equilibrium, relativity, and inertia showed that the motion could be repre­ sented as a sequence of states of rest (equilibrium). The formalized analy­ sis of equilibrium models was mostly carried out in the 18th century, first of all in the works by Euler and Lagrange. Lagrange using his equilibrium equation gave a single mathematical description for the entire Newton mechanics (Lagrange, 1997). Revealing the interrelations between the models of motion and rest resulted in creation of mathematical disciplines and methods related to the solution of variational and extreme problems: the method of multipliers, the theory of optimal equilibrium trajectories—calculus of variations and, later, the modern mathematical theory of extreme equilibrium states—mathematical programming (MP). In the 19th century the variational principles of mechanics were extended to the analysis of nonconservative, nonholonomic, and nonscler­ onomous systems. However, the greatest progress in equilibrium model­ ing in the century before last is certainly connected with the science about equilibrium—the thermodynamics—created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Owing to thermodynamics the extreme principle—the principle of entropy increase (the second law of thermo­ dynamics) came into physics. It is more general compared with the prin­ ciples of virtual work and of least action that were formulated in mechanics. Boltzmann explained this law in two ways (Boltzmann, 1878; Polak, 1987): (1) from the motion trajectory analysis (kinetic) formulated as the H-theorem and (2) from the immediate consideration of possible states of a system and determination of the most probable among them. This explanation facilitated further analysis of interrelations between the mod­ els of motion and rest as interrelations between kinetics and thermody­ namics. The assumptions on the Markov random behavior of processes of motion toward entropy maximum and on existence of the thermodynamic Lyapunov functions (without using the corresponding terms, of course), which had been made by Boltzmann in his research even before Markov and Lyapunov, became the foundation for the development of equilibrium modeling of diverse processes including irreversible natural ones. However, with successful penetration of equilibrium models into physics, chemistry, biology, and social sciences in the 20th century, largely because of the need to study various nonlinear effects (self-oscillations,

4

B.M. Kaganovich et al.

self-organization), special sciences started to develop: the theory of dynamic systems (Arnold, 1989; Katok and Hasselblatt, 1997), nonequili­ brium thermodynamics (Glansdorff et al., 1971; Kondepudi et al., 2000; Prigogine, 1967), synergetics (Haken, 1983, 1988) and others which are intended either completely or partly for the analysis of nonequilibrium irreversible processes. The “seizure” of a considerable part of the applica­ tion area of equilibrium thermodynamics by other sciences was fostered by two conditions: First, is it due to the contradiction in the Boltzmann explanation of the second law which lies between the following supposi­ tions: on the one hand, reversibility of the individual interactions among micro particles and, on the other hand, irreversibility of the final result of all these interactions in the aggregate (the Boltzmann paradox). Second, it is because of a wide discussion of the mentioned contradiction which unfolded at the turn of the 20th century. Now the opinion that “Classical thermodynamics gives a complete quantitative description of equilibrium (reversible) processes,… for nonequilibrium processes it establishes only the inequalities which indicate the direction of these processes (for exam­ ple, the Clausius inequality)” (Zubarev, 1998) has become widespread. The MEIS developers relying on the capabilities of modern computers and computational mathematics started the work whichresulted in an essential expansion of the application area of “good, old” classical thermo­ dynamics and in the possibility to study (using thermodynamics) any states on all possible motion trajectories of a nonequilibrium system. In other words, they put forward the goal to use the models of equilibrium not only to determine the directions of irreversible processes but to esti­ mate the attainability of desired and undesired states on these directions. The works on equilibrium modeling of dissipative systems include four natural components: 1. 2.

3.

4.

substantiation of the possibility to describe irreversible processes in terms of equilibrium; creation of quite a representative set of models (modifications of a general equilibrium model) to enable the analysis of a wide range of problems interesting in terms of theory and application; comparison of advantages and disadvantages of equilibrium and nonequilibrium models and differentiation of the areas of their effective applications; solving as many as possible specific problems and analysis of the modeling experience gained.

The above four topics of studies are subsequently discussed below. In creation of MEIS its authors have used as a base the ideas of their collea­ gues in studies of equilibria—those by Bykov, Gorban, and Yablonsky (Gorban, 1979, 1984, 1986; Yablonsky et al., 1991), as well as works by Horn (1964; Horn and Jackson, 1972), Feinberg (1972, 1999; Feinberg and

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

5

Hildebrant, 1997; Feinberg and Horn, 1974), Shinnar (1988; Shinnar and Feng, 1985) and other scientists who dealt with thermodynamic analysis of macroscopic systems kinetics and attainable states and areas research. In development of the latest MEIS modifications intended for modeling of irreversible processes we have utilized the equilibrium trajectory interpre­ tations suggested by Gorban (2007; Gorban and Karlin, 2005; Gorban et al., 2001), which in turn are based on elaboration of Erenfests’ idea on coarse-graining phase spaces (Ehrenfest, 1959). Besides, the study being presented has been strongly influenced by the works by Gorban on “Model Engineering”—a new scientific discipline formulated in Gorban and Karlin (2005) and further unfolded in Gorban (2007; Gorban et al., 2007). Whereas mathematicians often use formalized statements of pro­ blems that were suggested in other sciences and, then, based on the study of mathematical features develop the methods of their solution, the “Model Engineering” supposes the choice of the initial models which are most suitable in terms of both analysis and computation. The research area presented below fits naturally into this new discipline and represents a limit case of reduction, i.e., the transformation of models of motion into models of rest (equilibrium).

2. SUBSTANTIATION OF THE EQUILIBRIUM THERMODYNAMICS CAPABILITIES FOR DESCRIBING IRREVERSIBLE PROCESSES 2.1 The experience of classics Great experience in equilibrium modeling of irreversible processes was gained even in the classical mechanics. Lagrange, analyzing the specific features of the equilibrium search problems (Lagrange, 1997), stated that if the left-hand side of his equation of the mechanical system equilibrium represents a total differential of some function, then the solution to the problem of determining the equilibrium corresponds to the solution to the problem of finding the extremum of this function. Thus, he assumed that the cases were possible where the equilibria are attained between nondifferentiable variables. The assumption on the equilibrium of mechanical systems that was made by him in a formalized description of the Newton mechanics appeared to be more general than the assumption on the conservatism of these systems that was in fact used by Newton and Leibniz for creation of the differential calculus. While the conservatism is a sufficient but not necessary condition of the possibility to describe the system behavior with the help of differential equations, the equilibrium is a necessary condition for admissibility of such a description. Indeed, the nonequilibrium systems cannot be described by intensive macroscopic

6

B.M. Kaganovich et al.

parameters which are the functions of states and the notion of the function differential for such systems loses its sense. In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduc­ tion, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of “equili­ brium” and “reversibility,” “equilibrium processes” and “reversible pro­ cesses” started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltz­ mann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the “time arrow.” The principle of entropy increase was explained by Boltzmann in two ways: (1) by analyzing the feasible paths (H-theorem) and (2) by consider­ ing the possible attainable states of thermodynamic system and searching for the most probable ones among them. In both explanations he made assumptions on the independence of the considered states from the attain­ ment prehistory and the possibility of their full description on the basis of functions, determined exclusively by the probability of fulfillment, and changing monotonously in the process of transitions from state to state. Using modern terms we can say that Boltzmann presupposed the Markov behavior of processes taking place and the existence of the Lyapunov functions. In turn the possibility to represent the states by a set of quantity related only to the probability of attainment implies that such a state can be interpreted as partial or complete equilibria. Indeed, fixing some quan­ tity (function) can be easily explained by equilibrium of the forces tending to change it. This makes clear both the equilibrium of the Boltzmann trajectories of attaining the entropy maximum and the “equilibrium” of the Boltzmann descriptions of irreversible processes. Boltzmann’s expla­ nations reveal to a great extent the interrelations between thermodynamics and kinetics and the possibilities of thermodynamic equilibrium analysis of kinetic equations, i.e., the equations of motion. Gibbs in his system explanation of macroscopic thermodynamics (Gibbs, 1876), which had been made before he and Boltzmann formulated the principles of statistical mechanics, relied on the Lagrange equilibrium description of mechanical systems. However, instead of the single Lagrange equilibrium equation, which, according to Krylov, included all Newton mechanics, Gibbs, in order to derive all thermodynamic relation­ ships, used four fundamental equations written for different combinations of independent parameters. Thus, not ordinary but partial differential equations became the mathematical apparatus of thermodynamics unlike mechanics. Owing to thermodynamics, a more general, compared with the

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

7

mechanical principles, the extreme principle, i.e., the second law of ther­ modynamics, appeared in the science. For trajectories it determined the entropy nondecrease instead of action minimization (according to the least action principle) and for states—instead of the virtual work principle the entropy maximum: global for complete equilibrium and conditional (under the condition of braking all the processes that can continue after the given state is attained)—for partial equilibria. Gibbs conducted the specific studies on the basis of mathematical relations obtained. These studies focused on the complex systems, sub­ jected to the action of various forces (chemical, electrical, gravitational, surface tension, and elastic), and the systems in which there can be sub­ stance transformations and phase transitions along with energy transfor­ mations. In these studies Gibbs used the method of potentials which supposed equilibrium trajectories of attaining the sought equilibrium states. Gibbs, without using time variable, distinguished the approximate equilibria which settle fast and the final equilibrium which is slowly attained. Not considering the computational problems proper he foresaw the relations between physical stability and uniqueness of the final equili­ brium point. In terms of the art of equilibrium modeling of irreversible processes the analysis of the process of hydrogen burning in oxygen (Gibbs, 1876) is particularly impressive. Gibbs, without information on thermodynamic properties of substances and without computers, mana­ ged to draw a complete qualitative picture of this process. Discussing potential solutions to the system of equations he explained the decrease in reaction temperature due to water dissociation and the presence of constrained explosibility and ignition regions. Certainly, he could not find the chain mechanism of the considered reaction but the probable results of the studied processes for different conditions under which these processes occur Gibbs showed absolutely correctly. Discussing the period when the thermodynamic equilibrium descrip­ tions of various irreversible phenomena started one cannot but point out the papers on the theory of electric circuits that are of explicitly thermo­ dynamic character. These had been written by Kirchhoff (1848, 1882) before the works by Clausius and Boltzmann, which made the second law the property of science. In his work (Kirchhoff, 1848)(yet in 1848!) Kirchhoff proved the theorem on minimum heat production in the open passive (without electromotive force sources) electric circuit for the case of isothermal motion of charges. It is easy to ascertain that this theorem represents a particular case of the nonequilibrium thermodynamics theo­ rem on minimum entropy production that was proved by Onsager and Prigogine approximately 100 years after Kirchhoff. In the 1870s Rayleigh suggested the principle of the least energy dissipation (Rayleigh, 1873). Following the founders of thermodynamics, Planck and Einstein pre­ sented vivid illustrations of the possibilities to analyze irreversible

8

B.M. Kaganovich et al.

processes in terms of equilibrium. Taking into consideration the condition of equilibrium between substance and radiation and the condition of equilibrium energy transfer, they derived the laws of light radiation, propagation, and absorption. As is known their works were the develop­ ment and brilliant completion of the radiation thermodynamics works by Kirchhoff and Boltzmann. In the papers dedicated to the Brownian motion Einstein, proceeding from the equations of equilibrium between the Brownian particle and carrying fluid, derived the law of Fick for a princi­ pally irreversible process, i.e., diffusion. In the theories of fluctuations and opalescence Einstein “broke into” the fields of applications of modern nonequilibrium thermodynamics. He introduced the notion of partial equilibria (in his terms “incompletely determined in the phenomenologi­ cal sense of the word”) in the analysis of opalescence phenomenon and used this notion to explain the irreversible process of light diffusion, considering, in fact, a set of attainable states. In the second half of the 20th century it is precisely the classical equilibrium thermodynamics that became a basis for the creation of numerous computing systems for analysis of irreversible processes in complex open technical and natural systems as applied to the solution of theoretical and applied problems in various fields. The methods of MP, i.e., the mathematical discipline that emerged from the Lagrange interpre­ tation of the equilibrium state, were a main computational tool employed for the studies.

2.2 New interpretations of equilibrium and reversibility In order to clearly explain the possibilities of describing nonequlibrium irreversible processes in terms of equilibrium it is certainly necessary to define quite accurately the notions of equilibrium and reversibility, nonequilibrium and irreversibility. It is clear that their interpretation, as well as the interpretation of other scientific notions, changes with the development of respective theories, models, and methods. Since the work touches upon the issues of interrelations between the competing models in a historical profile it is desirable that the appropriateness of various interpretations of the said notions be assessed in this profile. Making no pretence of the systematic presentation of the issue we will only touch upon some points that are important for understanding the text1 below. Mechanics emerged as a science studying reversible processes that are symmetrical relative to time. Euler, in his “thesis” on the least action 1

A rich material for the comprehension of the evolution of basic notions in the course of development of variational principles and principles of equilibrium and extremality in physics can be found in remarkable books by L.S. Polak (1987, 1960).

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

9

principle, wrote that this principle did not hold for the systems with friction. Lagrange took the equilibrium equation as a basis for the for­ malization of mechanics and simultaneously made equilibrium a most important property of mechanical systems as well. This property appeared to be essentially more general than conservatism which is still believed to be the main feature that determines the obeyance of studied system to the classical mechanics laws. Relying on the Lagrange idea of equilibrium the mechanics (Hamilton, Gauss, Ostrogradsky et al.) started to gradually expand the area of their science applications to nonconser­ vative systems, allowing, obviously, some analysis errors. Boltzmann related the possibilities of equilibrium descriptions with the possibilities of describing irreversible processes. The Boltzmann trajectories of motion to the entropy maximum that meet the conditions of Markov behavior of the studied processes and the existence of the Lyapunov functions, as well as the Euler–Lagrange–Hamilton trajectories, can be represented by a continuous series of equilibrium states. It is convenient to characterize these states (points on trajectories) with the help of local potentials and to describe the trajectories themselves by autonomous equations of the form x_ ¼ f ðxÞ:

However, unfortunately, the definitions of equilibrium processes that relate them with the notion of irreversibility did not arise from the Boltzmann description of irreversible processes by equilibrium trajec­ tories. This was possibly to some extent a result of the discussion related to Boltzmann’s paradox, mentioned in the “Introduction.” To the contrary, in the 20th century the idea about the identity of equilibrium and rever­ sible processes grew strong. In the workbooks on macroscopic thermo­ dynamics equilibrium processes were interpreted as infinitely slow, in the course of which, at each time instant, equilibrium has time to settle within the system and between the system and the environment. If to implement such a process after attaining the final state in the reverse direction to the initial state, nothing will change in the system and in the environment, i.e., the results of the direct process will appear to be reversible. Such inter­ pretation of equilibrium processes fits harmoniously into the theories and models associated with the efficiency analysis of various technical sys­ tems. Indeed, any deviation of the system parameters from the equili­ brium values leads to additional potential differences between the system and the environment and additional work or heat loss, i.e., to a decrease in the working (target) process efficiency. However, the presented interpretation of equilibrium processes turns out to be unsatisfactory for the analysis of possibility to use equilibrium descriptions for irreversible phenomena. The interpretation of interrelations between equilibrium and reversibility that was given by

10

B.M. Kaganovich et al.

Gorban et al. in their works (Gorban, 2007; Gorban et al., 2001, 2006) seems to be more comprehensive for our discussion. The works unfolded the idea of the Ehrenfests (1959) on the isolated system tending toward the Boltzmann equilibrium trajectory as a result of “agitations.” Figure 1, taken with some changes from Gorban et al. (2001), illustrates the processes that occur in the isolated systems where the number of particles is so large that the statistical regularities take place. Closed curves represent the levels of entropy S. Dotted straight lines show the sets of states with constant values of macroscopic parameters. The contact points of the curves with straight lines are the equilibrium points that meet the equilibrium distributions. At these points entropy has its maximum value on a corresponding tangent. The set of these points forms the equilibrium trajectory S
, along which the system moves toward the point of global maximum of entropy Smax. Curve arrows stand for isentropic (reversible) processes that occur as a result of reversible (elastic) interactions of parti­ cles. Straight arrows show the system “agitations” that are explained by deviations of part of the interactions from reversibility and that push the system toward an equilibrium trajectory. According to the given interpretation of the equilibrium processes they differ principally from the reversible ones and represent at the limit (with the intervals between agitations and, hence, the distances S
1  S
2 ; S
2  S
3 ; etc., tending to zero) a continuous sequence of local entropy maxima. The above statement on the identity of equilibrium and reversible processes is also consistent to some extent with the Gorban interpretation only in the assumption on the limiting coincidence of nonequilibrium states, located on the trajectories S = const, with equilibrium ones—on the Boltzmann trajectory. In this case the entire set of possible states in Figure 1 is reduced to the curve S
1  Smax .

S = const S*1 S*2

S*3

S*4 S max

Figure 1 Dynamics of a system with periodic agitation.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

11

It will apparently be possible to provide coordination between the capabilities of equilibrium models in (1) the analysis of perfection of the energy and substance transformation processes and (2) the analysis of different irreversible phenomena on the basis of dual interpretation of equilibrium processes as being both reversible and irreversible at a time. In the first case they are convenient for interpretation as reversible in terms of the system interaction with the environment and in the second case—as irreversible in terms of their inner content according to Gorban. It is clear that to explain the dual interpretation it is necessary to extend the analysis by Gorban to the nonisolated thermodynamic systems with other charac­ teristic functions to be used along with entropy. Concurrently with the joint analysis of the notions “equilibrium” and “irreversibility” we must determine in our discussion the meaning of and the extent to which the notion “nonequilibrium” and word combination “far from equilibrium” affect the requirements for the models employed. There are three meanings assigned to the indicated word combination in different contexts in Gorban (1984). Firstly, it is assigned to the systems in which distribution of some microscopic variables (for example, energy of particle translational motion) differs from the equilibrium distribution so much that the evolution of macroscopic variables cannot be described by the first-order differential equations (autonomous, if the environment is stationary). Secondly, the closed system with equilibrium environment (or isolated one) is supposed to be far from the equilibrium if its relaxation from the given state toward a small neighborhood of the equilibrium continues for a long time, during which various nonlinear effects can be observed (self-oscillations, spatial ordering, etc.). The third use of the word combination “far from equilibrium” refers to the open systems that exchange mass and energy with the environment which is not in the state of thermodynamic equilibrium. Normally the apparatus of equilibrium thermodynamics can be used for the remoteness in the second and third sense and a corresponding choice of space of variables, though in each specific case this calls for additional check. Because for the spaces that do not contain the functions of state (in the descriptions of nonequilibrium systems these are the spaces of work–time or heat–time) the notion of differential loses its sense, and transition to the spaces with differentiable variables requires that the holonomy of the corre­ sponding Pfaffian forms be proved. The principal difficulties in application of the equilibrium models arise in the case of remoteness from equilibrium in the first sense when the need appears to introduce additional variables and increase dimensionality of the problem solved. In some cases where it is impossible to strictly substantiate the feasibility of equilibrium descriptions we have to be content with equili­ brium approximations. Such approximations are considered below in Section 2.4.

12

B.M. Kaganovich et al.

2.3 Equilibrium interpretations of the basics of nonequilibrium thermodynamics We can specially show that the main principles of nonequilibrium thermo­ dynamics (the Onsager relations, the Prigogine theorem, symmetry princi­ ple) and other theories of motion (for example, theory of dynamic systems, synergetics, thermodynamic analysis of chemical kinetics) are observed in the MEIS-based equilibrium modeling. In order to do that, we will derive these statements from the principles of equilibrium thermodynamics. First of all relying directly on the second law we will try to give the interpretation of the Prigogine theorem. Taking into account that the tradi­ tional variables of equilibrium thermodynamics are the parameters of state and, wishing to reveal the formalized relations between both thermody­ namics, let us consider two situations sequentially: (1) when some para­ meters of interaction that hinder the attainment of final equilibrium between the open subsystem and other parts of the isolated system that contains this subsystem are set; (2) when flows are taken constant for the flow exchange between the open subsystem and the environment. It is obvious that both situations can be reduced to the case of fixing individual forces which is normally considered in the nonequilibrium thermodynamics. Studying the first situation let us assume, for example, that tempera­ ture T and pressure P are set. The state of the final equilibrium of the isolated system corresponds to � � Hin  Hos max Sis ¼ Sos þ os þ const ¼ Seq ; T

ð1Þ

where H—enthalpy; indices “is,” “os,” “in,” and “eq” refer to isolated system, open subsystem, initial state, and final equilibrium, respectively. The second term in the right-hand side of the equality in square brackets stands for entropy transferred from the open subsystem to the environment. Taking into account that Sos ¼

Hos  Gos ; T

where G—the Gibbs energy, and multiplying (1) by T, we obtain a trans­ formed criterion of equilibrium � � in max TSis ¼ Gos þ Hos : in Since Hos is constant, we find that max Sis corresponds to max(Gos) and, hence, to min Gos. In turn, attainment of the minimum possible value of the Gibbs energy means the largest feasible useful transformation and the minimum dissipation of the total energy, i.e., the minimum (in this case a zero one) entropy production in the open subsystem.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

13

If owing to the external conditions the point of final equilibrium in this subsystem is not attained and the subsystem passes into one of possible partial equilibria the change in the Gibbs energy reduces as compared to (GinGeq) but remains maximum for the newly established conditions, which is explained by the reasoning similar to those presented above. Accordingly, the entropy production appears to be minimum. Derivation of the expression for the minimum production of S in the systems with constant T and V (volume) differs from the one above only by replacement of enthalpy by internal energy (U) and the Gibbs energy by the Helmholtz energy in the equations. When we set S and P or S and V dissipation turns out to be zero according to the problem statement. In the case of constant U and V or H and P, the interaction with the environment does not hinder the relaxation of the open subsystem toward the state max Sos. Let us discuss the second situation taking the isolated system shown in Figure 2 as an example. The open system (os) exchanges flow J with the environment through boundaries 1 and 2. The constant value of the flow is maintained owing to the source of thermodynamic potential 1 situated to the left of 1. With the increasing energy dissipation and constant 1 the value of thermodynamic potential on boundary 2 decreases (to do the same amount of the effective work (useful effect) requires greater differ­ ence in potentials). Entropy production in the isolated system (is) can be expressed by the equation Tð Z 2Þ

DSis ¼ Tð1 Þ

’ðÞ dT þ TðÞ

is

1

J

Teq Z

T ð 2 Þ

’ðÞ dT; TðÞ

2

J OS

Figure 2 Isolated system (is) and open subsystem (os) with minimum entropy production.

ð2Þ

14

B.M. Kaganovich et al.

where the first and second integrals in the right-hand side denote produc­ tions of S in os and part of its environment located to the right of boundary 2, respectively (production of S to the left of boundary 1 can be assumed constant). With decrease in the difference (12) the second integral increases faster than the first decreases. Indeed, to the right of boundary 2 entropy is generated at lower values of thermodynamic potential than in os. However, the flow with a larger value of  heats a certain amount of substance to the higher temperature than the flow with lower . Therefore, if the infinitely small change in the potentials of these flows occurs the dissipation of energy and, hence, entropy generation J

d dq ¼ T T

appears to be lower for larger , i.e., @ðDSÞ  0: @

ð3Þ

From this follows the tendency of the isolated system toward the distribu­ tion of energy dissipation among its parts so that the share of the total dissipation in the open system with fixed flows was ultimately small. The simplest and vivid example of such a distribution of entropy generation is the case of fixed heat flow from the open subsystem to the environment. For this case equality (2) takes the form: ZT2 DSis ¼ T1

dq þ T

Teq Z

dq : T

T2

It is clear that even at partial transformation of heat into work in open subsystem the maximum entropy of the isolated system will be reached at the largest value of T2. Probably the presented equilibrium interpretation of the Prigogine theorem cannot be considered as its strict or general proof. At the same time this interpretation reveals the possibilities to automatically observe the principle of the least entropy production at equilibrium modeling of a wider spectrum of physicochemical processes. From a satisfactory, to a certain extent, explanation based on the second law of the Prigogine theorem we can pass to an absolutely macro­ scopic explanation of the Onsager reciprocal relations by changing the order of proofs accepted in the nonequilibrium thermodynamics (in the nonequilibrium thermodynamics the Prigogine theorem is derived from the Onsager relations).

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

15

Let us consider an elementary system with two acting forces F1 and F2 and two flows J1 and J2 caused by them. If the forces are not constant then in the equilibrium state both the flows and the Onsager kinetic coeffi­ cients—L11, L12, L21, L22—turn out to be equal to zero. The equality to zero of these coefficients follows from the absence of flows at the initial moment of applying the forces that cause deviation of the system from its equilibrium. If either force (for example, F2) is fixed then the equalities hold: ðDSÞeq ¼ L11 F21 þ ðL12 þ L21 ÞF1 F2 þ L22 F22 ;

ð4Þ

@ ðDSÞeq ¼ 2L11 F1 þ L12 F2 þ L21 F2 ¼ 0; @F1

and L11 F1 þ L12 F2 ¼ J1 ¼ 0;

since J1 is caused by the nonfixed force and is absent at equilibrium. Hence, L11 F1 þ L21 F2 ¼ 0

and L12 ¼ L21 :

ð5Þ

It is obvious that using the properties of homogeneity and additivity of thermodynamic functions it is easy to obtain the Onsager relations in a general form Ljk ¼ Lkj :

We have managed to interpret the theorem of minimum entropy generation and the Onsager relations on the basis of the second law; therefore, we can additionally explain the Curie symmetry principle in terms of equilibrium. Let us suppose that far from the equilibrium between flows and forces there are nonlinear relationships Jjk ¼ Ljk Fk

ð6Þ

(a change in the form of this relationship does not affect the result of the reasoning below). Then for the case of two forces and two flows 1 DS ¼ L11 F1 þ1 þ L12 F1 F1 þ L21 F1 F2 þ L22 Fþ 2 ;

@ðDSÞ ¼ F2 F1 @L12

and

@ ðDSÞ ¼ F1 F2 : @L21

16

B.M. Kaganovich et al.

It is clear that equality (5) with available relationship (6) will be attained only in the limit at the tendency of a toward unity near the point “eq.” Hence, in the course of relaxation toward equilibrium the number of system symmetry elements should increase (or at least not decrease). In order to reveal the generality of equilibrium thermodynamic models it seems to be useful to interpret the least action principle (PLA) as a corollary to the second law. Each equilibrium flow that occurs in the isolated system between states (or time instants) 1 and 2 can be considered as an open subsystem. At any infinitely small time interval energy dis­ sipation in this subsystem takes minimum possible value (the flow goes through a continuous sequence of equilibrium states). Accordingly, inte­ gration with respect to time from the initial to the final state determines the minimum of a quantity with a dimension of the product of energy by time, i.e., action. Historical interrelations between the PLA and the second law of thermodynamics and futile efforts of deriving the latter from the former are considered in detail in Polak (1960). Besides, let us note the automatic observance (certainly with correctly set initial data) and, hence, needlessness of the formalized descriptions in equilibrium modeling of such important regularities of macroscopic sys­ tem behavior as the Gibbs phase rule, the Le Chatelier–Brown principle, mass action laws, the Henry law, the Raoult law, etc.

2.4 Equilibrium approximations In Section 2.2 we mentioned the impossibility to strictly substantiate the equilibrium descriptions for all cases of life and the need to apply equili­ brium approximations in some situations. The vivid examples of the cases, where the strongly nonequilibrium distributions of microscopic variables are established in the studied system and the principal difficulties of its description with the help of intensive macroscopic parameters occur, are fast changes in the states at explosions, hydraulic shocks, short circuits in electric circuits, maintenance of different potentials (chemical, electric, gravity, temperature pressure, etc.) in some spatial regions or components of physicochemical composition; interaction with nonequilibrium and sharply nonstationary state environment. A known method of overcoming these difficulties and passing to the equilibrium terms is the introduction of additional variable forces balan­ cing the differences in potentials in the description of the modeled phe­ nomenon. A good example of using similar method is given in Einstein’s paper related to the quantum theory of radiation (Einstein, 1914) in which he presented a chemically homogeneous gas as a mixture of different components that are characterized by their values of mole energy, found the law of energy distribution among them from the condition of chemical

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

17

equilibrium, and relying on this distribution, derived the Planck formula for the monochromatic radiator. Einstein’s derivation demonstrates the efficiency of equilibrium descriptions and also shows nontriviality of the method of their construction. Indeed, the constraints on attainability of the final complete equilibrium can be caused both by different processes of transfer or chemical transformations within the system and by specific features of the system interaction with the environment. The diverse conditions of nonequilibria appearance make it hardly possible to imagine the development of a single algorithm for transition from nonequilibrium to equilibrium modeling. Along with the difficulties associated with searching for a principal idea of this algorithm there are complications related to the “damnation of dimension.” A rapidly growing dimension of the solved problem certainly affects the time and stability of the computational process convergence, however, even to a greater extent, it affects the volume of the initial data preparation and difficulty to formalize the numerous constraints to be set. Unfortunately, so far the authors have managed to overcome the problems of equilibrium approximations only in solving several specific problems. Some of them are discussed in Sections 4 and 5.

3. MODELS OF EXTREME INTERMEDIATE STATES 3.1 MEIS with variable parameters Currently the authors are developing three classes of models of extreme intermediate states (MEIS): (1) with variable parameters; (2) with variable flows, and (3) those describing spatially inhomogeneous systems. All these classes of the models are formulated and analyzed in terms of MP, which, in the authors’ opinion, can be defined as a mathematical theory of equilibrium states. It is natural to start the analysis of the created modifications with the MEIS with variable parameters, which is the closest in character to the traditional equilibrium thermodynamics models. With fixed T, P, and initial composition of the components y of physi­ cochemical system this model will take the form: find 2 max4FðxÞ ¼

X

3

� � cj xj 5 ¼ F xext

ð7Þ

j2J ext

subject to Ax ¼ b;

ð8Þ

18

B.M. Kaganovich et al.

Dt ðyÞ ¼ fx : x  yg;

ð9Þ

’r ðxr ; zr Þ  cr ; r 2 Rlim ;

ð10Þ

GðxÞ ¼

X

Gj ðxÞxj ;

ð11Þ

j

xj  0;

ð12Þ

where x = (x1,…,xn)T—a vector of mole quantities of the system compo­ nents; the vector of the initial composition y  x; cj —a coefficient, ranging the property of the j-th component x (usefulness or harmfulness) of interest for a researcher; Jext—a set of indices of the components, with the extreme concentration of their mixture to be determined; A—the (m  n) matrix of element contents in the system components; b—a vector of mole quantities of elements; Dt(y)—the region (the set) of thermodynamic attainability from y; jr and cr—the constrained kinetic function of the r-th component x or (and) any other parameter zr and its limit value, respectively; Rlim—a set of indices of constraints on macroscopic kinetics; G and Gj —the Gibbs energies of the system and its j-th component; xext—the point with extreme value of the system property of interest to a researcher. The sign £ in expression (9) is understood in the thermodynamic sense suggested by Gorban: x £ y, if it is possible to pass from y to x along the continuous trajectory, along which G(x) does not monotonously increase. The objective function (7) in accordance with the general purpose of MEIS that was mentioned in the introduction, i.e., finding the state with extreme value of the system property of interest to a researcher, in this case determines the extreme concentration of the given set of substances. Equality (8) represents a material balance. Expression (9) represents the region of thermodynamic attainability from point y. It is obvious that in Dt(y) the inequalities are satisfied: G(xeq) £ G(x) £ G(y), where xeq—the final equilibrium point. Inequalities (10) are used to set the constraints on macroscopic, including irreversible, kinetics. Presence of this constraint makes up principal difference of the model (7)–(12) from previous mod­ ifications of parametric MEISs. The choice of equations for the calculation of individual terms under the sign of sum in the right-hand side of equality (11) depends on the properties of the considered system. Writing model (7)–(12) we use two main assumptions following from the previous text: 1.

all points of the set Dt(y) are the points of equilibria; partial equilibria within Dt(y) possess all the properties of the complete final equilibrium provided any processes to occur in them are inhibited.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

2.

19

each of these points can be attained from y along the equilibrium, in a general case irreversible, trajectory along which the condition of monotonous change in the characteristic thermodynamic function is met (as applied to the described case—the Gibbs energy).

An important specific feature of model (7)–(12) consists in the fact that, unlike the models of chemical kinetics, nonequilibrium thermodynamics or equilibrium thermodynamics, in the case of applying the law of mass action, this model, does not employ the complete representation of mechanism (a list of stages) of the studied process (chemical reactions and transfers of energy, impulses, substance, charges) but sets the lists of components of physicochemical composition x and parameters z. The need to indicate individual stages may arise only in the description of constraints (10) on the rates of change in some components of the vectors x and z. Eliminating or considerably reducing the list of stages in the model description and retaining in the description only the list of sought vari­ ables make essentially easier the preparation of initial information, which is particularly important for solution of the applied large-dimensional problems. Note that when setting the list xj, j = 1,…,n, the authors deviate from the classical Gibbs definition, understanding by the system components not individual substances but their quantities contained in a certain phase. For example, if the water in reaction mixture is in gaseous and condensed phases, its corresponding phase concentrations represent different para­ meters of the studied system. Such expansion of the space of variables of the problem solved facilitates its reduction to the problems of convex programming (CP). System (7)–(12) does not include the formalized condition of meeting the Gibbs phase rule. This is related to the fact that this rule may appear to be untrue, for example, when finding part of the studied system components in the states close to critical, however, if it is true it should be observed automatically (with accuracy depending on the errors in calculations) due to meeting the equilibrium and conservation princi­ ples. The requirement for equality of phase potentials is also not set in the MEIS version presented due its obvious automatic satisfiability. The mentioned “omissions” in the description of the modeled phenomena reflect general advantages of the extreme approach that are well known in physics and reveal additional advantages of the equilibrium models discussed. Along with the model with fixed T and P the authors also suggested MEIS versions for other classical conditions of interaction between sys­ tems and the environment, i.e., for fixed: T and V, S and V, S and P, U and V, H and P. The model has been created to simultaneously search for xext and optimize the initial set of reagents y.

20

B.M. Kaganovich et al.

The MEIS written in form (7)–(12) does not determine unambiguously the character of the mathematical problem solved, whether it belongs to convex or concave programming. However, to date when specifying this model as applied to the solution of numerous theoretical and applied problems it has always been reduced to the CP problem, which facilitated enormously to the development of computational algorithms and the computational experiments. The examples of specifying the representation of individual expres­ sions in system (7)–(12) and application of MEIS with variable parameters are presented in Section 5.

3.2 MEIS with variable flows Development of the “flow” MEIS with the form reminding the models of nonequilibrium thermodynamics seems to be a very promising direction in equilibrium modeling of physical and chemical systems. Application of these models opens prospects for simpler analysis and solution of many complex problems related to the calculations of processes considered to be irreversible in principle. Certainly the flows in MEIS are interpreted stati­ cally as the coordinates of states. Thermodynamic interpretations are natu­ rally extended to the kinetic coefficients that relate these flows with forces. Correctness of such interpretations is confirmed by the application of MP, being the theory of equilibrium states, as the terms for MEIS description. The flow modifications created can be divided into two groups: (1) the models of systems with real flows that are distributed on the schemes in the form of graphs and (2) the models of systems with conditional flows that undergo some certain chemical transformations or transfer processes. The modifications of the first group, in turn, are divided into parts that are related, respectively, to stationary and nonstationary flow distribution. The main object of modeling on the basis of the first group of the flow MEIS are hydraulic (heat-, water-, oil-, gas supply, etc.) (Gorban et al., 2001, 2006; Kaganovich et al., 1997) and electric networks. However, there can be other applications of such models as well. For example, Kaganovich et al. (1997) show their use to describe the distribution of harmful sub­ stances in the vertical column of the atmospheric air. MEIS of stationary isothermal flow distribution in a closed (without sources and sinks) multi-loop hydraulic circuit has the form: find n X Pbr max i xi

ð13Þ

Ax ¼ 0;

ð14Þ

i¼1

subject to

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

n X i¼1

Pmov xi  i

n X i¼1

Pbr i xi ¼ 0;

21

ð15Þ

jr ðxÞ  cr ; r 2 Rlim ;

ð16Þ

Pbr i ¼ f i ðxi Þ; i ¼ 1; … ; n:

ð17Þ

mov In (13)–(17) Pbr —friction pressure loss and effective pressure i and Pi (created by a pump or gravitational), respectively, on the i-th branch of the circuit; x = (x1,…,xn)T—the vector of volumetric flows in branches; A = [aij]—the (m1)  n—matrix of incidence of independent nodes and branches; aij = 1, if the flow in the i-th branch in accordance with the direction set in advance nears the j-th node; aij = 1, if the i-th flow goes from the j-th node, and aij = 0, when node j does not belong to the branch i; j = 1,…,m; functions jr and their limiting values cr in this case can be determined by setting values of regulated pressures at individual nodes or flow rates in the individual branches of the circuit. The objective function (13) representing the total dissipation of kinetic energy of the flows at isothermal motion of fluid is proportional to the entropy production in the circuit and its transfer to the environment, i.e., proportional to the entropy accumulated by the isolated system (inter­ connection of the circuit and environment). The matrix equation (14) describes the first Kirchhoff law, which, as applied to hydraulic circuits, expresses the requirement for mass conservation. Equality (15) repre­ sents a balance between the energy generated and consumed in the circuit. Using system (13)–(17) it is possible to describe the hydraulic circuits with lumped, with regulated, and with distributed parameters (Gorban et al., 2001, 2006). It stands to reason that depending on the type of circuits the types of functions fi(xi) in equalities (17) (closing relations) will change. In Gorban et al. (2001, 2006) and Kaganovich et al. (1997) system (13)–(17) was modified as applied to the description of flow distribution in the heterogeneous circuits in which flows in branches undergo chemical transformations and phase transitions. In the analysis of such circuits the extreme thermodynamic approach reveals to a greater extent its advan­ tages relative to the use of closed systems of equations. In particular, it turns out to be the simplest for practical implementation. Using model (13)–(17), it is possible to identify the extremality criteria for different cases of interaction between the circuit and environment and reveal the reducibility of the problem of calculating the stationary flow distribution to the CP problem. Let us suppose that for the circuit with lumped parameters the closing relations have the form:

22

B.M. Kaganovich et al.

 Pbr i ¼  i xi ;

ð18Þ

where  r—a constant coefficient and exponent   1. In this case the Lagrange function of the circuit is L¼

n X i¼1

 i xþ1  i

m 1 X

j

j¼1

X

n X

aij xi þ m

i¼1

i2Ij

 i xþ1  i

n1 X

Pmov xi ;

ð19Þ

i¼1

where j and m—the Lagrange multipliers; Ij —a subset of branches incident to node j. It can be shown (Gorban et al., 2001, 2006) that the second partial derivatives of L with respect to xi @2L  0: ¼ ð þ 1Þ i x1 i @x2i

ð20Þ

Hence, the solution to problem (13)–(17) in this case corresponds to the maximum L = f(x) and maximum of the objective function (13). The possibility of existence of the maximum point at the function convexity and nonlinearity of the system of constraints is illustrated in Figure 3a. The extreme thermodynamic model of passive circuit (without effec­ tive pressure sources) is obtained by transforming model (13)–(17). Toward this end let us mentally isolate a passive fragment with np

(a) f(x)

(b) f(x)

5

1

x2

3 2

Q

4

4

a′

F max

x2

1 Q

3 2

a′ f(x) b′

f(x) a

a

b′

b x1

F min

x1+x2=Q b x1

Figure 3 An objective function and extreme points on nonlinear (a) and linear (b) set of constraints.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

23

branches and mp nodes from the active circuit. The variables xi(i = np=1,…,n) on the branches of the rejected circuit part include flows that determine the directions and magnitudes of sources and sinks Qj(j = 1,…,mp) in the iso­ lated fragment. Then condition (15) is excluded and model (13)–(17) is replaced by the model that corresponds to the Kirchhoff theorem of mini­ mum heat production (Kirchhoff, 1848): find min

X

Pbr i xi

i

subject to  Ax ¼ Q; Pbr i ¼  i xi ;

i ¼ 1; … ; np :

g

ð21Þ

In this case the second derivatives of the Lagrange function L¼

np X i¼1

 i xþ1 i



m 1 X

0 j @

j¼1

X

1 aij xi A

j2Ij

with the above supposition on the form of closing relations @2L  0: ¼ ð þ 1Þ i x1 i @x2i

ð22Þ

Hence, the extremum L(x) is the point of minimum. Thus, the problem of entropy maximization is transformed into the problem of heat minimiza­ tion; and the Kirchhoff and Prigogine theorems result from the extension of the second law to the passive isothermal circuits. The graphical inter­ pretation of problem (21) is given in Figure 3b. In the work by Gorban et al., (2001, 2006) the extremality criteria and corresponding MEIS modifications were presented for different cases of interaction between hydraulic circuits and the environment. Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, fric­ tion, and inertia forces (Gorban et al., 2006). find � � mov ext Ps ¼ ePm þ Pbr q  Pq

ð23Þ

Axk ¼ 0;

ð24Þ

subject to

24

B.M. Kaganovich et al.

n X i¼1

Pmov xki  i

( Dt ðyÞ ¼

x : k

n X i¼1

Pbr;k i

n X i¼1

�

xki li ¼ 0; Ps:br;k i

s:br;k  br i P

jr ðxkr Þ  cr ;



f r � �2  i xki

r 2 Rlim ;

� 1 � mov P ¼ ePm  A q Pbr ; q  Pq

ð25Þ

�

) 0

ð26Þ

ð27Þ

ð28Þ

where P—a vector of pressures at nodes; Pm—the fixed pressure at node mov j = m; Ps.br—the specific (per length unit) pressure loss; Pbr — q and Pq vectors of pressure losses and effective pressures in branches of the “cir­ cuit tree” q comprising the paths from nodes j = 1, …, m – 1 to the node m, Aq 1 —a matrix of “paths” corresponding to this tree, which is obtained by inversion of the submatrix of matrix A for branches which belong to this fr tree;  br r and i —coefficients; e—a unit vector; s—an index of the node for which we find the extreme pressure; k—the index of computational pro­ cess iteration. Further development of MEIS for nonstationary flow distribution is related to the possibility of many new problem statements and their attractiveness. The obvious objects to be studied in the future are changes and deviations from the required values of sources and sinks at nodes and flow rates in the branches of circuits under normal and emergency operat­ ing conditions. The problem of calculating the emergency operating con­ ditions seems to be the most important. The emergency operating conditions result from the disturbances that are too fast for the friction forces to manifest themselves, and we have to consider the propagation of shock waves in the ideal fluid. The starting points for the thermodynamic description of this problem can be the “equilibrium” derivation of the formula for hydraulic shock in an individual pipe (Gorban et al., 2001, 2006) and the work of several authors on modeling of hydraulic shocks in pipeline systems on the basis of traditional (nonthermodynamic) methods of hydraulic circuit theory (for example, Balyshev and Kaganovich 2003; Balyshev and Tairov 1998). The experience gained with construction of the flow models of hydrau­ lic systems was applied to create the models that are based on the graph representation of mechanisms of chemical reactions (sets of elementary reactions) and transfer processes. In the work by Kaganovich et al. (1989, 1993) and Kaganovich and Filippov (1995), the advantages of setting a list

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

25

of substances (components of a system) versus setting mechanisms as initial information have been discussed. Setting of substances appears to be immeasurably simpler, for example, in the analysis of coal combustion processes where the list of reactions includes thousands of units. However, in many cases, the use of the mechanism notion can be useful either for a quite complete elucidation of specific features of the studied process or for the choice of the analyzed mechanism which can be implemented by selecting catalysts or by specially arranging the course of reactions that constitute the process. Construction of the flow models of the second group (with conditional flows) will be exemplified by the MEIS of chemical system with constant T, P, and y that has the form: find 2 max4FxððÞÞ ¼

X

3

� � �� cj xj ðÞ5 ¼ F x ext

ð29Þ

j2J ext

subject to xj ¼ yj þ

X

 ij i ;

j ¼ 1; … ; n;

i ¼ 1; … ; m;

ð30Þ

i

Dt ðyÞ ¼ f : xðÞ  yg;

ð31Þ

jr ðxr ðÞ; zr Þ  cr ; r; 2 Rlim ;

ð32Þ

GðxðÞÞ ¼

X

Gj ðxðÞÞxj ;

ð33Þ

j

xj  0;

0  i  1;

ð34Þ

where  = (1,…,m)T; i —the degree of completeness (a coordinate) of the i-th reaction; —a stoichiometric coefficient. Equation (30) describes the material balance of transformations of the j-th system component. The kinetic constraint (32) is similar to (10), but it includes the relationships between the constrained functions (rates of reactions, the most attainable concentrations of reagents, etc.) and the degrees of completeness of reactions. Model (29)–(34) determines the chemical process mechanism which is optimal from the stand point of the formation � � of a sought extreme con­ centration of the given set of substances cj xj that are ranked in terms of

26

B.M. Kaganovich et al.

importance. Here a specific formulation of constraint (32) seems to be even simpler than that of condition (10) since “kinetics” determines the rates of reactions that are already contained in the “prekinetic” system (29)–(31), (33), and (34) and requires a comparatively small amount of additional information for its description. Currently the flow MEISs are less developed and used than the MEISs with variable parameters. Some examples of their use were presented in the works by Gorban et al. (2001, 2006) and Kaganovich et al. (1997). The problems of further equilibrium modeling evolution on the basis of flow models are discussed below.

3.3 MEIS of spatially inhomogeneous systems A generalized description of spatially inhomogeneous systems seems to be rather complicated. Indeed, various natural and technical systems can possess a very diverse specific inhomogeneity. In some cases the inhomo­ geneous system can be divided into parts with fixed spatial coordinates that differ from one another by the values of intensive parameters, phase, and component composition. In the other cases parts of the system with macroscopic nonzero volumes that possess different thermodynamic prop­ erties get constantly mixed up with each other and change spatial coordi­ nates. The original modifications of MEIS with spatial inhomogeneity were constructed as applied to the first cases and, in particular, to describe the distribution of harmful substances in the atmospheric air (Gorban et al., 2001, 2006; Kaganovich et al., 1997) and combustion of fuels in fixed-bed, fluidized-bed, and torch furnaces (Kaganovich et al., 2004, 2005a, 2006a, 2006c). Modeling of the exchange processes between some zones was based on the construction of flow distribution graph (Figure 4). Let us write the MEIS of spatially inhomogeneous system with inten­ sive parameters changing only along the vertical axe, and with fixed y and, in each k-th zone, T and P has the form: find 2 max4FðxÞ ¼

X

3

� � cjk xjk 5 ¼ F xext

ð35Þ

j;k2J ext

subject to

Ax ¼ b;

ð36Þ

Ain m ¼ Q;

ð37Þ

Dh1 P1 k Tk  Dhk Pk 1 T1 ¼ 0

ð38Þ

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

27

k

k−1

Δ h k−1

h k−1

1

Figure 4

A graph of spatially inhomogeneous system.

Dt ðyÞ ¼ fx : x  yg; ’r ðxrk ; zrk Þ  cr ; EðxÞ ¼

X

ð39Þ

r 2 Rlim ;

Ejk ðxÞxjk ;

ð40Þ ð41Þ

j;k

�

EjðqÞk ¼

G0jk ðTk Þ

xjk þ RTk ln Pk k

EjðcÞk ¼ G0jðcÞk ðTk Þ þ

xjk  0;

� þ Mj ghk ;

2j  j ; rjr

zrk  0;

ð42Þ

ð43Þ

ð44Þ

28

B.M. Kaganovich et al.

where Ain—a matrix of incidences (connections) of independent nodes to branches (arcs)� of the graph mapping the system structure; � m T m ¼ m ; m 1 ; … ; k k ¼  xjk j ; j —the mole mass of the j-th system j component; Q—the vector of external sources and sinks; hk and Dhk—an average level and thickness of the k-th zone, respectively; k—the mole quantity of gaseous components of the k-th zone; E and Ejk—energy functions of the system and its jk-th component, respectively; R—the universal gas constant; g—the free fall acceleration; —the surface tension; —molar volume; r—a radius of condensed particle; index 1 refers to the zone with minimum h; indices g and c refer to the gaseous and condensed phases, respectively. Model (35)–(44) includes two material balances: the first of them (36) represents the condition of conservation of element quantity in chemical reactions and phase transitions; the second balance (37) is the expression of the first Kirchhoff law (in this case the law of mass conservation at substance motion along branches of the system graph). Equation (38) is based on the assumption that the gas phase in each zone is ideal. Expres­ sion (40) is the constraint on macroscopic kinetics. Energy functions E represent the sums of chemical (the Gibbs energy), gravitational, and surface (related to interphase formation) components. Derivation of for­ mulas (42, 43) is given in Kaganovich et al. (1997). Here we will only note that Equation (43) unlike (42) does not include the member Mjghk that reflects the action of the gravity forces that are balanced for the condensed phase by resistance forces whose field is not a potential. Therefore it is difficult to associate the solution to the equilibrium problem of these two types of forces with the solution to the extreme problem. At the same time their equilibrium does not affect the other equilibria that occur in the heterogeneous system and can be excluded from consideration. On the whole model (35)–(44) includes the features of both parametric (the main sought variables are system parameters) and flow (the flow distribution graph is used) MEISs and can be considered as their combina­ tion in some respects. For the time being the MEIS of spatially inhomogeneous systems (with­ out kinetic constraint (40)) has found the use only in solving the problem of harmful substance distribution in the vertical air column of isothermal atmosphere (Gorban et al., 2001, 2006; Kaganovich et al., 1997). Extension of this model application to every new problem even provided that the system can be divided into zones with different spatial coordinates is asso­ ciated with considerable difficulties due to specific character of the studied object. It is natural that these difficulties increase greatly with inclusion of the macroscopic kinetics constraints into the model. The main difficulties are: division of a modeled system into zones, determination of a set of limiting processes, and choice of mathematical formulations for correspond­ ing constraints taking into account sensible accuracy of calculations.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

29

3.4 Variants of kinetic constraints formalization The section addresses the problem of specifying constraints (10), (16), (27), (32), and (40) on macroscopic kinetics as applied to various problems. Formalization of these constraints as well as constructions of MEIS are on the whole based on the Boltzmann assumption on the equilibrium of “kinetic” trajectories of motion toward point xeq and the possibility to describe them by autonomous equations of the form x_ ¼ f ðxÞ: The problems of including kinetic blocks into thermodynamic models vary in their significance and, hence, differ by the level of complexity for the MEIS types considered. In the flow and “spatial” models even initial, basic relations reflect to a certain degree the kinetics, and additional description of special kinetic constraints can affect insignificantly the computational process. The experience in applying the models of form (7)–(12) and preliminary analysis of problems to be studied in the future show that taking into account kinetics in MEIS with variable parameters, in many cases, leads to a sharp change in the mathematical character of the studied problems and, hence, to the need to modify the computational algorithms employed. Therefore, the discussion presented below refers largely to the parametric MEISs. The “translation” of kinetics into thermodynamic terms which is neces­ sary for constraint (10) to be organically included in the MEIS and which suggests exclusion of time variable from this constraints is, of course, a nontrivial problem. The optimal solution of the latter, which determines the formalized problem statement providing comparative simplicity and accuracy of computing experiments, refers to the newly formed scientific discipline “Model Engineering” (Gorban, 2007; Gorban and Karlin, 2005; Gorban et al., 2007). Three approaches can be outlined to choose the formalized thermo­ dynamic description of the kinetic block of model (7)–(12): (1) the thermo­ dynamic approach, with additional thermodynamic relations that limit some stages of the studied process mechanism, to be written; (2) the approach related to the transformation of right-hand sides of the kinetic equations and transition from the space of sought variables of the solved problem to the space of thermodynamic potentials, and (3) the approach based on direct use of these sides. Applicability of the first approach suggested by Keiko and Zarod­ nyuk is based on the unity of thermodynamics and kinetics which explain differently the same physical regularities. As was said above this unity was brilliantly revealed by Boltzmann in his “kinetic” and “thermodynamic” explanations of the second law. In our case, setting, for example, a constraint on the equilibrium constant value of an indi­ vidual reaction   j xj ¼ 0 within complex chemical process and writing j this constraint in one of the possible forms:

30

B.M. Kaganovich et al.



Kp ¼

 # pj j j

# xj j � � DG0 j ¼ exp  ¼  ; RT # yj j

ð45Þ

j

we indirectly impose the constraints on the rate of this reaction which determines attainable concentrations of its final products. (In (45) DG0— the difference between the total Gibbs energies of products and initial substances.) The constraints similar to (45) allow the mechanism of pro­ cesses to be taken into account in the thermodynamic studies and do not require the complete knowledge of the mechanism and corresponding formalized description. Instead of the equalities of type (45) one can compose an auxiliary MEIS intended to search for the extreme values of constrained variables and substitute them into the initial model. It is obviously most expedient to set thermodynamic constraints on individual stages as applied to fast variables whose formation to a great extent determines further course of the process studied. For example, the fast formation of harmful substances can complicate the production of target products. Derivation of formulae for additional thermodynamic con­ straints disregarding the permissible time of chemical reactions and trans­ fer processes narrows the area of effective application of the given approach. Nevertheless, some possibilities of its effective application are demonstrated in Section 5.3. The usefulness of this idea undoubtedly deserves further scrutiny. The second method of excluding time variable from (10) is based on thermodynamic analysis of kinetic equations suggested by Horn (1964; Horn and Jackson, 1972), Feinberg (1972, 1999; Feinberg and Hildebrant, 1997; Feinberg and Horn, 1974), Gorban, and other authors. We consider the technique used in the work by Gorban (1984), which implies the transformation of right-hand sides of kinetics equations, i.e., replacement of coordinates by potentials and further substitution of the transformed sides into the expression for the derivative of the total characteristic function of the considered system with respect to time . In the work by Gorban (1984) according to Boltzmann it was supposed that this function possessed the properties of the Lyapunov functions. We will explain this method on the example of setting the constraint on the rate of the i-th chemical reaction. Let the rate equation of this reaction have the form: wi ¼

dxi  ¼ ki # xj j ; d

j

ð46Þ

where k—the rate constant. Independence of the right-hand side of Equation (46) from makes possible the transformation (47) and the

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

31

representation of derivative of the characteristic function (48) with respect to time in the form (49): 0  ki # xj j j

! ’i exp @

X

1  j j A ;

ð47Þ

j

G¼

XZ

dx;

ð48Þ

i

G_ ¼

XX i

0 ji j exp @

X

j

1  j j A ;

ð49Þ

j

where —a chemical potential. The set of thermodynamic attainability Dt(y) can be described in MEIS by either using the nonpositivity condition of the right-hand side of Equation (49), or writing the constraint on the sign of derivative of (48) with respect to x. Both methods as applied to MEIS are associated with great difficulties that can be explained by the fact that the con­ straints on rates are set only for part of the stages of the studied process mechanism. Therefore, the representations of the components of vector x, which take part in the constrained reactions and influence the values of this function only through their concentrations in the sequential states, should be matched in the formulation of monotonicity condition of the system characteristic function (in description of the set Dt(y)). Due to these difficulties to date the authors, despite the theoretical effectiveness of the second approach, have used mainly the third of the above methods for excluding time variable from MEIS to specify the kinetic constraints. The simplest situation in the use of the third method is when the constraint on the process rate is determined only by one reaction, for example, of form (46). In this case to find the limiting concentration (or another parameter of the r-th component) we can write the inequality: 

dxr  kr # xj j d :

ð50Þ

j

When the mechanism of formation xr includes several reactions, (50) is replaced by a more complex expression dxr 

X i

!  kir # xj j j

d :

ð51Þ

32

B.M. Kaganovich et al.

With integration of (50) or (51) the considered time interval b is taken equal to either the duration of the components stay in whole reactor or in a zone where the limiting stage of the process occurs. The values xj, here, are very often set constant. Depending on the statement of the problem solved, sometimes they can be taken equal to the values of corresponding components of vector y, and sometimes they can be calculated on the basis of search for the extremum of the objective function on the auxiliary MEISs. It is clear that by replacing all variables in the integrals of the form Z X 0

i

!  kir # xj j j

d

by constant constraints (10) reduce to linear inequalities

xr 

X i

!  kir # xj j j

d

ð52Þ

and their inclusion into MEIS does not affect the reducibility of the latter to the CP problems. Unfortunately it is not always possible to use only linear inequalities. In further studies we will have to include into the kinetic constraints both the equations of nonlinear chemical kinetics and the nonlinear equations of transfer processes. Nonconvexity of the problem solved and possible multi­ valuedness of its solutions, in case the constraints on radiant heat exchange are included into MEIS, are shown in the work by Kaganovich et al. (2005a). The MEIS modifications including the constraints on macroscopic kinetics have already revealed their high efficiency in the analysis of environmental characteristics of the fuel combustion processes (Kaganovich, 2002; Kaganovich et al., 2004, 2005a, 2006a, 2006b, 2006c, Shamansky, 2004). Their application enriched the explanations of the equilibrium model capabilities for studying the irreversible phenomena of different nature with vivid examples. Simultaneously it is shown that the account taken of the macrokinetic constraints reduces appreciably the thermodynamic attainability region studied with the help of MEIS and hence enhances the accuracy of thermodynamic estimations of the limiting characteristics of processes. Specific examples of formulation and use of kinetic blocks of MEIS are considered in the Section 5. Along with the merits of the new equilibrium model modifications serious difficulties of their construction and application were revealed. First of all these difficul­ ties are related to the above change in the mathematical character of the

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

33

problems solved: convexity violation of the system of constraints or an objective function, multivaluedness of solutions, etc. The difficulties emerging are certainly surmountable. However, thor­ ough studies are necessary to overcome them. The discussion of prospects for MEIS application includes, in addition, the issue on the optimally complete description of constraints on macroscopic kinetics. A normal desire here is to include into the unified model the constraints on max­ imum possible number of stages that limit the results of the total process. The “uniting” tendency should undoubtedly manifest itself in thermody­ namic modeling. At the same time creating comprehensive descriptions of chemical kinetics and transfer processes one should remember that overcomplication of thermodynamic models leads to the loss of their compara­ tive advantages over kinetic ones: loss of comparative easiness of setting the initial information (first of all about the process mechanism) and simplicity of the mathematical apparatus employed.

3.5 Geometrical interpretations Graphical interpretation of MEIS first will be given for a parametric model of elementary reaction of isomerization. Let us suppose that the reaction proceeds with constant T and P, y = (1,0,0), the maximum value of the third isomer is found and MEIS has the form: find max x3

ð53Þ

x1 þ x2 þ x3 ¼ y;

ð54Þ

Dt ðyÞ ¼ fx : x  yg;

ð55Þ

subject to

GðxÞ ¼

3 � X 1

� x �� j xj ; G0j þ RT ln P xj  0:

ð56Þ ð57Þ

Model (53)–(57) does not include kinetic constraint that corresponds to constraint (10) in the general model (7)–(12). Graphical illustrations of the efficiency of including constraints on macroscopic kinetics into MEIS are given in Section 5. This section focuses on the geometrical explanation of

34

B.M. Kaganovich et al.

comparative advantages of using traditional space of kinetic variables (rates, flows, and time) and thermodynamic space of characteristic func­ tions and parameters of state in physicochemical modeling. Let us explain the specific features of model (53)–(57) using Figure 5. The equilateral triangle A1A2A3 in Figure 6a is a material balance polyhedron Dt(y), which is determined by equality (54) and inequality (57). The vertices correspond to the states in which the mole content of one of the components equals an absolute value of y, which for simplicity and without loss of generality can be assumed equal to unity and the remaining two—equal to zero. Index at the symbol of vertex A coincides with the index of the corre­ sponding component. The interior points of the edges represent the reaction mixture compositions in which the concentration of only one reagent is zero and the total mole quantity of the remaining two makes up unity (in case the above possible supposition is assumed). On the area of the triangle you can see the points xeq and xext, lines G = const as well as dashed zones of thermo­ dynamic unattainability from y by condition (55) near the vertices A2 and A3. Correspondingly the nondashed part of the triangle represents Dt(y). The two trajectories of motion from y to xext are shown: the one that meets (a continuous line) and the one that does not meet (a dot-and-dash line) the requirement for the Gibbs energy monotony. The point ~x ext represents approximately the maximum concentration of x3. It is obtained at motion from y along the edge A1A3 to the point minimum value of Gibbs energy and succeeding transfer to the curve G ¼ Gmin A1 A3 tangent to the edge. Figure 5b presents the surface (in this case it is plane) of the objective function F(x) = x3 and two closed sets that represent a feasible set of solutions on the surface of function G(x) and in the space of variables x. The set x is represented by the projection of the triangle A1A2A3 (Figure 5a) to the horizontal plane x2Ox3. Point O, the beginning of coordinates, coincides with the projection of vertex A1 to this plane, which corresponds to the corresponding initial composition of reagents y. The points xeq and xext, the lines G = const, and the feasible and unfeasible trajectory of transition from y to xext are shown on the plane of compositions x2Ox3, and on the surface of function G(x), and on the objective function plane. Even Figure 5 prompts some thoughts about the convenience of using thermodynamic variables. The form (topology) of the surface of function G (x) helps find the feasible directions of motion to the point G(xext), which maps the point F(xext) in the thermodynamic space. These directions are invariant with respect to the second law of thermodynamics and lead to the extremum of the characteristic thermodynamic function of the system (in case, shown in Figure 5, to the minimum G, i.e., to G(xeq)). The projection of the motion trajectories to the manifolds that are invariant with respect to the second law represents one of the components of the method for reducing the physical and chemical kinetics models, which is developed by Gorban and Karlin (2005). The specific feature of

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

A 1, X in

(a)

x eq x∼ ext x ext A2

A3

(b) F(x) F (x ext) F(x eq) 0 G(x) G(x ext)

0

x

G(x eq) A3 x eq x ext

x3

A2 x2

Figure 5 A graphical interpretation of the model of extreme intermediate states.

35

36

B.M. Kaganovich et al.

1 (a)

y

(−421.034)

4

6

(−424.118)

(−425.950) x∼2 ext

• eq

x2 ext

x3 ext

2 (−423.620)

5

(b) x mat 1 y

1

x2 mat x3

3

x eq

−420.255 −421.034

2

ext

x2 ext

3 (−420.255)

(−425.678)

4 5 eq

−423.620 −424.118 −425.678 −427.483

Figure 6 Polyhedron of material balance (a) and thermodynamic tree (b) of hexane isomerization reaction. T = 600 K, P = 0.1 MPa.

the MEIS-based approach lies in the fact that it envisages the projection (mapping) of sets of possible states rather than trajectories. A remarkable advantage of the optimization models based on the use of thermodynamic space consists in the possibility, in case of reducing these models to CP problems, to transform the region of feasible solutions into a one-dimensional set (a graph in the form of a tree) and to study the specific features of the studied system behavior on this graph—“a thermo­ dynamic tree.” The notion of thermodynamic tree (the graph, each point of which represents the set of thermodynamically equivalent states) was introduced by Gorban (1984) where he also revealed the possibilities of applying this notion for analysis of the chemical kinetics equations. In the work by Gorban et al. (2001, 2006) the authors consider the problems of employing thermodynamic tree to study the physicochemical systems using MEIS.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

37

Let us exemplify the effectiveness of the idea of tree in equilibrium thermodynamic modeling again by isomerization, using the calculations of transformations of three hexane isomers: n-hexane (x1), 2-methylpen­ tane (x2), and 3-methylpentane (x3) at T = 600 K and P = 0.1 MPa. Graphi­ cal interpretation of the analysis is presented in Figure 6a and b. Figure 6a presents the same triangle of material balance as in Figure 5a, but with some additional details included. It shows the feasible trajectories of motion toward the points of extreme values of both x3 and x2. Different shadings denote five components of the arcwise connectedness (Gorban, 1984), i.e., regions, in each of which any two points can be connected by thermodynamically feasible trajectories. Figure 6b shows a thermody­ namic tree with the branches connected with the regions highlighted in Figure 6a by one-to-one correspondence. Each point on the tree represents a section of the curve G = const that belongs to a corresponding compo­ nent of connectedness (a set of thermodynamically equivalent states). The points of this section obey the linear balance: ∑Gjxj = G = const. The constructed tree which replaces the thermodynamically attainable set Dt(y) allows one to study the behavior of both the characteristic thermodynamic function (in this case the Gibbs energy) and the objective function F(x). The feasible trajectories of the motion from point y = (1,0,0) ext to the points xext 3 and x2 are shown in Figure 6a represented on the tree by paths 1–4 and 1–4–5, respectively. Motion from point 4 to point 3 (the maximum feasible value x3 according to the condition of material balance mat xmat 3 ) and from point 5 to point 2 (x2 ) turns out to be impossible due to the Gibbs energy increase. Points 4 and 5 are levels of G (isopotential surfaces: G = –424.118 kJ/mol and G = –425.672 kJ/mol), at which the ext extreme compositions xext 3 and x2 should be located. Though in formulations of MEIS of type (7)–(12) or the particular form (53)–(57) the possibility of projecting the space of thermodynamic vari­ ables to a tree is not shown, the knowledge of principal possibility to reduce the set Dt(y) to the tree makes the analysis of capabilities and comparative merits of the model of extreme intermediate states essentially easier, clearer, and more convincing. Using the tree we can analyze the situations when the solution of the problem posed appears to be degenerate. As applied to the considered example of isomerization such a situation occurs when we search for max (x2 + x3). In this case all points of edge 2–3 of the material balance triangle that belong to Dt(y) are the points of the objective function maximum. Let us make a natural supposition that the two extreme cases (1) x2 = 1, x3 = 0 and (2) x2 = 0, x3 = 1 are equally satisfactory and, hence, we can seek to attain both vertex 2 and vertex 3. In the first case the sought level of the Gibbs energy will be G = G5, and in the second—G = G4. In Figure 6a the found range of levels G4–G5 corresponds to the part of edge 2–3, which is located between the point of its intersection with the curve G = G4 and the

38

B.M. Kaganovich et al.

point of contact with the curve G = G5. In Figure 6b the range of possible solutions is represented by the branch of tree 4–5. The idea of tree is rather effective for solving and analyzing the pro­ blem of determining G(xext) and when its solution is unique. The difficulty of developing relevant computational algorithms is to a great extent related to the implicit form of setting the constraints on the Gibbs energy in MEIS (expressions (9), (26), (39)) (Gorban et al., 2001, 2006; Kaganovich and Filippov, 1995; Kaganovich et al., 1989, 1993). The methods of over­ coming the difficulty that have been used so far lead often to the algo­ rithmic (not related to the accuracy of computer computations) error in computations. The use of the tree notion allows it to be used to develop accurate algorithms for the calculation of G(xext), or substantiate the applicability and assess the accuracy of alternative algorithms. The issues of constructing the algorithms intended for solving the problem of search for G(xext) and related to the direct application of thermodynamic tree were considered in the work by Gorban et al. (2001, 2006). When setting the constraints on macroscopic kinetics in MEIS the idea of tree is useful even from the viewpoint of interpreting the applied method for formalization of these constraints. It (the idea) can help repre­ sent even the deformation of the region of feasible solutions in the thermo­ dynamic space and the deformation of extremely simple representation of this region (a thermodynamic tree), and the projection of limited kinetic trajectories on the tree. In other words the use of the tree notion helps reveal the interrelations between kinetics and kinetic constraints, on the one hand, and thermodynamics, on the other. Geometrical illustrations of the efficiency of thermodynamic descrip­ tion of the stationary flow distribution problems as applied to the analysis of closed active and open passive hydraulic circuits were already pre­ sented in Section 3.2. The geometrical interpretation of the general models for the nonstationary flow distribution in the hydraulic circuit ((23)–(28)) and chemical systems with the set redundant mechanism of reaction ((29)– (34)) is still to be carried out which will obviously require a number of nontrivial problems to be solved.

4. COMPARISON OF MEIS WITH THE MODELS OF NONEQUILIBRIUM THERMODYNAMICS 4.1 Introductory notes Feasibility of applying the models of equilibrium thermodynamics to the analysis of nonequilibrium irreversible processes were described in Sec­ tion 2 of this chapter. This section discusses the comparative efficiency of such application to solve diverse theoretical and applied problems.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

39

Nonequilibrium thermodynamics was chosen as a main object for com­ parison, though an essential part of conclusions drawn below is useful in MEIS comparison with the models of chemical kinetics, synergetics, theory of dynamic systems and other models, model engineering and theories of motions. Comparison is made from two standpoints: (1) a scope of areas of effective applications and (2) simplicity and fruitfulness of computing experiments.

4.2 On the areas of effective applications of equilibrium and nonequilibrium thermodynamics First of all, we will touch a widely believed misunderstanding about impossibility of using the second law of thermodynamics in the analysis of open systems. Surely, the conclusion on inevitable degradation of iso­ lated systems that follows from the second law of thermodynamics cannot be applied to open systems. And particularly unreasonable is the supposi­ tion about thermal death of the Universe that is based on the opinion of its isolation. The entropy production caused by irreversible energy dissipa­ tion is, however, positive in any system. Here we have a complete analogy with the first law of thermodynamics. Energy is fully conserved only in the isolated systems. For the open systems the balance equalities include exchange components which can lead to the entropy reduction of these systems at its increase due to internal processes as well. The courses of chemical and technical thermodynamics, as the whole applied thermodynamics, are devoted, in their vast majority, to the effi­ ciency analysis of open systems (heat engines, chemical reactors, metal­ lurgical furnaces, etc.) and based in this analysis primarily on the second law. The fundamental Gibbs equations that describe behavior of open and closed systems for different cases of interaction with the environment are devised from the second law. And correspondingly numerous computing algorithms and systems (employing the equations) that were developed in the late 20th century to solve different problems of energy, chemical technology, metallurgy, cosmonautics, geology, ecology, and other spheres of science and technology satisfy it. From the viewpoint of our discussion it is worth noting that all these computing systems are based on the concepts of exactly equilibrium thermodynamics. Therefore, the state­ ment that the nonequilibrium thermodynamics forms a “theoretical base for studying open systems” (Zubarev, 1998) arouses surprise. The next sphere of competition between equilibrium and nonequili­ brium thermodynamics is the analysis of irreversible trajectories. A pop­ ular opinion about the possibility for the equilibrium thermodynamics only to determine admissible directions of motion for nonequilibrium processes was already mentioned in Introduction. However, the more

40

B.M. Kaganovich et al.

than 20-year experience of MEIS application has revealed the possibility of analyzing any probable states in the admissible directions. This possibility follows directly from two approaches of the second law substantiation by Boltzmann (see “Introduction”): (1) from the analysis of trajectories and (2) from the analysis of states. It is natural to assume that permutation of axioms and theorems allows the methods for search and analysis of any attainable states of the thermodynamic system to be devised from the second law. Reduction of the models of motion to the models of rest, analysis of trajectories to analysis of states is the specific feature of the approach developed by the authors, which determines its role in “Model Engineering” (Gorban, 2007; Gorban and Karlin, 2005; Gorban et al., 2007). This specific feature stipulates to a great extent its comparative computa­ tional simplicity and efficiency. The third, and probably the most complex, area with respect to com­ parative analysis of equilibrium and nonequilibrium approaches is the modeling of mechanisms of the studied processes. As was described above, the parametric MEISs were constructed on the basis of the lists of substances (system components) rather than the lists (mechanisms) of reactions. However, there is a wealth of experience gained in considera­ tion of both individual stages with the help of parametric models and the complete mechanisms based on the flow models of hydraulic and chemi­ cal systems. For the hydraulic systems it is possible to choose not only physical mechanism (flow distribution over branches of the given redun­ dant scheme that provides the minimum energy consumption for fluid transportation), but also “technical” (distribution of resistances of branches depending on their technical characteristics) and “economic” mechanisms (distribution of economic expenditures over branches that minimizes production of “economic” entropy, being a measure of useless irreversible spending of money). A partial inclusion of mechanisms (their individual stages) by means of Equations (10), (16), (27), (32), and (40) is clear to some extent from the above said and its efficiency is illustrated by examples in Section 5. Here we explain the possibilities of choice by the equilibrium models of com­ plete mechanisms. Prior to discussion of modeling the complete mechanisms of chemical systems on the basis of MEIS (29)–(34), we will present capabilities of the approximate analysis of the efficiency of such mechanisms by MEIS (7)– (12) with variable parameters. This possibility was studied in Kaganovich and Filippov (1995; Kaganovich et al., 1993). Let us consider a process during which some initial composition a should be used to get the max­ imum quantity of products b. Let the point y in Figure 7 denote an initial state of the reactive system. The point m corresponds to the maximum thermodynamically admissible concentration of b. The points l and k represent states of a chemical system with the use of catalysts that provide

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

41

x cmat l

y •

x eq

m

x bmat

k x dmat

Figure 7 Catalyst impact on the attainable state.

appropriate mechanisms of processes. Figure 7 shows that the mechanism connected with motion through the point l is more effective, since the Gibbs energy monotonically decreases on the curve ylm. From the point k that is passed at the competing mechanism of the process the state m proves to be unattainable. Comparative assessment of the indicated mechanisms can be obtained on the basis of the multivariant calculations on model (7)–(12). At first we must solve the problems of maximization of sets of the substances c and d that correspond to the main components of compositions of the reactive mixtures l and k. Then taking the obtained extremal states as initial the problems of maximizing b must be solved. The use of MEIS (29)–(34) allows the comparison of mechanisms based on the single-variant calculation. For this purpose a redundant graph of the process (complete mechanism) is constructed and the “unnecessary” branches are automatically excluded from the scheme during optimiza­ tion. In this case the constraints on the reaction rates (inequality (32)) are also taken into consideration. An example of the redundant graph as applied to hexane isomerization (see Section 3.5) is given in Figure 8. Preliminary analysis of the efficiency of using the flow MEIS to study mechanisms of chemical reactions (on the basis of the final equilibrium model only and without kinetic constraints) is described in the works by Gorban et al. (2001, 2006), Kaganovich and Filippov (1995), and Kagano­ vich et al. (1993). Possibility of equilibrium thermodynamic modeling of fluid transpor­ tation mechanisms will be discussed on the example of optimal synthesis problem of multiloop hydraulic systems that was stated by Khasilev, the founder of the theory of hydraulic circuits (Khasilev, 1957, 1964, 1966; Merenkov and Khasilev, 1985) and was studied in many works (see, for example, Kaganovich (1978); Kaganovich and Balyshev (2000); Merenkov et al. (1992); Sumarokov (1976)). We will formulate this problem as a MEIS

42

B.M. Kaganovich et al.

1 A1 ← → A1

7 ← A3 → A2

3

8 ← A3 → A1

A2 ← → A3 6 1 A1

2

4

9 A3 ← → A3

A1, A2, A3,

3

2

A2 ← → A2

A1 ← → A2

4 A2 ← → A1 5 A2 ← → A3

Figure 8 A graph of hexane isomerization reaction.

modification that represents development and generalization of the model of stationary isothermal flow distribution (13)–(17): find " min Fðx; P Þ ¼ br

X

# Fi ðxj ðxÞ; Pbr i Þ

¼ Fðxext Þ

ð58Þ

i

subject to

Ax ¼ Q; n X i¼1

Pmov xi  i

n X i¼1

Pbr i xi ¼ 0;

Dt ðyÞ ¼ fx : x  yg; ’r ðxÞ  cr ;  Pbr i ¼  i xi ;

ð59Þ

ð60Þ

ð61Þ

r 2 Rlim ;

ð62Þ

i ¼ 1; … ; n;

ð63Þ

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

in  = 2 (turbulent flow) i ¼

8l0 ; 2 d5

br  Fi ¼ axi ðxÞPbr i þ bxi ðxÞ= ðPi Þ þ c;

43

ð64Þ

ð65Þ

where F and Fi—the cost (economic) characteristics of the whole network and its i-th branch; —a coefficient of the i-th branch resistance; —a coeffi­ cient of friction; —the fluid density; li and di—the pipeline length and diameter of the i-th branch, respectively; a, b, and c—coefficients; and — exponents depending on the exponent value at xi in (63) k ¼ ak þ bk dz :

ð66Þ

The first term in the right-hand side of (65) is proportional to the energy consumption to move fluid, the second is proportional to capital invest­ ments (pipe diameters for the pipeline network), and the third represents a fixed part of expenditures. Figure 9 gives an insight into the potential objects of studies by model (58)–(65). It shows a scheme of the main double-pipe water heat network of the heat supply system for a large urban district. The optimal synthesis problem for this network consists in the determination of flow distribution

1 2 3 4 5

Figure 9 The scheme of heat supply system in a “double-line” representation 1, 2—sections of supply and return pipelines, 3—heat source, 4—nodes of consumer connection; 5—pumping station.

44

B.M. Kaganovich et al.

over the scheme branches (the zero-flow branches are excluded from the scheme) and pipeline diameters. Hence, it includes a hydrodynamic (phy­ sical) and a technico-economic component. Herewith, in addition account can be taken of the requirements to heat supply reliability (e.g., by specify­ ing the condition of two-sided supply of individual consumers) and ecol­ ogy (e.g., by specifying the constraints on loading a less environmentally sound heat source). Interrelations between the simultaneously solved problems of hydro­ dynamic (calculation of xi) and technico-economic (choice of diameters di) optimization of the network are revealed by taking as initial the empirical Darcy–Weisbach equation Pbr ¼

w2 8x2 l ¼ 2 5 l; 2d d

ð67Þ

from which formula (64) was obtained to determine . In (67) w is the fluid speed. When Khasilev studied mathematical properties of the problems on choosing the vectors x and Pbr (d) (Khasilev, 1957), he varied the type of equations (67) based on the exponent change at w and x from unity (a laminar mode of fluid flow) to two (a turbulent mode). He established that the functions F and Fi are concave along the axes xi and convex along the directions Pbr (on the assumption that 0 £  £ 1 and 0 £ £ 1). Hence, i problem (58)–(65) does not belong to the CP. Since with the fixed value of vector Pbr and the lack of constraint (62) the admissible region of solutions is a polyhedron, F reaches its minimum at one of its vertices. With the rank of matrix A equal to m – 1 and n unknowns the reference solution contains no less than n – (m – 1) zero components, which equals the number of chords of the system of inde­ pendent loops of the network graph. In this case the graph tree is a polyhedron vertex and the optimal variant should be among the set of trees of a redundant scheme. The problem of choosing Pbr with the fixed x is a problem of CP. Khasilev carried out an interesting analysis of the problem with respect to the properties of F(Pbr) (disregarding (62) Khasilev, 1957). To do this he applied dimensionless characteristics e¼

Pbr ext ðPbr Þ

;

¼

ðF  Fext Þ : ðF  cÞ

ð68Þ

Transforming (65) based on (68) Khasilev devised the equation ¼

j" 1 þ  1; ð1 þ jÞ Ej ð1 þ jÞ

ð69Þ

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

45

where the value of the constant j is determined by the exponent value at x in (63) and the relationship between capital investments in pipelines and their diameters (66). Analysis of (69) in the context of our discussion is of interest, first of all because it reveals independence of , i.e., the relative change of economic cost characteristics of the system, from the coefficients a, b, and c and hence, independence from such technico-economic indices determining these coefficients as specific costs of electricity used for fluid pumping, specific capital investments in pipelines with different dia­ meters, depreciation and repair charges, etc. (naturally, these indices determine absolute values of costs and their variations at pressure loss deviation from the optimum). The only factor that influences the shape of the curve = f("), i.e., mathematical features of the economic optimization problem of pressure loss distribution over the hydraulic network branches in the form of a tree, is the exponent value at x in the hydrodynamic relation (67). Specifically, from Equation (69) follows the property of exceptionally great flatness of near the optimum point (" = 1). For example, for the turbulent fluid flow ( = 0.19) a twofold pressure loss in comparison to the optimal value increases transportation cost by 4.6% and a twofold reduc­ tion of loss decreases the cost only by 3.8%. For the linear electric networks (Equation (69) is also true for them) the corresponding figures are much higher and account for 8.3 and 25.0%. The revealed property of economic function flatness allows a reasonable simplification of the pressure loss optimization methods. Before discussing of the general method to solve problem (58)–(65) (joint optimization of x and Pbr it should be noted that the pressure losses and pipe diameters in branched networks with different constraints, including those of type (62), can be effectively optimized by the dynamic programming method (Kaganovich, 1978; Merenkov and Khasilev, 1985; Merenkov et al., 1992). It is applicable to parameter optimization only in the tree-like schemes. For the closed multiloop networks xi = f(x) and corre­ spondingly, the cost characteristics of individual branches Fi = c(x), i.e., the minimized economic characteristic of the network as a whole, prove to be nonadditive, which does not allow the use of dynamic programming. The method of coordinatewise optimization was proposed for simul­ taneous choice of flow rates and pressure losses on the closed redundant schemes (Merenkov and Khasilev, 1985; Merenkov et al., 1992; Sumaro­ kov, 1976). According to this method motion to the minimum point of the economic functional F(x, Pbr) is performed alternately along the concave (F(x)) and convex (F(Pbr)) directions. The convex problem is solved by the dynamic programming method and the concave one reduces to calcula­ tion of flow distribution. The pressure losses in this case are optimized on the tree obtained as a result of assumed flow shutoff at the end points of some branches. The concave problem is solved on the basis of entropy

46

B.M. Kaganovich et al.

maximization of an isolated system including a hydraulic network and its environment. As was shown in Section 3.2, the sought maximum corre­ sponds to the minimum of entropy production (the minimum energy dissipation) in the open subsystem, i.e., in the network. From the “physico-economic” standpoint convergence of the cho­ sen method can be explained by the fact that it naturally represents the tendency of an open system with fixed conditions of interaction with the environment to equilibrium, which corresponds to minimum production of both physical and economic entropy. Optimization for the obtained “technico-economic mechanism” determines flow distri­ bution corresponding to the minimum energy consumption, i.e., a physical mechanism. Thus, in this case the model of equilibrium thermodynamics—MEIS solves the problem of self-organization, ordering of the “physico-economic” system that is referred as a rule to the area of applications of nonequilibrium thermodynamics or synergetics. Note that the coordinatewise optimization method has already found numerous practical applications to optimization of heat, oil, water, and gas supply systems (Merenkov and Khasilev, 1985; Merenkov et al., 1992; Sumarokov, 1976). As a matter of fact, in the algorithms used for applied problems the flow distribution was calcu­ lated not on the base of entropy maximization, but with the help of the closed system of equations of the first and second Kirchhoff laws. However, because of equivalence of approaches that are based on the principle of conservation and equilibrium (extremality) the Kirchhoff equations can be strictly replaced by thermodynamic relations. And the extreme thermodynamic approach in many cases should be prefer­ able owing to the known low sensitivity of the extremal methods to variation of the space of variables.

4.3 Comparison of computational efficiency of equilibrium and nonequilibrium approaches Comparative simplicity of MEIS-based computing experiments is due primarily to the simplicity of the main initial assumption of its construc­ tion on the equilibrium of all states belonging to the set of thermodynamic attainability Dt(y) and the identity of their physico-mathematical descrip­ tion. These states belong to the invariant manifold that contains trajec­ tories tending to the extremum of characteristic thermodynamic function of the system and satisfying the monotonic variation of this function. The use of the mentioned assumption consistent with the second thermody­ namics law allows one, as was noted, not to include in the formulation of the problem solved different more particular principles, such as the Gibbs

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

47

phase rule, Raoult’s and Henry’s laws for diluted solutions, principles of the linearity of motion equations, etc. The lacking special description of the Gibbs phase rule in MEIS that should be met automatically in case of its validity is very important for solution of many problems on the analysis of multiphase, multicomponent systems. Indeed, without information (at least complete enough) on the process mechanism (for coal combustion, for example, it may consist of thousands of stages), it is impossible to specify the number of independent reactions and the number of phases. Prior to calculations it is difficult to evaluate, concentrations of what substances will turn out to be negligibly low, i.e., the dimensionality of the studied system. Besides, note that the MEIS application leads to departure from the Gibbs classical definition of the notion of a system component and its interpretation not as an indivi­ dual substance, but only as part of this substance that is contained in any one phase. For example, if water in the reactive mixture is in gas and liquid phases, its corresponding phase contents represent different para­ meters of the considered system. Such an expansion of the space of vari­ ables in the problem solved facilitates its reduction to the CP problems. Errors in the description of nonequilibrium processes in the linear nonequilibrium thermodynamics (Glansdorff et al., 1971; Kondepudi et al., 2000; Prigogine, 1967; Zubarev, 1998) are caused primarily by the assumptions (unnecessary at MEIS application) on the linearity of motion equations. One of the main equations of this thermodynamics has the form Jj ¼ Ljk Fk ;

ð70Þ

that relates flows with the forces creating them. The assumption on linear­ ity of differential equations describing fluctuations underlies derivation of the Onsager relations and the Prigogine theorem. These assumptions cause inaccuracy of the formulas for the Onsager kinetic coefficients L. Below are some of them (Kondepudi et al., 2000): Lqq ¼ kT 2 ;

Lee ¼

L11 ¼ D1 T

T ; r

�� � �  1 n1 @1  1 1þ ;  2 n2 @2

L11 ¼

D1 n1 ; R

ð71Þ

ð72Þ

ð73Þ

ð74Þ

48

B.M. Kaganovich et al.

L¼

Rf ; eq Rr ; eq ¼ : R R

ð75Þ

The formulas refer: (71) to thermal conductance; (72) to electric current; (73) to diffusion in the two-component mixture; (74) to dissolution in the ideal solution; (75) to the single-stage chemical reaction. In the formulas k is a coefficient of thermal conductance; r is the electric resistance per unit of conductor length; D1 is a coefficient of diffusion of substance soluble;  and n are the specific volume and the number of moles, respectively; Rf,eq and Rr,eq are the rates of the forward and reverse reactions in the equili­ brium state. Analysis of formulas (71)–(75) shows that the Onsager coefficients are connected with the constants of the corresponding processes (k, r, D) through the quantities (T, v, n, , Rf, Rr) that vary during relaxation to the equilibrium state. It is clear that T will change essentially in the thermal conductance process; v, n, and —during diffusion and dissolution; Rf and Rr—in chemical reactions. Therefore, the Onsager coefficients are constant only close to the equilibrium state. This fact causes errors in calculations even in the quantitative analysis of the simplest ideal systems. Analysis of complex large-dimensional real systems firstly, requires that the whole mechanism of the process modeled be known (which is usually impossi­ ble) to derive formulas similar to (71)–(75) and construct the equality of type (4). Secondly, it proves to be feasible only at the essential approxima­ tion of the obtained analytical relations. This makes clear the difficulty of applying the nonequilibrium thermodynamics models to solution of sophisticated computational problems because of inevitable low accuracy of the obtained results in many cases. Surely, despite the absence of requirements to linearity of any relations the application of linear approximations providing convenient (from the computational viewpoint) statement of the problem solved at MEIS-based modeling is admissible. Simplicity of the initial assumptions in MEIS construction stipulates to a great extent both comparative simplicity of the mathematical apparatus applied and easiness of initial information preparation. Simplification of mathematical descriptions concerning kinetics and nonequilibrium ther­ modynamics is seen first of all in the transition from differential to alge­ braic and transcendent equations that provides sharp decrease in the number of used complex analytical dependences (for example, similar to (73)). Decrease of the initial information volume directly and largely depends on the fact that there is no need to know a complete detailed mechanism of the studied process. The use of MEIS with variable para­ meters (7)–(12) calls for the information on individual limiting stages only. Substitution of the assignment of the list of reactions by the assignment of

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

49

the list of substances is a main computational advantage of this modifica­ tion of MEIS. Certainly, determination of the composition of vectors x and y is also connected with solution of a series of complex problems. Appreciable errors in calculations can be caused by the incomplete list of the vector x components. If the modeled system has a gas phase, theoretically this list can reach astronomic sizes, since in this case xeq is an interior point of the polyhedron of the material balance2, which in a general case implies a complete set of substances formed from the elements of components y. Composition and sizes of the assigned list determine the possibility for revealing superequilibrium contents of the sought set of substances, which is shown in (Gorban, 1984). Quantitative estimates of errors in the determined numerical values of substance concentrations at points xeq and xext as a function of the struc­ ture and dimension of x and y are obtained with great difficulty. It is only clear that when we are interested in the detailed composition of products, it is desirable to increase this dimension with thorough choice of the set of components xj and yj based on the whole preliminary knowledge about specific features of the studied process. Such an increase will be limited by the possibility to analyze numerous results. However, despite the great sophistication of the problem of specifying a list of substances, it is solved much easier than the problem of specifying a process mechanism. Both the list of elementary reactions (that can include many hundreds and even thousands of elements) and the constants of their rates are hard by far to determine than the list and thermophysical properties of reactants of the studied system. Simplification of the solution or complete exclusion of the problem of dividing the variables into fast and slow is a great computational advan­ tage of MEIS in comparison with the models of kinetics and nonequili­ brium thermodynamics. The problem is eliminated, if there are no constraints in the equilibrium models on macroscopic kinetics. Indeed, the searches for the states corresponding to final equilibrium of only fast variables and states including final equilibrium coordinates of both types of variables with the help of these models do not differ from one another algorithmically. With kinetic constraints the division problem is solved by one of the three methods presented in Section 3.4, which are applied in the majority of cases to slow variables limiting the results of the main studied process. On the whole, simplicity of the initial assumptions and correspond­ ingly comparative simplicity of the mathematical formulation of MEIS allows one to include in it rather easily descriptions of the most diverse conditions of great influence on the results of the studied process. In 2

If the components of x are substances, rather than their phases.

50

B.M. Kaganovich et al.

particular, it becomes possible to take into consideration comprehen­ sively enough constraints on kinetics; transfer and exchange of energy, mass and charges; fixing the parameters of the environment and in different zones of the modeled system. More detailed representation of the model, in turn, makes the comprehensive and deep analysis on its base more feasible and enriches both theoretical understanding of the considered phenomena and applied knowledge for a technologist, designer or constructor.

5. EXAMPLES OF MEIS APPLICATION 5.1 Introductory notes In the works devoted to study and development of MEISs numerous examples on their application to the analysis of various problems were certainly presented. They are formation of harmful substances during fuel combustion and cleaning of combustion products from these components, fuel processing, atmospheric pollution with anthropogenic emissions, sta­ tionary and nonstationary flow distribution in hydraulic systems, etc. These examples should illustrate practical efficiency of MEISs, their cap­ abilities for revealing specific features of the modeled process and deter­ mining directions of its improvement. This Section deals with the problem of MEIS comparison with the models of motion that was studied in the previous Section. However, whereas comparison was performed there on the basis of purely theore­ tical analysis, here it was made on the examples of specific objects. Com­ pared are the attainable completeness and significance of the results of computing experiments, and the possibility of using these results, accu­ racy of the obtained estimates for the sought characteristics of the modeled system, laboriousness of calculations and preparation of initial information.

5.2 Isomerization In the works devoted to MEISs isomerization became a “through” exam­ ple for explanation of their specific features and efficiency. The example is simple and very obvious, since the isomerization reaction at any mechan­ ism is described by the same material balance because of invariable amounts of substances and elements. This fact essentially facilitates both analytical and graphical interpretations. Here the comparative analysis of MEIS characteristics will be made on the example of the simplest mechanism: x1 ! x2, x2 ! x3 that was consid­ ered in Section 3.5. But the applied model is supplemented with the

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

51

constraint on the rate of the second reaction stage, i.e., the studied problem is represented in the form: find max x3

ð76Þ

x1 þ x2 þ x3 ¼ 1;

ð77Þ

Dt ðyÞ ¼ fx : x  yg;

ð78Þ

kx2  c;

ð79Þ

xj  0:

ð80Þ

subject to

Obviously, model (76)–(80) does not require any comments. The problem of determining the constant c was discussed in Section 3.4 and is not treated additionally here. The stated problem is presented graphically in Figure 10. The thermo­ dynamically unattainable zones from y subject to (78) are indicated by the hatched area of the triangle A1A2A3 of the material balance. Besides, the triangle contains the points xeq for final equilibrium, ~x ext for maximum concentration of x3 without constraint (79) and xext for the largest attain­ able value of the third isomer concentration with a complete system of constraints (77)–(80). The straight line kx2=c representing kinetic con­ straint (79) makes unattainable the part of Dt(y) to the left of it. Owing to

A1 (y) k2x2=Ψ

G = const

G = const

•x

eq

x ext

∼ X ext A2

Figure 10

Graphical interpretation of isomerization process.

A3

52

B.M. Kaganovich et al.

this constraint solution to problem (76)–(80) shifts from x~ ext to xext (inter­ section of the straight line kx2 = c with the boundary of the unattainability zone near the vertex A3). It is seen by sight that xext is somewhat distant from the vertex A3 (the point of maximum concentration of x3 with the only constraint of the material balance) than ~x ext and hence the mole content of the target product in xext is lower than in ~x ext . Even very short interpretation of the problem allows a most important advantage of MEIS to be indicated, namely its capability to choose and determine the value of the subjective parameter of order (Klimontovich, 1997) of the modeled system. In this case the state with the maximum possible content of the target product of the process—x3 and correspond­ ingly, with the lowest content of “waste”, i.e., useless substances contam­ inating a produced required “valuable” commodity, is naturally thought to be the most ordered one. In parallel with assessment of the maximum concentration the computing experiments on MEIS determine conditions for its achievement and reveal the factors having the greatest influence on the results of modeled process. This becomes possible owing to the MEIS description in MP language (the theory of extremal problem solution) and capabilities of the computational mathematics as a whole (in particular, capabilities of making multivariant calculations with variation of both the values of initial parameters, and applied dependences between the para­ meters, including those specified in the nonanalytic form, and presenting calculation results in a convenient tabular and graphical forms). The example of MEIS-based analysis of the “physico-economic” self-organiza­ tion problem was treated in Section 4.2. The MEIS advantages in detailed analysis of attainability of the ordered states and limiting values of the order parameters are examined in the next Section on an example of nitrogen oxides formation during fuel combustion. Isomerization will be used as an example to explain to some extent the issues of comparing laboriousness of computing experiments and their accuracy. For the assumed process mechanism the kinetic model has the form: x1 þ x2 þ x3 ¼ 1; dx1 ¼ k1 x1 ; d

dx2 ¼ k2 x2 þ k1 x1 ; d

x  0:

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

53

Solution to this system (integrals of differential equations) is: x1 ¼ exp ðk1 Þ; x2 ¼

x3 ¼ 1 

k1 ½exp ðk1 Þ  exp ðk2 Þ; k2  k1

k2 k1 exp ðk1 Þ þ exp ðk2 Þ; k2  k1 k2  k1

In Figure 11 the curves corresponding to these equations at k1 = 1c1 and k2 = 0.5c1(the values are chosen for the purposes of illustration, the real values of rate constants for the monomolecular reactions can be by many orders of magnitude higher). As is seen from Figure 11, at ! 1 x3 ! 1 and x1 ! 0, which is not allowed by thermodynamics (see Figure 10). The results of kinetic and thermodynamic analysis could surely be coordinated by including the reverse reactions in the considered mechanism: x2 ! x1 and x3 ! x2 and assigning for both stages the values of rate constants strictly corresponding to thermodynamics. For complex problems con­ cerning the studies of multistage processes such a growth of dimension­ ality can sharply increase laboriousness of computing experiments and cause great difficulties in preparation of initial information. The men­ tioned difficulties will increase still further, if the constraints on rates of transfer and exchange processes are inserted in MEIS in parallel with the constraints on chemical kinetics. In this case it will be necessary to harmo­ nize the values of reaction rate constants and the values of constant coefficients in the equations of Fourier, Fick, Navier-Stokes, etc.

1.0 0.8

xi, mole x1

x3

0.6 0.4 x2 0.2 τ, s

0 3

Figure 11

6

9

12

Curves of the kinetic equations for isomerization process.

15

54

B.M. Kaganovich et al.

If we refuse to excessively increase dimensionality and laboriousness of kinetic descriptions, their accuracy can turn out to be lower than the accuracy of the MEIS-based estimates. This was just the case for the considered example, when the results of solving kinetics equations proved to be contradictory to the thermodynamics laws. Needless to say that the accuracy of thermodynamic modeling can be improved unlimitedly by increasing the number of constraints on the macroscopic kinetics. Figure 10 shows that solely constraint (79) in model (76)–(80) sharply decreased Dt(y). However, it should be understood that the increase in accuracy leads to partial or complete loss of such a traditional advantage of thermo­ dynamics as simplicity and possibility of constructing geometrical inter­ pretations of the models applied. While assessing comparative advantages of the equilibrium thermo­ dynamic modeling, one should remember that with any possible and obligatory expansion of the area of thermodynamics applications and the increasing fruitfulness of thermodynamic modeling it can never sub­ stitute and make useless the motion models which determine the rate and time of process course in physicochemical and engineering systems. Kinetics will be always a significant element in designing and constructing diverse engineering objects and studying natural processes.

5.3 Formation of nitrogen oxides during coal combustion This example belongs to a highly complex physicochemical system. It reveals capabilities of equilibrium thermodynamic modeling of such a purely irreversible process as coal combustion. The calculation using the traditional MEIS (Gorban et al., 2001, 2006; Kaganovich et al., 2006c) shows that the global equilibrium reached by such a system gives rise to formation of a great amount of NO. In practice, however, such amounts of nitrogen oxides are not formed and this fact is indicative of the system transition only to the intermediate equilibrium state because of kinetic factors (bonds). Hence, model (7)–(12) is applicable to study this system. In the model condition (10) for kinetic constraints should be written in the thermodynamic form by using time as a parameter that is determined by technological characteristics of the system (length of the reaction path, flow rates) and the scales of microscopic inhomogeneities of the reaction space. For this purpose it is sufficient to analyze only limiting stages of all basic mechanisms out of numerous chemical reactions that participate in NO formation during coal combustion. Note that by virtue of their sig­ nificance these processes are studied in sufficient detail and the informa­ tion on kinetic coefficients is most reliable (Warnatz et al., 2001). The basic mechanisms of NO formation will be described below in short. Fuel Nitrogen Oxides, according to the current views, are produced from nitrogen-containing compounds of coal in the initial section of torch

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

55

at the temperature 900–1,000 K. Transformations of the fuel nitrogen in the process of thermal destruction of the nitrous substances of the organic coal mass (OCM) can be represented by a simplified scheme: NðcarbonÞ ! CN ! NCO ! NH ! N ! NO:

ð81Þ

All these transformations take place in the region beyond the surface layer of the coal particle, where the gas phase is enriched with oxygen owing to intensive turbulent mixing. The final stage in the chain is NO formation from the monatomic nitrogen by the reactions: N þ OH ! NO þ H;

ð82Þ

N þ O2 ! NO þ O:

ð83Þ

In the oxidizing medium they are more preferable than other reactions bonding active nitrogen (for example, N + NH ! N2 + H). Therefore, according to some data in (Warnatz et al., 2001) up to 70% of fuel nitrogen is converted to NO by this scheme, which makes up on the average 5–7 kg per ton of fuel burnt. The competitive reaction decreasing nitrogen oxide formation in this region is: N þ NO ! N2 þ O:

ð84Þ

The process, in the course of which volatile nitrogen-containing compo­ nents of OCM leave the coal particle and break down to the nitrile radicals, is a limiting stage of this mechanism. This process is presented in scheme (81) as N(carbon) ! CN. Thermal Nitrogen Oxides start to form virtually in the same reaction space region as the fuel ones during fuel combustion. According to the Zeldovich mechanism formation of fuel nitrogen oxides includes elemen­ tary reactions (82) and (83) and reactions of active nitrogen generation in this region from atmospheric nitrogen: O þ N2 ! NO þ N:

ð85Þ

Reaction (85) is limiting in this mechanism and has high activation energy (about 318 kJ/mole (Warnatz et al., 2001) ) because of the strong triple bond in the nitrogen molecule. For this very reason the probability of reaction (84) reverse to (85) is high (activation energy is some 27 kJ/mole Warnatz et al., 2001) ), which causes the NO content in the low tempera­ ture zones to decrease. Prompt Nitrogen Oxides emerge because of the lack of oxidizer in the reaction medium. Their formation (Fenimore mechanism) is based on the following reactions:

56

B.M. Kaganovich et al.

CH þ N2 ! HCN þ N;

ð86Þ

HCN þ 2O ! NO þ CO þ H:

ð87Þ

When burning coal and volatiles, by virtue of diffusion limitations on delivery of oxygen molecules to the reaction surface the oxidizer deficit occurs close to the coal particle surface, which leads to formation of a considerable amount of CH radicals. In parallel the atmospheric nitrogen appears in this region, which favors the course of reaction (86). In the case of strong turbulization of the reaction space the rate of molecular diffusion limits oxygen (as well as nitrogen) access into the turbulent vortex of volatile hydrocarbons, whose size is 104 m. Therefore, reaction (86) can be supposed to proceed basically in the surface layer, where generation of the CH particles from the volatile components of OCM to the reaction region will be a limiting stage. The products of this reaction pass to the gas phase, where hydrogen cyanide is oxidized in accordance with (87) with a low potential barrier and the nitrogen radical N can participate in reac­ tions (82)–(84). NO Formation from Dinitrogen Oxide (Nitrous Oxide) takes place during combustion of gaseous hydrocarbons of volatiles in the case of lean mixtures. In accordance with this mechanism at first the dinitrogen oxide N2O is formed by the termolecular reaction: N2 þ O þ M ! N2 O þ M;

ð88Þ

(M—any particle), then the molecule N2O interacts with the oxygen atom: N2 O þ O ! 2NO:

ð89Þ

This mechanism of NO formation is believed to be basic for burning lean mixtures, when the Fenimore mechanism is already inefficient because of absence of CH radicals. Reaction (88), being termolecular, notably accelerates at high pressures and is considered to be limiting in this case. Relatively low activation energies of reactions (88) and (89) make this mechanism responsible for nitrogen oxides formation at low temperatures and pressure of several MPa, when the thermal nitrogen oxides are not virtually formed. Since coal is burnt, as a rule, at the pressure close to atmospheric, this mechanism may not be considered below. The presented brief survey of basic mechanisms of NO formation during coal combustion allows the MEIS construction with their simulta­ neous inclusion in the kinetic constraints. Kinetic constraints can be for­ mulated according to the third way among those considered in Section 3.4.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

57

However, several general principles in derivation of multifactor con­ straints in MEIS should be underlined. Different limiting processes in the form of kinetic constraints in one thermodynamic model can be taken into account by representing the modeled system as two subsystems—slow and fast. A slow subsystem naturally includes all the limiting stages of different mechanisms and the related processes in the form of a closed system of autonomous kinetic equations. As far as there are usually few limiting processes even in complex physicochemical or other modeled objects, construction of such a system and its solution (analytical or numerical) generally causes no difficulties. Moreover, since we consider conditions for limitation of the thermodynamic attainability region, part of kinetic equations can be sub­ stituted by algebraic ones based on the upper (lower) estimates of some variables. This procedure is even necessary, if the kinetic curves of any components are nonmonotonic. Separate points of the phase trajectory of the slow subsystem (i.e., its states) that is obtained by solving the equa­ tions correspond to some time parameters of a real object, e.g., the time of passage of the reaction mixture through the flow reactor or the time of diffusion through the boundary layer in the heterogeneous process. If we take these time parameters on the phase trajectory as constant, i.e., limit it on the path to attaining a global equilibrium as a system of inequalities (10) in model (7)–(12), the region of thermodynamic attainability will be constrained for the whole system. In this case several points of constraints for different components of the slow subsystem that correspond to differ­ ent time scales of the limiting processes can be analyzed simultaneously on one phase trajectory. Thus, for the kinetic limitation of the thermody­ namic attainability region MEIS applies separate states of the slow sub­ system that belong to its phase trajectory. Based on the above said, MEIS intended for study of nitrogen oxides formation in the process of torch combustion of coal at constant P and T can be written as follows: find max xNO

ð90Þ

Ax ¼ b;

ð91Þ

Dt ðyÞ ¼ fx : x  yg;

ð92Þ

subject to

G¼

X j

Gj xj ;

ð93Þ

58

B.M. Kaganovich et al.

xj  ci ;

ð94Þ

xj  0:

ð95Þ

Expression (94) in this model is a system of kinetic constraints for compo­ nents of the reaction medium. Constraints (94) will be determined through the rates of NO and N formation on the basis of reactions (82)–(87). Here account will taken of the above assumption that in the mechanism of forming prompt nitrogen oxides reaction (86) proceeding under oxidizer deficit in the surface layer of the coal particle is limiting and further all the hydrogen cyanide converts to NO beyond this region at oxygen excess. This assumption is sound owing to the low activation barrier of reaction (87) (E 25.6 kJ/mole). dxNO ¼ k ð82 Þ xN xOH þ k ð83 Þ xN xO2 þ k ð85 Þ xO xN2 þ k ð86 Þ xCH xN2  k ð84 Þ xN xNO ; ð96Þ d

dxN ¼ k ð85 Þ xO xN2 þ k ð86 Þ xCH xN2  k ð82 Þ xN xOH  k ð83 Þ xN xO2  k ð84 Þ xN xNO ; ð97Þ d

Since generation of the atomic nitrogen is a limiting stage in the process of NO formation in these conditions, we can suppose that in terms of other faster reactions the equilibrium condition dxN =d ¼ 03 is satisfied for it (xN here is some equilibrium value of nitrogen radical content in the reaction region). Then the rate of nitrogen oxide formation (96) can be written as the equality: � � dxNO ¼ 2x N k ð82 Þ xOH þ k ð83 Þ xO2 : d

ð98Þ

System (97)–(98) comprises the following unknowns: xNO, xN , xOH, xCH, xO2, xN2. Hence, four equations more should be added to make it closed. The relations for the molecular nitrogen and oxygen are most easily represented as an estimate of the upper boundary of their amount: xO2  yO2

ð99Þ

x N2  y N2 :

ð100Þ

and

Assuming that under high humidity of the reaction mixture the OH content is controlled by the reaction: H þ OH>H2 O; 3

This condition is also true for other short-lived radicals: O, ON, SN, etc.

ð101Þ

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

59

that belongs to the “fast” subsystem, one more equality can be added to Equations (97)–(98): � �0:5 xOH ¼ Kp ð101 Þ xH2 O ;

ð102Þ

xH2 O  yH2 O :

ð103Þ

and the relation

Let us note that Equations (102), (103), and (107) supplement the immedi­ ate use of kinetic equations for formulation of inequality (94) with a “thermodynamic approach” (see Section 3.4). The quantities xN and xCH will be determined based on the following facts. In the sequence of fuel nitrogen oxides formation (81) all the stages, except for the first, proceed without the activation barrier and as a first approximation they can be taken as fast with respect to the rate of volatiles burning. Assuming that under these conditions according to reactions (85) and (86) the key supplier of active nitrogen to the gas phase is nitrogencontaining components of coal organic matter and CH radicals, we will write the relations for the rates of forming xN and xCH4: dx N dxNðcarbonÞ ¼ þ k ð85 Þ xO xN2 þ k ð86 Þ xCH xN2 ; d

d

ð104Þ

dxCHðcarbonÞ dxCH ¼ : d

d

ð105Þ

In these expressions the quantities xN(carbon) and xCH(carbon) are active nitrogen and CH radicals supplied to the reaction from organic matter of coal, the second term in the right-hand side of (104) corresponds to the mechanism of forming the thermal (85) and prompt (86) nitrogen oxides. Since one more unknown xO appears in Equation (104), the system should be sup­ plemented with the relation determining it. It can be obtained from the fast reaction: O þ O>O2 ;

ð106Þ

which can limit the number of O radicals as a first approximation. Then: � �0:5 xO ¼ Kp ð106 Þ xO2 :

ð107Þ

Integration of (98), (104), and (105) with an account of (99), (100), and (103) results in the following system of inequalities:

4

These rates are not equal to zero at the time intervals less than D (see below).

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� �

xNO  2x N k ð82 Þ xOH þ k ð83 Þ yO2 j0f � � x N  xNðcarbonÞ þ k ð85 Þ xO yN2 þ k ð86 Þ xCH yN2 j 0D � �0:5 � �0:5 xOH  Kp ð101 Þ yH2 O ; xO  Kp ð106 Þ yO2

) :

ð108Þ

xCH  xCHðcarbonÞ

Here f —the time of complete burning of volatile components of coal (Ots, 1977):

f ¼ kf1 exp

�

� � � Ef  ln ; RT  þ 0:01V daf

ð109Þ

where —a coefficient of excess air; Vdaf—content of volatiles in coal; kf and Ef—variables depending on the statistic characteristics of coal particle sizes. At the values of T = 1500 K, a = 1.2 and Vdaf = 47% the time interval

f accounts for approximately 0.02 s. The parameter D characterizes the time of diffusion of pyrolysis components from the coal particle surface into the turbulent reaction region and along with the chemical kinetics reflects the possibility of accounting for limiting processes of transfer in MEIS like it is done in (Ots, 1977). For the assumed initial parameters of the model D 3 104 s the “thermal” summand in (104) will be small because of diffusion-limited supply of oxygen atoms to the surface layer and mainly because of high concentration of such active reducing radicals in this layer as H, CH, CH2, etc. These radicals have no activation barrier, when they interact with oxygen, and will reduce the probability of course for reaction (85) practically to zero. Therefore, this summand can be neglected in calculations. The quantities xN(carbon) and xCH(carbon) can be determined from the following relations: daf xNðcarbonÞ  NðcarbonÞ kN Kf

ð110Þ

daf xCHðcarbonÞ  HðcarbonÞ kCH Kf

ð111Þ

daf where N(carbon) and H(carbon)—the number of these elements in coal; kCH daf and kN —coefficients for the composition of volatiles; Kf—a coefficient that determines the share of volatile substances moving from the coal particle to the gas phase for the time interval D. The quantity Kf can be calculated from the relation (Ots, 1977):

� � � ��� Kf =0:01V daf 1  exp kf D exp Ef =RT :

ð112Þ

Conceivably, in the limit all nitrogen of coal organic matter in the daf burning zone of volatiles turns into the active state, i.e., kN ¼ 1. The

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

61

daf quantity kCH can be determined using an auxiliary MEIS which determines max xCH and the composition of volatiles as an initial vector y. Then daf ¼ xext kCH CH =HðcarbonÞ . Use of the auxiliary model as the use of Equations (102), (103), and (107) supplements the kinetic deduction (94) with a thermodynamic one. System (108) determines constraints on nitrogen oxide formation on the basis of the three indicated mechanisms. Formation of thermal NO, however, continues after the burning of volatile components of coal up to some decrease in the reaction medium temperature5. Therefore, the righthand side of the first inequality of system (108) should be supplemented with the quantity xterm NO —the amount of thermal NO formed in the whole high temperature region of the torch:

b xterm NO ¼ 2k ð85 Þ xO yN2 j f ;

ð113Þ

where b—the mean time of reaction mixture passage through the com­ bustion chamber. Thus, the key kinetic constraints of the model that simultaneously describe three basic mechanisms of NO formation are determined. Finally, based on the aforesaid the system of constraints (94) is transformed in the following way: � � � � �� xNO  2 x N k ð82 Þ xOH þ k ð83 Þ yO2 f þ k ð85 Þ xO yN2 b  f ; � � daf x N  NðcarbonÞ kN Kf þ k ð85 Þ xO yN2 þ k ð86 Þ xCH yN2 D ; � � �0:5 �0:5 xOH  Kp ð101 Þ yH2 O ; xO  Kp ð106 Þ yO2 ;

)

ð114Þ

daf xCH  HðcarbonÞ kCH Kf :

It is easily seen that this system is linear with respect to the variables xj. The studies on NO formation by the traditional MEIS have been performed at Melentiev Energy Systems Institute for a long time. In parallel with MEIS the use was made of kinetic models and full-scale experiments that assisted in turn to gain information for variant calcu­ lations on MEIS. The results of these calculations allowed the condi­ tions for nitrogen oxides formation by different mechanisms to be determined and the ways for improvement of coal combustion technology to increase environmental safety of boiler units to be outlined. Figure 12 presents the results of calculations on model (90)–(95) (curve 7) in comparison with those performed earlier (Gorban et al., 2001, 2006) and experimental data. As is seen from the figure the calculations on the new model were in good agreement with the earlier results and proved to be even closer to experimental volumes of NO emissions by pulverizedcoal boilers. 5

The quantity of thermal oxides sharply falls at T below 1400 K.

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x, g/m3 2 1

1

2

3 7

5

6

4

B C A

800

1400

2000

T, K

Figure 12 Theoretical and experimental NO emissions at coal combustion that were calculated by model (90)(95) (curve 7) and presented in the work by Gorban (2001, 2006): equilibrium (1), maximum (2); actual (36): fluidized bed combustion (3), low-temperature combustion of brown coals (4), high-temperature combustion of hard coals (5), averaged for coal-fired boilers (6); A—“prompt” NO, B—“fuel” NO, C—“thermal” NO.

Somewhat overestimated calculation results in comparison with the in-situ measurements in a low-temperature region are explained probably by the fact that the model does not take into account NO reactions with nitrogen of reduced forms such as NH, NH2, NH3, etc., for example, NO+NH2!N2+H2O or NO+NH!N2+HO that are typical of compara­ tively low temperatures (Warnatz et al., 2001). At the same time calculations on the modified MEIS are possible with­ out additional kinetic models and do not require extra experimental data for calculations, which makes it possible to use less initial information and obviously reduces the time and labor spent for computing experiment. Furthermore, there arise principally new possibilities for the analysis of methods to mitigate emissions from pulverized-coal boilers, since at sepa­ rate modeling of different mechanisms of NO formation the measures taken can result in different consequences for each in terms of efficiency. Consideration of kinetic constraints in MEIS will substantially expand the sphere of their application to study other methods of coal combustion (fluidized bed, fixed bed, etc.) and to model processes of forming other pollutants such as polyaromatic hydrocarbons, CO, soot, etc. The advantage of thermodynamic models of such complex processes as coal combustion over kinetic ones can be clearly understood, if we briefly deal with the current problems of kinetic description of this process

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

63

(Warnatz et al., 2001) and compare it with the described thermodynamic approach. Coal burning is associated with three basic interrelated processes: coal pyrolysis, burning of volatiles and burning of coke. The kinetic model must include all these processes combined by the material and energy balances. In the thermodynamic model dealing with states it is sufficient to estimate limiting stages. To take into account, for example, pyrolysis it is enough to know only the rate of volatiles yield that can be calculated, as was shown, from the semi-empirical or even empirical relations. Kinetic description of pyrolysis requires that the chemical mechanism of the process and the diffusion coefficients on the coal particle surface and in the surface layer be known. Since the molecular chemical composition of coal used is not known exactly, even the listing of chemical reactions is a very complex scientific problem up to now. One can only guess the diffusion coefficient values, in so far as their measurement or theoretic description is extremely sophisticated and unreliable because of inhomo­ geneity and variation of coal particle surface during combustion. Note that this change should also be taken into consideration in the kinetic model. Hence even the first stage of coal combustion—pyrolysis—can be described only by the rough empirical (in the best case—semi-empirical) kinetic models that are true only in a narrow range of conditions typical of specific cases. In MEIS there is no need to describe the process of volatiles burning. Their preset composition is limited by the dimension of vector x, and can be increased to several hundreds of components, which virtually does not affect model complexity but somewhat increases the time of calculations. The results obtained allow the estimation and withdrawal from the vector x of the components of low impact on the calculation results. In the calculations we used 68 chemical components. In the kinetic model uncer­ tainty in the composition of volatile substances makes it impossible to describe in detail their combustion based on the elementary kinetics. The description in this case should also include processes of evaporation from the particle surface and diffusion. As a rule the parameters of these processes are unknown as well. And finally, the coke burning is a heterogeneous process. Its modeling includes description of the processes of molecule adsorption on the surface, surface reactions, desorption of reaction products, diffusion through the pores and diffusion to the particle surface. At present the majority of these processes for coke are relatively poorly known. The key distinction of surface reactions from reactions at the gas phase consists in the necessity to attract for description of their rates such notions as surface active centers and adsorbed particles. And in the kinetic models a different nature of active centers (different energy of dislocations) necessitates consideration of the same particles adsorbed on them as different compounds because of

64

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different constants of the surface rates. In MEIS different phase states are assigned to such particles, which leads only to the increasing dimension of component composition. In contrast to the constants of surface reaction rates, whose theoretical calculation is a rather complex problem and prac­ tical measurements have a low reproducibility, the thermodynamic para­ meters can be determined with a sufficiently high degree of accuracy from the spectral data and statistical calculations (Adamson and Gast, 1997). From the above said, it may be concluded that a detailed kinetic model of coal combustion process that combines all three basic processes can not virtually be constructed, as it is impossible to do for each process sepa­ rately. Therefore, the empirical models based on separation and experi­ mental study of the limiting stages are extensively used. Such models separately do not reveal general regularities and do not allow the general­ ized conclusions to be drawn. The thermodynamic model makes it possi­ ble to study the whole attainability region and hence to consider states of the considered system as a whole and to keep track of the variation in the amounts of any component as a function of some or other kinetic con­ straints. The latter are written, as was shown above, easily enough even for such complex processes as coal combustion. Advantages of the MEIS-based modeling of such complex chemical processes as nitrogen oxides formation at coal burning in comparison with the models of nonequilibrium thermodynamics prove to be even more clear and significant than its advantages compared with kinetic analysis. It is sufficient to mention only several facts. If the process mechanism is unknown, its analytic description required by the nonequilibrium thermo­ dynamics is impossible. Formalization of constraints on duration of indi­ vidual stages and concentrations of individual components seems to be highly difficult. For some transfer processes it is very hard to determine formulas for the Onsager coefficients such as for diffusion in the multi­ component medium. The assumption on linearity of motion equations will certainly adversely affect the accuracy of calculations.

5.4 Stationary flow distribution in hydraulic circuits The analysis of stationary and nonstationary flow distributions in multiloop hydraulic systems with lumped, regulated, and distributed para­ meters and in heterogeneous systems was given in (Gorban et al., 2001, 2006; Kaganovich et al., 1997). In the concluding section of Section 5 the abundant capabilities of the flow MEIS are illustrated by the simplest example of stationary isothermal flow distribution of incompressible fluid in the three-loop circuit. It is shown how the degrees of order (laminar or turbulent modes) on the branches of this circuit are deter­ mined from calculation of the final equilibrium.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

3

2

65

4

1

Figure 13 A scheme of the hydraulic circuit. 14—the numbers of nodes; the arrow in the circle—a source of effective pressure; the arrows specified directions of flows in the branches.

The design diagram of the hydraulic system is presented in Figure 13. The pressure generated by the pump Pmov (2 MPa) and characteristics of branches (pipe lengths and diameters, coefficients of resistances  i) are given. The model of final equilibrium (13)–(17) for the assumed conditions has the form:find max DS ¼ T

1

X

! Pbr i xi

ð115Þ

i

subject to Ax ¼ 0;

Pmov xi 

6 X i¼1

 Pbr j ¼  i xi ;

ð116Þ

Pbr i xi ¼ 0;

ð117Þ

i ¼ 1; … ; 6;

ð118Þ

where the exponent  in (118) is taken equal to unity for the laminar mode and to 2 for the turbulent mode. The sum in the parenthesis of the objective function equation (115) is the total kinetic energy of fluid flows that is converted into heat and then transferred to the environment. The results of flow distribution calculations are presented in Table 1. Table 1 shows that with the equilibrium stationary flow distribution that corresponds to the maximum entropy of an isolated system

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Table 1 Results of flow distribution calculation Branch

i

DPi, MPa

xi, m3/s



1–2 2–3 2–4 1–3 3–4 1–4

1.00  102 1.56  103 2.50  102 1.25  102 1.00  101 4.22  101

1 0.1 0.05 0.9 0.05 0.95

10 8 2 8.5 0.5 1.5

2 2 1 2 1 2

Note. The dimension of  is not indicated because of its dependence on the value .

(a combination of the circuit and the environment) and the minimum energy dissipation in a circuit (see Section 2.3) we have a quadratic closing relation (the turbulent flow mode) on four branches and a linear relation on two branches (the laminar mode). Thus, this example reveals the possibility of assessing the levels of order (self-organization) in individual elements (subsystems) of complex systems by means of the models of thermodynamic equilibria. As is known, in the nonequilibrium thermo­ dynamics and synergetics the turbulent mode is believed to be more organized than the laminar one. It should also be noted that the Prigogine theorem on the minimum entropy production is applicable to the circuit as a whole and for its individual branches (open subsystems). Actually, the maximum amount of entropy is formed in the environment owing to heat transfer to it from the hydraulic circuit. In the circuit itself the energy imparted to the fluid is entirely spent on its motion along the branches, i.e., on performance of effective work, and the entropy production at given conditions of interac­ tion with the environment takes its minimal value equal to zero. The minimality of DSi was shown in (Gorban et al., 2001, 2006).

6. CONCLUSION: WHAT WE HAVE AND WHAT WILL BE? In the first years of the current century a new direction was formed in development of the model of extreme intermediate states (Kaganovich, 2002; Kaganovich et al., 2004a, 2004b, 2005a, 2006a, 2006b, 2006c) that is defined in this paperchapter as equilibrium macroscopic modeling of nonconservative systems. The described attempts to generalize and develop the studies presented in (Kaganovich, 2002; Kaganovich et al., 2004a, 2004b, 2005a, 2006a, 2006b, 2006c) allow the progress achieved on this path to be assessed.

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67

The following problems were solved as a first approximation. The capabilities of equilibrium macroscopic modeling of irreversible processes in chemical transformations and mass and energy transfer, reduction of motion models to rest models (states) were revealed. The possibility to address kinetic constraints within a thermodynamic model (MEIS) unfolds the ideas first suggested in thermodynamic analysis of kinetic equations (Feinberg, 1972, 1999; Feinberg and Hildebrant, 1997; Feinberg and Horn, 1974; Gorban, 1984; Horn, 1964; Horn and Jackson, 1972). MEIS modifications with variable parameters and variable flows and also of spatially inhomogeneous systems were created. They include constraints represented in the thermodynamic form (without time vari­ able) on the irreversible macroscopic kinetics. Computational problems of devising methods on the basis of these modifications that reduce to CP were somewhat solved, which made it possible to construct some relevant computing algorithms. The efficiency of MEIS modifications was tested on the examples of modeling and analysis of fuel combustion and processing and flow dis­ tribution in multiloop hydraulic systems. The capabilities of MEIS and the models of kinetics and nonequili­ brium thermodynamics were compared based on the theoretical analysis and concrete examples. The main MEIS advantage was shown to consist in simplicity of initial assumptions on the equilibrium of modeled processes, their possible description by using the autonomous differential equations and the monotonicity of characteristic thermodynamic functions. Simpli­ city of the assumptions and universality of the applied principles of equilibrium and extremality lead to: the lack of need in special formalized descriptions that automatically satisfy the Gibbs phase rule, the Prigogine theorem, the Curie principle, and some other factors; comparative simpli­ city of the applied mathematical apparatus (differential equations are replaced by algebraic and transcendent ones) and easiness of initial infor­ mation preparation; possibility of sufficiently complete consideration of specific features of the modeled phenomena. At the same time the indicated valuable results may be treated only as a groundwork for further more versatile studies in comparison with the performed ones. The increasing versatility is due to more detailed com­ parison of MEIS with each of the basic macroscopic disciplines dealing with studies on the motion trajectories: chemical kinetics (Feinberg, 1972, 1999; Feinberg and Hildebrant, 1997; Gorban, 1984), theory of dynamic systems (Arnold, 1989; Katok and Hasselblatt, 1997), synergetics (Haken, 1983, 1988), nonequilibrium thermodynamics (Glansdorff et al., 1971; Kondepudi et al., 2000; Prigogine, 1967; Zubarev, 1998); finite time ther­ modynamics (Rozonoer L et al., 1973; Tsirlin, 2006) and with physico­ mathematical description and analysis of the main transfer processes; heat and mass exchange, electric current, radiation, including such phenomena

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as thermal diffusion, Dufour effect, electrokinetic and thermoelectric pro­ cesses; and with computational problems arising because of nonlinearity and nonautonomy of equations that describe constraints on the macro­ scopic kinetics; and with the solution of a large number of specific theore­ tical and applied problems. Special attention should be paid to determination of the role of the discussed scientific direction in the recently formulated more general direction—“Model Engineering” (Gorban, 2007; Gorban and Karlin, 2005; Gorban et al., 2007). The technology developed there makes it possible to choose an initial formalized problem statement that would be optimal from the standpoints of calculations and analysis. The best for­ mulations are searched for based on the reduction of the known equations of statistical physics, physical kinetics, or some macroscopic theories. Analysis of equilibrium thermodynamic modeling within the “technol­ ogy” in general requires that the possibilities of such modeling as the ultimate method of reduction—transformation of the models of motion to the models of rest be estimated. The list of the most important problems to be solved during further studies is presented in the following pages. Let us briefly comment upon this list, paying attention to the facts that are not described or almost not described in this chapter. The first group of problems (1–4) deals with the determination of fundamental capabilities of macroscopic models of equi­ libria in the study of irreversible processes. Whereas in the performed studies of MEIS the formalism of the motion and rest (equilibrium) the­ ories was compared exclusively at the macroscopic level, in the future it is intended to obligatorily reveal MEIS relationships with statistical physics and physical kinetics. Analysis of admissibility of equilibrium approxima­ tions should become important in the statistical microscopic substantiation of state models for devising macroscopic equations from the initial prob­ abilistic descriptions. Mathematical substantiation is based on the consid­ eration of a wide scope of problems: from application of differential calculus as a whole (note that the infinitesimal changes of heat and work are not in general differentials) to the possibility of using autonomous differential equations and MP. Special analysis of correctness of equili­ brium descriptions of explosions, hydraulic shocks, and other similar apparently irreversible processes seems to be needed, at least to be sure of the admissibility of equilibrium interpretations of “less nonequilibrium” phenomena. The second group of problems (5–8) is associated directly with MEIS construction. Here the main goal for the future is to expand directions of analysis of the formalized descriptions of transfer processes, whose neces­ sity was underlined above. Of particular interest are the studies on inter­ relations between thermal, electric, and chemical phenomena (the cross effects). Greater attention to transfer processes should be paid in

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

69

geometrical interpretations of MEIS. The idea of using a thermodynamic tree in the analysis of both chemical and macroscopic kinetics as a whole is also attractive. The computational problems of the third group (9–16) inevitably stem from the conditions of solving the problems of the first and second groups. Analysis of situations, when a high irreversibility level compli­ cates application of the notion of function differential and especially the CP methods (items 10 and 13), is considered as the greatest extent of novelty here. Certainly, the list of specific problems of the fourth group may be extended unrestrictedly. The authors favored the problems concerning the energy research area they are engaged in. The problem of MEIS creation for an economic system (item 21) that was dwelt on in Section 4.2 is undoubtedly of interest from the viewpoint of assessing the capabil­ ities of equilibrium thermodynamic modeling and revealing the identity in description of physical and socio-economic regularities. All 29 enumerated problems can be solved on the basis of long-stand­ ing studies of many experts. However, we hope that even partial perfor­ mance of the stated tasks will make the models and methods of the present-day equilibrium thermodynamics the property of a wide circle of researchers and engineers and they would find extensive application in the basic and applied science.

7. PROBLEMS OF EQUILIBRIUM THERMODYNAMIC DESCRIPTIONS OF MACROSCOPIC NONCONSERVATIVE SYSTEMS 7.1 Substantiation of the capabilities of equilibrium descriptions and reduction of the models of irreversible motion to the models of rest 1.

Comparison of MEIS capabilities (equilibrium descriptions) with capabilities of kinetics, theory of dynamic systems, nonequilibrium thermodynamics, synergetics, thermodynamic finite time, and thermodynamic analysis of motion equations.

2.

Statistical substantiation of MEIS. MEIS relations with equilibrium and nonequilibrium statistical thermodynamics and physical kinetics. Choice of the mathematical apparatus of macroscopic equilibrium descriptions. Problems in modeling the nonholonomic, nonscleronomous, and nonconservative systems. Possibilities for using differential equations (autonomous and nonautonomous) and MP.

3.

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4.

B.M. Kaganovich et al.

Reduction of the motion models to the rest models and determination of their role in the general model engineering. Transformation of the equations of irreversible macroscopic kinetics. Equilibrium description of explosions, hydraulic shock, short circuit, and other “supernonequilibrium” processes.

7.2 Analysis and development of MEISs 1.

Comparison of MEISs with traditional methods of equilibrium thermodynamics. Initial physico-mathematical assumptions. Physico-mathematical characteristics. Admissible and efficient spheres of application: physics, chemistry, engineering systems, biology, and socio-economic systems.

2.

Classification of MEISs. Models with variable parameters: with variable flows and spatially inhomogeneous systems; with constraints on the macroscopic kinetics and without them. Specific features of modifications and their comparative capabilities. MEISs and macroscopic kinetics. Formalization of constraints on chemical kinetics and transfer processes. Reduction of initial equations determining the limiting rates of processes. Development of the formalization methods of kinetic constraints: direct application of kinetics equations, transition from the kinetic to the thermodynamic space, and direct setting of thermodynamic constraints on individual stages of the studied process. Specific features of description of constraints on motion of the ideal and nonideal fluids, heat and mass exchange, transfer of electric charges, radiation, and cross effects. Physicochemical and computational analysis of MEISs with kinetic constraints and the spheres of their effective application. Geometrical interpretations of MEISs. Kinetic and thermodynamic surfaces. Representation of kinetics in the space of thermodynamic variables. Thermodynamic tree. Graphs of chemical reactions, hydraulic flows, and electric currents.

3.

4.

7.3 Analysis of computational problems in MEIS application and MEIS-based devising of methods, algorithms, and computing system 1.

Computational problems of setting kinetic constraints. Development of methods for transformation and approximation of the motion equations applied.

Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems

2. 3.

4. 5.

6. 7.

8.

71

Convex analysis of MEIS and determination of the areas of admissible and effective application of the CP methods. Development of methods of searching for the optimal level of a characteristic thermodynamic function of the system G(xext). Interpretation of the proposed methods on the basis of a thermodynamic tree. Development of optimization methods for systems with a variable composition of reagents (y = var). Development of optimization methods for MEIS with variable flows of a substance participating in chemical reactions and transfer processes of heat, mass, and electric charges. Development of optimization methods for MEIS of flow or current distribution in circuits. Devising the methods for analysis of spatially inhomogeneous systems, applied first of all to nonisothermal natural systems and installations for fuel combustion and processing. Creation of the computing system to perform laborious multivariant computing experiments with the maximum automation of man– machine interface.

7.4 Solution of specific theoretical and applied problems on MEIS 1.

Modeling the processes of energy transfer by electromagnetic field.

2.

Modeling the transfer processes with available phase transitions, sorption, dissolution, etc. Description of nonstationary kinetics and transfer in spatially inhomogeneous systems. Construction of models of electrochemical processes. Construction of MEIS for an economic system. Modeling the macroscopic kinetics of forming harmful substances in the processes of fuel combustion and processing. Modeling the processes of pollution of air, soil, and water bodies. Construction of models of equipment corrosion. Modeling the slag and scale formation in energy installations. Construction of thermodynamic models of stationary and nonstationary operating mode of pipeline systems for energy carrier transportation. Construction of thermodynamic models of stationary and nonstationary operating mode of electric power systems. Modeling the air conditioning systems including release and transformation of harmful substances. Modeling the fires in buildings and installations.

3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

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ACKNOWLEDGMENTS The authors have been happy to cooperate with many remarkable specialists: L.S. Polak, V. Ya. Khasilev, A.P. Merenkov, E.G. Antsiferov, S.V. Sumarokov, L.I. Rozonoer, G.S. Yablonsky, V.I. Bykov, and S.P. Filippov. Each of them had influenced greatly development of the ideas presented in this chapter. We highly acknowledge all of them. Unfortunately L.S. Polak, V.Ya Khasilev, A.P. Merenkov, E.G. Antsiferov, and S.V. Sumarokov passed away. We specially acknowledge the contribution of A.N. Gorban. His ideas on thermodynamic analysis of equilibrium irreversible trajectories and model engineering in general along with the works of the thermodynamics founders make up the base for our studies. Alexander Gorban has immediately inspired translation of this chapter into English and fostered invalu­ able its submission for publication. This work was supported by the Russian Foundation of Basic Research (grant 09-08­ 00245-a).

REFERENCES Adamson, A. W. and Gast, A. P., “Physical Chemistry of Surfaces”, 6th ed., 784 p. John Wiley & Sons, Inc, New York (1997). Antsiferov, E. G., Kaganovich, B. M., Semeney, P. T. and Takayshwily, M. K., “Search for the Intermediate Thermodynamic States of Physicochemical Systems. Numerical Methods of Analysis and their Applications”, pp. 150–170. SEI SO AN SSSR, Irkutsk (1987). (in Russian). Arnold, V. I., “Mathematical Methods of Classical Mechanics”, 2nd ed., 416 p. SpringerVerlag, New York (1989). Balyshev, O. A. and Kaganovich, B. M. Izvestiya RAN. Energetika 5, 116–120 (2003). (in Russian). Balyshev, O. A. and Tairov, E. A., “Analysis of Transient and Nonstationary Processes in Pipeline Systems (Theoretical and Experimental Aspects)”, 164 p. Nauka. Sib. pre­ dpriyatie RAN, Novosibirsk (1998). (in Russian). Boltzmann, L. Wien. Akad. Sitzungsber. Bd. 76, S. 373–S. 435(1878). Ehrenfest, P., Ehrenfest T. “The conceptual foundations of statistical approach in mechanics”, The Cornell University Press, Ithaca, NY (1959). Einstein, A. Dtsch. Phys. Ges. 16, S. 820–S. 828 (1914). Feinberg, M., Arch Rat. Mech. Anal. 46 (1), 1–41 (1972). Feinberg, M., Chem. Eng. Sci. 54 (7), 2535–2544 (1999). Feinberg, M. and Hildebrant, D., Chem. Eng. Sci. 52(10), 1637–1665 (1997). Feinberg, M. and Horn, F. Chem. Eng. Sci. 29, 775–787 (1974). Gibbs, J. W. Trans. Connect. Acad. 3, 108–248 (1876) 3, 343–524 (1878). Gibbs, J.W. Elementary principles in statistical mechanics, developed with essential reference to the rational foundation of thermodynamics. N.Y. (1902). Glansdorff, P. and Prigogine, I. R., “Thermodynamics of Structure, Stability and Fluctua­ tions”, p. 280. Wiley, New York (1971). Gorban, A. N. and Karlin, I. V., Invariant Manifolds for Physical and Chemical kinetics in “Lecture Notes in Physics” Vol. 660. Springer, Berlin, Heidelberg (2005). Gorban, A. N., Chislennye Metody Mekhaniki Sploshnoi Sredy. 10 (4), 42–59 (1979). Gorban, A. N., “Equilibrium Encircling: Equations of Chemical Kinetics and their Thermo­ dynamic Analysis”, 226 p. Nauka, Novosibirsk (1984). (in Russian). Gorban, A. N., in “Model Reduction and Coarse-Graining Approaches for Multiscale Phe­ nomena” (A. N. Gorban, N. Kasantzis, I. G. Kevrekidis, H. C. Öttinger, C. Theodor­ opoulos, Eds.). pp. 117–176, Springer, Berlin, Heidelberg (2007).

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Gorban, A. N., Bykov, V. I. and Yablonsky, G. S., “Sketches on Chemical Relaxation”, 236 p. Nauka, Novosibirsk (1986). (in Russian). Gorban, A. N., Kaganovich, B. M. and Filippov, S. P., “Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibria in Physicochemical and Technical systems”, 296 p. Nauka, Novosibirsk (2001). (in Russian). Gorban, A. N., Kaganovich, B. M., Filippov, S. P., Keiko, A. V., Shamansky, V. A. and Shirkalin, I. A., “Thermodynamic Equilibria and Extrema Analysis of Attainability Regions and Partial Equilibria”, 282 p. Springer, Berlin, Heidelberg, New York (2006). Gorban, A. N., Karlin, I. V., Ottinger, H. C. and Tatarinova, J. J. Phys. Rev. E. 63, 1–6 (2001). Gorban, A. N., Kasantzis, N., Kevrekidis I. G., Öttinger, H. C. and Theodoropoulos, C., “Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena”, 561 r. Springer, Berlin, Heidelberg (2007). Haken, H., “Advance Synergetics. Instability Hierarchies of Self-Organizing Systems and Devices”. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, (1983). Haken, H., “Information and Self-organization“. Springer, Berlin, Heidelberg, New York (1988). Horn, F., “Attainable regions in chemical reaction technique // The Third European Sympo­ sium on Chemical Reaction Engineering”, pp. 1–10. Pergamon Press, London (1964). Horn, F. and Jackson, R., Arch. Rat. Mech. Anal. 47 (2), 81–116 (1972). Kaganovich, B. M., “Discrete Optimization of Heat Networks”, 88 p. Nauka, Novosibirsk (1978). (in Russian). Kaganovich, B. M., “On the Feasibility of Equilibria in Physicochemical Systems”. ISEM SO RAN, Irkutsk (2002). (Preprint N º 6. 42 p. (in Russian)). Kaganovich, B. M. and Balyshev, O. A., “Development of the Theory of Multiloop Hydraulic Systems. Papers for the XV Session of the International School on the Models of Continuous System Mechanics”, pp. 15–25. NIIKh SPbGU, Saint Petersburg (2000). (in Russian). Kaganovich, B. M. and Filippov, S. P. “Equilibrium Thermodynamics and Mathematical Programming”. 236 p. Nauka, Novosibirsk. Sibirskaya Izdatelskaya Firma RAN (1995). (in Russian). Kaganovich, B. M., Filippov, S. P. and Antsiferov, E. G., “Efficiency of Energy Technologies: Thermodynamics, Economy, Forecasts”, 256 p. Nauka, Novosibirsk (1989). (in Russian). Kaganovich, B. M., Filippov, S. P. and Antsiferov, E. G., “Modeling of Thermodynamic Processes”, 101 p. Nauka, Novosibirsk (1993). (in Russian). Kaganovich, B. M., Filippov, S. P., Shamansky, V. A. and Shirkalin, I. A. Izv. RAN. Energetika, 5, 123–131 (2004). (in Russian). Kaganovich, B. M., Keiko, A. V. and Shamansky, V. A., “Thermodynamic Studies of Fuel Combustion in Connection with the Use of Gas Turbine Technologies in Small-Scale Energy”. ISEM SO RAN, Irkutsk (2006). (Preprint N º 6. 37 p. (in Russian)). Kaganovich, B. M., Keiko, A. V., Shamansky, V. A. and Shirkalin, I. A. Izv. RAN. Energetika, 3, 64–75 (2006). (in Russian). Kaganovich, B. M., Keiko, A. V. and Shamansky, V. A. Novosibirsk 3, 229–236 (2006). (in Russian). Kaganovich, B. M., Keiko, A. V., Shamansky, V. A. and Shirkalin, I. A. “Analysis of the Feasibility of Thermodynamic Equilibria in Physico-Technical systems”. ISEM SO RAN, Irkutsk, (2004a). Preprint N º 10. 61 p. (in Russian) Kaganovich, B. M., Keiko, A. V., Shamansky, V. A. and Shirkalin, I. A. “Application of Equilibrium Feasibility Models in the Analysis of Efficiency of Gas Turbine Technol­ ogies in Small-Scale Energy”. ISEM SO RAN, Irkutsk, (2005). Preprint N º 5. 59 p. (in Russian) Kaganovich, B. M., Kuchmenko Ye., V., Shamansky B. A. and Shirkalin, I. A. Izv. RAN. Energetika, 2, 114–121 (2005). (in Russian).

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Kaganovich, B. M., Merenkov, A. P. and Balyshev, O. A., “Elements of the Theory of Heterogeneous Hydraulic Circuits”, 120 p. Nauka, Novosibirsk (1997). (in Russian). Katok, A. B. and Hasselblatt, B., “Introduction to the modern theory of dynamical systems”, 802 p. Cambridge Univ. Press, Cambridge (1997). Khasilev, V.Ya. Teploenergetika. N º 1, 28–32 (1957). (in Russian). Khasilev, V. Ya. Izv. AN SSSR. Energetika i transport. N º 1, 69–88 (1964). (in Russian). Khasilev, V.Ya. Elements of the Hydraulic Circuit Theory: Abstract of Doctoral thesis. Novosibirsk, 98 p. (1966). in Russian Kirchhoff, G. R. “Ueber die Anwendbarkeit der Formeln für die Intensitäten der galvani­ chen Ströme in einem Systeme linearer Leiter auf Systeme, die zum Theil aus nicht linearen Leitern bestehen”, Ges. Abhandl, S. 33–49. Johann Ambrosius Barth, Leip­ zig, (1848). Kirchhoff, G. R., “Ueber eine Ableitung der Ohm’schen Gesetze, welche sich an die Theorie der Elektrostatik anschliesst. Ges. Abhandl”, S. 49–55. Johann Ambrosius Barth, Leipzig (1882). Klimontovich, Yu. L., “Statistical Theory of Open Systems”, 624 p. Kluwer Academic Publ., Amsterdam (1997). Kondepudi, D., Prigogine, I. R. and Thermodynamics., M., “From Heat Engines to Dissipative Structures”, 460 p. John Wiley and Sons, Chichester, New York (2000). Lagrange, J., “Analytical Mechanics.” Kluwer, Dordrecht (1997). Merenkov, A. P. and Khasilev, V. Y., “The Theory of Hydraulic Circuits.”, 278 p. Nauka, Moscow (1985). (in Russian). Merenkov, A. P., Sennova, E. V., Sumarokov, S. V. et al., “Mathematical Modeling and Optimization of Heat, Water, Oil and Gas Supply Systems”, 407 p. VO Nauka. Sibirskaya Izdatelskaya Firma, Novosibirsk (1992). (in Russian). Ots, A. A., “Processes in Steam Generators when Burning Shales and Kansk-Achinsky Coals”, 312 p. Energiya, Moscow (1977). (in Russian). Polak, L. S., “Variational Principles of Mechanics, their Development and Application in Physics”, 599 p. GIFML, Moscow (1960). (in Russian). Polak, L. S., “Ludwig Boltzmann. 1844–1906”, 208 p. Nauka, Moscow (1987). (in Russian). Prigogine, I. R., “Introduction to Thermodynamics of Irreversible Processes”. John Wiley, New York (1967). Rayleigh, I. Proceedings of Mathematical Society London, (1873), pp. 357–363. Rozonoer, L. I. and Tsirlin, A. M. Optimal Control of Thermodynamic Processes. 1–3. Avtomatika i Telemehanika. (1973). N º 5. pp. 115–132; N º 6. P. 65–79; N º 8. P. 82–103. (in Russian) Shamansky, V. A., “Thermodynamic Modeling of Slag Formation on the Heating Surfaces of Boiler Units”. ISEM SO RAN, Irkutsk (2004). (Preprint N º 2. 70 p. (in Russian)). Shinnar, R Chem. Eng. Sci. 8, 203–2318 (1988). Shinnar, R. and Feng Ch. A., Ind. Eng. Chem. Fund. 24 (2), 153–170 (1985). Sumarokov, S. V. Ekonomika i mat. metody. 12 (5), 1016–1018 (1976). in Russian. Tsirlin, A. M., “Mathematical Models and Optimal Processes in Macrosystems”, 500 p. Nauka, Moscow (2006). (in Russian). Warnatz, J., Maas, U. and Combustion, D. R., “Physical and Chemical Fundamentals, Model­ ing and Simulations, Experiments, Pollutants Formation”, 352 p. Springer, Berlin, Heidelberg, New York (2001). Yablonsky, G. S., Bykov, V. I., Gorban, A. N. and Elokhin, V. I., “Kinetic Models of Catalytic Reactions”, 400 p. Elsevier, Amsterdam (1991). Zubarev, D.N., “Thermodynamics of Nonequilibrium Processes. Physical Encyclopedia”, pp. 87–89. Bolshaya Rossiiskaya Entsyklopedia, Moscow, Vol. 5 (1998). (in Russian).

CHAPTER

2 Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering Miroslav Grmela�

Contents

1. Introduction 2. Multiscale Equilibrium Thermodynamics 2.1 Classical equilibrium thermodynamics 2.2 Mesoscopic equilibrium thermodynamics 3. Multiscale Nonequilibrium Thermodynamics 3.1 Single scale realizations 3.2 Combination of scales 4. Multiscale Nonequilibrium Thermodynamics of Driven Systems 4.1 Example: a simple illustration 4.2 Example: Chapman�Enskog reduction of kinetic theory to fluid mechanics 5. Concluding Remarks Acknowledgment References

76 78 78 79 91 95 111 116 120 122 127 128 128

Ecole Polytechnique de Montreal, Montreal, QC H3C 3A7, Canada � Corresponding author. E-mail address: [email protected]

Advances in Chemical Engineering, Volume 39 ISSN: 0065-2377, DOI 10.1016/S0065-2377(10)39002-8


2010 Elsevier Inc. All rights reserved.

75

76

Abstract

Miroslav Grmela

An important challenge brought to chemical engineering by new emerging technologies, in particular then by nano- and biotechnologies, is to deal with complex systems that cannot be dealt with and cannot be fully understood on a single scale. This review introduces a unifying thermodynamic frame­ work for multiscale investigations of complex macroscopic systems.

1. INTRODUCTION Recent interest of chemical engineers in combining nano, micro, and macro scales (called nano-engineering and bioengineering) provides a renewed motivation for investigating macroscopic systems simulta­ neously on several different levels (scales) of description. Such investi­ gation needs a setting that unifies levels such as the level of classical thermodynamics, the level of hydrodynamics, and the level of particle theory. While each level has its own unique flavor, an investigation of the relations among the levels shows universal features. These features are then suggested to constitute the framework for multiscale investiga­ tions. We argue that the framework obtained in this way is in fact a framework of an abstractly formulated thermodynamics. The path lead­ ing to such abstract theory begins with the Gibbs formulation of classi­ cal thermodynamics (see, e.g., Callen, 1960). The first step toward more microscopic (mesoscopic) analysis is made by recognizing the maximum entropy principle as an essence of thermodynamics and as the universal passage to more macroscopic levels (Jaynes, 1967; Jaynes et al., 1978). The subsequent step is a realization that minimization of a convex function subjected to constraints is, from the mathematical point of view, a Legendre transformation and that the natural mathematical setting for Legendre transformations is contact geometry (Arnold, 1989; Her­ mann, 1984). Finally, in this geometrical environment we introduce the time evolution representing the approach to a more macroscopic level of description as a continuous sequence of Legendre transformation. This is then the passage from equilibrium to nonequilibrium thermody­ namics in the setting of multiscale analysis. The viewpoint sketched above has been so far developed and applied mainly in the context of mechanics and thermodynamics of complex fluids (Grmela, 2009 and references cited therein, also Section 3.1.6 of this review). The coupling between macroscopic (hydrody­ namic) flow behavior and the behavior of a microstructure (e.g., macromolecules in polymeric fluids or suspended particles or mem­ branes in various types in suspensions) is naturally expressed in the multiscale setting. In this review we shall include in illustrations also

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multicomponent systems undergoing chemical reactions. The motiva­ tion for this type of applications comes from three sources: i. ii.

iii.

Importance of such systems in chemical engineering. Inclusion of multicomponent systems into equilibrium thermodynamic considerations played an important role in arriving at the Gibbs formulation of equilibrium thermodynamics. In fact, the extension from viewing a system (e.g., air) as a single component system to viewing it as a multicomponent system (e.g., viewing air as a mixture of several gases) can well be the first example of the multiscale analysis (recall, e.g., the so-called Gibbs paradox arising in such consideration). Chemically reacting systems have also played an important role in the development of nonequilibrium thermodynamics. One of the key new concepts introduced in nonequilibrium thermodynamics is the concept of a thermodynamic force driving macroscopic systems to equilibrium. Chemical affinity is one of the first and one of the most important examples of such force. Having identified the thermodynamic forces, the next step that has to be made in nonequilibrium thermodynamics is to construct with them the time evolution agreeing with the observed approach to equilibrium. The usual linear (Onsager) construction is not satisfactory in chemical kinetics since it does not lead to the mass action law that generates the well-established governing equations of chemical kinetic. This difficulty has certainly been one of the reasons for a rather low standing of nonequilibrium thermodynamics in chemical kinetics in particular and in chemical engineering in general. But this apparent difficulty is not really a difficulty at all. It can be overcome by a simple mathematical construction that does not require any additional physical insight and that has moreover been suggested long time ago by Marcelin and de Donder in their viewpoint of chemical kinetics (de Donder et al. 1936; Feinberg, 1972; Gorban and Karlin, 2003, 2005). As we shall see below (Section 3.1.2), the formulation of chemical kinetics that appears naturally in multiscale nonequilibrium thermodynamics does not only include the mass action law as a particular case but it extends it to systems with complex equilibrium behavior and to more microscopic levels.

The review is organized as follows: In Section 2 we present the multiscale equilibrium thermodynamics in the setting of contact geometry. The time evolution (multiscale nonequilibrium thermodynamics) representing approach of a mesoscopic level Lmesol to the level of equilibrium thermo­ dynamics Leth is discussed in Section 3. A generalization in which the level Leth is replaced by another mesoscopic level Lmeso2 is considered in Section 4. The notion of multiscale thermodynamics of systems arises in the analysis of this type of time evolution.

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2. MULTISCALE EQUILIBRIUM THERMODYNAMICS In this section we limit ourselves to equilibrium. The time evolution that is absent in this section will be taken into consideration in the next two sections. We begin the equilibrium analysis with classical equilibrium thermodynamics of a one-component system. The classical Gibbs formu­ lation is then put into the setting of contact geometry. In Section 2.2 we extend the set of state variables used in the classical theory and introduce a mesoscopic equilibrium thermodynamics.

2.1 Classical equilibrium thermodynamics The variables characterizing complete states at equilibrium (we shall use hereafter the symbol y to denote them) are y ¼ ðe; nÞ

ð1Þ

where (e,n) are respectively the energy and the number of moles per unit volume. The two-dimensional space with y as its elements will be denoted by the symbol N. The fundamental thermodynamic relation is a function N ! R; ðe; nÞ
sðe; nÞ, where s denotes entropy per unit volume. The function s(e,n) as well as all other functions introduced below are assumed to be sufficiently regular so that the operations made with them are well defined. We can see the fundamental thermodynamic relation s = s(e,n) geometrically (as Gibbs did, see Gibbs, 1984) as a two-dimen­ sional manifold, called a Gibbs manifold, imbedded in the threedimensional space with coordinates (e,n,s) by the mapping: (e,n),!(e, n,s(e,n)). In view of the importance of Legendre transformations in equilibrium thermodynamics, we shall make, following Hermann (Hermann, 1984), an alternative formulation of the fundamental thermodynamic relation. We introduce a five-dimensional space (we shall use hereafter the symbol N to denote it) with coordinates (e,n,e�,n�,s) and present the fundamental thermodynamic relation as a two-dimensional manifold imbedded in the five-dimensional space N by the mapping � � @s y ,! y; ðyÞ; sðyÞ @y

ð2Þ

We shall call this manifold a Gibbs–Legendre manifold and denote it by the symbol N. The advantage of this formulation is that the space N is naturally

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equipped with 1-form ds � e�de � n�dn and this 1-form is preserved in Legendre transformations. In the standard thermodynamic notation, the variables y� = (e�,n�) are denoted 1/T, –/T, respectively, where T is the temperature and  the chemical potential. The Legendre transformation of the fundamental thermodynamic relation s = s(e,n) into its dual form s� = s�(e�,n�), where s�� =P/T, P is the pressure, is made in the following three steps: (i) we introduce thermodynamic potential ’(e,n;e�,n�)=�s(e,n) + e�e + n�n; (ii) we solve equations @’/@e = 0, @’/@n = 0, let their solution be (eeth(y�),neth(y�)); and (iii) s�(y�) = ’(eeth(y�),neth(y�); y�). The Legendre � manifold N is the image of the mapping image � N of the Gibbs–Legendre � �

@s � � � y� ,! y� ; @y � ðy Þ; s ðy Þ :

2.2 Mesoscopic equilibrium thermodynamics Question: Where does the fundamental thermodynamic relation s = s(y) come from? Given a physical system, what is the fundamental thermodynamic relation representing it in equilibrium thermodynamics? We shall give two answers in equilibrium theories and another answer in nonequilibrium theories (discussed in Sections 3 and 4). Answer 1: The only way to find the fundamental thermodynamic relation s = s(y) inside the classical equilibrium thermodynamics is by making experi­ mental measurements. Results of the measurements are usually presented in Thermodynamic Tables. The second answer is found by taking a more microscopic (i.e., more detailed) view, called a mesoscopic view, that we shall now present. Let the state variables (1) be replaced by a state variable x[M corre­ sponding to a more detailed (more microscopic) view than the one taken in classical equilibrium thermodynamics. Several examples of x are dis­ cussed below in the examples accompanying this section. Following clo­ sely Section 2.1, we shall now formulate equilibrium thermodynamics that we shall call a mesoscopic equilibrium thermodynamics. The mesoscopic fundamental thermodynamic relation is now a function M ! R; x
hðxÞ. In order to avoid possible confusion, we use in meso­ scopic formulations the symbol h instead of s and call h(x) an eta-function instead of entropy function. In analogy with the geometrical representation of the fundamental thermodynamic relation introduced in Section 2.1, we present the mesoscopic fundamental thermodynamic relation as a manifold (denoted now by the symbol }), imbedded in the space (denoted now by the symbol M) with coordinates (x,x�,h) as the image of the mapping � � @h x ,! x; ðxÞ; hðxÞ @x

The 1-form with which M is naturally equipped is dh – x�dx.

ð3Þ

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Miroslav Grmela

Now we want to pass from the fundamental thermodynamic relation h = h(x) in M to the fundamental thermodynamic relation s = s(y) in N. First, we have to know how y is expressed in terms of x. Formally, we introduce y ¼ yðxÞ

ð4Þ

Having the fundamental thermodynamic relation h = h(x) and the relation (4), we can pass to the fundamental thermodynamic relation s = s(y). This gives us the second answer: Answer 2: The fundamental thermodynamic relation s = s(y) in the space N is inherited from the fundamental thermodynamic relation. h = h(x) in a more microscopic space M and from the relation y = y(x). Now we explain how the passage h(x) ! s(y) is made. The passage is a Legendre transformation. In order to provide an appropriate setting to formulate it, we introduce a new space MN combining the spaces M and N. Elements of MN have the coordinates (x,y�,x�,y,’). The 1-form with which MN is naturally equipped is d’ � x�dx � y dy�. The fundamental thermodynamic relation in MN that combines the fundamental thermo­ dynamic relation h = h(x) in M and the way y [ N is expressed in terms of x [ M (see (4)) takes now the form ’ ¼ ’ðx; y� Þ ¼ �hðxÞ þ hy� ;yðxÞi

ð5Þ

where hy� ;yðxÞi¼e� eðxÞ þ n� nðxÞ. The Gibbs–Legendre manifold (denoted by the symbol }N ) representing it is a manifold imbedded in the space MN by the mapping � � @’ @’ � � ðx; y� Þ ,! x; y� ; ðx; y� Þ; ðx; y Þ; ’ðx; y Þ @x @y�

ð6Þ

Let }N jx � ¼ 0 be the intersection of }N with the plane x� ¼0. We note that its restriction to the plane x is a manifold of the states that we denote xeth ðy� Þ and call equilibrium states. These are the states for which ’ reaches its extremum if considered as a function of x. We shall denote the manifold of equilibrium states by the symbol Meth . Restriction of }N jx � ¼ 0 to the plane y is the manifold representing y expressed in terms of x (i.e. the function (4)), and its restriction to the plane ðy� ;’Þ is the Gibbs–Legendre manifold N � representing the dual form s� ¼s� ðy� Þ of the fundamental thermodynamic relation s ¼ sðyÞ in N that is implied by the fundamental thermodynamic relation h = h(x) in M. This completes our presentation of the passage hðxÞ ! sðyÞ: The setting of contact geometry that we have used in the presentation following Hermann (1984) is very useful at least for three reasons: (i) It puts the calculations involved in thermodynamics on a firm ground. The formulation presented above remains in fact the same as the original Gibbs

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formulation of thermodynamics except that it naturally incorporates the “Maximum entropy principle” and the Legendre transformations; (ii) It unifies the formulations of equilibrium theories on all levels of description; and (iii) It prepares the setting for investigating the multiscale time evolu­ tion (discussed in Sections 3 and 4). Answer 2 given above invites, of course, another question: Where do the fundamental thermodynamic relation h = h(x) and the relation y = y(x) come from? An attempt to answer this question makes us to climb more and more microscopic levels. The higher we stay on the ladder the more detailed physics enters our discussion of h = h(x) and y = y(x). Moreover, we also note that the higher we are on the ladder, the more of the physics enters into y = y(x) and less into h = h(x). Indeed, on the most macroscopic level, i.e., on the level of classical equilibrium thermodynamics sketched in Section 2.1, we have s = s(y) and y = y. All the physics enters the fundamental thermo­ dynamic relation s = s(y), and the relation y = y is, of course, completely universal. On the other hand, on the most microscopic level on which states are characterized by positions and velocities of all (�1023) microscopic particles (see more in Section 2.2.3) the fundamental thermodynamic rela­ tion h = h(x) is completely universal (it is the Gibbs entropy expressed in terms of the distribution function of all the particles) and all physics (i.e., all the interactions among particles) enters the relation y = y(x). The geometrical setting for the Legendre transformation hðxÞ ! sðyÞ that has been introduced above is illustrated below on a few examples. A particularly simple illustration (but still containing all the structure) is developed in Section 3.1.1. The other examples presented below in Sec­ tions 2.2.1–2.2.5 deal mostly with well-known and well-studied physical systems. The geometrical setting is demonstrated to provide a unified framework for their investigation on all scales. Before starting to develop the physical illustrations we shall briefly turn to an illustration which does not contain all the structure but it does allow us to make a graphical representation of a part of the geometry described above. In Figure 1 we illustrate the Gibbs and Gibbs–Legendre manifolds. We take the space M to be one dimensional (i.e., x  R) and hðxÞ ¼ �x ln x: In Figures 2 we illustrate the manifold }N and in Figure 3 the manifold N } jx � ¼ 0 . We take both M and N as one dimensional, y(x) = x, and ’ðx; y� Þ ¼ x ln x þ y� x:

2.2.1 Example: equilibrium kinetic theory (ideal gas) In this illustration we choose x ¼ nðr; vÞ

ð7Þ

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Miroslav Grmela

0.4

Gibbs manifold

0.35

h

0.3 0.25 0.2 Gibbs-Legndre manifold 0 0.8

−0.2 −0.4 x*

0.6

−0.6

x

−0.8 0.4

Figure 1 Gibbs and GibbsLegendre manifolds for ’ðx; y� Þ ¼ x ln x þ y� x. (b) 0.4

1.5

0.2

1

0

x*

ϕ

(a)

−0.2

0.5 0

−0.4

1

−0.5 1

0.8

1

0.6 y * 0.4

0.5 0.2 0

1

0.8 0.6

x

y*

0.5

0.4 0.2 0

x

Figure 2 Manifold }N for ’ðx; y� Þ ¼ x ln x þ y� x.

and y is given in (1). By (r, v) we denote the position coordinate and momentum of a particle, n(r, v) is a one-particle distribution function. The projection (4) is given by ð ð 1 dr dv nðr; vÞ V ð ð 1 v2 e¼ dr dv nðr; vÞ V 2m

n¼

ð8Þ

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

83

0 Meth

−0.05

ϕ

−0.1

h*(y *) −0.15 −0.2 =0 −0.25 0.1 1

0.2 X

0.8

0.3

0.6 0.4

0.4

y*

Figure 3 Manifolds }N jx � ¼ 0 , Meth, and the conjugate Gibbs manifold h� ¼h� ðy� Þ for ’ðx; y� Þ ¼ x ln x þ y� x.

where V is the volume of the fixed region in the position space in which the system under consideration is confined, m is the mass of one particle. The first line in (8) is simply the chosen normalization of the distribution function n(r, v); the second line means that the only energy in the gas that we investigate is the kinetic energy of the particles. The particles do not interact among each other. The system under consideration is thus an ideal gas. Next, we turn to the eta-function h(x). Following Boltzmann, we choose hðnÞ ¼ �

ð ð 1 kB dr dv nðr; vÞln nðr; vÞ V

ð9Þ

where kB is the Boltzmann constant. Now, we proceed to make the Legendre transformation leading from hðnðr; vÞÞ to s(e,n). The thermodynamic potential (5) becomes ð ð 1 dr dv nðr; vÞln nðr; vÞ V ð ð ð ð 1 1 v2  1 þ dr dv nðr; vÞ � dr dv nðr; vÞ TV 2m TV

’ðn; e� ;n� Þ ¼ kB

ð10Þ

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where we put e� ¼1=T and n� ¼�=T. The equation ’n = 0 becomes � � 1 1 v2  � ¼0 kB ln nðr; vÞ þ kB þ V T 2m T

ð11Þ

Its solutions, i.e., equilibrium states, are � neth ðr; vÞ ¼ exp

� � �  v2 � 1 exp � kB T 2mkB T

ð12Þ

This is the well-known Maxwell distribution function. If we now evaluate the thermodynamic potential (10) at the equilibrium states (12) we obtain � s

�

1  ; T T

�

ð ð P 1 ¼� dr dv neth ðr; vÞ V kB T � �� � �3=2  m ¼ exp �1 2kB T kB T ¼�

ð13Þ

� � that is the dual form s� ¼s� T1 ; T of the fundamental thermodynamic relation s ¼ sðe; nÞ. In order to obtain s ¼ sðe; nÞ itself, we make another Legendre transformation. We introduce ’ðy� ;yÞ ¼ �s� ðy� Þ þ hy; y� i, solve ’y � ¼ 0, and insert the solutions to ’ðy� ;yÞ. In this way we arrive at sðe; nÞ ¼ const: n þ Rn ln

�� � � �� e 3=2 1 n n

ð14Þ

where R ¼ kB NAv , NAv is the Avogadro number. The P � V � T relation P = nRT implied by the fundamental thermodynamic relation (14) is already seen ð ð directly in the second equality in (13) (note that nNAv ¼

1 V

dr dv neth ðr; vÞ).

2.2.2 Example: equilibrium kinetic theory (van der Waals gas) In this example we keep the same state variables (7) as in the previous example but change the projection (8). The new relation y = y(x) is ð ð 1 dr dv nðr; vÞ V ð ð 1 v2 1 e¼ dr dv nðr; vÞ þ !long ð%Þ V 2m V

n¼

ð15Þ

where !long ð%Þ ¼

ð ð 1 dr1 dr2 "ðjr1 � r2 jÞ%ðr1 Þ%ðr2 Þ 2

ð16Þ

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is a contribution to the energy due to the long-range interactions among the particles; "ðjr1 � r2 jÞ is the potential generating attractive interaction between two particles at the points with coordinates r1 and r2. The sub­ script “long” denotes “long range.” By the symbol %ðrÞ we denote ð %ðrÞ ¼ dv nðr; vÞ

ð17Þ

The short-range (hard-core type) interactions will be expressed in the following modification of the eta-function: hðnÞ ¼ �

ð ð 1 1 kB dr dv nðr; vÞln n ðr; vÞ � kB !short ð%Þ V V

ð18Þ

where !short ð%Þ remains at this point unspecified. The thermodynamic potential ’ðn; e� ;n� Þ becomes now ð ð 1 1 dr dv nðr; vÞln nðr; vÞ þ kB !short ð%Þ V V ð ð 1 1 v2 1 1 þ dr dv nðr; vÞ þ !long ð%Þ TV 2m TV ð ð  1 � dr dv nðr; vÞ TV

’ðn; e� ;n� Þ ¼ kB

ð19Þ

The equation ’n ¼ 0 implies � � 1 1 v2 1  kB ln nðr; vÞ þ kB þ þ kB ð!short Þ % ðr Þ þ ð!long Þ % ðr Þ � ¼0 V T 2m T T

ð20Þ

Its solutions are � neth ðr; vÞ ¼

m 2kB T

�3 = 2

� %eth ðrÞexp

�

v2 2mkB T

� ð21Þ

where the number density %eth ðrÞ is a solution of ln %ðrÞ þ ð!short Þ % ðr Þ

1 þ ð!long Þ % ðr Þ þ ln kB T

�

m 2kB T

�3 = 2

þ1�

 ¼0 kB T

ð22Þ

To complete the Legendre transformation we evaluate (19) at (21) and arrive at � � ð ð � � PV ¼ dr %eth ðrÞ � !short ð%eth Þ � dr %eth ðrÞ ð!short Þ % ðr Þ % ðr Þ ¼ % ðr Þ eth kB T � � ð � � 1 � !long ð%eth Þ � dr %eth ðrÞ ð!long Þ % ðr Þ % ðr Þ ¼ % ðr Þ eth kB T

ð23Þ

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In order to make the fundamental thermodynamic relation (23) explicit we have to solve Equation (22). We now proceed to do it. We begin by looking for a solution in the form %eth ðrÞ ¼ %eth ¼ const: ¼ n

ð24Þ

We note that if we insert this solution into (23) we obtain the well known van der Waals fundamental � thermodynamic relation � sðe; nÞ ¼ const: n þ Rn ln ð ne þ an Þ 3 = 2 ð n1 � bÞ provided !short ð%Þ and !long ð%Þ are chosen as follows: 1 !long ¼ �an2 V 1 !short ¼ �n ln ð1 � bnÞ V

ð25Þ

where a > 0 and b > 0 are the two parameters appearing in the famous van der nRT Waals P–V–T relation P ¼ 1�bn � an2 . We note that for a = 0 and b = 0 both !long and !short disappear and the projection (15) becomes the same as (8), which then means that the van der Waals fundamental thermodynamic rela­ tion reduces to the fundamental thermodynamic relation (14) of the ideal gas. In this example we can also illustrate how the violation of the con­ vexity of the thermodynamic potential  (i.e., existence of multiple solu­ tions of (22), in particular then solutions that are not independent of r) is physically interpreted as an appearance of two or more phases.

2.2.3 Example: Gibbs equilibrium statistical mechanics In this example we take the most microscopic viewpoint of macroscopic systems. We regard them as being composed of np � 1023 number of particles. Gibbs (1902) has realized that the state variable of classical mechanics, namely x ¼ ðr1 ; r2 ; … ; rn ; v1 ; v2 ; … ; vnp Þ

ð26Þ

where (ri, vi) are the position coordinate and momenta of i-th particle is not a good choice of state variables if one wants to investigate thermody­ namics. With ðr1 ; r2 ; … ; rn ; v1 ; v2 ; … ; vnp Þ we cannot express the eta-func­ tion. In the search for an alternative state variables, we may recall that a standard tool used in mathematics to investigate a set D � R 6np is to investigate the set of functions D !R. This suggests to replace (r1, r2,…, rn, v1, v2,…, vnp ) by functions f : ðr1 ; r2 ; … ; rn ; v1 ; v2 ; … ; vnp Þ
R. This is indeed the state variable suggested by Gibbs x ¼ f ðr1 ; r2 ; … ; rn ; v1 ; v2 ; … ; vnp Þ

ð27Þ

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In fact, Gibbs (following Maxwell) has proposed (27) on the basis of another type of consideration. The function f ðr1 ; r2 ; … ; rn ; v1 ; v2 ; … ; vnp Þ is physically interpreted as a distribution function. This means that the functions f have some particular properties (e.g., f ðr1 ; r2 ; … ; rn ; v1 ; v2 ; … ; vnp Þ is always positive) and that, from the physical point of view, we are starting to see macroscopic systems as statistical ensembles of macroscopic systems. From this interpretation comes then the name (coined by Maxwell) “statistical mechanics.” However useful is the statistical viewpoint, it is important to realize that it is not the only viewpoint that one can adopt in statistical mechanics (of course, we are obliged to keep the well established name “statistical mechanics” even if we choose not to see anything “statistic” in it). In the illustration discussed in this section we shall consider the number of particles np as well as the volume V in which the macro­ scopic system under consideration is confined as fixed. We constrain thus f by ð

ð ð d1 d2… dnp f

ðV ; np Þ ð1; 2; … ; np Þ

¼1

ð28Þ

and the projection (4) becomes e¼

ð ð ð 1 d1 d2… dnp enp ð1; 2; … ; np Þf V

ðV ; np Þ ð1; 2; … ; np Þ

ð29Þ

where enp ð1; 2; … ; np Þ ¼

� np � 2 X vi þ "ðr1 ; r2 ; … ; rnp Þ 2m i¼1

ð30Þ

The subscript (V, np) indicates the constraints V = const, and np = const. We are also using the abbreviated notation 1 ¼ ðr1 ; v1 Þ; 2 ¼ ðr2 ; v2 Þ; … ; np ¼ ðrnp ; vnp Þ, m is the mass of one particle (all particles are assumed to be identical), and "ðr1 ; r2 ; … ; rnp Þ is the interaction potential among the particles. The constraint (28) is the normalization (if we are interpreting f ðV ; np Þ as a distribution function), the first term in (29) is the kinetic energy and the second the potential energy. Next, we turn to the eta-function h(f(V, np)). Gibbs proposed hðf

ðV ; np Þ Þ ¼ �kB

ð ð ð 1 d1 d2… dnp f V

ðV ; np Þ ð1; 2; … ; np Þln

f

ðV ; np Þ ð1; 2; … ; np Þ

ð31Þ

In the setting (27), (28), (29), and (31) that we have just presented, the individual features of the macroscopic system under consideration are expressed solely in (29). Everything else is universal; it remains the same

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for all systems. This is of course consistent with classical mechanics where all information about the individual features of a system is expressed in the energy. The rest is now routine. The thermodynamic potential is � ’ f

ðV ; np Þ; e

�

�

ð ð ð 1 d1 d2… dnp f ðV ; np Þ ð1; 2; … ; np Þln f ðV ; np Þ ð1; 2; … ; np Þ V ð ð ð 1 d1 d2… dnp f ðV ; np Þ ð1; 2; … ; np Þenp ð1; 2; … ; np Þ þe� V ð32Þ

¼ kB

The equation ’f �

� f

ðV ; np Þ

eth

ðV ; np Þ

¼ 0 thus becomes

ð1; 2; … ; np Þ ¼

�! np � e� X v2i exp � þ "ðr1 ; r2 ; … ; rnp Þ kB i ¼ 1 2m Z ðnp ; V Þ ðe� Þ

ð33Þ

where Z ðnp ; V Þ ðe� Þ, called a partition function, is given by � � � ð ð e Z ðnp ; V Þ ðe Þ ¼ d1 d2… dnp exp � enp ð1; 2; … ; np Þ kB �

ð

ð34Þ

Consequently h� ðe� ;np ; VÞ ¼ ’

��

� f

ðV ; np Þ

� 1 ; np ; e� ¼ �kB ln Z ðnp ; V Þ ðe� Þ eth V

ð35Þ

is the fundamental relation of classical equilibrium thermodynamics implied by (27), (28), (29), and (31). The above realization of the abstract mesoscopic equilibrium thermo­ dynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. In addition to keeping the volume V and the number np of particles fixed, we shall also keep fixed the energy (29). This means that only the subspace of the 6np-dimensional phase space corresponding to a fixed energy e constitutes the space M. Its elements are f

ðV ; np ; e Þ ð1; 2; … ; np Þ

ð ð ð ¼ d1 d2… dnp ðe � enp ð1; 2; … ; np ÞÞf

ðV ; np Þ ð1; 2; … ; np Þ

ð36Þ

The eta-function is the same as (31) except that f(V, np) is replaced by f(V, np, e).

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89

We can now proceed to the Legendre transformation. The potential generating it is now � ’ f

� ðV; np ; e Þ

¼ kB

ð ð ð 1 d1 d2… dnp f V

ðV ; np ; e Þ ð1; 2; … ; np Þln

f

ðV ; np ; e Þ ð1; 2; … ; np Þ

ð37Þ

Solution to the equation ’f ðV ; np ; e Þ ¼ 0 is �

� f

ðV; np ; e Þ

eth

ð1; 2; … ; np Þ ¼

1 Z ðV; np ; e Þ

ð38Þ

where ð ð ð Z ðV; np ; e Þ ¼ d1 d2 dnp ðe � enp ð1; 2; … ; np ÞÞ

ð39Þ

is a uniform distribution on the energy shell in the 6np-dimensional phase space. Finally, we obtain h� ðV; np ; eÞ ¼ �kB

1 ln Z ðV; np ; e Þ V

ð40Þ

We note that the classical equilibrium entropy (i.e., the eta-function eval­ uated at equilibrium states) acquires in the context of the Microcanonical Ensemble an interesting physical interpretation. The entropy becomes a logarithm of the volume of the phase space that is available to macroscopic systems having the fixed volume, fixed number of particles and fixed energy. If there is only one microscopic state that corresponds to a given macroscopic state, we can put the available phase space volume equal to one and the entropy becomes thus zero. The one-to-one relation between microscopic and macroscopic thermodynamic equilibrium states is thus realized only at zero temperature.

2.2.4 Example: multicomponent isothermal systems

Let the system under consideration be composed of k0 components. Let the mixture moreover be kept under constant temperature T. We thus replace (by making an appropriate Legendre transformation) the state variable e by the temperature T and then, since T = const., we shall, in order to simplify the notation, completely omit it. Consequently, we have in this illustration: x ¼ n 0 ¼ ðn 10 ; n 20 ; … ; n k0 0 Þ

ð41Þ

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Miroslav Grmela

The system under consideration is at equilibrium. Let us suddenly switch on new interactions among the components. The components start to react chemically. They start to transform one into another. This new type of interactions brings the system initially out of equili­ brium but eventually the system will reach a new equilibrium, called a chemical equilibrium. What is the fundamental thermodynamic relation of the system involving chemically reacting components at new chemi­ cal equilibrium? First, we need to identify the state variables y describing the chemical equilibrium. Following the notation of Gorban and Karlin (2003), the conservation laws in chemical reactions introduce k linearly independent vectors b1, b2,…, bk. The state variables describing the system at the chemical equilibrium are y ¼ n ¼ ðn1 ; n2 ; … ; nk Þ

ð42Þ

where nj ¼ hbj ; n 0 i;

j ¼ ¼ 1; 2; … ; k

ð43Þ

the relation (43) serves in this particular example the role of the projec­ tion (4). Let us assume now that we know the eta-function h = h(x). We have thus everything that we need to follow the general theory and derive the fundamental thermodynamic relation at the chemical equilibrium (we 0 introduce ’ðn 0 ; n� Þ ¼ �hðn 0 Þ þ hn� ;nðn 0 Þi, identify n eth that are solutions 0 0 to ’n = 0, and evaluate ’ at n eth ). The time evolution bringing the system from n0 to the chemical equilibrium described by n will be discussed in Section 3.1.3. Here we shall only recall the standard notation for chemical reactions. Let the species be denoted by the symbols A 1 ; A 2 ; … ; A k 0 . The chemical reac­ tions are ðjÞ

ðjÞ

ðjÞ

ðjÞ

1 A 1 þ …þ k 0 A k 0 >  1 A 1 þ … þ k 0 A k 0

ð44Þ

where j enumerates reaction steps. The quantities defined by ðjÞ

ðjÞ

ðjÞ

i ¼  i � i ;

j ¼ ¼ 1; 2; … ; J; i ¼ ¼1; 2; … ; k 0

ð45Þ

are called stoichiometric coefficients. As a particular example we take a two-step, four-component reaction with one catalyst A 2 : discussed in Gorban and Karlin (2003): A1 þ A2 > A3 > A2 þ A4

ð46Þ

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In this example we thus have k0 = 4 in (41) and k = 2 in (42) with n ¼ ðhb1 ; n 0 i;hb2 ; n 0 iÞ; b1 ¼ ð1; 0; 1; 1Þ; b2 ¼ ð0; 1; 1; 0Þ

2.2.5 Example: multicomponent nonisothermal systems In this example we consider the same situation as in the previous example except that the temperature is left unregulated. The state variables x are thus x ¼ ðe 0 ; n 0 Þ

ð47Þ

The state variables describing chemical equilibrium are y ¼ ðe; nÞ

ð48Þ

The projection y = y(x) becomes nj ¼ hbj ; n 0 i; e ¼ e0 þ

k0 X

 j0

j¼1

j ¼ 1; 2; … ; k ! k X 1� blj n j0

ð49Þ

l¼1

where the first line is the same as in (43), blj is the j-th component of the vector bl. The second line is implied by the conservation of energy between the state before switching on the chemical reactions and at the state of chemical equilibrium, j is the energy of formation of one mole per unit volume of the species i, where i ¼ 1; 2; … ; k 0 :

3. MULTISCALE NONEQUILIBRIUM THERMODYNAMICS The most fundamental experimental observation on which equilibrium thermodynamics is based on is the observation that all externally unforced macroscopic systems (with some exceptions, namely glasses, that we shall mention later in Section 4) can be prepared in such a way that their behavior shows some universal features. Subsequent investigation of these features leads then to the formulation of equilibrium thermody­ namics. The preparation process consists of letting the macroscopic sys­ tems evolve sufficiently long time without external influences. The states reached when the preparation process is completed are called equilibrium states. The approach to equilibrium states is thus a primary experience; the behavior at equilibrium states is the secondary experience. An investiga­ tion of the secondary experience leads to equilibrium thermodynamics. We may expect that an investigation of the primary experience (i.e., an

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Miroslav Grmela

investigation of the approach to equilibrium) will possibly contribute in an important way to equilibrium thermodynamics. Indeed, we shall argue below that the question asked at the beginning of Section 2 can be answered as follows: Answer 3: The fundamental thermodynamic relation h ¼ hðxÞ and the rela­ tion y = y(x) in the space M emerge in an investigation of the time evolution of x representing the approach to equilibrium. Let xeth [ M denote the state variable corresponding to the equili­ brium state. We shall now investigate the time evolution bringing x [ M to xeth as t ! 1. Roughly speaking, we look for a time evolution for which y(x) are constants of motion, xeth are the fixed points, and h (x) the Lyapunov function associated naturally with the approach to the fixed points. In addition, the time evolution has to be somewhat compatible with the multiscale equilibrium thermodynamics discussed in Section 2. In fact it is this last requirement with which we begin. We shall argue as follows: We have seen in Section 2.2 that the passage from a mesoscopic equilibrium theory to the classical equilibrium theory is made by a Legendre transformation taking the mesoscopic Gibbs–Legendre manifold }N to the classical Gibbs–Legendre manifold N � that is a submanifold of }N jx � ¼ 0 Now, we extend the mesoscopic equilibrium theory to a mesoscopic nonequilibrium theory by requiring that the passage}N ! }N jx � ¼ 0 is made by mesoscopic time evolution taking place in the space MN and preserving the 1-form with which the space MN is naturally equipped (see Section 2.2.2). We shall call such time evolu­ tion a Legendre time evolution since Legendre transformations pre­ serve the 1-form, and consequently, the time evolution in which the 1­ form is preserved can be seen as a continuous sequence of Legendre transformations. In the rest of this section we shall identify the Legendre time evolution, By applying well-known results of differential geometry (see, e.g., Appendix 4 in Arnold (1989)) to the space MN we arrive at the following equations governing the Legendre time evolution: x_

¼

Yx �

x_

�

¼

�Yx þ x� Y’

y_

�

¼

Yy

y_

¼

�Yy � þ yY’

’_

¼

�Yþhx� ;Yx � iþhy; Yy i

ð50Þ

where Y is a real-valued function (called a contact Hamiltonian) of ðx; y� ;x� ;y; ’ÞMN such that YjMN ¼ 0. The symbol <, > denotes the inner product. We shall use hereafter the shorthand notation: Yx ¼ @Y @x ; …:

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

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Our problem now is to find the contact Hamiltonian Y for which the time evolution generated by (50) has the following properties: [Y 1]: the Gibbs–Legendre manifold MN is invariant, i.e., the time evolution that starts on }N remains there in all times, and ½Y 2� the time evolution generated by (50) carries }N into }N jx � ¼ 0 as t ! 1 It is easy to verify that Yðx; x� ;y� Þ ¼ �Hðx; x� ;y� Þ þ

1 � � �� E x; x ;y e�

ð51Þ

� � where Hðx; x� ;y� Þ ¼ ðx� Þ � ð’x Þ and E x; x� ;y� ¼ hx� ;L’x i with L and  appearing in (55) is such contact Hamiltonian provided the operator L is skew symmetric and degenerate in the sense that Lhx ¼ 0 and Lnx ¼ 0;

ð52Þ

the dissipation potential  is degenerate in the sense that hex ; x � jx � ¼ hx i ¼ hhx ; x � jðx� ¼ex Þ i ¼ 0

ð53Þ

hnx ; x � jx � ¼ hx i ¼ hhx ; x � jðx� ¼nx Þ i ¼ 0

and the inequality hhx ; x � jx � ¼ hx i>0

ð54Þ

holds. In many applications, the operator L is, besides being skew sym­ metric, also a Poisson operator, which means that hAx ; LBx i is a Poisson bracket denoted hereafter by the symbol fA; Bg; A and B are real-valued functions (sufficiently regular) of x. We recall that the Pois­ son bracket, in addition to satisfying the skew symmetry fA; Bg ¼ �fB; Ag, satisfies also the Jacobi identity fA; fB; Cgg þ fB; fC; Agg þfC; fA; Bgg ¼ 0: If the time evolution Equations (50) with the contact Hamiltonian (51) are restricted to the invariant manifold MN then they become dx @e @ ¼L þ � � @h dt @x @ @x

ð55Þ

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Indeed, E j}N ¼ 0 due to the skew symmetry of L (skew-symmetric multi­ plication of two identical terms gives always zero) and Hj}N ¼ 0 as a result of subtraction of two identical terms. The first line in (50) evaluated on }N is clearly identical to (55). The second line evaluated on }N is (55) multiplied by ’xx. The third and the fourth lines evaluated on }N become y_ � ¼ 0; y_ ¼ 0. The fifth line evaluated on }N becomes _ ¼ h’x ; Y’x i>0. The inequality sign is a consequence of the requirements put on the dissipation potential . Solutions to (55) have the following properties: de ¼ 0; dt

dn ¼ 0; dt

dh > 0 dt

ð56Þ

which then implies that d’ dt � 0. If the thermodynamic potential ’ is a convex function of x, it can serve thus as a Lyapunov function for the approach x ! xeq as t ! 1. Summing up, we have arrived at the mesoscopic time evolution Equation (55) by extending the geometrical structure of equilibrium thermodynamics to the time evolution. A few observations are now in order: 1.

2.

3.

In order to prove that (55) is the only possible equation generating the Legendre time evolution we would have to prove that (51) is the only possible contact Hamiltonian. While we cannot prove its uniqueness, we cannot also find other possible contact Hamiltonians satisfying the properties [Y 1] and [Y 2] introduced above in this section. We note that the Jacobi identity for the bracket fA; Bg¼ðAx ; LBx Þ is not needed for the manifold }N to be invariant and for (50) evaluated on it to become equivalent to (55). The skew symmetry of L suffices to guarantee both these properties. If however we begin with the time evolution generated by (50) and define the manifold }N as the manifold on which (51) equals zero then the Jacobi identity is the integrability condition for }N (see Courant (1989)). There is also another way the mesoscopic time evolution Equation (55) can be introduced. We collect a list of well-established (i.e., well tested with experimental observations) time evolution equations on many different levels of description and try to identify their common features. This is indeed the way the time evolution Equation (55) has been first introduced. The Hamiltonian structure of the nondissipative part has been discovered first in the context of hydrodynamics by Clebsch (1895). Equations of the type (55) have started to appear in Dzyaloshinskii and Volovick (1980) and later in

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

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95

Grmela (1984), Kaufman (1984), Beris and Edwards (1994), and Morrison (1984). In the form (55) and with the name GENERIC (an acronym for general equation for nonequilibrium reversible– irreversible coupling), the abstract time evolution Equation (55) has appeared first in Grmela and Ottinger (1997) and Ottinger and Grmela (1998). GENERIC has been then further developed in Grmela (2001, 2002, 2004, and 2010) and in a different direction in Ottinger (1998, 2005). There is now a large list of new and very useful time evolution equations of complex fluids that have been introduced first as particular realizations of (55) (some of them are listed in Section 3.1.6). It has been noted in Grmela (2002) that the total eta-function produced in the course of the approach to equilibrium governed by the GENERIC Equation (55) reaches its extremum. The proof of this statement proceeds as follows: First, we note that with the choice (51) the second term on the right-hand side of the second equation in (50) equals zero. Next, we introduced a function ð � � � � � � I ¼ dt Yðx; x ; y Þ � hx xi

ð57Þ

and note that the Euler–Lagrange equations corresponding to the variations with respect to x� and x are the same as the first two @I equations in (50). Indeed, @x is the first equation and � ¼ 0 � @I � d @I � dt @x_ þ @x ¼ 0 is the second equation in (50). Moreover, we � note that on the Gibbs–Legendre ð ð ðmanifold (i.e., x = ’x), the function I ¼ � dt ’ x� ¼ � dt ’� ¼ dt hðxÞ, which is the etax

function produced in the course of the approach to the equilibrium state xeth. This concludes the proof.

3.1 Single scale realizations We recall a terminology used routinely in mathematics. Let us have an abstract mathematical object G ; gG (e.g., G is a group). We say that G real ; G real G real is a particular realization (or representation) of G if the elements greal of G real assume a concrete identity (e.g., they are linear transformations in the space R n ) but the structure (e.g., the group structure if G is a group) of G and G real remain the same. Analogically, we say that a particular realization of the abstract Equation (55) is the time evolution Equation (55) in which all the building blocks have been specified. The building blocks are state variables x  M, their Hamilto­ nian kinematics L, dissipation potential , the eta-function hðxÞ, the

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energy e(x), and the number of moles n(x). The point of departure of every construction of a particular realization of (55) is a physical insight that arises from a combination of an experience collected in experimen­ tal observations and a hypothesis of the physical mechanisms involved. The insight is then expressed in the building blocks. We shall now present some examples. In this subsection we shall consider singlelevel realizations. Multilevel realizations are illustrated in the next subsection.

3.1.1 Example: a simple illustration The main purpose of this example is to provide a very simple but still physically meaningful illustration of the Legendre time evolution intro­ duced above. The physical system that we have in mind is a polymeric fluid. We regard it as Simha and Somcynski (1969) do in their equilibrium theory but extend their analysis to the time evolution. As the state vari­ ables we choose x ¼ ðq; p; ; Þ

ð58Þ

and the projection y = y(x) e¼ n¼

ð59Þ

The quantity q has the physical interpretation of the free volume. It is the state variable used in the Simha–Somcynski equilibrium theory of polymeric fluids (Simha and Somcynski, 1969). The new variable p that we adopt has the meaning of the velocity (or momentum) associated with q. The polymeric fluid that we investigate with the state variables (58) is thus static (i.e., without any macroscopic flow) and spatially homoge­ neous. The only time evolution that takes place in it is the evolution of the internal structure characterized by two scalars q, p. The physical insight involved in the Simha–Somcynski theory and an additional insight that we need to extend it to the time evolution will now be expressed in the building blocks of (55). We shall construct a particular realization of (55). We begin with the state variables. They have already been specified in (58). Now we proceed to specify the kinematics of (58). In order that the etafunction h (that remain unspecified at this point) and the number of moles n be preserved in the nondissipative time evolution (i.e., the time evolu­ tion governed by the first term on the right-hand side of (55)), the matrix L has to be such that Lhx ¼ 0 and Lnx ¼ 0. This degeneracy can be discussed more easily if we pass from the state variables x to a new set of state

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97

variables  ¼ ðq; p; h; vÞ. Using the terminology of equilibrium thermody­ namics (see Callen, 1960), x are the state variables in the entropy repre­ sentation and  in the energy representation. The passage x $  is assumed to be one-to-one since the fundamental thermodynamic relation h(x) is assumed to satisfy he > 0. Since h" has the physical inter­ pretation of the inverse of temperature, this requirement means that we require the temperature to be always 0 positive. The1matrix L expressing 0 1 0 0 B �1 0 0 0 C C kinematics of  is chosen to be L ¼ B @ 0 0 0 0 A. This matrix mani­ 0 0 0 0 festly satisfies the degeneracy requirement (note that h ¼ ð0 ; 0 ; 1 ; 0Þ T and n ¼ ð0; 0 ; 1 ; 0 Þ T ; ðÞ T denotes the transpose operation). If this L is transformed (by using the one-to-one0transformation x $ 1 ) into the h

0 1 � hp 0 B C hq C B �1 0 0 B C h state variables x it takes the form L ¼ B C. We are @ hq � hp A 0 0 h h 0 0 0 0 using hereafter the notation hp ¼ @h=@p; …. We directly verify that Lhx ¼ Lnx ¼ 0. The kinematics of q and p is chosen to be the same as if q is the position coordinate and p the momentum associated with it. In other words, whatever is the physical interpretation of q (e.g., free volume in the Simha–Somcynski theory), p is the momentum associated with it. The thermodynamic force X that generates the dissipation will be assumed to be a standard friction force. This means that X ¼ hp . For the sake of simplicity, we shall limit ourselves only to small forces and introduce the quadratic dissipation potential  ¼ 21 LX2 , where > 0 is a kinetic coefficient. [We can easily consider also non-quadratic potentials as, for example,  ¼ Lðexp X þ expð�XÞ � 2Þ]. The energy e(x) appearing in (55) is chosen to be eðxÞ ¼ ðsee ð59ÞÞ and h(x) can be, for example, the fundamental thermodynamic relation arising in the Simha–Somcynski theory supplemented by a new term proportional to p2 representing the contribution of the internal structure (characterized by (q,p)) to the kinetic energy. In this illustration we leave the function h(x) unspecified. The thermodynamic potential ’ � � becomes in this case ðx; y Þ ¼ �hðq; p; ; Þ þ e  þ n� and the contact Hamiltonian (51) �

�

�

Yðq; p; ; ; q� ; p� ; � ; ; e ; n Þ ¼

� � � 1� � � � �p hq � q hp þ L hp2 � ðp Þ 2 h

ð60Þ

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Miroslav Grmela

Consequently Equations (50) look as follows: q_ ¼ �

hp h

hq � Lp� h _ ¼ 0 p_ ¼

_ ¼ 0 _ ¼ 0

_ ¼ 0

� � 1� � p hq � q� hp q h � � � 1� � p hq � q� hp q p_ � ¼ �Lhp hpp � h � � � � � 1 p hq � q� hp  _ � ¼ �Lhp hp � h � � � 1� � p hq � q� hp

_ � ¼ �Lhp hp � h e_ � ¼ 0 �

�

q_ ¼ �Lhp hpq �

ð61Þ

n_ � ¼ 0 e_ ¼ 0 n_ ¼ 0 � 1 � hp hq 1 � ’_ ¼ L ðp� Þ 2 � h2p � ðp gq � q� hp Þ � q� þ p� � Lðp� Þ2 2 h h h

If this time evolution is restricted to the invariant Gibbs–Legendre mani� � � � � fold }N [recall that on this manifold x ¼ ’x ðx; x ; y Þ and ’ ¼ ’ðx; x ; y Þ], we obtain q_ ¼ �

hp ; h

p_ ¼

hq þ 2Lhp ; h

_ ¼ 0;

v_ ¼ 0;

’_ ¼ �Lðhp Þ 2

ð62Þ

Equations (62) represent indeed a particular realization of GENERIC (55) in which our insight into the physics that is behind the model discussed in this illustration is expressed. It is easy to verify that solutions to (62) have de dh the following properties: dt ¼ 0; dn dt ¼ 0; dt > 0; x ! xeth as t ! 1, where xeth is the equilibrium state at which the thermodynamic potential ’(x, y�) reaches its minimum.

3.1.2 Example: chemically reacting isothermal systems The physical systems considered in this example are the same as in Example 2.2.4. We also choose the same state variables (41). What is new now is that we shall follow explicitly the time evolution initiated by

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99

switching on the chemical reactions. The final state reached as t ! 1 will be the equilibrium state (chemical equilibrium state) characterized by the state variables (42) discussed in Section 2.2.4. The mass action law corresponding to the reaction (44) leads to the following time evolution equations: dn1 ð1Þ ð2Þ ðJÞ ¼ � 1 J ð1 Þ � 1 J ð2 Þ � � � � � 1 J ðJ Þ dt dn2 ð1Þ ð2Þ ðJÞ ¼ � 2 J ð1 Þ � 2 J ð2 Þ � � � � � 2 J ðJ Þ dt .. . dnk 0 ð1Þ ð2Þ ðJÞ ¼ � k 0 J ð1 Þ � k 0 J ð2 Þ � � � � � k 0 J ðJ Þ dt

where ðJ

ð1 Þ

;…;J

ðJ Þ

J ðjÞ

ð63Þ

Þ are the fluxes given by

ðj Þ

ðjÞ



ðjÞ



ðjÞ



!ðjÞ ðjÞ ðjÞ ðjÞ

ðjÞ

¼ k n1 1 n2 2 …nk 0k 0 � k n1 1 n2 2 nk 0k 0

ð64Þ

!ðjÞ

are the rate coefficients of the backward and the forward j-th k ;k reaction step, respectively. An obvious disadvantage of this formulation of the time evolution is that we do not see in it any connection with the equilibrium analysis that we have done in Section 2.2.1. In order to establish such connection we now reformulate (63) as a particular realization of (55). The state variables are (41). The time evolution (63) does not involve any nondissipative part and consequently the operator L, in which the Hamiltonian kinematics of (41) is expressed, is absent (i.e., L � 0). Time evolution will be discussed in Section 3.1.3. We now continue to specify the dissipation potential . Following the classical nonequilibrium ther­ modynamics, we introduce first the so-called thermodynamic forces ðX ð1 Þ ; … ; X ðJ Þ Þdriving the chemically reacting system to the chemical equilibrium. As argued in �nonequilibrium thermodynamics, they are lin­ � � � � ear functions of n1 ; … ; nk 0 (we recall that n ¼ ’ni ; i ¼ 1; 2; … ; k 0 on the Gibbs–Legendre manifold) with the coefficients ðiÞ

j ¼

@X ði Þ � ; @nj

i ¼ 1; 2; … ; J; j ¼ 1; 2; … ; k 0

ð65Þ

that are the stoichiometric coefficients defined in (45). We note that the thermodynamic forces X ðj Þ ; j ¼ 1; 2; … ; J indeed disappear at the chemi­ cal equilibrium. The next problem is to find the dissipation potential that is a function @ of the thermodynamic forces, and � @n � is identical with the right-hand j

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Miroslav Grmela

of i-th equation in (63). Since the thermodynamic forces involve �side � � � n1 ; … ; nk 0 and (63) does not, we see that we have to begin with the specification of the thermodynamic potential ’(x) so that we can express � � � � the conjugate variables n1 ; … ; nk 0 in terms of the state variables (41). We choose it ’ðxÞ ¼

k0 X

ðnj ln nj þ Qj nj Þ

ð66Þ

j¼1

where Q1 ; … ; Qk 0 are parameters. Now we proceed to the specification of the dissipation potential X(i). J @ @ @X ði Þ First, we note that � @n � ¼ � � , which, in view of (65), i¼1 @X ði Þ @n j

j

implies that the dissipation potential  that we look for has to satisfy J

ðl Þ

¼

@ @X ðl Þ

ð67Þ

The most obvious candidate with which we satisfy the properties (53) is J X ¼ Wj ðxÞ ðX ðj Þ Þ 2

ð68Þ

j¼1

provided Wj ðxÞ > 0; j ¼ ¼1; … ; J. But with this dissipation potential we have J ðl Þ ¼ 2Wj ðxÞX ði Þ that are not the mass action law fluxes (64). The expression 2Wj ðxÞX ði Þ is however related to (64). When we appropriately ðjÞ

!ðjÞ

relate the rate coefficients k and k to Wj we see that 2Wj ðxÞX ði Þ is the flux (64) linearized about the state of chemical equilibrium. With the potential (68) in (55) we are thus describing the time evolution in a small neighborhood of equilibrium. The following question now arises: What is the dissipation potential  that implies the mass action law fluxes? In other words, we look for  that (i) satisfies (53), (ii) satisfies (67) with the fluxes given in (64), and (iii) in a small neighborhood of the equilibrium it reduces to (68). The answer is the following: ¼

J X

� 1 ðl Þ � 1 ðl Þ W ðl Þ ðxÞ e 2 X þ e � 2 X � 2

ð69Þ

l¼1

If we insert this dissipation potential into the right-hand side of (67) and use (66) then indeed we get the flux (64) with

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

ðjÞ

k

� � ðjÞ ðjÞ ðjÞ 1 1 ¼ W ðj Þ e 2 ðQ1 þ1Þ 1 þðQ2 þ1Þ 2 þ��� þðQK þ1Þ K 2 � ðjÞ ðjÞ �1 ðjÞ ðjÞ ðjÞ ðjÞ       � n1 1 n2 2 � � � nk 0k 0 n1 1 n2 2 � � � nk 0k 0 2

101

ð70Þ

and ðjÞ

k !ðjÞ

� � ðjÞ ðjÞ ðjÞ ¼ e ðQ1 þ1Þ 1 þðQ2 þ1Þ 2 þ��� þðQK þ1Þ K

ð71Þ

k

We also immediately see that (69) turns into (68) if we neglect the terms � Xm ; m � 3 and also that (53) holds. We recall that in order that the inequality (54) holds we need a function  with the following properties:  = 0 at 0,  reaches its minimum at 0, and  is convex in a neighborhood of 0. All these three properties clearly hold for  given in (69). We have just demonstrated that the multiscale nonequilibrium ther­ modynamics includes the mass action law of chemical kinetics as a parti­ cular case. The form (69) of the dissipation potential has been, at least implicitly, introduced already by Marcelin and de Donder (de Donder et al., 1936; Feinberg, 1972; Bykov et al., 1977; Gorban and Karlin, 2003, 2005). In the case when the thermodynamic potential does not have the specific form (66), the fluxes (67) are not exactly the same as the fluxes (64) given by the mass action law. Marcelin and de Donder have suggested that for the modified free energy the fluxes (67) should replace the fluxes given by the mass action law.

3.1.3 Example: kinetic theory of chemically reacting systems We take in this example a more microscopic viewpoint of chemically reacting systems. In order to be able to take into account more details, we change the state variables (41). The number of moles ni; i ¼ 1; 2; … ; k 0 are replaced by distribution functions. We begin with the case k0 = 1, i.e., with a one-component system and choose the kinetic theory state variables (7), i.e., the number of moles n becomes the one-particle distribution function n(r,v). First, we turn our attention to the Hamiltonian kinematics of the distribution function (7). Hamilton’s equations governing the time evolution of one particle with the position coordinate r and momen­ tum v are r_ v_

! ¼ Lmech

er ejv

! ð72Þ

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Miroslav Grmela

�

� 0 1 . The (canonical) �1 0 Poisson bracket fa; bg expressing the particle Hamiltonian kinematics is thus where emech ðr; vÞ is the energy and Lmech ¼

mech

fa; bg

¼ ðar ; av ÞL

br bv

! ¼ ar bv � br av

ð73Þ

where a and b are sufficiently regular real-valued functions of (r,v). Instead of (r,v) we now take the distribution function n(r,v) as the state variable. The time evolution of n(r,v) is governed by the Liouville equation corresponding to (72). Can the Liouville equation be cast into the form @n ¼ Len @t

ð74Þ

The answer is yes. This can indeed be done with ð ð � �mech fA; BgLiouville ¼ dr dv nðr; vÞ An ðr ; v Þ ; Bn ðr ; v Þ

ð75Þ

and eðnÞ �

ð ð 1 dr dv nðr; vÞemech ðr; vÞ V

ð76Þ

It is easy to verify that time evolution (74) is the Liouville equation corresponding to the particle time evolution governed by (72). Since the verification is made in essentially the same way also for all other Poisson bracket formulations that will be discussed below, we shall write down the details. The easiest way to pass from the Poisson bracket (75) to the time evolution (72) is by noting that (74) can also be written as dA dt ¼ fA; eg holds for all A(n). This equation, if written explicitly, becomes ð

ð @nðr; vÞ dr dv An ðr ; v Þ @t � � ð ð � � @ � � @ mech nðr; vÞemech þ nðr; vÞe ¼ dr dv An ðr ; v Þ � v r @r @v

In the calculation needed to arrive at the right-hand side we have used integrations by part and boundary conditions that make to disappear the integrals over the boundaries that arise in the calculations. This equation can hold for all A(n) only if

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

103

� @ � � @nðr; vÞ @ � ¼� nðr; vÞemech þ nðr; vÞemech v v @t @r @v

which is indeed the Liouville equation corresponding to (72). Both the Poisson bracket (75) and the energy (76) are averages of the mechanical Poisson bracket (73) and the mechanical energy. While this observation helps to understand the physical interpretation of the formu­ lation (74), it does not constitute the proof that (75) is indeed a Poisson bracket (i.e., in particular that it satisfies the Jacobi identity). Such proof can be made by a direct verification of the Jacobi identity. An alternative proof is based on a general association of Poisson brackets with Lie algebras (e.g., Marsden et al., 1983). This way of seeing the Poisson bracket is physically very meaningful since the Lie group involved is the group representing the kinematics. In the case of n(r,v), the Lie group implying the bracket (75) is the group of canonical transformations of (r,v). As for the degeneracy of the bracket (75), we easily verify that a function ð ð C ¼ dr dvFðnÞ, where F : nðr; vÞ ! R is a sufficiently regular function is a Casimir function. That is, f A ; C g Liouville ¼ 0 for all A. Thus, in parti­ cular, f A ; hg Liouville ¼ 0 for h given in (9). Having established the nondissipative time evolution of n(r,v), we now turn to the dissipative part. Following the insight of Boltzmann, we shall regard collisions as a source of the emergence of the time irreversibility and the dissipation. The act of collision (we restrict ourselves only to collisions of two particles) is, of course, a part of the time-reversible motion (graphically represented by trajectories) of the particles. The tra­ jectory of a colliding particle remains smooth and time reversible. The changes of the trajectory are however more pronounced. Following Boltz­ mann, we exaggerate the singular nature of the act of collision, make it into a real singularity, and give it a role of forgetting (except the informa­ tion needed to conserve the momentum and the energy) everything that went on with the colliding particles before the collision. Having this picture in mind, we can treat collisions as chemical reactions with the momentum v playing the role of the label of species. We consider the particles with a different momentum as belonging to a different species. In the following discussion we shall consider the particles to be points, which then means that the position coordinate r is not affected by collisions. In the discussion of collisions we can thus omit the position coordinate r altogether. This brings us to the setting discussed in Section 3.1.2 with n1 ; n2 ; … ; nk 0 replaced by n(v). The chemical reaction representing the collision is V þ V 1 > V 0 þ V 10

ð77Þ

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Miroslav Grmela

where V denotes the species with the label v; V 1 with the label v1 ; V 0 with the label v 0 , and V 10 with the label v10 . Following Section 3.1.2, we introduce the thermodynamic force (chemical affinity)











X ¼ n ðvÞ  n ðv1 Þ þ n ðv 0 Þ þ n ðv 10 Þ

ð78Þ

and the dissipation potential ð ð ð ð  1  ¼ dv dv1 dv 0 dv 10 W ðn; v; v1 ; v 0 ; v 10 Þ e 2 X þ e 

1 2

X

 2

ð79Þ

where W  0, symmetric with respect to the interchange of v . v1 and the interchange of ðv; v1 Þ þ ðv 0 ; v10 Þ. Moreover, W 6¼ 0 if and only if v þ v1 ¼ v 0 þ v 10

ð80Þ

v2 þ ðv1 Þ 2 ¼ ðv 0 Þ2 þ ðv 10 Þ 2

ð81Þ

and

The dissipative time evolution of n(r,v) is thus governed by


@n @t

 dissip

ð ¼

ð ð dv1 dv 0 dv 10 W Boltzmann ðn; v; v1 ; v 0 ; v 10 Þ

ð82Þ

ðnðr;v 0 Þnðr; v10 Þ  nðr;vÞnðr; v1 ÞÞ

where 1 1 W Boltzmann ¼ W ðn; v; v1 ; v 0 ; v10 Þðnðr; vÞnðr; v1 Þnðr; v 0 Þnðr; v 10 ÞÞ 2 2

ð83Þ

provided h(n) is chosen to be (9). The right-hand side of (82) is indeed the famous Boltzmann collision term ð in ð the Boltzmann equation. If we choose the energy (76) to be eðnÞ 

1 V

dr dv

v2 2m

and the eta-function (9) then the

time evolution of n(r,v) is governed by the Boltzmann equation ð ð ð vj @nðr; vÞ @nðr; vÞ þ dv1 dv 0 dv10 W Boltzmann ðn; v; v1 ; v 0 ; v10 Þ ¼ m @rj @t

ðnðr; v

0

Þnðr; v 10 Þ

ð84Þ

 nðr; vÞnðr; v1 ÞÞ

In the rest of this section we shall briefly describe a few variations on the theme of the Boltzmann equation (presented above as a particular realization of (55)) that are or could be of interest to chemical engineers.

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

105

3.1.3.1 Multicomponent systems with binary chemical reactions. The setting discussed above can be extended in an obvious way to multicomponent gases and also to multicomponent gases involving binary chemical reactions. 3.1.3.2 Spatially nonlocal collisions. The collisions discussed above are all local, i.e., they take place at one point and the position coordinates of the incoming and the outcoming particles remain unchanged. This is the consequence of considering the particles as points. Following Enskog, we can extend the collision transformation ðv; v1 Þ ! ðv 0 ; v10 Þ to ðv; r; v1 ; r1 Þ ! ðv 0 ; r 0 v10 ; r10 Þ. By doing it we change the Boltzmann kinetic equation into the Enskog kinetic equation or, depending on the choice of the nonlocal collision transformation, into an Enskog-like kinetic equation. This type of modification of the time evolution provides a dynamical setting for the equilibrium theory discussed in Section 2.2.2 (see Grmela, 1971) 3.1.3.3 Exchange-of-identity collisions. Instead of the collision ðv; v1 Þ ! ðv 0 ; v10 Þ we can try a particularly weak collision consisting only of the exchange of identities of the colliding particles, i.e., v 0 ¼ v1 and v10 ¼ v. This type of collision makes the Boltzmann collision term to disappear. However, in the context of two point kinetic theory (i.e., n(r,v) is replaced by nðr1 ; v1 ; r2 ; v2 Þ) the corresponding collision term is nontrivial. It drives the gas to states at which nðr1 ; v1 ; r2 ; v2 Þ ¼ nðr1 ; v1 Þnðr2 ; v2 Þ: 3.1.3.4 Inelastic collisions. If the particles composing the gas have an internal structure, in particular an internal energy ", we replace n(r,v) by n(r,v,") and replace the elastic collisions ðv; v1 Þ ! ðv 0 ; v10 Þ for which (80) and (81) hold with inelastic collisions ðv; ;v1 ; 1 Þ ! ðv 0 ; 0 ; v10 ;  10 Þ for which (80) still holds but (81) is replaced by v2 þ ðv1 Þ 2 þ  þ 1 ¼ 0 0 2 0 2 0 ðv Þ þ ðv1 Þ þ  þ  1 . In this way we arrive at a modified version of the kinetic theory that is suitable for discussing the time evolution of granular gases.

3.1.4 Example: fluid mechanics At least in the traditional domains of chemical engineering and in the traditional core of instructions that chemical engineers receive during their education, fluid mechanics (transport phenomena) has played a key role. Also one of the principal motivations for creating nonequilibrium thermo­ dynamics was an attempt to make fluid mechanics manifestly compatible with equilibrium thermodynamics. Even the noncanonical Hamiltonian structures that play such an important role in the multiscale nonequili­ brium thermodynamics presented in Section 3 have been first discovered

106

Miroslav Grmela

by Clebsch (1895) for the Euler equation. We shall therefore recall below how fluid mechanics arises as a particular realization of (55). The state variables (called hydrodynamic fields) chosen in fluid mechanics are x ¼ ð ðrÞ;eðrÞ;uðrÞÞ

ð85Þ

where (r) is the field of mass per unit volume, e(r) the field of the total energy per unit volume, u(r) the field of momentum also per unit volume, and r denotes the position vector. Except for the field of momentum, the state variables (85) are the same as those used in classical equilibrium thermodynamics (1) but they are let to depend on the position coordinate r. Consequently, the projection (4) is given by ð 1 dr eðrÞ V ð 1 n¼ dr ðrÞ VMmol e¼

ð86Þ

where Mmol is the molecular weight and V is, as in the previous examples, the volume of the region of the space confining the fluid under consideration. The eta-function h ¼ hð ; e; uÞ

ð87Þ

will be left in the discussion below unspecified. We only recall that the choice of the most used eta-function is based on the following argument. We regard the fluid as being locally at equilibrium. This means that we choose h( ,e,u;r) to be the function s(e,n) with e replaced by the internal u ðrÞ energy eðrÞ � 2 ðrÞ and n replaced by M ðrÞ , Examples of spatially nonlocal mol eta-functions (needed, e.g., when dealing with fluids involving large spa­ tial inhomogeneities as, e.g., fluids in the vicinity of gas–liquid phase transitions) can be found in Grmela (2008). The next step in the construction of the particular realization of (55) is the specification of the kinematics of (85) (i.e., specification of the operator L). The necessity to satisfy the degeneracy requirement (52) suggests to begin the search for the operator L by passing to new state variables 2

^x ¼ ð ðrÞ; hðrÞ; uðrÞÞ

ð88Þ

We have already used the same strategy in Section 3.1.1. The transforma­ tion x ! ^ x is one-to-one because he, having the physical interpretation of the inverse of the temperature, is always positive. We need now to find L expressing kinematics of (88). The physical insight on which we shall base

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

107

our search is the following: The motion of a fluid is seen as a continuous sequence of transformations R 3 ! R 3 . These transformations form a Lie group and the field u(r) is an element of the dual of the Lie algebra corresponding to the group. The general relation (see, e.g., Marsden et al. (1983)) between the Lie group structure and the Poisson bracket in the dual of its Lie algebrað(that we have already used in kinetic theory— � � see (75)) implies fA; Bg ¼ dr ui @j ðAui ÞBuj � @j ðBui ÞAuj . The two remaining scalar fields (r) and h(r) in (88) are assumed to be passively advected (in other words, Lie dragged) by the flow generated by u(r). Consequently, the Poisson bracket expressing kinematics of (88) is given by ð h � � fA; Bg ¼ dr ui @j ðAui ÞBuj � @j ðBui ÞAuj � � � � � � þ @j A Buj � @j B Auj � �i þ h @j ðAh ÞBuj � @j ðBh ÞAuj

ð89Þ

It is easy to verify that the degeneracy requirements (52) are indeed satisfied. ^x $ x on (89) we obtain a By applying the one-to-one transformation � � ð new Poisson bracket H ¼ dr hðxÞ �� � ð � � � huj hu Buj � Be fA; Bg ¼ dr ui @j Aui � Ae i he he � �� � huj hu � @j Bui � Be i Auj � Ae he he � � �� � huj h þ @ j A � Ae Buj � Be he he � �� � huj h � @j B � Be Auj � Ae he he � � �� � � �� �� huj huj

1 1 þ h @j Ae Buj � Be � @j Be Auj � Ae he he he

he

ð90Þ

expressing kinematics of x (see (85)). We turn our attention now to the dissipative part of the time evolution. We shall discuss it with the state variables (85). Following Navier, Stokes, and Fourier, the thermodynamic forces driving the fluid to equilibrium are

108

Miroslav Grmela

ðFÞ

�

Xi ðe Þ ¼ @i e

�

ð91Þ

� � �� � � �� uj u 1 ðNSÞ Xij ðu� ; e� Þ ¼ ∂j �i þ ∂i � 2 e e � �� uj X ðNSvol Þ ðu� ; e� Þ ¼ ∂j � e

ð92Þ ð93Þ

where @i ¼ @@ri . The force (91) generates the Fourier heat flow, the force (92) the Navier Stokes friction, and (93) the Navier–Stokes friction in the volume deformation. From these three thermodynamic forces we now construct a dissipation potential . If we limit ourselves to small thermo­ dynamic forces (recall that the thermodynamic forces disappear at equili­ brium) then we can choose the following quadratic potential: ð

�

1 ðFÞ ðFÞ 1 ðNSÞ ðNSÞ 1  ¼ dr Xi Xi þ Xij Xij þ vol X ðNS vol Þ X ðNS vol Þ 2 2 2

� ð94Þ

where the three coefficients > 0;  > 0, and vol > 0 are material para­ meters (or material functions) in which the individual features of the fluids under consideration are expressed; is called a coefficient of heat conduc­ tion,  a coefficient of viscosity, and vol a coefficient of volume viscosity. We can now write explicitly (55) @ ¼ �@j ð vj Þ @t @ui ðNSÞ ¼ �@j ðui vj Þ � @i p � @j ij @t � � @e ðNSÞ ðFÞ ¼ �@j ðevj Þ � @j ðpvj Þ � @j vj ij � @j qj @t � ð � ðFÞ ðNSÞ h_ ¼ dr Xi XðFÞ þ Xij XðNSÞ þ X ðNS vol Þ X ðNS vol Þ i

ð95Þ

ij

where  ¼ h1e is the local hydrodynamic temperature, v = –hu is the fluid velocity. ðNSÞ

ij

¼  @j vj

ð96Þ

is the Navier–Stokes extra stress tensor ðFÞ

qi

¼ @i he

ð97Þ

is the Fourier heat flow, and p ¼ �e � h þ h þ uj vj

ð98Þ

the local hydrodynamic pressure. Equations (95) are indeed the familiar Navier–Stokes–Fourier equations.

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109

The computations involved in the passage from (94) and (89) to the time evolution equations are straightforward. Here we shall only indicate the calculations. We note, as we already did in Section 3.1.3 (see the text following Equation (76)), that the equation x_ ¼ L’x can also be written as A_ ¼ fA; Fg required to hold for all sufficiently regular functions A(x). This formulation provides a systematic method for passing from the bracket to the time evolution equation. We shall illustrate it below on the example of the bracket (89), but the method can obviously be used for any bracket. We proceed as follows: (i) we replace in (89) ð B by ’, (ii) we write (if necessary � � using integration by parts) fA; ’g as dr A ð�Þ þ A ð��Þ þ Au ð���Þ . The equation x_ ¼ L’x is then @t ¼ ð�Þ; @t  ¼ ð��Þ; @t u ¼ ð���Þ:

3.1.5 Example: particle dynamics In this example we shall comment about the time evolution corresponding to the Gibbs equilibrium statistical mechanics (see Section 2.2.2). We shall use the same notation as the one used in Section 2.2.2. The macroscopic system under consideration is regarded as being composed of np � 1023 particles. The state variables describing it are (26). Their time evolution is governed by (72) extended in an obvious way from one to np particles. If we pass from (26) to the state variable (27), the time evolution (72) transforms into the Liouville equation corresponding to (72), i.e., to Equation (74) with n replaced by (27) and the Poisson bracket that extends ð (75) ð in an obvious way to np particles (i.e., � � h @Af @Bf @Bf @Af i np d1… dnp f 1; … ; np fA; Bg ¼ i¼1 @vi � @ri @vi ). @ri But this is only the starting point. In order to see the approach to L eth (in other words in order to provide an appropriate dissipative part to the Liouville equation), we have to enter the process of solving the Liouville equation. Roughly speaking, the process is a pattern recognition in the phase portrait that corresponds to the Liouville equation (i.e., a pattern recognition in the set �of all trajectories generated by the Liouville equation � with a family of ep 1; … ; np ). The pattern recognition process may involve, for example, an investigation of various ergodic and/or chaotic type properties of the particle trajectories. An alternative and a very interesting strategy to proceed in the pattern recognition consists of trans­ forming first the Liouville equation governing the time evolution of (27) into�a hierarchy of coupled equations governing the time evolution of � �� x ¼ f 1 ð1Þ; f 2 ð1; 2Þ; … ; f np 1; 2; …; np , where f ni ð1; …; ni Þ; ni < np are reduced distribution functions, and then entering the process of reduc­ tions that we shall discuss in some detail in Section 4 below. The famous BBGKY hierarchy is an example of the hierarchy of equations governing

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Miroslav Grmela

� the time evolution of

� �� f 1 ð1Þ; f 2 ð1; 2Þ; … ; f np 1; 2; …; np . Another

hierarchy of this type has been introduced in Grmela (2001). In the particular case of np ¼ 1 (i.e., the state variable is the one-particle distribution function), the Boltzmann collision term discussed in Section 3.1.3 is an example of the dissipative term that, if attached to the Liouville part, provides a time evolution equation describing the approach to Leth . The pattern recognition process is in this case a separation of particle trajectories into collisionless and collision parts and treating them in a different way. The term “coarse-graining” is often used to denote the process leading to reduced descriptions. Indeed, one possible strategy that can be applied to recognize patterns is to coarse grain. However, because the coarsegraining is only one among the many ways to proceed in the pattern recognition process, because it is important to emphasize the complexity involved in the reduction process (by invoking our innate ability to recog­ nize patterns in everyday life), and because the coarse-graining is obviously very coordinate dependent and as such very nongeometrical in its nature, it is always preferable to visualize the micro ! macro passage as a “pattern recognition in the phase portrait” rather than a “coarse­ graining.”

3.1.6 Examples: complex fluids Complex fluids are the fluids for which the classical fluid mechanics discussed in Section 3.1.4 is found to be inadequate. This is because the internal structure in them evolves on the same time scale as the hydro­ dynamic fields (85). The role of state variables in the extended fluid mechanics that is suitable for complex fluids play the hydrodynamic fields supplemented with additional fields or distribution functions that are chosen to characterize the internal structure. In general, a different internal structure requires a different choice of the additional fields. The necessity to deal with the time evolution of complex fluids was the main motivation for developing the framework of dynamics and thermodynamics dis­ cussed in this review. There is now a large amount of papers in which the framework is used to investigate complex fluids. In this review we shall list only a few among them. The list below is limited to recent papers and to the papers in which I was involved. A simple fluid like, e.g., water, that is in the vicinity of the point of phase transition becomes a complex fluid. Large spatial inhomogeneities and long range pair correlations become the internal structure whose time evolution cannot be ignored. An extension of the setting of Section 3.1.4 to this type of fluids is discussed in Grmela (2008). We can see this extended fluid mechanics also as an attempt to provide the time evolution corre­ sponding to the equilibrium theory discussed in Section 2.2.2.

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Suspensions of various types are all complex fluids. We list a few examples: suspensions of macromolecules (polymeric fluids) (Eslami and Grmela, 2008), suspensions of nanoparticles in polymeric fluids (Eslami et al., 2007, 2009), suspensions of membranes (immiscible blends and biological fluids) (Gu and Grmela, 2008, 2008a, 2009; Gu et al., 2008), and suspensions of rigid spherical particles (Zmijevski et al., 2005, 2007) Interesting examples of complex fluids are superfluids. The time evo­ lution of these fluids is dominated by the nondissipative part. The frame­ work (55) is thus particularly well suited to deal with this type of the time evolution (Holm et al., 1987; Grmela, 2008a))

3.2 Combination of scales In this subsection we shall construct two independent realizations of (55) on two independent levels and then we combine them.

3.2.1 Example: direct molecular simulations In chemical engineering as well as in physics and in science in general, direct molecular simulations became very popular. The enormous com­ puting power that is now routinely available allows to make a direct passage from molecules to macroscopic properties. The idea is simple. We calculate trajectories of all the molecules composing the macroscopic system and then extract from them the macroscopic properties of our interest. When trying to follow this path, we meet however some difficul­ ties. Leaving aside the technical challenges in calculations (the problem of solving numerically a very large system of ordinary differential equa­ tions), there are three important issues that have to be addressed: (Problem 1) The actual number of molecules in macroscopic systems is �1023 but computers can deal at most with �106. (Problem 2) The macroscopic systems of interest are often subjected to external forces that are defined and have a clear meaning only on some mesoscopic and macroscopic levels of description. For example, the macroscopic system that we regard in direct molecular simulations as composed of, say, 106 particles can be subjected to a temperature gradient. How do the particles feel such force? (Problem 3) The results coming out of computers are trajectories of the particles. How shall we extract from this information the information that we can compare with results of mesoscopic or macroscopic measurements (e.g., the measurements done in fluid mechanics)? We shall now use the setting developed above in this chapter to address these three problems. We begin with the first problem (Problem 1). Let np be the number of particles whose trajectories are actually followed in computers. Since np � 106 << 1023 , we shall choose to describe states of the particles by

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Miroslav Grmela

the np -particle phase space distribution function rather than by points in � � the 6np-dimensional phase space with coordinates r1 ; v1 ; … ; rnp ; vnp as it is done in particle mechanics. Next, we shall proceed as follows: First, we construct the time evolution equation (the np-particle kinetic equation) and establish the passage from the np-particle distribution functions to quantities measured in macroscopic measurements. Second, we solve the kinetic equa­ tion by regarding it (approximately) as a Liouville equation corresponding to time evolution equations of np particles whose trajectories we then find with the assistance of computers. Following this route, the equations used in the simulations (a system of 6np or more ordinary differential equations) are not constructed directly from Newton’s mechanics of np particles but indir­ ectly by writing down first a kinetic equation and only then arriving at the time evolution equations of particles as a result of choosing a particular method of solving the kinetic equation. In the text below, we shall address mainly the first step (i.e., formulation of the kinetic equation and its coupling with other scales). The second step (passage from the kinetic equations to particle equations) will only be indicated. We proceed now to the problems (Problem 2) and (Problem 3). At least two levels of description are involved in direct molecular simulations. The first one is the level of the np-particle kinetic theory and the second is the level of fluid mechanics on which the external forces and the final results that we seek are formulated. We shall use the multiscale formulation developed above and combine the two levels. The two levels that we consider in this section are L 1–classical fluid mechanics with the state variables x1 given in (85) or (87) whose kinematics is expressed in the bracket (89) and L 2–np-particle kinetic theory with the state variables � � x2 ¼ f np r1 ; v1 ; � � � ; rnp ; vnp

ð99Þ

whose kinematics is expressed in the Poisson bracket ð fA; Bg ¼

ð dr1

f np

ð ð dv1 � � � drnp dvnp

np � X @Af np @Bf np i¼1

@ri

@vi

�

@Bf np @Af np @ri

@vi

� ð100Þ

which is based on the same physics as the bracket (75) except that one particle is replaced by n particles.

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

113

From the above two levels we construct now a third level as their combination: L 1&2–classical fluid mechanics and np-particle kinetic theory The level L 1&2 is a simple superposition of the two levels L 1 and L 2. This means that the state variables on the levels L 1&2 are x3 ¼ ðx1 ; x2 Þ

ð101Þ

with the kinematics given by the sum of the brackets (89) and (100). Next, we introduce new coordinates on the level L 1&2. The coordinates ðx1 ; x2 Þare replaced by � � �� u rÞ; g fnp r1 ; v1 ; … ; rnp ; vnp x ¼ ~ðrÞ; ~ðrÞ; ~ð

ð102Þ

that are related to ðx1 ; x2 Þ by ð ð ð ð n

X ~ ðrÞ ¼ ðrÞ � kB dr1 dv1 … drnp dvnp ðri � rÞ f np ln f np

ð

ð ð n X dv1 … drnp dvnp ðri � rÞ f np

ð

ð ð n

X dv1 … drnp dvnp vi ðri � rÞ f np

~ ðrÞ ¼ ðrÞ þ dr1

ð

i ¼ 1

ð

~ ðrÞ ¼ uðrÞ þ dr1 u

i¼1

�

�

ð103Þ

i¼1

� � g fnp r1 ; v1 ; … ; rnp ; vnp ¼ f np r1 ; v1 ; … ; rnp ; vnp

We note that the transformation ðx1 ; x2 Þ ! x defined in (103) is one-to­ ~ as overall (i.e., large-scale) hydrody­ ~ u one. We interpret physically ~; ; namic fields on which there are superposed “L 2-fluctuations” (i.e., smallf np : scale quantities) characterized by the np -particle distribution function g We apply now the one-to-one transformation (103) on the bracket expressing kinematics of ðx1 ; x2 Þ (i.e., on the sum of the brackets (89) and (100)). To prepare the calculations we note that A ðr Þ ¼ A~ ðr Þ A ðr Þ ¼ A~ ðr Þ Af n

p

Au ðr Þ ¼ A~u ðr Þ � � ~ r1 ; v1 ; … ; rnp ; vnp ðr1 ;v1 ;… ; rn ;vn Þ ¼ Afnp p

p

�kB

np h i� � X 1 þ ln f np r1 ; v1 ; … ; rnp ; vnp A~ ðri Þ i¼1

þ

np np X X A ~ ðri Þ þ vi A~u ðri Þ i¼1

i¼1

ð104Þ

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Miroslav Grmela

Using (104) in the brackets (89) and (100), whose sum is the bracket expressing the kinematics of ðx1 ; x2 Þ, we arrive, after somewhat lengthy but completely straightforward calculations, to (we are omitting hereafter the tilde) ð ð ð ð fA; Bg ¼ f A; B g ðhyd Þ þ f A ; Bg ðKTnp Þ þ dr1 dv1 … drnp dvnp " f np

� np � � X @ � @ � � Af np Buðri Þ � Bf n Auðri Þ @ri @ri i¼1

þ f np

np � � �� X � @ � � @ � @ � @ � A ðri Þ B f np � B ðri Þ Af np @ri @vi @ri @vi i¼1

þ f np

np � � �� X � @ � � @ � @ � @ � vi Au ðri Þ Bf np � vi Bu ðri Þ A f np @ri @vi @ri @vi i¼1

# np � � @ � �� X � @ � � @ � @ � B ðri Þ A ðri Þ Bf np � Af np � kB f np ln f np @vi @ri @vi @ri i¼1

ð105Þ

where fA; Bg ðhyd Þ is the Poisson bracket (89) expressing kinematics in classical hydrodynamics and fA; Bg ðKTnp Þ is the Poisson bracket (100) expressing kinematics in the np-particle kinetic theory. The nondissipative time evolution equations x_ ¼ e1� L’x (see (55)) cor­ responding to (105) are the following: @ ð ’uðrÞÞ @r ! ð ð ð ð @’f np @ X � r1 dv1 … drnp dvnp ðr � ri Þf n @r i @vi

@t ¼ �

r

@ ð ’uðrÞÞ @t  ¼ � @r ! ð ð ð ð @’f np @ X þ kB r1 dv1 … drnp dvnp ðr � ri Þf np ln f np @r i @vi r � � @ @pðrÞ @ðrÞ � u’u ðr Þ � @t u ¼ � @r @r @r " ! ! np @’f np

X @’f np @ @ @t f np ¼ � f np � f np

@vi @ri

@ri @vi i¼1 �

� � � @’u ðri Þ @ � @ f np vi

f np ’u ðri Þ þ @ri

@ri @vi

� � � �# @’ ðri Þ @’ ðri Þ @ @ � kB f np ln f np þ f np @vi @ri @ri @vi

ð106Þ

115

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

where p(r) is the hydrostatic pressure given by pðrÞ ¼ �ðrÞ þ ðrÞ’ ðr Þ þ ðrÞ’ ðr Þ þ uðrÞ’u ðr Þ ð ð ð Xð þ dr1 dv1 … drnp dvnp f np ðr � ri Þ’f np

ð107Þ

ir

and ðrÞ ¼ �

Xð

ð ð ð @’f np dr1 dv1 … drnp dvnp f np ðr � ri Þvi

@vi

ir ð ð ð ð XX @’f np þ dr1 dv1 … drnp dvnp f np ðr � ri Þðri � rj Þ @ðri � rj Þ i j r

ð108Þ

r

By ir we denote sum over the particles that are in a neighborhood of the point with the coordinate r. The next steps are the following: Step 1: Passage to the entropy repre­ sentation and specification of the dissipative thermodynamic forces and the dissipative potential . Step 2: Specification of the thermodynamic potential . Step 3: Recasting of the equation governing the time evolution of the np-particle distribution function f np into a Liouville equation corre­ sponding to the time evolution of np particles (or np0 quasi-particles, np0 > np —see the point 4 below) that then represent the governing equa­ tions of direct molecular simulations. We shall leave to interested readers to make these three steps (see also Grmela (1993) where a similar approach is taken but only for isothermal fluids). We end this section with a few observations about Equations (106). 1.

We note that the np-particle kinetic equation in (106) implies that r_i ¼

@’f np @vi

þ ’u ðr Þ

ð109Þ

ð ð ð ð if ’ involves the kinetic energy dru2 =2 þ dr1 dv1 � � � drnp ð np dvnp f np  v2i =2m then (109) shows that the velocities (v1,…, vnp ) i¼1

2.

are the peculiar velocities (i.e., the velocities modulo the overall velocity) of the simulated particles. If we omit the last two terms in the kinetic equation (involving gradients of and ) then (106) implies v_ i ¼ �

@’f np @ri

� vi

@’u @r

ð110Þ

116

3.

4.

Miroslav Grmela

The first term on the right-hand side of (110) is the gradient of the particle potential energy and the second term the force induced by the imposed gradient of the overall flow. We note that the second term in (108) is the familiar Kirkwood expression for the stress tensor in terms of the n-particle distribution function. The kinetic equation (the last equation in (106)) is, in general, a nonlinear equation and as such it cannot be seen as being exactly equivalent to a Liouville equation corresponding to ordinary differential equations representing the time evolution of np particles. The nonlinearity of the kinetic equation becomes apparent in particular in terms of involving the entropy (or alternatively the temperature). In order to proceed with recasting the np-particle kinetic equation into a set of 6np ordinary differential equations governing the time evolution of the np particles, we have to make an approximation which, in general, will bring into the picture other particles (called quasiparticles or also “ghost” particles) representing collective features. Such features are described by the distribution function but are not described by the position coordinate and momenta of the np particles. They can however be brought (but only approximately) into the formulation as the time evolution of some kind of quasiparticles. How to make this type of approximation and keep at the same time the GENERIC structure (55) remains an open problem.

4. MULTISCALE NONEQUILIBRIUM THERMODYNAMICS OF DRIVEN SYSTEMS The time evolution that has been investigated in Section 3 is the time evolution seen experimentally in externally unforced macroscopic systems approaching, as t ! 1, equilibrium states at which their behavior is seen experimentally to be well described by classical equilibrium thermody­ namics. In other words, in Section 3, we have investigated the time evolu­ tion L meso 1 ! L eth

ð111Þ

where L meso 1 is a symbol denoting a mesoscopic level (e.g., the level of kinetic theory) on which x1 M1 serves as a state variable and L eth denotes the level of classical equilibrium thermodynamics (discussed in Section 2) on which y [ N serves as a state variable. We recall the main features of (111) that are all displayed in (50) and in the associated with it Gibbs Legendre manifold }N1 :

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

ðIÞ

117

ð112Þ

�

�

The thermodynamic potential ’ðx1 ; y Þ ¼ �hðx1 Þ þ ; hðx1 Þ is the eta-function; yðx1 Þ is the projection M1 ! N of x1 on y. The potential ’ plays the role of the Lyapunov function associated with the approach (111). ðIIÞ

ð113Þ

The Invariant manifold Meth � M1 composed of equilibrium states (x1)eth(y�) [ M1. This manifold is invariant with respect to the time evolution taking place on � M1, and its elements are states on which the thermodynamic potential ’ðx1 ; y Þ reaches its minimum if considered as a function of x1 M1 : ðIIIÞ

ð114Þ

No time evolution takes place on Meth. ðIVÞ

ð115Þ �

�

�

The fundamental thermodynamic relation ’ ðy Þ ¼ ’ð ðx1 Þ eth ðy Þ; y� ÞÞ We recall that both the manifold Meth and the fundamental thermo­ dynamic relations ’� are displayed on the manifold }N |x�=0 (see the text following (6)). Experimental observations of the time evolution of externally unforced macroscopic systems on the level L meso 1 show that the level L eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if L meso 1 is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here L eth ) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for L eth (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description: (i) }slow , (ii) L meso 2 ! L eth , (iii) L meso 1 ! L meso 2 , and (iv) L meso 1 ! L meso 2 ! L eth . The first two are the same as (111). We now turn our attention to the third one, that is, L meso

1

! L meso 2

ð116Þ

We shall call the time evolution involved in (116) a fast time evolution and the time evolution taking place on L meso 2 a slow time evolution. We can use

118

Miroslav Grmela

this terminology also for L meso 1 ! L eth . The slow time evolution is in this case indeed very slow. It is no (or we can also say still) time evolution. Before starting to discuss (116), we make an observation. The fast time evolution (116) is also observed in driven systems that cannot be described on the level L eth . For example, let us consider the Rayleigh–Bénard system (i.e., a horizontal layer of a fluid heated from below). It is well established experimentally that this externally driven system does not reach thermo­ dynamic equilibrium states but its behavior is well described on the level of fluid mechanics (by Boussinesq equations). This means that if we describe it on a more microscopic level, say the level of kinetic theory, then we shall observe the approach to the level of fluid mechanics. Con­ sequently, the comments that we shall make below about (116) apply also to driven systems and to other types of systems that are prevented from reaching thermodynamical equilibrium states (as, e.g., glasses where inter­ nal constraints prevent the approach to L eth ). We propose that the fast time evolution (116) is governed by the same equations as those governing the time evolution involved in L meso 1 ! L meso 2 (discussed in Section 3) with only a few necessary modifications. ðIÞ

ð117Þ

� �� � The thermodynamic potential ’ x1 ; x2 ¼ �hðx1 Þ þ hx2 ; x2 ðx1 Þi; hðx1 Þ is the eta-function, x2 ¼ x2 ðx1 Þ is the projection M1 ! M2 of x1 on x2. The potential  plays the role of the Lyapunov function associated with the approach L meso 1 ! L meso 2 : ðIIÞ

ð118Þ

The quasi-invariant manifold Mslow � M1 composed of slow states � �� ðx1 Þslow x2 M1 . This manifold is quasi-invariant with respect to the time evolu­ tion taking �place on M1, and its elements are states on which the thermodynamic �� potential ’ x1 ; x2 reaches its minimum if considered as a function of x1 M1 : As we have already pointed out in Section 3, our primary knowl­ edge of macroscopic systems comes from observing their time evolution. To provide the space M with time evolution means to provide it with a vector field. We can thus visualize M equipped with a vector field as M filled with arrows providing the orders to move. The manifold Meth is the manifold of fixed points, i.e., the points to which there is no arrow attached. It is much more difficult to recognize Mslow than to recognize Meth. Roughly speaking, in order to recognize Mslow, we look for a manifold (a submanifold of M1, imagine it as a surface imbedded in M) on which the arrows filling the space M1 will appear to be attached in such a way that they are as much as possible tangent to the surface

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

119

(if they are exactly tangent then the surface is exactly invariant with respect to the time evolution) and relatively small (guaranteeing that the time evolution taking place on Mslow will be slow). We have to emphasize that the whole picture of M filled with arrows is coordinate dependent. It changes by changing the way the elements x [ M are represented (i.e., by making one-to-one nonlinear transformations of x). The recognition of an appropriate Mslow is thus a sort of pattern recog­ nition process involving a play with different views (in different coor­ dinate systems) of M. In addition, in order to be able to see more clearly the pattern and the extent of the invariance of Mslow , the passing from arrows to trajectories or at least to pieces of trajectories (i.e., the govern­ ing equations have to be solved or at least partially solved) has to be made. The manifold Mslow emerging in the process is not anymore an invariant manifold as Meth but a quasi-invariant manifold. It is difficult to give a precise definition of a “quasi-invariant manifold”. Examples (some of which are presented below) provide the best way to explain this important concept. ðIIIÞ

ð119Þ

Slow time evolution takes place on Mslow : While there is, of course, no interest in the slow time evolution in the discussion of L meso 1 ! L eth , since the slow time evolution is in this case no time evolution, this point is the main focus of the investigation of reduced descriptions (see, e.g., Gorban and Karlin, 2005; Yablonskii et al., 1991). The lack of interest in the point (III) (see (114)) in the context of L meso 1 ! L eth then also means the lack of interest in the manifold Meth, i.e., in the point (II) (see (113)). ðIVÞ

ð120Þ

The slow fundamental thermodynamic relation � �� � � �� �� � The main difference between ’slow x2 ¼ ’ ðx1 Þslow x2 ; x2 Þ and L meso 1 ! L meso 2 is that the focus in the investigation of L meso 1 ! L eth is put on the points (III) and (II) (see (119),(118)) while the focus in the investigation of L meso 1 ! L meso 2 is put on the points (I) and (IV) (see (112),(115)). The most important message that we want to convey in this section is that attention should be paid also to the points (I) and (IV) in the investigation of the reduction L meso 1 ! L eth . Even if the slow time evolution is certainly the main output of the investigation of L meso 1 ! L meso 2 , the fast time evolution complementing the slow time evolution provides an additional useful information. It provides the fundamental thermody­ namic relation in the space in which the slow time evolution takes place. We shall call it a “slow fundamental thermodynamic relation.” The slow

120

Miroslav Grmela

dynamics and the slow fundamental thermodynamic relation are two features of the reduced description inherited from the more microscopic space L meso 1 ! L meso 2 . If seen only on the reduced level Mmeso 1 , these two features are independent of each other. Since L meso 2 applies also to driven systems, the slow fundamental thermodynamic relation intro­ duces thermodynamics into the investigation of driven systems (Grmela, 1993a) We shall not enter here into the general formulation in the context of contact geometry. Instead, we shall only work out two examples.

4.1 Example: a simple illustration We take the level L meso 1 to be the level discussed in Section 3.1.1 (we shall now use x1 M1 instead of xM to denote the state variables) and the level L meso 2 on which x2 ¼ ðq; ; Þ

ð121Þ

serve as state variables and q_ ¼ L2 hq _ ¼ 0

ð122Þ

_ ¼ 0

govern their time evolution; Lð; Þ > 0 is a material parameter. The phy­ sical system that we can think of being represented by (121) and (122) is a glass. The state variable q is a kind of free volume. Due to internal constraints, the material parameter L2 ð; Þ becomes very small for some ð; Þ which then means that the approach to equilibrium (i.e., to the state at which hq ¼ 0) is prevented and the system “freezes” in a state out of equilibrium. The projection x1 ! x2 is given by: ðq; p; ; Þ
ðq; ; Þ

ð123Þ

As for the eta-function hðq; p; ; Þ, we choose it to be 1 hðq; p; ; Þ ¼ hðq; ; Þ � ap2 ; 2

ð124Þ

where hðp; ; Þ is the eta-function on the level L 2 , and a > 0 is a material parameter. The time evolution Equations (62) thus become

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

ap h hq p_ ¼ � Lap h _ ¼ 0

121

q_ ¼

ð125Þ

_ ¼ 0

In order to identify the manifold Mslow � M1 we assume that in (125) is large and that p has already reached its stationary value determined by equating the right-hand side of the second equation in (125) to zero. We note that if we interpret p as a momentum corresponding to q then the above assumption has the physical meaning of applying a large friction force and neglecting inertia. As a consequence of the assumption we obtain Mslow ¼

� � 1 hq ðq; p; ; ÞM1 j p ¼ La h

ð126Þ

The slow time evolution on M1 (i.e., the time evolution on M1 restricted to Mslow) is governed by 1 hq L ðh Þ 2 _ ¼ 0

q_ ¼

ð127Þ

_ ¼ 0

If we compare (127) with (122), we see that the reduction process brought us the following relation between the material parameters introduced on the level L meso 1 and 2 introduced on the level L meso 2 : L2 ¼

1 L ðh Þ 2

ð128Þ

Everything that we have done so far in this example is completely stan­ dard. The next step in which we identify the slow fundamental thermo­ dynamic relation in the state space M2 (i.e., we illustrate the point (IV) (see (120))) is new. Having found the slow manifold Mslow in an analysis of the time evolution in M1, we now find it from a thermodynamic potential. � � We look for the thermodynamic potential ’ðq; p; ; ; q ; � ; Þ so that the manifold Mslow arises as a solution to ’ ¼ 0;

’ ¼ 0;

’q ¼ 0;

’p ¼ 0

ð129Þ

We look for ’ in the form �

�

�

�

�

�

�

�

�

’ðe; q; p; q ; e Þ ¼ �hð; ; q; pÞ þ e þn þ q q þ p ðq ; e ; n Þp

ð130Þ

122

Miroslav Grmela

�

�

�

�

�

where p ¼ p ðq ; e ; n Þ is a function to be specified. We easily verify that with �

�

p ¼�

q Le�

ð131Þ

solutions to (129) is the slow manifold Mslow. Finally, the fast time evolution (i.e., the time evolution governing the approach to the slow time evolution on Mslow governed by (122), (128)) is governed by p_ ¼ �L’p

ð132Þ

and the slow fundamental thermodynamic relation (i.e., ’ evaluated on the slow manifold Mslow) becomes �

�

�

�

�

�

�

�

hslow ðe ; n ; q Þ ¼ h ðe ; n ; q Þ � �

�

�

1 2aL2

�

�

q e�

�2 ð133Þ

�

which h ðe ; n ; q Þ is the dual form of the eta-function h(e, n, q).

4.2 Example: ChapmanEnskog reduction of kinetic theory to fluid mechanics In this illustration we take the level L1 to be the level of kinetic theory (see Section 3.1.3), the level L2 the level of fluid mechanics (see Section 3.1.4), and the projection x2 ¼ x2 ðx1 Þ is given by ð

ðrÞ ¼ dv m nðr; vÞ ð uðrÞ ¼ dv v nðr; vÞ

ð134Þ

ð v2 eðrÞ ¼ dv nðr; vÞ 2m

Reduction of kinetic theory to fluid mechanics is historically the first example of a successful reduction of a mesoscopic dynamical theory to a more macroscopic dynamical theory. The method (called the Chapman– Enskog method) that was invented by Chapman and Enskog for this particular reduction remains still a principal inspiration for all other types of reduction (see, e.g.,. Gorban and Karlin, 2003, 2005), Yablonskii et al., 1991). In this example we briefly recall the geometrical viewpoint of the Chapman–Enskog method. We shall also illustrate the point (IV)

123

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

(see (120)). This illustration is a new result in the Chapman–Enskog reduction. The setting for our discussion is the space M1 equipped with the vector field ðv : f : Þ1 and the mapping pr1 ! 2 : M1 ! M2 . Our problem is to equip the space M2 with a vector field (v.f.)2 that is inherited from the vector field (v.f.)1 in M1. The elements x1 of M1 are the distribution functions (7); the elements x2 of M2 are the hydrodynamic fields (85). We shall organize the Chapman–Enskog reduction into four steps: ð0Þ (Step 1) Initial suggestion Mslow � M1 for the slow manifold Mslow � M1 is made. It is a manifold that has a one-to-one relation with the space M2. We can regard it as an imbedding of M2 in M1. (Step 2) The vector ð0Þ ð0Þ field (v.f.)1 is projected on Mslow (i.e., ðv:f:Þ1 denoting the vector field (v. ð0Þ

f.)1 attached to a point of Mslow , is projected on the tangent plane of ð0Þ

Mslow at that point). The projected vector field is denoted by the symbol ð0Þ

Mslow

ðv:f:Þ1

ð0Þ

ð1Þ

. (Step 3) The slow manifold Mslow is deformed into Mslow in ð1Þ

such a way that the difference between ðv:f:Þ1

attached to a point of

ð1Þ

ð1Þ Mslow

M and its projection ðv:f:Þ1 slow becomes smaller (as small as possible) ð0Þ than for the slow manifold Mslow . (Step 4) The slow fundamental ther­ ð0Þ modynamic relations associated with the slow manifolds Mslow and ð1Þ Mslow are identified.

Step 1 The dominant term in the Boltzmann Equation (84) is assumed to be the collision term, i.e., the second term on the right-hand side of (84). This then implies that, as t ! 1, solutions to Equation (82), denoted n ð0 Þ ðr; v; ; u; eÞ represent a good approximation to the asymptotic solution to (84). Con­ sequently, we choose n o ð0Þ Mslow ¼ nðr; vÞM1 jnðr; vÞ ¼ n ð0 Þ ðr; v; ; u; eÞ ð0Þ

ð135Þ

Equivalently, Mslow ¼ fnðr; vÞM1 jX ¼ 0g, where X is the thermody­ ð0Þ namic force given in (79). The elements of Mslow are the local Maxwell distribution functions. They are denoted hereafter by the symbol n ð0 Þ ðr; v; ; u; eÞ:

Step 2

ð0Þ

ð0Þ

Mslow

The vector fields ðv :f : Þ1 ðv:f:Þ1 ðv:f:Þ1

ð0Þ

and ðv:f:Þ2 are given by

124

Miroslav Grmela

ðv:f:Þ1 is given by the right hand side of ð84Þ � � 1 @n ðv:f:Þ1ð0Þ ¼ � vi ð0 Þ m @ri n ¼ n �� � � � � � � ð0Þ ð Þ ð0Þ ðu Þ ð0Þ ðe Þ ðv:f:Þ2ð0Þ ¼ ðv:f:Þ2 ; ðv:f:Þ2 ; ðv:f:Þ2

ð0Þ

ðv:f:Þ1Mslow

! � � � ð � ð � ð � 2 @n @n v @n ; � dv vvj ; � dv vj ¼ � dv vj @rj n ¼ n ð0 Þ @rj n ¼ n ð0 Þ 2m @rj n ¼ n ð0 Þ � ð0 Þ � � � � � � � ð0 Þ @n ð0 Þ @n ð0Þ ð Þ @n ð0Þ ðu Þ ð0Þ ðe Þ ðv:f:Þ2 ¼ ðv:f:Þ2 ; ; ðv:f:Þ2 @ ðrÞ @uðrÞ @eðrÞ ð136Þ ð0Þ Mslow

ð0Þ

The vector field ðv:f:Þ2 is the pull back of ðv:f:Þ1 by the mapping ðeðrÞ; ðrÞ; uðrÞÞ ! n ð0 Þ ðr; v; ; u; eÞ: It is easy to verify that the slow time evolution on M2 corresponding to the choice (135) of the slow manifold (i.e., the time evolution governed by ð0Þ the vector field ðv:f:Þ2 ) is the Euler time evolution (see (95) with ¼ 0;  ¼ 0; vol ¼ 0).

Step 3 ð0Þ

If the vector field ðv:f:Þ1 (i.e., the vector field (v.f.)1 evaluated on the slow ð0Þ

ð0Þ

Mslow

manifold Mslow is identical to the vector field ðv:f:Þ1

then the slow

ð0Þ

manifold Mslow is invariant (i.e., any trajectory that starts on it remains ð0Þ

Mslow

on it). However, ðv:f:Þ1

ð0Þ

Mslow

6¼ ðv:f:Þ1

ð0Þ

ð0Þ

and consequently ðv:f:Þ1 sticks out

ð0Þ

of the slow manifold Mslow and Mslow is thus not invariant. Following ð0Þ

Chapman and Enskog we shall now deform the slow manifold Mslow into ð1Þ

ð1Þ

ð0Þ

Mslow with the objective to make Mslow “more invariant” than Mslow (i.e., the vector field (v.f.)1 evaluated on the deformed manifold will stick less ð0Þ ð0Þ out of it than (v.f.)1 evaluated on Mslow sticks out of Mslow . The deforma­ tion is made by passing from nð0Þ ðr; v; ; u; eÞtonð1Þ ðr; v; ; u; eÞ that is a solution to ð0Þ

M ð0Þ ðv:f:Þ1 slow �ðv:f:Þ1

ð

¼

ð

0

ð

dv1 dv dv10 W Boltzmann ðn; v; v1 ; v 0 ; v 10 Þ � � � nð1Þ ðr; v 0 Þnð1Þ ðr; v10 Þ � nð1Þ ðr; vÞnð1Þ ðr; v1 Þ

ð137Þ

The new slow manifold will be n o ð1Þ Mslow ¼ nðr; vÞM1 jnðr; vÞ ¼ nð1Þ ðr; v; ; u; eÞ

ð138Þ

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

125

How shall we solve Equation (137)? Again, following Chapman and Enskog, we assume that the deformation from nð0Þ ðr; v; ; u; eÞ to nð1Þ ðr; v; ; u; eÞ We thus� introduce nð1Þ ðr; v 0 ; ; u; eÞ ¼ nð0Þ ðr; v 0 ; ; u; eÞ �is small. ð1Þ 1 þ ðr; v; ; u; eÞ , consider ð1Þ ðr; v; ; u; eÞ as small, and replace (137) by its linearized version ð0Þ

Mslow

ðv:f :Þ1

ð0Þ

�ðv:f :Þ1 ¼

ð

ð ð dv1 dv 0 dv 0 1 W Boltzmann ðn; v; v1 ; v 0 ; v 10 Þ

� nð0Þ ðr; v 0 ; ; u; eÞnð0Þ ðr; v 0 1 ; ; u; eÞ � � ð1Þ ðr; v 0 ; ; u; eÞ þ ð1Þ ðr; v 0 1 ; ; u; eÞ � � ð1Þ ðr; v; ; u; eÞ � ð1Þ ðr; v1 ; ; u; eÞ

ð139Þ

This is a linear integral equation for the unknown function ð1Þ ðr; v; ; u; eÞ. To find its solution is a technical problem that we shall ignore here and consider hereafter ð1Þ ðr; v1 ; ; u; eÞ and thus nð1Þ ðr; v1 ; ; u; eÞ and conse­ ð1Þ quently (see (138)) the slow manifold Mslow to be known. ð1Þ The slow time evolution corresponding to the slow manifold Mslow is ð1Þ ð1Þ governed by the vector field ðv:f :Þ1 that is the same as ðv:f :Þ1 introduced in (136) except that ð0Þ ðr; v1 ; ; u; eÞ is replaced by ð1Þ ðr; v1 ; ; u; eÞ. When explicitly calculated, the slow time evolution equation turns out to be the Navier–Stokes–Fourier Equation (95) with the kinetic coefficients ; ; vol expressed in terms of W Boltzmann : With this result, the investigation of the reduction usually ends. There are, of course, many questions that remain unanswered. For example, what would be the next improvement in the search for the slow manifold ð2Þ (i.e., what is the manifold Mslow )? How shall we address the fact that we have succeeded to relate two theories belonging to two different levels of description but with the domains of applicability that hardly have a common intersection? Indeed, the domain of applicability of fluid mechanics, regarded as an autonomous theory, are fluids (e.g., water) and the domain of applicability of the Boltzmann kinetic theory are gases (e.g., air). We shall not discuss here these questions but proceed to illustrate the point (IV) (see (120)) of the reduction process. The observa­ tions made in the rest of this section are new.

Step 4 We shall make the fourth step. We shall make it first for the slow manifold ð0Þ ð1Þ Mslow and then for its deformation Mslow : ð0Þ

The slow manifold Mslow ð0Þ Let the slow manifold be Mslow (see 135) and the slow time evolution the ð0Þ Euler equations generated by the vector field ðv:f :Þ2 (see 136)). We look

126

for ð0Þ Mslow

Miroslav Grmela

the thermodynamic (0) such � n o potential �� ð0Þ � ¼ nðr; vÞM1 � ’ nðr;vÞ ¼ 0 . We easily verify that � � � � � ’ð0Þ nðr; vÞ; ðrÞ; u ðrÞ; e ðrÞ ¼ �hð0Þ ðnÞ ð ð v2 � þ dr dv e ðrÞ nðr; vÞ 2m ð ð � þ dr dv ðrÞnðr; vÞ ð ð � þ dr dv uj ðrÞvj nðr; vÞ

that

ð140Þ

where h(0)(n) is the Boltzmann eta-function (9). ð0Þ The thermodynamic potential evaluated on Mslow (i.e., ’ð0Þ jn  Mð0Þ ) slow

becomes the fundamental thermodynamic relation of an ideal gas which differs from (13) only by the local nature of the state variables (i.e., the state variables depend on the position coordinate r) and by the presence of the momentum u(r). This is the slow fundamental thermodynamic relation ð0Þ corresponding to the slow manifold Mslow : The fast time evolution, i.e., the time evolution describing the approach ð0Þ the slow manifold, is governed by (82) (i.e., Mslow with the dissipation @ potential  given by (79) and with @@tn ¼ @n � ). � ð 0Þ @h The slow manifold n ðr; vÞ ¼ @nðr;vÞ ð1Þ Now we turn to the slow manifold Mslow and repeat the analysis that ð0Þ we have just made above for Mslow . We begin by looking for the thermo­ �� n o � � ð1Þ dynamic potential ’(1) such that Mslow ¼ nðr; vÞM1 � ’ð1Þ nðr;vÞ ¼ 0 . We easily verify that � � � � � � � � � ’ð1Þ nðr; vÞ; ðrÞ; u ðrÞ; e ðrÞ ¼ �hð1Þ n; ; u ; e Þ ð ð v2 � nðr; vÞ þ dr dve ðrÞ 2m ð ð � þ dr dv ðrÞnðr; vÞ ð ð � þ dr dvuj ðrÞvj nðr; vÞ

ð141Þ

where ð ð � � � � � � � � h ð1 Þ n; ; u ; e Þ ¼ h ð0 Þ ðnÞ þ dr dv ð1 Þ r; v; ; u ; e Þ nðr; vÞ

ð142Þ

is such a potential provided the terms quadratic and higher order in (1) are neglected. The slow fundamental thermodynamic relation is thus

Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

127

’ ð1 Þ jn  Mð1Þ . The fast time evolution, i.e., the time evolution describing the slow

ð1Þ

@n @ @t ¼ @n� ð1 Þ @h ¼ @n ðr;vÞ :

approach to the slow manifold Mslow , is governed by �

dissipation potential  given by (79) and with n ðr; vÞ

with the

5. CONCLUDING REMARKS The titles of the first two papers (published in 1873) in which Gibbs (1984) completed the formulation of classical equilibrium thermodynamics () are: “Graphical methods in the thermodynamics of fluids” and “A method of geometrical representation of thermodynamic properties of substances by means of surfaces.” This by itself clearly indicates the importance that Gibbs gave to geometry in thermodynamics and the importance that it played in his way of thinking about thermodynamics. In this review we have attempted to follow his lead. We have continued his geometrical viewpoint and extended it to multiscale formulations and to the time evolution. The resulting framework is intended to provide a basis for dealing with new challenges brought to chemical engineering by new emerging technologies (in particular then the nano- and biotechnologies). The framework embraces all scales and both equilibrium and nonequili­ brium. In the illustrations developed in this review we have addressed multiscale fluid mechanics of complex fluids and multiscale chemical kinetics. Among the systems that are not included in the illustrations but that clearly need the multiscale framework for their investigation are heterogeneous systems (for a review of the research in this field that remains inside the classical nonequilibrium thermodynamics see Kjelstrup and Bedeaux (2008)). The multiscale viewpoint is essential in heteroge­ neous systems since the physics taking place on the boundaries and in the bulk requires typically different scales for their formulations. Other inter­ esting lines of research that can be followed and further developed are indicated in Sections 3.1.3, 3.2.1, and 4. Finally, we make a comment about the somewhat more-than-usual abstract character of the formalism used in this review. Any multiscale and combined equilibrium and nonequilibrium investigation has to be conducted in a setting that unifies the scales and is suitable for both statistics and dynamics. Since the formulations on every scale have been developed separately and in large extent independently of each other, the single scale formulations are typically very different one from the other (compare, e.g., the settings of fluid mechanics, kinetic theory, and classical mechanics of particles). Consequently, a setting allowing a unified formu­ lation has to be inevitably abstract. Gibbs’ success with using abstract geometrical tools in classical equilibrium thermodynamics is, of course,

128

Miroslav Grmela

encouraging. Also another pioneer of chemical engineering, Leonhard Euler, encourages us to take more general viewpoints by saying: La gén­ éralité que j’embrasse, au lieu d’éblouïr nos lumiéres, nous découvrira plutôt les véritables loix de la Nature dans tout leur èclat, et on y trouvera des raisons encore plus fortes, d’en admirer la beauté et la simplicité.

ACKNOWLEDGMENT The author acknowledges the financial support provided by the Natural Sciences and Engi­ neering Research Council (NSERC) of Canada.

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CHAPTER

3 Application of Mesoscale Field-Based Models to Predict Stability of Particle Dispersions in Polymer Melts Prasanna K. Jog
,1, Valeriy V. Ginzburg1, Rakesh Srivastava1, Jeffrey D. Weinhold2, Shekhar Jain3, and Walter G. Chapman4

Contents

1. Introduction 2. Theory 2.1 iSAFT model 2.2 Extension of iSAFT model to grafted polymer chains 2.3 Self-consistent field theory 3. Applications 3.1 Structure of grafted polymer monolayers in the presence of a polymer melt 3.2 Interaction between two grafted monolayers in the presence of free polymer melt with both the grafted and the free polymer chains having equal segment sizes 3.3 Interaction between two grafted monolayers in the presence of free polymer melt with the grafted and free polymer chains having different segment sizes

132 135 135 140 141 146 147

148

154

1 The Dow Chemical Company, Midland, MI 48674, USA 2 The Dow Chemical Company, Freeport, TX 77515, USA 3 Shell Technology India Pvt. Ltd., Bangalore 560048, India 4 Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005, USA
Corresponding author. E-mail address: [email protected]

Advances in Chemical Engineering, Volume 39 ISSN: 0065-2377, DOI 10.1016/S0065-2377(10)39003-X


2010 Elsevier Inc. All rights reserved.

131

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Prasanna K. Jog et al.

3.4 Interaction between two grafted monolayers in the presence of attractive chains 3.5 Interaction between two grafted monolayers in the presence of end-functionalized chains 4. Summary and Outlook Acknowledgment References

Abstract

154 156 161 162 162

We review recent results in theoretical modeling of particle dispersions in polymer melts. In particular, we focus on mesoscale field-based theories (interfacial statistical association fluid theory (iSAFT) and com­ pressible self-consistent field theory (SCFT)). To demonstrate the application of these theories, we calculate “dispersion phase maps” for particles with grafted oligomer brushes mixed into polymer melts. The systems used here are generic and can be considered as model systems for a wide class of application areas. Examples include disper­ sion of nanoparticles in a polymer matrix and stabilization of polymer blends by addition of a block copolymer. It is demonstrated that, as expected, the quality of particle dispersion depends strongly on the brush grafting density and the ratio of the oligomer length to the matrix polymer length. The results obtained using two different meth­ ods (iSAFT and SCFT) show remarkable agreement for the case of the so-called “athermal” mixtures. Applying the analysis to the case of polymer�clay nanocomposites, we propose a new hypothesis for the ubiquity of the so-called “intercalated” morphologies. Overall, this example demonstrates the utility of mesoscale field-based models like iSAFT and SCFT for chemical engineering applications.

1. INTRODUCTION The balance in modern chemical industry is gradually shifting from large integrated plants and commodity-based products to high-value specialty products involving micro- and nanostructured materials using advanced material science, high-throughput technology, and sophisticated comput­ ing tools. A common practice in making these specialty products is to disperse organic or inorganic particles in polymer melts to provide enhanced toughness, chemical resistance, or other desirable properties. Applications of such materials range from performance polymers to pharmaceutical suspensions, agricultural and environmentally benign chemicals, microelectronics, and biological systems. Polymer adsorption is important to commercial applications such as inks, paints, and coatings, where there is considerable interest in understanding and controlling the segregation to the surface of one component from a blend of polymers. Water-soluble polymers acting as colloidal stabilizers are the enabling

Application of Mesoscale Field-Based Models

133

component in the development of environment-friendly water-based coatings that replace older formulations employing organic solvents. Adsorption-based surface treatments comprise an inexpensive means of surface modification for DNA separations and recognition chips. Surface modification strategies that accommodate molecules with a complex dis­ tribution of hydrophobicity and charge will facilitate extensions to protein separations. Among the naturally occurring systems, the mechanical prop­ erties of biological cell membranes depend upon the fraction of cholesterol molecules and conformational degrees of freedom of the lipids forming the layer. Porous materials with pore sizes on the order of a few nan­ ometers are widely used in chemical, oil and gas, food, and pharmaceu­ tical industries for pollution control and mixture separation and as catalyst and catalyst supports for chemical reactions. The design of such processes is largely empirical at the present time. In these applications the local microstructure or heterogeneity deter­ mines the properties of the macromolecular system. In general the experi­ mental measurement of thermodynamic or interfacial properties of these systems is often not practically feasible. Therefore it is not a coincidence that recent journal publications in these areas have focused on newer theoretical and modeling approaches for these systems. Modeling such a system is a multi-scale problem that can be solved for simple fluids given a large parallel computer. For polymeric systems, the problem is more com­ plex for several reasons. First, for polymeric systems, a modest surface– fluid interaction per segment translates to a large surface–fluid interaction per molecule. Second, unlike in simple fluids that are predominantly enthalpy driven, the structure of a complex fluid has a strong entropic driver. Finally, if we consider a polyatomic molecule to be a chain of beads, the range of the heterogeneity is often only over several bead diameters while the size of the polymer may be thousands of bead diameters. In this case a polymeric molecule could exist in the interfacial and bulk regions simultaneously. Therefore, the very notion of “local” molecular density in the inhomogeneous region loses its significance. As bulk polymer theories analyze the system in terms of molecular density, any extension of their arguments to inhomogeneous systems cannot be expected to paint an accurate picture. Traditionally, modeling of polymeric fluids has been motivated by the reasoning that most of the properties of interest depend largely on the long-range structure (Yethiraj, 1996), allowing one to neglect chemical details on short length scales. Recently, it has become increas­ ingly clear that the short-range structure plays an important role in many applications such as coatings of polymer blends (Yethiraj, 1996, 2002). To be able to design and control such applications with confidence, a success­ ful model must incorporate molecular features on all length scales while remaining computationally tractable. Even the most sophisticated existing theories fail to simultaneously meet these criteria (Tripathi, 2005).

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To understand the nanoscale features of many polymer-based materi­ als, then, one needs a new class of theoretical methods—something between the traditional atomistic/molecular simulations (length scales of 1–10 Å) and the traditional continuum mechanics or hydrodynamics (length scales usually greater than 100 mm). These new methods—often called mesoscale approaches—cover length scales between 1 and 100 nm and generally deal not with individual atoms (C, H, N, O, etc.) but with “repeat units” or monomers (CH2, styrene, etc.) as their “building blocks.” Tailoring the interactions between these monomers based on the results of atomistic simulations is one way to establish a connection between these two length scales (Baschnagel et al., 2000). For convenience, one often distinguishes between “particle-based” mesoscale simulations (dissipative particle dynamics (DPD), coarse-grained Monte Carlo (CG-MC), and coarse-grained molecular dynamics (CG-MD)) and “field-based” mesos­ cale theories (density functional theory (DFT) and polymer self-consistent field theory (SCFT)). This chapter mainly deals with the latter methods. We will highlight the strengths of these theories by focusing on their application to predicting the stability of particles in polymer melts. Nano-size organic or inorganic particles are dispersed in polymer melts to produce polymer nanocomposites. As mentioned, addition of these nano­ fillers can significantly improve the mechanical, thermal, electrical, and optical properties as compared to the pure polymer or conventional microand macrocomposites (Giannelis et al., 1999; Sinha Ray and Okamoto, 2003) without increasing the bulkiness of the host polymer. Polymer/clay nano­ composites are classical examples. Polymer–clay nanocomposites have attained substantial interest in the past two decades, since the successful development of a Nylon-6/montmorillonite hybrid by Okada et al. (1990) at Toyota research laboratories. Since then, hybrid polymer–clay nanocompo­ sites have been successfully prepared using polyethylene (PE), polypropy­ lene (PP), polymethylmethacrylate (PMMA), polystyrene (PS), epoxy, polyurethane, and other matrices (see, e.g., review articles by Sinha Ray and Okamoto 2003 and Alexandre and Dubois 2000). It has been shown that under some conditions, such nanocomposite materials could provide 2–4 times increase in the elastic modulus, order of magnitude decrease in gas permeability, as well as substantial reduction of flammability com­ pared to the pristine polymer matrix, at clay loadings of less than 10 wt.%. However, one of the challenges in the synthesis of polymer/clay nanocomposites is dispersing the broad clay sheets in the polymer matrix. This depends upon the polymer-mediated interactions between the dispersed particles. The van der Waals and/or electrostatic interactions between the particles are usually attractive, leading to aggregation/ flocculation of these particles. Furthermore, it has been shown that for unmodified particles, there is a strong polymer-mediated attraction lead­ ing to nanoparticle aggregation (Balazs et al., 1998a, 1999a; Vaia and

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Giannelis, 1997a, 1997b); this attraction can be attributed to the fact that polymer chains tend to lose their configurational entropy when confined between colloidal surfaces. The property enhancement, if any, depends strongly on the morphology of the nanocomposite and the quality of nanoclay dispersion in the matrix. In many cases, clay platelets fail to disperse and remain aggregated into large (micron-sized) “stacks” because of polymer-mediated attraction; in this case, modulus and strength of the composite could be comparable to those of the matrix polymer, while toughness and ultimate elongation could even worsen. One way of stabilizing the dispersion is to end-graft polymers onto the particle surfaces. In that case, positive changes in the configurational entropy of grafted polymer chains (or polymer brushes) can, under certain conditions, counterbalance the negative changes in the configurational entropy of the free polymer (melt) chains and prevent the particles from coming close together. The optimum separation between the grafted par­ ticles depends upon the profile of the interaction force. If the force is purely repulsive, the particles are well dispersed into the polymer melt leading to an exfoliated state. In other cases, the force can have an attrac­ tive minimum at a finite separation, H
. Here, the clay sheets will be stacked at separations H
, leading to an intercalated morphology. Thus, entropic contributions together with enthalpic factors determine the equilibrium morphology of the polymer/clay nanocomposites. Despite the multidisciplinary interests that surround these systems, many challenges still remain for both experimentalists and theoreticians to understand the interplay of forces and microstructure with the multiple length scales and broad parameter space involved. The interaction between particles in polymer melt, as well as the surrounding fluid struc­ ture, is dictated by a number of molecular parameters, including the chain lengths of the grafted and free polymer, grafting density, polymer melt concentration, sizes of the polymer segments, and the nature of the poly­ mer segment–segment interactions. We attempt to investigate the effect of some of these parameters using iSAFT and SCFT, and explore the implica­ tions of these effects on modeling of nanocomposite thermodynamics.

2. THEORY 2.1 iSAFT model Classical DFT is an efficient theoretical tool for prediction of microscopic structure, thermodynamics, and phase behavior of bulk and inhomoge­ neous fluids, both simple (atomic) and molecular (polymeric) (Evans, 1992; Hansen and McDonald, 1986; Wu, 2006; Wu and Li, 2007). The approach has roots in quantum DFT developed by Hohenberg and Kohn

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(1964). Application of DFT as a general methodology to classical systems was introduced by Ebner et al. (1976) in modeling the interfacial properties of a Lennard–Jones (LJ) fluid. The basis of all DFTs is that the Helmholtz free energy of an open system can be expressed as a unique functional of the density distribution of the constituent molecules. The equilibrium density distribution of the molecules is obtained by minimizing the appro­ priate free energy. DFTs retain monomer or statistical segment length-level information rather than a more coarse-grained representation of polymers. Thus, DFT provides an approach that is intermediate between macroscopic thermody­ namic approaches and truly all atom molecular simulation-based methods. Although the theory incorporates molecular-level detail, calculation time is modest for many systems (Jain et al., 2007). DFT provides a single frame­ work for modeling interfacial, confined, and bulk systems. A thorough review of classical DFT is given by Evans (1992), while many applications of DFT to interfacial systems are described by Davis (1997) and Wu (2006). Multiple versions of DFT have been developed for polymeric systems (Chandler et al., 1986; Jain et al., 2007; Kierlik and Rosinberg, 1992; Phan et al., 1995; Tripathi and Chapman, 2005a, 2005b; Wu, 2006; Wu and Li, 2007; Yethiraj, 1996, 1998; Yu and Wu, 2002). The theory developed by Chandler, McCoy, and Singer (1986) uses a Taylor series expansion to describe the excess Helmholtz free energy relative to the homogeneous liquid state. The direct correlation function, determined for the homogeneous system using the polymer reference interaction site model (PRISM) (Schweizer and Curro, 1997), is then used to solve for the structure of the inhomogeneous system. This theory has been applied to determine the force between grafted mono­ layers in an implicit good solvent (McCoy and Curro, 2005). Owing to the success of Wertheim’s thermodynamic perturbation theory (TPT1) (Wertheim, 1984a, 1984b, 1986a, 1986b) for homogeneous systems, several DFTs based on TPT1 have been proposed. The central approximation of any DFT is an expression for the intrinsic Helmholtz free energy of the system. Considering the polyatomic system as a mixture of associating spherical segments in the limit of complete association, the intrinsic Helmholtz free energy functional can be derived from Wertheim’s TPT1 as shown in the development of the SAFT equation of state (Chapman, 1988; Chapman et al., 1989, 1990). In this chapter, we focus on iSAFT, a computationally simple, thermo­ dynamically consistent DFT that accurately predicts the structure and thermodynamics of inhomogeneous polymeric solutions and blends (Jain et al., 2007, 2008, 2009; Tripathi and Chapman, 2005a, 2005b). Like mole­ cular simulation, the DFT uses explicit models of molecules, but the DFT is not limited computationally in molecule size or number of components. The DFT shows excellent agreement with molecular simulation for local structure, compressibility effects, and the effects of molecular size.

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In addition to the free polymer and grafted polymer systems shown here, iSAFT has been developed for branched and associating molecules (Bymaster 2010). For a homogeneous fluid, the iSAFT DFT has the advan­ tage that it reduces to an accurate equation of state (SAFT) (Chapman, 1988; Chapman et al., 1989; Jackson et al., 1988; Mueller and Gubbins, 2001), which is widely applied in academia and industry. In the iSAFT approach, polyatomic molecules are modeled as flexible chains of tangentially bonded spherical segments. Each of the segments of the chains can be different. For simplicity, we present the derivation for a pure fluid of chain molecules with m segments, but the theory is in general applicable to mixtures. Additionally, we will only focus on the physical basis of the derivation leaving out the mathematical details. The details of the derivation can be found in one of the original iSAFT papers (Jain et al., 2007). Additionally, the details of the derivation for extending the theory to grafted polymer chains can be found in Jain et al. (2008, 2009). DFTs are commonly formulated for an open system in the grand canonical ensemble. Hence, the system is at fixed volume (V), temperature (T), and chemical potential (
) in the presence of an external field (Vext(R)). The appropriate free energy for the grand canonical ensemble is the grand free energy (
). The grand free energy functional for a system of chain fluid can be related to the intrinsic Helmholtz free energy functional (A) as
½ ðrÞ ¼ A½ ðrÞ 

m ð X ¼1

  dr 0  ðr 0 Þ
  Vext ðr 0 Þ ;

ð1Þ

where  is the density of segment ,
 is its chemical potential, and the sum is over all the m segments of the chain. For the system at equilibrium, the grand free energy has to be minimum. Minimization of the grand free energy with respect to density of the segments yields a system of varia­ tional equations, known as the Euler–Lagrange equations, A½ ðrÞ ¼
  Vext ðrÞ 8  ¼ 1;…;m:  ðrÞ

ð2Þ

Solution of the set of these equations gives the equilibrium density profile of all the segments in the polymer chain. Hence, the central aim of all DFTs is to come up with a formulation of the Helmholtz free energy functional. In iSAFT, we follow an approach similar to that used to develop the SAFT (Chapman et al., 1989, 1990) model for bulk polymer systems, wherein the polymer chains are considered to be a stoichiometric mixture of m asso­ ciating spherical segments in the limit of complete association, as shown in Figure 1. Following SAFT, we can now write down the free energy func­ tional of this mixture of associating spheres as a perturbation expansion about the free energy of the purely repulsive spheres except that the

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Associating sites

A

B

m

3

2

1

A

B

A

B

Figure 1 Schematic of the formation of a linear polymer chain of m segments from m associating spheres. For the middle segment , site A associates with the site B on the segment “ þ 1” and site B associates with site A on the segment “ þ 1.” End segments 1 and m have only one associating site, A or B.

various free energies are actually functionals of the segment density pro­ files. Hence, A½  ¼ Aid ½  þ AEX ; hs ½  þ AEX ; chain ½  þ AATT ½ ;

ð3Þ

where ideal free energy functional id plus EX, hs accounts for the excluded volume of the segments, and EX, chain for the chain connectivity. These three terms together account for the entropic interactions and ATT adds on the attractive interactions between the segments of the polymeric fluid mixture. The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld’s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler–Lagrange equations, are straightforward. The free energy functional contribution due to chain connectivity lies at the heart of the iSAFT. We start with the free energy functional of a mixture of associating spheres with finite association between them fol­ lowing the extension of the Wertheim’s first-order thermodynamic pertur­ bation theory (TPT1) (Wertheim, 1984a, 1984b, 1986a, 1986b) to mixtures of associating spheres (Chapman et al., 1986; Joslin et al., 1987; Segura et al., 1997; Tripathi and Chapman, 2005a, 2005b) and take the limit of complete association between them at a convenient point in the derivation. For finite association  ð seg m X X  X  ðr1 Þ 1 þ ;  ðr1 Þ ln XA ðr1 Þ  A AEX ; assoc ½  ¼ dr1 2 2 ð Þ ¼1 

ð4Þ

A2G

where the first summation is over all the segments , and second over all the association sites on segment , as G{} is the set of all the associating

Application of Mesoscale Field-Based Models

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sites on segment . XA denotes the fraction of segments  that are not 0 associated at their site A. XA depends upon XB by XA ðr1 Þ ¼

ð

1 0

1 þ dr2 XB ðr2 ÞD ðr1 ; r2 Þ 0 ðr2 Þ 0

seg

;

ð5Þ

where 0 denotes the segment with site B which associates with site A on segment , and D accounts for the strength of association between these neighboring segments  and 0 . In limit of complete association, XA ! 0. The limit is taken while computing the functional derivate of the association free energy functional which is required to solve the set of Euler–Lagrange equations. This functional derivative was derived in the original iSAFT paper (Jain et al., 2007): f gð m X X ln AEX ; chain 1X ¼ ln XA ðrÞ   ðr1 Þ  ðrÞ 2 0 ð Þ  ¼1  0

A2G

  0 y contact  ðr1 Þ dr1 ;  ðr1 Þ

ð6Þ

where {´} is the set of all segments bonded to segment , y´contact is the cavity correlation function at the point of contact of segments  and ´, and it is evaluated for the weighted density of segments {} at position r1. The long-range attractive interactions between the segments are included using the mean field approximation, ignoring the pair correlation between the segments: AATT ½  ¼

m X m 1X 2 ¼1  ¼1

ð

seg

j r2  r1 j > 

seg dr1 dr2 uATT  ðjr2  r1 jÞ ðr1 Þ ðr2 Þ;

ð7Þ

where uATT is the attraction potential between the segments. After substituting these functional derivatives, the set of the nonlinear Euler–Lagrange equations is solved for the density profile of the segments. For a linear chain, the density profile of a segment  is (Jain et al., 2007) ! m m 1 X Y 0  ðr Þ ¼ expð
M Þ dr1 …dr  1 dr þ 1 …drm exp ½D i ðri Þ Dði;iþ1Þ ðri ; ri þ 1 Þ; ð

i¼1

i¼1

ð8Þ

where
M is the bulk chemical potential of the polymer chains, and D0  (r) is given by D 0  ð rÞ ¼

  0 m f0 g  ln y 1 XX contact  ðr1 Þ dr1   ð r1 Þ 2 ¼1  0  ðrÞ AEX ; hs ½  AATT ½     Vext ðrÞ  ðrÞ  ðrÞ

ð9Þ

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As can be seen from the density profiles, there are two important features of iSAFT, which makes it applicable to a range of heterogeneous polymer systems. One is that iSAFT satisfies overall stoichiometry, which means that the average densities of all segments on a molecule in the system are equal. This constraint comes naturally during the derivation of the theory. The second feature is that each segment on the polymer chain knows about the other segments on the chain. Finally, the equilibrium grand free energy of the fluid mixture is given by 
½f ðrÞg ¼

m ð X i¼1

" dr ðrÞ D

0

 ðrÞ

þ

Vext ðrÞ

#   n G ð Þ þ  1 þ AEX ; hs þ AATT : 2 ð10Þ

2.2 Extension of iSAFT model to grafted polymer chains In the case of grafted polymer chains, one of the chain end segments is physically/chemically tethered to a surface. To extend the iSAFT model to such a case, this restraint is incorporated as an external field on the tethered segment of the polymer chains with the boundary condition being the number of chains grafted to the surface (grafting density, g). The surface is considered to be a flat hard wall and the chains are uniformly distributed over the area of the surface. Hence, this reduces to a one-dimensional (1D) problem, with the relevant dimension along the normal to the wall. The external field exerted by the wall on the tethered segment “1” is ( V1ext ðzÞ

¼

v

if z ¼ 0

ð11Þ

1 otherwise

For other segments, the external potential is just due to a planar hard wall. Furthermore, ð dz  ðz Þ ¼ g :

ð12Þ

Following these arguments, the density profile of segment  ( 6¼ 1) in the grafted polymer chains is ðð dz2 …dz1 dzþ1 …dzm exp  ðz Þ ¼ g

ðð

! m m 1 X Y ½D 0 i ðzi Þ Dð1;2Þ ð0;z2 Þ Dði;iþ1Þ ðzi ;ziþ1 Þ

i¼2 i¼2 ! m m 1 X Y dz2 …dzm exp ½D 0 i ðzi Þ Dð1;2Þ ð0;z2 Þ Dði;iþ1Þ ðzi ;ziþ1 Þ i¼2

i¼2

ð13Þ

Application of Mesoscale Field-Based Models

141

Clearly, this derivation is only for the polymer chains grafted to the wall at z = 0. For the case of two grafted surfaces, similar results can be obtained for the polymer chains grafted to the wall at z = H, where H is the separa­ tion between the two surfaces. The force of interaction between the two grafted surfaces (in the absence/presence of free polymer) at separation H is given by f ðH Þ ¼ A

    1 
1 
   ; A H H A H H ! 1

ð14Þ

where
is the equilibrium grand free energy, A is the surface area of the two surfaces, and H ! 1 implies the limit when the separation between the two surfaces is large enough that they do not interact with each other. If f is positive, the surfaces repel each other, and if f is negative, they attract. In the current work, the two hard surfaces are grafted with the same polymer chains at the same grafting density. Hence, the density profiles of the two grafted monolayers are symmetric. For such a sym­ metric system, the functional derivative of the grand free energy can be simplified as (Evans and Marconi, 1987) Xð 1 
dV ext;s ðzÞ dz; ¼  ðzÞ  A H dz 

ð15Þ

where Vext,s is the external field on segment  due to a single surface  at z = 0. For hard walls, this reduces to the sum of the contact densities of the grafted and free polymer chains at the surface at z = 0. Hence, the structures of the grafted polymer chains (and the free polymer) have to be calculated first before calculating the force of interaction between them.

2.3 Self-consistent field theory SCFT today is one of the most commonly used tools in polymer science. SCFT is based on de Gennes–Edwards description of a polymer molecule as a flexible Gaussian chain combined with the Flory–Huggins “local” treatment of intermolecular interactions. Applications of SCFT include thermodynamics of block copolymers (Bates and Fredrickson, 1999; Matsen and Bates, 1996), adsorption of polymer chains on solid surfaces (Scheutjens and Fleer, 1979, 1980), and calculation of interfacial tension in binary polymer blends compatibilized by block copolymers (Lyatskaya et al., 1996), among others. Over the past decade, SCFT was often applied to analyze the problem of particle dispersion in polymers (thermodynamics of nanocomposites). Vaia and Giannelis (1997a, 1997b) formulated a simple version of SCFT

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that provided qualitative description of exfoliated, intercalated, and immiscible morphologies by estimating the free energy of the polymer and organic modifier confined between two adjacent parallel surfaces separated by a gallery with a width H. Subsequently, Balazs and co­ workers (Balazs et al., 1998b, 1999b, 2000, 2008; Ginzburg and Balazs, 2000; Ginzburg et al., 2000; Zhulina et al., 1999) utilized a more elaborate Scheutjens–Fleer version of SCFT to generate various nanocomposite phase diagrams (e.g., how the composite morphology depends on the ratio of the grafted chain length to the matrix chain length) In particular, in a practically important case of organically modified clays in a homo­ polymer melt, it was shown that clay platelets could favor aggregation even when there is no enthalpic repulsion between the matrix polymer and the surfactant (Flory–Huggins parameter PS = 0), provided that the surfactant has low chain length and high grafting density. This prediction was indeed confirmed for cases like the mixtures of montmorillonite clay with C16 or C18 grafted chains, melt-compounded with polyethylene or polypropylene; such systems were demonstrated to be mostly immiscible, in agreement with theoretical predictions (Balazs et al., 2000; Lyatskaya and Balazs, 1998; Sinha Ray and Okamoto, 2003). Similarly, for the cases where end-functionalized chains (e.g., maleated polypropylene) were added to the melt, intercalated and exfoliated morphologies could be observed (Beyer et al., 2002; Hasegawa et al., 2000; Sinha Ray and Okamoto, 2003), as predicted by the theory. In our view, one area where SCFT predictions are not fully satisfactory is in explaining intercalated morphologies. Indeed, there are numerous nanocomposite systems exhibiting intercalated morphology with a typical gallery spacing H
 1–2 nm (10–20 Å). While conventional SCFT does predict intercalated morphologies in various systems (Balazs et al., 1998b, 1999a,), the gallery spacing corresponding to those structures is expected to be somewhat larger than observed (probably on the order of 4–10 nm, depending on the molecular weight of the functionalized chains). Further­ more, in recent experimental studies by Swain and Isayev (2007), inter­ calated structures have been reported even in systems without “stickers” (Cloisite 20A in high-density polyethylene) when they were subjected to strong ultrasound treatment in the melt state. The increase in the clay–clay spacing (compared to the pure organoclay), measured by wide-angle X-ray scattering (WAXS), was fairly small (from 2.4 to 3.56 nm). In a sense, it appears that intercalated structures are far more ubiquitous than originally predicted by the theory. We proposed (Ginzburg et al., 2009) that the origin for this discrepancy was due to the “incompressible” nature of the SCFT approaches. Accordingly, we formulated a “compres­ sible” extension of the SCFT model for nanocomposites and applied it to the specific case of organoclay mixed with a blend of homopolymer and a “one-sticker” polymer.

143

Application of Mesoscale Field-Based Models

Clay

Clay

Figure 2 Schematic depiction of the lattice model. Note that some of the lattice sites are not occupied by polymeric species—those are “voids.”

Our lattice model is schematically depicted in Figure 2. In its most general form, the grand canonical free energy per unit area, f, is written as 0 f ¼

1

B C X i C X i B Ng G b b B Cþ ¼ ln ln ’ þ  ln ’ 
i 
v v v i v B C 2 H Ma kB T Ni Ni  A @X i i Gg z; Ng

z ¼ 1

þ =2 1

H X X z¼1

;



  ðzÞ 

b



h  ðzÞi 

b



!! 

H X X z¼1



! u ðzÞ  ðzÞ ð16Þ

Here, a is the lattice unit dimension, M is the number of lattice units per clay platelet (so that the product Ma2 = A is the total area of the platelet),  is the grafting density of “surfactants,”  are the Flory–Huggins para­ meters between species  and ,
i is the chemical potential of the ith component, and Yi is the excess amount of the ith component in the system. The density profiles of various species, (z), and conjugate fields, u(z), are calculated as described below. Note that we introduce a separate species and component—voids (denoted as subscript v)—to account for density variation within the gallery. The bulk chemical potentials of all polymers and voids are described as  
i ¼ ln ’bi þ ð1  Ni Þ 1  ’bi ;

ð17aÞ

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Prasanna K. Jog et al.


v ¼ ln ’bv þ

X ’b i ð1  Ni Þ: Ni i

ð17bÞ

In Equations (17a) and (17b), densities with superscript b refer to the equilibrium densities in the bulk (for all components and all species). The excess amount of each component, Y, is given by i ¼

H X

i ðzÞ;

ð18aÞ

v ðzÞ:

ð18bÞ

z¼1

v ¼

H

X z¼1

The effective fields, u(z) and uv(z), are given by u ðzÞ ¼ u’ðzÞ þ

  1X  h  ðzÞi  ’b ; 2 

ð19aÞ

uv ðzÞ ¼ u’ðzÞ þ

n o 1X v h  ðzÞi  ’b : 2 

ð19bÞ

The effective hard core potential u0 (z) is a Lagrange multiplier that enforces the incompressibility condition X 

 ðzÞ þ v ðzÞ þ g ðzÞ ¼ 1:

ð20Þ

To complete the set of equations needed to calculate the free energy and density profiles, one needs the rules to evaluate density profiles. The local density of the voids, v(z), is calculated always from Equation (20), while the calculation of the density of the polymeric components is more com­ plicated and depends on the specific composition of the bulk polymer and the architecture of each polymeric component. We restrict ourselves to the systems in which the bulk contains two types of polymer: “free” or “matrix” homopolymer chains, and “active” or “end-functionalized” chains with one “sticker.” For each type of chain (including the grafted surfactants), one can evaluate the “propagators” G(z,s) and G
(z,s). (The propagators are evaluated assuming Markov statistical process for Gaussian chains occupying various lattice sites.) All the propagators obey the same recurrence equation Yðz; sÞ ¼ hYðz; s  1ÞiGt ðs Þ ðzÞ:

ð21Þ

Here, Y is a shorthand for one of the five propagators: Gg(z,s) and G
g(z,s) (for grafted chains), Ga(z,s) and G
a (z,s) (for “active” chains), and Gf(z,s) (for “free” chains; because of symmetry, G
f (z,s) = Gf(z,s)). The factor G(z) is

Application of Mesoscale Field-Based Models

145

the Boltzmann factor for the species of type  to be at position z compared to the bulk: G ðzÞ ¼ expðu ðzÞÞ:

ð22Þ

In Equation (21), index t(s) labels the species type of a monomer having position s in the chain. For the case of “free” chains, all monomers are the same (we label them as “F”):

F ðzÞ ¼

Nf X

’bf

s¼1

Nf

!

  Gf ðz; sÞGf z; Nf  s þ 1 : GF ðzÞ

ð23aÞ

For the grafted chains, all monomers are also the same (we label them as “G”): 0 Ng X @X

G ðz Þ ¼ s¼1

1 

 G z’;Ng z’¼1 g H

A

  Gg ðz; sÞG
g z; Ng  s þ 1 GG ðzÞ

:

ð23bÞ

Finally, for active chains, there are two types of monomers: “sticker” (at position s = 1) and the rest (positions s = 2 to Na), labeled as “S” and “A,” respectively:

S ðzÞ ¼

1  b  X ’ Ga ðz; sÞGa ðz; Na  s þ 1Þ a

s¼1

A ðzÞ ¼

Na

GS ðzÞ

Na  b  X ’ Ga ðz; sÞGa ðz; Na  s þ 1Þ a

s¼2

Na

GA ðzÞ

;

ð23cÞ

:

ð23dÞ

In Equations (16), (19), and (21), h…i denotes local averaging according to the prescription hYðzÞi ¼   1 Yðz  1Þ þ 0 YðzÞ þ 1 Yðz þ 1Þ

ð24Þ

Constants –1, 0, and 1 are determined by the choice of the lattice; in our calculations, –1 = 1 = 0.25 and 0 = 0.5. This choice corresponds to the simple cubic lattice, which is commonly used in the application of lattice SCFT to polymer–clay composites. To relate the lattice coordinate to “real” dimensions, we must also specify the value of the lattice size a; here, we set a = 0.4 nm. (Later, in Section 3.5, when describing our results, we will automatically convert all distances from lattice units to nanometers.) Recursive relation (21) must be solved for s = 1 to Nf for the free chain propagator Gf(z,s), for s = 1 to Ng for the grafted chain related propagators Gg(z,s) and G
g ðz; sÞ, and for s = 1 to Na for the active chain propagators Ga(z,s) and

G
a ðz; sÞ. The initial conditions are as follows: Gf(z,1) = GF(z), Gg(z,1) = GG(z)z1, G
g ðz; 1Þ ¼ GG ðzÞ; Ga ðz; 1Þ ¼ GS ðzÞ, and Ga
(z,1) = GA(z).

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Equations (16)–(24) are solved on a lattice in an iterative fashion until a convergence criterion (e.g., the difference between free energy calculations from the two successive iterations) is satisfied. The free energy profiles f(H) and the density profiles of various species and components (including the voids) are used to characterize the nano­ composite morphology and decide whether the system tends to be in exfoliated, intercalated, or immiscible state. Note that because of the “compressibility” (non-zero bulk volume fraction of voids), SCFT results could be compared to iSAFT results. We will attempt such a comparison in Section 3.2 for the “athermal” (all Flory–Huggins  = 0) case with no active chains. Additionally, in Section 3.4, we explore SCFT phase diagram for a system in which active “one-sticker” chains are present and compare our predictions with the earlier results of Ginzburg and Balazs that were based on the “incompressible” SCFT; it will be shown that the new approach dramatically expands the region corresponding to intercalated morphologies.

3. APPLICATIONS This section highlights the applications of iSAFT and SCFT to predict the stability of grafted particles dispersed in a polymer solution. We consider cases where the surfaces of the grafting particles are significantly larger than the height of the monolayers formed by the polymer chains grafted onto those particles. In this limit, the grafted surfaces can be treated as planar. The stability of these particles depends upon the effective force of interaction between such grafted (planar) surfaces. For example, one of the challenges in the synthesis of polymer/clay nanocomposites is dispersing the broad clay sheets in the polymer matrix. This depends upon the polymer-mediated interactions between the dispersed particles. The van der Waals and/or electrostatic interactions between the particles are usually attractive leading to aggregation/flocculation of these particles and worsening the properties of the nanocomposite. This attraction can be attributed to the fact that polymer chains tend to lose their configura­ tional entropy when confined between colloidal surfaces. One way of stabilizing the dispersion is to end-graft polymers onto the particle sur­ faces. In that case, positive changes in configurational entropy of grafted polymer chains can, under certain conditions, counterbalance the negative changes in the configurational entropy of the free polymer (melt) chains. Thus, the effective interactions between the grafted surfaces determine the stability of the nanocomposite. We will begin with the structure of unperturbed polymer monolayers in the presence of a solvent and then move on to the case of two polymer grafted surfaces in the presence of a polymeric fluid, discussing

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the structure of the grafted monolayers and the effective force of interac­ tion between the grafted surfaces. Wherever possible, we have directly compared iSAFT and SCFT predictions.

3.1 Structure of grafted polymer monolayers in the presence of a polymer melt Molecular dynamics simulations for grafted polymer monolayers in the presence of free polymer melt have been done by Grest and Murat (1993). The calculations are done for free polymer chains with few segments (Nf = 2, 5, and 10). Both the grafted and the free polymer chains are purely repulsive and have the same segment sizes (g = f = ), where g and f are the segment sizes of the grafted and free polymer chains, respectively. The overall density of the segments in the system is quite large, tot3 = 0.85. The number of segments in the grafted chains is fixed to Ng = 100 and the grafting density is fixed to g2 = 0.1. The density of free polymer for all the three cases, Nf = 2, 5, and 10, is f 3 = 0.682, based on the overall segment density in the system. Figure 3a compares the density profiles of the monolayers for these three cases with that in the presence of an implicit solvent (Nf = 0). As can be seen from the figure, the presence of an explicit solvent (free polymer) significantly affects the structure of the monolayers. Since the chains are purely repulsive, the origin of the effect is solely due to the entropic interactions. Due to volume exclusion, the explicit solvent molecules compress the grafted polymer chains causing them to partially collapse compared to the case 1

(b) 2.5

0.8

2

0.6

1.5

ρσ 3

ρσ 3

(a)

0.4

1

0.2

0.5

0

0

10

20

30 z /σ

40

50

0

0

5

10

15

20

25

30

z /σ

Figure 3 (a) Segment density profiles of grafted chains, Ng = 100 and rgsg2 = 0.1, in the presence of implicit solvent () and explicit free chains with Nf = 2 (~), Nf = 5 (&), and Nf = 10 (.). Symbols are the simulation results from Grest and Murat (1993) and curves are the predictions from iSAFT. (b) Solid line represents segment density profiles of grafted polymer chains (Ng = 100 and rgsg2 = 0.1) and Dashed line represents density profiles of the free polymer solvent (Nf = 10 and rfsf3 = 0.682) from iSAFT.

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of implicit solvent. This compression increases with the increase in the number of segments in the solvent polymer chains and the height of the monolayers decreases further. As the solvent density is large, the solvent chains penetrate into the monolayer, as seen in Figure 3b. However, near the wall, the segments of the grafted polymer chains dominate where there is a strong layering because of the presence of the hard surface.

3.2 Interaction between two grafted monolayers in the presence of free polymer melt with both the grafted and the free polymer chains having equal segment sizes Now, we consider two grafted surfaces. Both the grafted and the free polymer chains are purely repulsive and have the same segment sizes (g = f = ). For large separations between them, they do not interact and behave as two independent monolayers immersed in the free polymer melt. These monolayers and the free polymer interpenetrate each other and the monolayers are compressed. The degree of interpenetration depends upon the chain lengths of the grafted (Ng) and free (Nf) polymers, grafting density g, and the bulk free polymer density f. As all the polymer chains are purely repulsive, they are implicitly in good solvent condition. Figure 4a shows the segment density profiles of the grafted and free polymers for Ng = 101, g2 = 0.1, Nf = 100, and f 3 = 0.2, when the grafted monolayers are far apart. Each monolayer interpenetrates the free polymer to a certain extent after which the free polymer reaches its bulk density. The monolayers start interacting as the separation is reduced. In addition to penetrating the free polymer, the monolayers themselves interpenetrate each other. In doing so, they expel the free polymer between them and the density of the free polymer decreases, as shown in Figures 4b and 4c. Eventually, at low enough separation, almost all the free polymer leaves the gap between the monolayers, as shown in Figure 4d. Let us now turn to the discussion of the SCFT results and compare them with the iSAFT calculations. In our calculations, we use the following parameters: Nf = 100, Ng = 100, g
= 0.1. The statistical segment length a in SCFT calculations and the segment size  in iSAFT calculations are assumed to be equal and the volume fraction of the free polymer in the bulk, ’bf, in SCFT calculations equal to the free polymer density, f3, in iSAFT calculations, so that we can attempt exact comparison with the iSAFT analysis. Density profiles and grand canonical free energy, F(H), are evaluated as function of the separation between the plates, H (expressed in units of a). Taking a derivative of F with respect to H, one can calculate the normal force of interaction between the grafted mono­ layers and compare it with the one estimated using iSAFT. For calculations, we utilized the code Polymer developed by P. Linse (Lund University).

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Application of Mesoscale Field-Based Models

(c)

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40 z /σ

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(d) 0.5

ρσ 3(z)

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0.5

0.4

ρσ 3(z)

ρσ 3(z)

(a)

0

10 20 30 40 50 60 70 80 z /σ

0

30 40 50 60 z /σ

70

Figure 4 Segment density profiles of the two grafted monolayers (solid curves) and the free polymer (dotted curves) at separations (a) H = 90 s, (b) H = 80 s, (c) H = 75 s, and (d) H = 72 s. Parameters: Ng = 101, rgsg2 = 0.1, Nf = 100, rf sf3 = 0.1, and sg = sf = s.

In Figure 5a, we plot the density profiles for the free polymer and grafted polymer for H = 50, with bulk polymer volume fraction set to 0.75. The agreement between the densities predicted by the two methods is surprisingly good. The main difference is in the behavior near the hard surfaces. While SCFT predicts pronounced polymer depletion near the walls, iSAFT calculations show strong, almost diverging, density fluctua­ tions near the surfaces. In the bulk, the predictions of the two theories agree not just qualitatively, but even semi-quantitatively. In Figure 5b, we plot the density profiles for the system with bulk polymer volume fraction of 0.25 and H = 80. Here, the agreement between the two theories is much worse. SCFT appears to over-predict the amount of free polymer drawn into the gallery (compared to iSAFT); it also predicts more diffuse distribution of the free polymer and the grafted (brush) polymer, as compared to iSAFT. In Figure 6, we plot the effective force of interaction between the monolayers from iSAFT and SCFT as a function of the distance H between

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(a)

1 0.9 0.8 0.7

φ

0.6 0.5 0.4 0.3 0.2 0.1 0 0

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50

z (b) 0.5

0.4

φ

0.3

0.2

0.1

0 0

10

20

30

40 z

50

60

70

80

Figure 5 Density profiles of grafted (solid curves) and free polymer chains (dotted curves) as a function of the spatial coordinate z, calculated using iSAFT (gray curves) and SCFT (black curves) methods. (a) Free polymer bulk volume fraction ’bf = 0.75; separation between the platelets H = 50, and (b) free polymer bulk volume fraction ’bf = 0.2; separation between the platelets H = 80. Other parameters: Ng = 101, rg
= 0.1, and Nf = 100.

them. It is instructive to consider the differences between the curves corresponding to various free polymer volume fractions. iSAFT and SCFT curves for the highest bulk polymer volume fraction (’bf = 0.75) lie very close to each other. For that system, both models predict, effectively, purely repulsive behavior: when the two monolayers are far apart, normal force is practically zero; when the monolayers are

Application of Mesoscale Field-Based Models

151

0.0006

Normal force

0.0004 0.0002 0 20 −0.0002

30

40

50

60

70

80

90

H

−0.0004 −0.0006

Figure 6 Normal force per unit area between the plates as a function of their separation, calculated using iSAFT (gray curves) and SCFT (black curves). Free polymer bulk volume fractions: 0.2 (dotted-dashed curves), 0.4 (dashed curves), and 0.75 (solid curves). Other parameters: Ng = 101, rg
= 0.1, and Nf = 100.

beginning to interact (H < 40), repulsive force appears and rapidly grows as H is decreased. For the intermediate polymer volume fraction (’bf = 0.4), there is effective repulsion at small distances (H < H
) and attraction at large distances (H > H
). The critical separation, H
, is approximately equal to 50 (from both iSAFT and SCFT). We note that the attractive force predicted by iSAFT is stronger than that predicted by SCFT; however, the inherent error in determining the normal force in SCFT is relatively high because of the need for numerical calculation of the free energy derivative. Finally, for the smallest free polymer bulk volume fraction (’bf = 0.2), there is some discrepancy in the prediction of H
(68 from SCFT, 72 from iSAFT); the attractive force estimated by iSAFT is once again stronger than that predicted by SCFT. Altogether, considering the differences between the two methods, the results appear to be in a remarkably good agreement. It is interesting that the best agreement—qualitative and quantitative—between the two meth­ ods corresponds to the high polymer volume fraction conditions (’bf = 0.75). It is well documented that the accuracy of SCFT is the highest for the “incompressible melt” conditions (’bf !1) and decreases as the fraction of “voids” goes up; on the other hand, iSAFT is expected to work best at low to intermediate polymer concentrations but fail when ’bf becomes too high. Our analysis shows that—at least for the athermal systems considered here—one could accomplish a smooth, continuous transition from iSAFT description (low polymer concentration) to SCFT description (high polymer concentration). Whether this continuous transi­ tion could be accomplished in other (especially non-athermal) systems will be the subject of future studies. The results depicted in Figure 6 have interesting implications for nanocomposite thermodynamics. If the “density” (bulk polymer volume

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fraction) is sufficiently high, the equilibrium nanocomposite morphology is expected to be exfoliated (normal force between the platelets is always repulsive). However, at intermediate to low bulk polymer volume frac­ tions, a pronounced attractive force region appears, indicating preference for intercalated morphology. This is due to the fact that voids could easily enter the “gallery” between the two surfaces, while the “free” polymer is pushed out of the gallery where its entropy is lower than in the bulk. Thus, let us consider the changes in the gallery as we reduce its width, H. At first, free polymer is squeezed out; the intra-gallery space is enriched by voids, free energy decreases, and normal force is attractive. However, as the two brushes come closer, the system can no longer expel the free polymer because of the growing imbalance between the intra-gallery and bulk compositions; at that point, the free energy would begin to increase, and normal force would change sign from attractive to repulsive. Thus, includ­ ing melt compressibility in the analysis of nanocomposite thermody­ namics provides a potential new mechanism favoring intercalated morphologies over exfoliated ones. Similar behavior for the interaction force is observed on changing the chain length of the free polymer while keeping its bulk density fixed, as shown in Figure 7. The interaction force is purely repulsive for the smallest free polymer chain length, while at larger chain lengths, the force is attractive at intermediate separations and repulsive at lower separations. There is a higher entropic advantage for the free ends of the two mono­ layers to mutually overlap each other rather than the two monolayers overlapping with the longer free polymer chains. Hence, the attractive region increases with the increase in the free polymer chain length.

0.0006

Normal force

0.0004 0.0002 0 20 −0.0002

30

40

50

60

70

80

90

H

−0.0004 −0.0006

Figure 7 Normal force per unit area between the plates as a function of their separation, calculated using iSAFT (gray curves) and SCFT (black curves). Free polymer chain lengths: 50 (dotted curves), 100 (solid curves), and 150 (dashed curves). Other parameters: Ng = 101, rg
= 0.1, and ’bf = 0.6.

Application of Mesoscale Field-Based Models

153

Furthermore, the attractive region occurs only when the free polymer exceeds a certain minimum chain length. If the free polymer chain length is lower than this value, then the force is purely repulsive. For Ng = 101, g2 = 0.1, and f3 = 0.6, this critical value of Nf lies somewhere between 50 and 100, as shown in Figure 7. Again, the agreement between iSAFT and SCFT is remarkable. The agreement is best (qualitatively as well as quantitatively) at the highest free polymer chain length, Nf = 150. For both Nf = 150 and 100, the attrac­ tive regions predicted by iSAFT are stronger than that predicted by SCFT. In fact, for Nf = 100, the attractive region predicted from SCFT is so small that it can be hardly be seen in the figure. Hence, the critical value of Nf, at which the force of interaction between the grafted monolayers turned from purely repulsive to having an attractive minimum, calculated using SCFT lies closer to 100 than that from iSAFT. However, this value still lies between 50 and 100. Using iSAFT, we further calculated the critical chain lengths of the free polymer for a number of cases at different bulk free polymer densities, grafting densities, and chain lengths of the grafted polymer. We found that it is the ratio of the chain lengths of the free and grafted polymers,  = Nf/Ng, that determines whether the force of interaction between the monolayers is purely repulsive or has an attractive minimum. Figure 8 plots the loci of the critical values of  at which the force of interaction between the grafted monolayers turned from purely repulsive to having

1

ρg σg2(Ng)1/2

0.9 Attractive

0.8 0.7

Repulsive

0.6 0.5 0.4

0.8

1

1.2

1.4

α

1.6

1.8

2

Figure 8 Locus of the critical values of  at which the interaction between the grafted monolayers becomes attractive for different bulk free polymer densities: rfsf3 = 0.6 (&) and rfsf3 = 0.75 (). Ng = 101 and sg = sf = s.

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an attractive minimum, for different bulk free polymer densities. The locus (at a fixed free polymer density) calculated using iSAFT agrees with the scaling relation obtained from previous theoretical studies using Strong Segregation Theory (SST) (Leibler et al., 1994) and numerical SCFT (Fer­ reira et al., 1998; Matsen and Gardiner, 2001): gHNg / , where  is the scaling exponent. From iSAFT,  = 2.

3.3 Interaction between two grafted monolayers in the presence of free polymer melt with the grafted and free polymer chains having different segment sizes Next, we consider the case where the segment sizes of the polymer in the monolayers and the free polymer are not identical, which is not easily accessible to SCFT. To our knowledge, other theoretical models have not previously addressed this case as well. Using iSAFT, we investigate the effect of changing the relative sizes of the segments in the free and grafted polymers,  = f/g, on the interaction force between the grafted mono­ 6 g, the repulsion/attraction boundaries layers. We found that even for f ¼ follow the same scaling relation, gHNg /  (at fixed free polymer den­ sity). The boundaries shift toward the attractive domain as  decreases, or, in other words, the critical value of  increases as the relative segment size of the grafted polymers increases. This is due to the fact that the steric hindrance is higher for grafted monolayers with bigger segments. The steric hindrance between the two monolayers depends upon the volume of the segments of grafted polymers; hence, intuitively, different repul­ sion/attraction boundaries for different segment sizes of the grafted poly­ mer (relative to the size of free polymer segments) may scale by g3. In fact, this is the case as shown in Figure 9. The figure shows that for cases where the relative segment sizes of the grafted and free polymers are different, the critical value of  where the interaction force between the two mono­ layers turns from purely repulsive to attractive follows the scaling relation, gHNg 3 / 2.

3.4 Interaction between two grafted monolayers in the presence of attractive chains Now we consider the attraction between the segments of the tethered and the free polymer chains. To make it realistic, the attractive LJ energy is based on that of polypropylene segments. The energy parameter "/k = 235 K is calculated from the correlation given by Dominik et al. (2006) using its value from perturbed-chain (PC)-SAFT (Gross and Sadowski, 2002, 2003; Tumakaka et al., 2002) (PC-SAFT "/k = 217 K). For the polypropylene melt at T = 170C, the dimensionless parameters are

Application of Mesoscale Field-Based Models

155

1 0.9

ρg σg2(Ng)1/2β 3

0.8

Attractive

0.7 0.6

Repulsive

0.5 0.4 0.3 0.2 0.5

1

1.5

2

α

Figure 9 Scaling relation for the locus of the critical values of  at which the interaction between the grafted monolayers becomes attractive for different relative segment sizes of the grafted and free polymer: Ng = 101 and sg = sf (&), Ng = 101 and sg = 1.1 sf (), Ng = 101 and sg = 1.2 sf (~), and Ng = 151 and sg = sf (^). The bulk free polymer density rfsf3 = 0.6. sf = s.

f3 = 0.75 and "/kT = 0.53. This energy parameter was used for both tethered chains and free chains. The values of c for this system as a function of the grafting density g are calculated using a sort of bisection scheme. At a fixedg, the forces of interaction are calculated as a function of the separation for two extreme values of , such that the lower value 1 shows pure repulsion and the higher value 2 shows the attractive minimum in the force curve. {1,2} is the initial set such that c lies between 1 and 2. Then the force curve is calculated at the midpoint, (1 + 2)/2. If the force is purely repulsive at this value of , then {(1 + 2)/2,2} is the new set otherwise {1,(1 + 2)/2} forms the new set. These steps are repeated till att  rep = 0.5, such that the force is purely repulsive for rep and shows a minimum foratt. Figure 10 shows these values of rep and att for different values of g. There are two reasons for reporting this band which contains c rather than exactly pinpointing the value of c. Although there is a minimum for values of  greater than c, for values of  closer to c the minimum is so shallow that it is difficult to ascertain it numerically. However, for   att, the attractive minimum is significant. To sum up the result shown in Figure 10, the dispersion is stable if the value of  lies on the left side of solid curve in the figure and it is unstable (causes agglomeration) if the value of  lies on the right side of the dashed curve. Hence, the values of  on the dashed curve give the optimum chain length of the tethered chain to create a stable dispersion.

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0.1 0.09 0.08 ρg σt 2

Repulsive

Attractive

0.07 0.06 0.05 0.04

0.4

0.6

0.8

1

1.2

1.4

1.6

α Figure 10 Stable (repulsive) versus unstable (attractive) dispersion of tethered chains as a function of the ratio of the polymer chain length to tethered polymer chain length ().

3.5 Interaction between two grafted monolayers in the presence of end-functionalized chains Here, again, we start from compressible SCFT formalism described in Section 2.2 and consider a model system in which bulk polymer consists of “free” matrix chains (Nf = 300) and “active” one-sticker chains (Na = 100). Flory–Huggins interaction parameters between various species are summarized in Table 1. This corresponds to the scenario in which surfactants, matrix chains, and functionalized chains are all hydrocarbon molecules (e.g., surfactant is a C12 linear chain, matrix is a 100,000 Da molecular weight polyethylene, and functionalized chain is a shorter poly­ ethylene molecule with one grafted maleic group). The nonzero interaction parameter between voids and hydrocarbon monomers reflects the non­ zero surface tension of polyethylene. The interaction parameter between the clay surface and the hydrocarbon monomers, c = 1.0 ( = G, F, A), reflects a very strong incompatibility between the nonpolar polymers and Table 1 FloryHuggins interaction parameters used in the calculation

Grafted (G) Free matrix (F) Active (A) Sticker (S) Void (V)

G

F

A

S

V

0.0 0.0 0.0 0.0 1.0

0.0 0.0 0.0 0.0 1.0

0.0 0.0 0.0 0.0 1.0

0.0 0.0 0.0 0.0 1.0

1.0 1.0 1.0 1.0 0.0

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Application of Mesoscale Field-Based Models

the polar clay surface. The interaction parameter between the clay surface and the voids—related to the surface tension of the bare clay—was set to 2.0 to reflect the fact that the surface energy of the clay should be much higher than that of the polymers. Finally, the interaction parameter between the stickers and the clay surface, SC, was varied to study the influence of the adhesion strength on the phase behavior. (The range of SC investigated here, between 0 and -100, roughly corresponds to sticker–surface adhesion energy, " =  SC/6, being bracketed between 0 and 10 kcal/mol, which is typical for hydrogen-bonding interactions.) We also varied the volume (or weight) fraction of the active chains. Finally, all the calculations were repeated for two densities: bv = 0 (incompressible melt, the model equiva­ lent to earlier studies of Balazs et al.) and bv = 0.4 (compressible melt). The length of the grafted chains, Ngr = 5, and the grafting density, gr = 0.2, were chosen to match the earlier study of Ginzburg and Balazs (2000). (One difference between the current study and that of Ginzburg and Balazs (2000) is that in that earlier study, Flory–Huggins parameters between the clay and the polymers were set to zero.) In Figure 11, we plot the calculated free energies, f(H), for several values of SC, while keeping the active chain weight fraction constant at 0.05 (same as in the study by Ginzburg and Balazs, 2000). In all cases, the dashed lines correspond to the incompressible model, and the solid lines (a)

(b)

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0.15 Incompressible Compressible

0.1

Free energy

Free energy

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Incompressible Compressible

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H, nm

0.2

Free energy

0.15 Incompressible Compressible

0.1 0.05 0 −0.05 −0.1

0

4

8

12

16

H, nm

Figure 11 Calculated free energy (per unit area) profiles based on compressible (solid) and incompressible (dashed) lattice SCFT approaches: (a) SC = 0; (b) SC = 48; (c) SC = 96.

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are compressible model calculations. We can now carefully examine some of the results. At low adhesion strength between the sticker and the clay (Figure 11a and b), the free energy has a deep, sharp minimum at small separation. Note that for the incompressible system, this minimum corresponds to the gallery which is “completely closed” and, thus, the plate separation is the same as for the neat organoclay (“immiscible morphology”). For the compressible system, the gallery opens up slightly (the increase in the gallery height is approximately 0.8–1 nm), and the structure that forms is now likely to be interpreted as “intercalated.” For the system with very high adhesion strength between the clay and the stickers, the favorable enthalpy of the sticker–clay interaction domi­ nates the unfavorable entropic contributions. Accordingly (Figure 11c), the composite now becomes exfoliated, as demonstrated by both compressible and incompressible models. Interestingly, within the compressible model, the old minimum at H  1.6–2 nm has not disappeared completely, indi­ cating the presence of a metastable intercalated morphology. What is the origin of this minimum? To understand that, we examined in more detail the case of SC = 84. Its compressible free energy profile is shown in Figure 12a, once again indicating the presence of a well-defined minimum at small separations. In Figure 12b, we plot the dependence of polymer density (sum of the volume fractions of all components except for voids) in the gallery on the gallery height. It can be seen that the free energy minimum coincides with the density minimum. Thus, the voids are the first to intercalate the gallery—this initial expansion simply because of the density reduction inside the gallery takes place even before the sticker attachment. The analysis of density profiles at different gallery heights helps understand this. At small separations (Figure 13a), the gallery is penetrated by some active chains, but their amount is relatively low, while the fraction of voids is slightly larger than in the bulk. The grafted chains form strongly overlapping brushes. At intermediate separations (Figure 13b), the brushes no longer overlap, and some matrix chains begin to penetrate the gallery, although end-functionalized chains still account for the majority of the intra-gallery polymer. Finally, at large separations (gallery height much greater than radius of gyration of the active chains), the composition in the center of the gallery is close to that of the bulk polymer, while near the surfaces there is excess of active chains (Figure 13c). The above discussion centered on a single active chain weight fraction x = 5 wt.%. We calculated free energy profiles as function of SC for several polymer compositions, varying the active chain content from 0 to 100%. The resulting phase diagram is shown in Figure 14. It can be seen that according to the “compressible” model, the range where intercalated structures could be found has dramatically expanded. Basically, the “initial” intercalation occurs when the plates are separated only slightly,

Application of Mesoscale Field-Based Models

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Free energy

0.1 0.05 0 −0.05 −0.1 0

(b)

4

8 H, nm

12

4

8 H, nm

12

16

1

Density

0.8 0.6 0.4 0.2 0 0

16

Figure 12 Free energy per unit area (a) and overall polymer density (b) as a function of the gallery height H for the SC = 84 system. Calculations are based on the compressible lattice SCFT model.

and the “voids” enter the gallery, thereby substantially decreasing the density of the grafted surfactant. The amount of active polymer entering the gallery is still relatively low. As the gallery height is increased further, the addition of the active chains requires that density go up, thus resulting in the loss of translational entropy for the “voids” (alternatively, this could be thought of simply as additional entropic penalty for higher polymer density). This increase in the free energy could overcome the enthalpic gain from adding a few “stickers” attached to the clay surfaces. Only when the separation is increased substantially, and the number of “stickers” attached to the surface becomes sufficiently high, free energy can decrease to its global minimum. The above predictions are certainly dependent on the choices of the Flory–Huggins interaction parameters, relative molecular weights of

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1 Grafted Matrix Voids Active Chains

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1 Grafted Matrix Voids Active Chains

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Prasanna K. Jog et al.

0.4 0.2

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Grafted Matrix Voids Active Chains

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4 Z, nm

0.6 0.4 0.2 0 0

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24

Z, nm

Figure 13 Calculated density profiles for the SC = 84 system at various gallery heights H: (a) H = 2.4 nm (6 lattice units); (b) H = 7.2 nm (18 lattice units); (c) H = 24 nm (60 lattice units). 100

Clay-sticker attraction

80

60

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Wt. Fr. active chains

Figure 14 Calculated nanocomposite phase diagram (in the limit of small organoclay loadings). Solid line is the phase boundary between weakly intercalated stacks and partially exfoliated, partially intercalated phase, as calculated using compressible SCFT. Dashed line is the phase boundary between immiscible and exfoliated phases, as calculated by incompressible SCFT.

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matrix and sticker chains, matrix polymer density, etc. Even so, the main predictions of the “compressible model”—the shifting of the global free energy minimum toward higher H for the “immiscible” systems and the persistence of the metastable small-H local free energy minimum for the “exfoliated” systems—appear to be universal and robust, at least within the parameter space investigated here. What are the practical implications of this new model? First, it suggests that intercalated morphologies should be observed, indeed, for majority of nanocomposites. In fact, it is likely that in the melt, almost all nanocompo­ sites exhibit gallery expansion compared to the room-temperature organo­ clays (see, e.g., Jacobs et al., 2006; Vaia et al., 1994). As nanocomposite is cooled down from the melt to room temperature, the average gallery spa­ cing could shrink back or stay expanded, depending on various kinetic factors. Second, it suggests caution in assuming that intercalated structures must lead to physical or mechanical property improvements. If intercalation proceeds because of the “frozen” density reduction in the galleries, rather than because of the true incorporation of the matrix polymer, the interface between clay and polymer will remain weak, and it would likely adversely impact properties such as toughness. Finally, it is important to note that even for the supposedly exfoliated nanocomposites, the presence of a local free energy minimum means that there is a strong barrier to exfoliation. As discussed by Ginzburg, Gendelman, and Manevitch (Balazs et al., 2008; Gendelman et al., 2003; Ginzburg et al., 2001), if the free energy profile has a double-well structure, the transition from intercalated to exfoliated mor­ phology takes place only when a sufficiently strong shear force is applied to overcome the barrier. The transition would occur via a so-called “kink” mechanism, and the kink could appear if—and only if—the clay stack is subjected to fairly large shear forces. Thus, it is easy to understand why— even if thermodynamically the nanocomposite should be exfoliated—a large fraction of the clay platelets still remains in the metastable, intercalated state. Yet another implication of the model—and a potentially testable pre­ diction—is that all other things being equal, one could find a higher degree of intercalation in nanocomposites with lower-density matrix polymers, compared to ones with higher density. We are not aware of experimental studies directly testing this proposition (although studies by Vaia and co­ workers (1994; Jacobs et al., 2006) appear to provide indirect evidence, as discussed above); we hope that our theoretical analysis would stimulate new investigation in this direction.

4. SUMMARY AND OUTLOOK In this chapter, we described the application of mesoscale field-based theories (iSAFT and SCFT) to describe polymer-mediated interaction

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between two brush-covered flat surfaces. This is a problem that is critical to the successful preparation of polymer–inorganic nanocomposites. While thermodynamics of particles in the polymer matrix is not the only issue determining nanocomposite morphology (dynamics of polymer intercalation and rheology of the particle-filled melt are extremely impor­ tant as well), its understanding is necessary to properly design a formula­ tion that could potentially lead to the exfoliated and dispersed nanocomposite. That (exfoliation and dispersion) is, in turn, necessary to achieve the desired improvement in mechanical and physical properties. Of course, nanocomposites are not the only area where mesoscale the­ ories are being used to predict nanostructure and morphology. Other appli­ cations include—but are not limited to—block copolymer-based materials, surfactant and lipid liquid crystalline phases, micro-encapsulation of drugs and other actives, and phase behavior of polymer blends and solutions. In all these areas, mesoscale models are utilized to describe—qualitatively and often semi-quantitatively—how the structure of each component and the overall formulation influence the formation of the nanoscale morphology. Despite all the successes of mesoscale theories, there are some limita­ tions, such as the following:

  

Determination of various interaction parameters is difficult and sometimes arbitrary. It is still very difficult to properly account for electrostatic interactions (there have been some attempts to incorporate electrostatics into SCFT (Popov et al., 2007), but their applicability is still rather limited). It is still difficult to apply mesoscale theories to describe aqueous solutions, as the description of hydrogen bonding is not yet very successful.

We anticipate that as these mesoscale approaches are developed and their shortcomings are overcome, their application in chemical engineering will become more and more widespread. Eventually, it is our expectation that the use of these models will become as common as the use of continuum hydrodynamics and quantum chemistry models is today.

ACKNOWLEDGMENT WGC and SJ acknowledge financial support provided by the Robert A. Welch Foundation (Grant No. C1241) and by the National Science Foundation (CBET-0756166).

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CHAPTER

4 Principles of Statistical Chemistry as Applied to Kinetic Modeling of Polymer-Obtaining Processes Semion Kuchanov


Contents

1. Introduction 2. Main Peculiarities of the Description of Polymers 2.1 Statistical approach 2.2 Microstructure parameters 2.3 Chemical correlators 3. Specificity of the Description of Branched Polymers 3.1 Stochastic branching process 3.2 Gelation 4. Kinetic Models of Macromolecular Reactions 4.1 Ideal kinetic model 4.2 Models allowing for the deviations from ideality 5. Methods of Calculations 5.1 Statistical method 5.2 Kinetic method 5.3 Extension of statistical and kinetic methods 6. Some General Theoretical Results 6.1 Polycondensation 6.2 Conventional free-radical copolymerization 6.3 Chemical modification of polymers 7. Effect of Hydrodynamic Stirring in a Reactor on Some

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Physics Department, Lomonosov Moscow State University, Moscow 119991, Russia A.N. Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Moscow 119991, Russia


Corresponding author. E-mail address: [email protected]

Advances in Chemical Engineering, Volume 39 ISSN: 0065-2377, DOI 10.1016/S0065-2377(10)39004-1

� 2010 Elsevier Inc. All rights reserved.

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7.2 Radical polymerization 7.3 Polycondensation 7.4 Effect of stirring on composition inhomogeneity 7.5 Polymer-analogous reactions 7.6 Microsegregation Acknowledgments References

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The subject matter of statistical chemistry of polymers consists in revealing quantitative theoretical dependencies of the characteristics of chemical structure of polymers on the conditions of their obtain­ ing. The fundamental concepts of this branch of macromolecular science are outlined in simple language, bypassing any mathematics. A classification is presented of the kinetic models of macromolecular reactions currently used under mathematical modeling of diverse processes of the synthesis and chemical modification of polymers. A comparative analysis is carried out of theoretical methods engaged in calculations of both the kinetics of such processes and the com­ position and chemical structure of macromolecules formed. The efficiency of application of general approaches of statistical chem­ istry is illustrated using the processes of polycondensation, radical polymerization, and chemical modification of polymers as example. A special section is devoted to the discussion of the peculiarities of the effect of hydrodynamic stirring in a reactor on chemical inhomo­ geneity of obtained polymer products.

1. INTRODUCTION Kinetic modeling plays an ever-increasing role in the design and optimi­ zation of the processes of polymer manufacturing. The efficiency of this method as applied to chemical engineering is predetermined to a great extent by the choice of an appropriate kinetic model underlying the modeling of a particular process. Such a model is supposed to correctly take into account the main physicochemical peculiarities proceeding from the current level of knowledge in polymer science. Only then the model chosen can be used for an extrapolation procedure, ensuring thus a scien­ tifically based employment of the results of the kinetic laboratory research for the optimization of industrial reactors. When choosing the kinetic model of a particular polymerization pro­ cess an engineer-researcher inevitably faces the necessity to proceed from two opposite considerations. On the one hand, he is interested in the maximal simplicity of this model bearing in mind the analysis of the results obtained on the basis of such a model and the subsequent solution of optimization problems. On the other hand, he is perfectly aware of the

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impossibility for too oversimplified a model to correctly predict the per­ formance properties of polymer products, which results in essentially reduced efficiency of the application of mathematical modeling. Hence, the question is to which extent a kinetic model may be simplified to retain the ability to properly predict the physical and mechanical properties of the polymers obtained. The answer depends on a number of factors, among which are the reliability of the knowledge on the kinetics and mechanism of the process of the synthesis of polymer, experimental pos­ sibilities for its characterization, and other factors. Their appropriate allowance predetermines an optimal strategy of the choice of the kinetic model and necessary extent of its detailing. The present review aims at formulating the general principles of such a choice based on the knowl­ edge of current state of art in polymer science. Here there is no discussion of particular works, and the list of references, not pretending to be com­ plete, contains just some key publications. Choosing a kinetic model of a polymer-obtaining process, one is sup­ posed to take into account some specific features absent under modeling of chemical processes in which only low molecular weight compounds are involved. Among such features, the most important is the polydispersity of the products of synthesis and chemical modification of polymers. As a rule, a great variety of macromolecules differing in molecular weight, composition, and chemical structure are simultaneously present in a reac­ tor at any time. Clearly, an exhaustive description of such a reaction system suggests the recourse to statistical approaches. The Nobel Prize winner Paul Flory (1953) was the first who introduced these approaches in polymer chemistry. Later, his ideas gained wide recognition in the theore­ tical description of many processes of polymer obtaining. The problem of crucial importance here is revealing the relation between the process mode and the chemical structure of polymer products obtained for this process. The elaboration of general approaches to the solution of this fundamental problem falls within the realm of the statis­ tical chemistry of polymers (see, for example, Kuchanov, 1978, 2000), whose mathematical apparatus is based on the employment of some fields of the applied mathematics. Among them, for instance, is the theory of random processes, the graph theory, the theory of dynamic systems, and some other mathematical disciplines. Their application in modeling of polymerization processes is perfectly indispensable to provide a complete description of polymer products formed. Regretfully, such level of the description of polymers has not been properly addressed so far in the literature devoted to polymerization engineering and polymerization process modeling (Asua, 2007; Beisenberger and Sebastian, 1983; Dotson et al., 1996; Gupta and Kumar, 1987; Meyer Keurentjes, 2005; Rudin, 1982; Seavey and Liu, 2008). This circumstance substantially restricts the possi­ bilities of theoretical prediction of some of their important service

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properties, especially when polymer products are polydisperse specimens composed of macromolecules varying in chemical composition and topo­ logical structure. Success in a quantitative description of the molecular structure of any such specimen and, consequently, the efficiency of the prediction of its properties are predetermined to a great extent by an appropriate choice of the kinetic model of polymer-obtaining process. In this brief review an attempt is undertaken to discuss in the context of the statistical chemistry of macromolecular reactions the strategy of the choice of such a model keeping in mind its employment under mathematical modeling of polymerization processes.

2. MAIN PECULIARITIES OF THE DESCRIPTION OF POLYMERS Polymer products formed in the course of the synthesis and chemical modification represent an ensemble of individual chemical compounds, the number of types of which is virtually infinite. Macromolecules of these products can differ in the degree of polymerization, tacticity, the lengths and number of branchings, as well as in other characteristics describing the configuration of a macromolecule. In case of copolymers, their macro­ molecules vary also in composition and the pattern of arrangement of different types of monomeric units. A complete quantitative description of such a system specifying the concentration of all individual compounds of a particular chemical (i.e., primary) structure is mostly impossible. However, for many practical purposes, it is often enough to know only partial distributions of molecules in some of their principal characteristics, such as molecular weight or composition, neglecting a more detailed description of their chemical structure. In other cases, the character of the structure of polymer chains essentially governs some of their physico­ chemical properties. Here the determination of quantitative characteristics of primary structure of macromolecules becomes indispensable. Generally speaking, the more minute is the quantitative description of a polymer system due to increased number of attributes responsible for the difference between individual macromolecules, the larger is the number of macro­ scopic properties of this system that can be theoretically predicted. This is because the chemical structure of macromolecules characterized by their composition and configuration carries a specific information on their possible conformations, secondary and supermolecular structures, and, consequently, their physical properties. Finding of quantitative correla­ tions between these properties and the characteristics of the primary structure of macromolecules belongs to the area of the statistical physics of polymers and constitutes one of the three central problems whose solution is indispensable to provide a scientifically grounded prediction

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for obtaining polymers with desired set of properties. The second problem relevant to the chemical kinetics of macromolecular reactions consists in ascertaining their mechanism and determining the constants of elementary reactions of the processes of polymer obtaining and transformation. Once this problem has been solved, one should choose an appropriate kinetic model of the process. Based on this model, one can calculate the quanti­ tative dependencies of the statistical characteristics of the primary struc­ ture of macromolecules on the reactivities of the species involved and the process mode. This review discusses just this, the third problem, which is typical for the statistical chemistry of polymers. Let us first address linear polymers, whose topological structure is the simplest. Depending on the number m of types of monomeric units, homo­ polymers (m = 1) and copolymers (m  1) are distinguished. In the most trivial case, homopolymers’ molecules are identified only by number l of constituent monomeric units, whereas the composition of a copolymer macro­ molecule is characterized by vector l, whose components l1,…,l
,…,lm are equal to the numbers of units of each type. Having identical composition, these molecules can vary in microstructure characterized by the pattern of arrangement of different units along a copolymer chain. Since typical values of the degree of polymerization l = l1 + … + lm in synthetic copoly­ mers are 102104, the number of possible types of isomers differing in microstructure is considered to be virtually infinite. Evidently, a quanti­ tative description of any such polymer specimen consisting of macromo­ lecules with enormous number of configurations is feasible only through statistical methods.

2.1 Statistical approach In the framework of a statistical approach, every macromolecule of such a specimen can be unambiguously associated with a certain realization of the stochastic process of conventional movement along the copolymer chain. This movement can be thought of as a succession of stochastic transitions from a unit of the chain to the neighboring one. The type of monomeric unit on each step is determined therewith in accordance with the statistics of the stochastic process describing the polymer specimen of interest. In order to consider the set of the trajectories associated with realizations of infinite length, it is convenient to believe that after the terminal unit of a molecule a trajectory falls into absorbing state to remain there forever. Therefore, to any particular specimen of linear copolymer with m types of units there corresponds a certain stochastic process with discrete time, m regular states, and one absorbing state. In mathematics such a stochastic process is referred to as a chain. The best known among them are the Markov chains (Lowry, 1970), for which the probability 
 to reach at any step a -type state is controlled exclusively by type
of the state at the preceding step.

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When the attempt to formulate the algorithm of finding the probability of any trajectory of the above-mentioned stochastic process is a success, a statistical description of a copolymer specimen will be exhaustive. For those of them in molecules of which the alteration of units is described by a Markov chain, such an algorithm is trivial. It enables one proceeding from the general theory of these chains to immediately find any statistical characteristics of the Markovian copolymers in terms of elements 
 of the transition matrix of the corresponding Markov chain. In this case, the peculiar features of particular processes of the synthesis of such copoly­ mers are taken into account only when finding the dependencies of matrix elements 
 on time, stoichiometric, and kinetic parameters of a reaction system. Hence, mathematical modeling of the products of the synthesis of copolymers with Markovian statistics of units’ alteration is trivial. That is why when choosing an appropriate kinetic model of a particular process of obtaining a copolymer, an engineer-researcher should know whether this copolymer is Markovian or not. This question is answered for many practically significant cases (Kuchanov, 1978, 2000). What is the strategy of the mathematical modeling for non-Markovian copolymers, where the attempt to formulate a general algorithm of the construction of the probability of any individual macromolecule fails? Here the necessity arises to find the statistical characteristics of the chemi­ cal structure of a polymer specimen in the framework of a chosen kinetic model for each particular manner of its synthesis. These characteristics can be divided into two groups. Those belonging to the first group describe macromolecules’ inhomogeneity in the degree of polymerization l (chemi­ cal size) and composition, while the characteristics pertaining to the second group describe their chemical structure. Among the characteristics constituting the first group is the size– composition distribution (SCD) function f(l), which is equal to the prob­ ability that a unit chosen at random pertains to the macromolecule with fixed value of vector l. This distribution is indispensable for constructing a phase diagram of solution or melt of a copolymer (Kuchanov and Panyukov, 1996). This is of crucial importance for the prediction of some of its properties, for example, the transparency. Its loss in the course of the synthesis of polymers can occur as a result of the phase separation in a reaction system induced by chemical transformations. Such a transforma­ tion is due to increasingly more pronounced composition inhomogeneity of the copolymer being formed. This assertion is rigorously theoretically substantiated (Kuchanov and Panyukov, 1996, 1998) and ensues from the formula relating the heat of mixing of a copolymer solution with the elements of the covariance matrix of composition distribution (CD). The latter can be measured by chromatographic methods (Glockner, 1991) and compared with the distribution calculated in the framework of the model chosen.

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Statistical characteristics belonging to the second group describe the microstructure of copolymer molecules. If these are linear, such character­ istics are fractions P{Uk} (probabilities) of sequences {Uk} involving k monomeric units (Koenig, 1980). The simplest here are dyads {U2}, the overall set of which for a binary copolymer comprises four pairs of units M1M1,M1M2,M2M1,M2M2. The number of k-ad types in chains of m-com­ ponent copolymer grows exponentially as mk, so that in practice a researcher has to confine his consideration to sequences {Uk} with small values of k. Their calculation is highly useful under mathematical model­ ing of any processes of copolymer obtaining by two reasons: Firstly, some of their important service characteristics are expressed through the frac­ tions of these sequences. So, it is a common practice to use semiempirical relationships connecting the glass transition temperature of copolymers with the fractions of dyads in their macromolecules (Guillot and Emelie, 1991). Secondly, probabilities P{Uk} can be determined for k = 2–4 by spectroscopic methods with high degree of accuracy (Tonelli, 1989). The comparison of these experimental data with the results of mathematical modeling calculated in the framework of the kinetic model permits jud­ ging its adequacy for the description of the process under consideration. Along with the isomerism of linear copolymers due to various distri­ butions of different monomeric units in their chains, other kinds of iso­ merisms are known. They can appear even in homopolymer molecules, provided several fashions exist for a monomer to enter in the polymer chain in the course of the synthesis. So, asymmetric monomeric units can be coupled in macromolecules according to “head-to-tail” or “head-to­ head”—“tail-to-tail” type of arrangement. Apart from such a constitu­ tional isomerism, stereoisomerism can be also inherent to some of the polymers. Isomers can sometimes substantially vary in performance prop­ erties that should be taken into account when choosing the kinetic model. The principal types of such an account are analogous to those considered in the foregoing. The only distinction consists in more extended definition of possible states of a stochastic process of conventional movement along a polymer chain.

2.2 Microstructure parameters The microstructure of polymer chains is frequently more convenient to characterize by parameters representing some combinations of probabil­ ities P{Uk}, rather than by such probabilities themselves. As an example, considering a binary linear copolymer it is convenient to use as such a parameter the coefficient of microheterogeneity KM. For copolymers con­ sisting of long blocks, the value of KM is close to zero, whereas for regularly alternating copolymer KM = 2. Between these two extreme cases corre­ sponding to a perfectly ordered distribution of monomeric units in

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copolymer macromolecules, there fall all possible values of the coefficient of microheterogeneity. Value KM = 1 corresponds to the most disordered (random) sequence distribution, described by the Bernoullian stochastic process. Consequently, the modulus of the deviation of KM from unity allows one to quantitatively estimate the degree of ordering of the sequence distribution in chains of a binary copolymer. Essentially, the sign of this deviation testifies to the inclination of units either to regular alternation (if KM > 1) or to the formation of long blocks of both types of units (if KM < 1). The coefficient of microheterogeneity has been introduced for the description of the microstructure of binary copolymers with symmetric units (Korshak et al., 1976). At larger number of types of units and/or when the structure isomerism is taken into account, the role of KM will be played by other analogous parameters. A general strategy of the choice of these latter is developed in detail (Korolev and Kuchanov, 1986), while their values are measured by the nuclear magnetic resonance (NMR) spectroscopy technique for a number of polycondensation polymers (Vasnev et al., 1997). It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the prob­ abilities of any sequences {Uk}, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composi­ tion. As for the CD of the Markovian copolymers, for any fraction of l-mers it is described at l >> 1 by the normal Gaussian distribution with covar­ iance matrix, which is controlled along with l only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical model­ ing of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics.

2.3 Chemical correlators Along with the traditional manner of the description of the chemical structure of linear copolymers by means of the hierarchy of probabilities P{Uk} of sequences of units Uk(k = 1,2,…), there is one more mode of the description of this structure. It is based on the consideration of the hier­ archy of chemical correlation functions (so-called chemical correlators) (Kuchanov, 1978, 2000). The simplest among them, Y
 (k), has the mean­ ing of probability to find two randomly chosen monomer units of types
and  divided along a macromolecule by any sequence Uk consisting of k units. This two-point correlator is of the utmost importance because its generating function enters into the expression for spinodal (Kuchanov,

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1992), where the spatially homogeneous state of a polymer liquid becomes absolutely unstable. An arbitrary representer of the hierarchy is the n-point chemical correlator Y
1…
2(k1,…,kn1), which equals the probabil­ ity of finding in a macromolecule n monomeric units whose types are
1, …,
n divided by (n1) sequences Uk1,…,Ukn1 comprising k1,…,kn1 units, respectively (Panyukov and Kuchanov, 1992). Generating functions of such correlators play a key role in the theory of microphase separation in solutions and blends of block copolymers (Kuchanov, 2007). Therefore, the approach commonly accepted in statistical chemistry, which rests on the consideration of chemical correlators, permits a theoretical prediction of the dependence of the phase behavior of heteropolymer liquids on chemical structure of their macromolecules.

3. SPECIFICITY OF THE DESCRIPTION OF BRANCHED POLYMERS Only polymers of linear structure have been discussed in the foregoing. Nevertheless, a variety of polymers exist with other topological configura­ tions of macromolecules. The most commonly encountered among them are comblike, starlike, treelike, and network polymers. The feature pecu­ liar to all of them is the presence of branching units and adjoining polymer chains. Any molecule of such a polymer can be schematically presented as a molecular graph with vertices and edges corresponding to branching units and polymer chains, respectively. If these latter consist of monomeric units of a single type, they can be marked by ascribing to each edge the number equal to the length of corresponding chain. Every macromolecule of branched or cross-linked homopolymer is associated with a marked molecular graph, whereas a set of such graphs corresponds to a particular macroscopic specimen. Its exhaustive statistical description is performed by specifying the probability measure on the set of all marked graphs. For starlike and comblike polymers, the finding of the above-mentioned prob­ ability distribution involves no difficulties of principal character, since the topology of their molecules is fixed unlike the statistically branched treelike and cross-linked network polymers with random topology of molecules. When calculating average geometric sizes of macromolecules of such polymers, their optical properties, viscosity, and some other char­ acteristics, the necessity arises to make an averaging procedure over topological configurations of macromolecules.

3.1 Stochastic branching process The probability measure on the set of such configurations can be con­ structed for some classes of statistically branched polymers whose

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molecules do not contain cycles. Each of them is associated with the molecular graph referred to as tree, while to the whole polymer specimen the set of such trees, termed molecular forest, corresponds. The latter can be transformed into the forest of rooted trees which are obtained from molecular trees by sequential sampling of each of their vertex as a root. Such a transformation retains the probability measure, so that the prob­ ability distribution of rooted trees is the only thing which remains to be found. Each of them can be viewed, in turn, as a genealogical tree describ­ ing the history of some family. Besides, a rooted tree can be associated with some realization of stochastic branching process describing “birth and death” of particles. The simplest among them is the Galton–Watson process, in which the distribution of the probability for a particle to give birth to a fixed number of “children” particles is the same in every gen­ eration, being independent of other particles (Harris, 1963). Gordon (1962) was the first who discovered that for some polycondensation branched polymers (which below will be referred to as Gordonian polymers) the probability distribution of the rooted trees is described by the probability measure on the set of genealogical trees representing the realizations of the Galton–Watson branching process. The theory of these stochastic processes has been thoroughly developed (Harris, 1963), which makes it possible in a relatively simple manner to express any statistical character­ istics of a Gordonian polymer in terms of the probability parameters of the corresponding branching process (Kuchanov, 1978, 2000). The only thing which remains to be done consists in establishing the dependencies of these parameters on time, constants of elementary reactions, and com­ position of the initial monomer mixture. The equations for finding the above dependencies have been derived for a variety of Gordonian poly­ mers (Kuchanov, 1978). Clearly, a special role of the latter among ran­ domly branched polymers resembles that of Markovian copolymers among linear ones. A simple algorithm (Kuchanov et al., 1988) enables one to determine the probability of any fragments of macromolecules of the Gordonian polymers. Their comparison with the NMR spectroscopy data permits estimating the adequacy of the chosen kinetic model of the process of synthesis of a particular polymer specimen. These probabilities also enter in the expressions for the glass transition temperature and some structureadditive properties of randomly branched polymers (Chompff, 1971).

3.2 Gelation A special feature of a process of branched polymer formation is the possibility of the appearance in the reactor of a macroscopic structure whose size is comparable with that of the reactor. The main peculiarity of such a phenomenon, known as gelation, is that the range of conversions

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at which an abrupt increase of the viscosity occurs is rather narrow. That is why gelation can be regarded as a phenomenon proceeding at a certain moment, referred to as gel point. After this moment, the increase of the weight fraction of gel !G in the course of the process is accompanied by the decrease of the weight fraction !S of sol molecules. The prime objectives of the mathematical modeling of branched polymerization and polyconden­ sation as well as of the processes of cross-linking of macromolecules are the calculation of the gel point and the determination of the dependence of !G = 1!S on conversion. If monomeric units of several types are involved, analogous dependencies for compositions of sol and gel should be additionally calculated. The gel is a polymer network normally exhibit­ ing elastic properties. According to the theory (Flory, 1976), the elasticity modulus of this network is governed by the cyclic rank of its molecular graph. This characteristic of a graph is equal, by definition, to the minimal number of its edges to be deleted in order to transform this cyclic graph into treelike one. Obviously, calculation of the cyclic rank, pertaining to the most essential characteristics of the topological structure of a polymer network, is among the most challenging problems of the mathematical modeling of network polymers. Its solution, along with that of the abovementioned problems of calculation of the statistical characteristics of the Gordonian polymers, can be found by means of the mathematical appa­ ratus of the theory of branching processes.

4. KINETIC MODELS OF MACROMOLECULAR REACTIONS When deriving a material balance equation, the rate of transformation of each component in a reactor is normally governed by the mass action law. However, unlike for the reactions in which only low molecular weight substances are involved, the number of such components in a polymer system and, consequently, the number of the corresponding kinetic equa­ tions describing their evolution are enormous. The same can be said about the number of the rate constants of the reactions between individual components. The calculation of such a system becomes feasible because certain general principle can be invoked under the description of the kinetics of the majority of macromolecular reactions. Let us discuss this principle in detail.

4.1 Ideal kinetic model The processes of the formation and transformation of polymer molecules proceed as a result of chemical reactions of their active centers, whose role can be played by functional groups, free valences in radicals, double bonds, and so on. Often, it may be suggested that the reactivity of the

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active center in a polymer molecule is controlled neither by its configura­ tion (i.e., molecular weight, composition, structure) nor by the location of this center in the molecule. This fundamental principle, advanced first by Flory (1953) and bearing his name, permits considering the rate constant of the elementary reaction of any active centers of given types as being the same. That is why the number of such elementary reactions in a particular process is usually small enough, which enables it to be characterized just by a few kinetic constants. So, the process of radical polymerization is described by the rate constants of elementary reactions of initiation, pro­ pagation, and termination of a chain. The Flory principle makes it possible in a simple way to relate the rate constants of the reactions of macromolecules (whose number is infinite) with the corresponding rate constants of elementary reactions. Since according to this principle all chemically identical active centers are kine­ tically indistinguishable, the rate constant of the reaction between any two molecules is proportional to the rate constant of the reaction between their active centers and numbers of these centers in reacting molecules. There­ fore, only a few rate constants of elementary reactions will enter in the material balance equations as kinetic parameters. The Flory principle is one of two main assumptions underlying the ideal kinetic model of any processes of synthesis and chemical modification of polymers. The second assumption is the neglect of the reaction between any active centers belonging to the same molecule. Clearly, in the absence of such intramolecular reactions, molecular graphs of all components of a reaction system will not contain cycles. The last affirmation applies just to sol molecules. As for the gel, in the framework of the ideal model, the cyclization reaction is admissible.

4.2 Models allowing for the deviations from ideality The Flory principle, whose validity has been established for a wide range of polyreactions, is rather good approximation for the description of the kinetics of many processes of polymer obtaining. However, a considerable body of experimental results is currently available concerning a number of macromolecular reactions where this principle is deliberately violated. Possible reasons for the deviations from this principle can be attributed to the short- or long-range effects (Kuchanov, 2000). Among the first of them are the so-called substitution effects due to steric, induction, catalytic, or some other types of influence of the reacted active centers on the reactivity of neighboring unreacted centers. In order to take account of such short-range effects it has been suggested (Kuchanov, 1978) to use under mathematical modeling an extended Flory principle. In line with this principle, the reactivity of any active center of a molecule is supposed to be controlled exclusively by local chemical structure of the

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fragment involving this center and changes in the course of the process due to the alteration of the local structure caused by the entry in the reaction of the neighboring active centers. The effect of other factors on the reactivity therewith may well be neglected. In the framework of the models of such a kind, the fragments of molecules comprising the active centers should be chosen as kinetically independent elements rather than these centers themselves, as it is normally done in case of the ideal model. Under such an approach employed for the mathematical modeling of the processes of copolymerization, co-polycondensation, and polymer-analogous transformations, the number of kinetic parameters characterizing the model chosen is rather small. The polymer nature of reagents is the most strongly pronounced in the long-range effects at which the reactivity of the active center of a macro­ molecule is acted upon by the fragments situated in all its parts. An example is intramolecular catalysis by functional groups spaced apart from the active center in a macromolecule but falling in the vicinity of this center as a result of spatial conformational rearrangements of the polymer chain (Plate et al., 1995). In the presence of such effects, the reactivity of its terminal active center can be controlled by the degree of polymerization of this macromolecule and the distribution along polymer chain of units containing catalytic groups. In case of fast reactions, like recombination of macroradicals, the dependence of the rate constant on their lengths owes its existence to the diffusion factors (Allen and Patrick, 1974). Sometimes thermodynamic reasons can be responsible for the appearance of the long-range effects in the course of macromolecular reactions. An example is the appearance at the initial stages of copolymer­ ization of the dependence of the rate of the growth of a macroradical on its length and composition. The origin of this phenomenon is connected with the fact that the monomer mixture composition differs inside and outside a macroradical due to the preferential sorption of monomers of different types (Harwood, 1987; Semchikov, 1996). The allowance for long-range effects in choosing a kinetic model of the processes of polymer synthesis, complicating to certain extent the calculations, proves to be sometimes indispensable for the treatment of experimental data.

5. METHODS OF CALCULATIONS Calculating the statistical characteristics of the primary structure of macro­ molecules, a researcher usually faces the problems of finding their average molecular weight, composition, molecular weight distribution (MWD), and SCD, as well as the characteristics of their chemical structure. To tackle these problems, two different approaches, namely, kinetic and statistical, are generally invoked. The first consists in deriving and solving the

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material balance equations for the concentrations of molecules of all types involved in the process under consideration. Under the second approach, the most general formulation of which has been proposed in a monograph by Kuchanov (1978), every macromolecule is conceived as an individual realization of a particular stochastic process of conventional movement along polymer chain. The probability of this realization is equal to the fraction of corresponding molecules in the reactor. In the framework of such a method, for the calculation of the statistical characteristics of poly­ mers the averaging procedure is performed over realizations of the corre­ sponding stochastic process instead of averaging over macromolecules. Methods of such an averaging for some of these processes are perfectly developed, which makes it possible to find the required characteristics of the Markovian and Gordonian polymers in a relatively simple way.

5.1 Statistical method Both statistical and kinetic methods of calculation of the parameters of a polymer’s primary structure naturally have advantages as well as short­ comings. Nobel Prize winner Paul Flory was the first who, as early as in late 1930s, proposed (Flory, 1953) to resort to the statistical method for the calculation of the kinetics of the reactions with participation of macromo­ lecules. This method has found an extensive application in the quantitative description of a variety of particular processes of obtaining and chemical modification of polymers. An indisputable advantage of the statistical method is its ability to exhaustively describe in a straightforward manner a detailed structure of macromolecules in terms of few probability para­ meters. However, the issue of specifying appropriate stochastic processes for the quantitative description of the products of a particular process cannot in principle be settled in the framework of the statistical method itself, whose application in all its modifications is, in essence, of a formal character. Expressions for the statistical characteristics of polymers there­ with are traditionally obtained by speculative probabilistic reasoning, whose correctness is predetermined to a great extent by the author’s intuition. Naturally, adhering to such a formal statistical approach, a scientist cannot rigorously establish a strict correspondence between the kinetic model of the process of polymer synthesis and the type of the stochastic process adequately describing the statistical characteristics of macromolecules formed. This, generally speaking, makes the issue of the areas of applicability of the statistical method an open question. Only the kinetic approach provides an answer to this question, enabling one to simultaneously express the probability parameters of this stochastic process through kinetic constants, reagents’ concentrations, and other variables describing the process at hand.

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With this in mind, a general strategy has been developed (Kuchanov, 1978, 2000) implying a rigorous (in the framework of the models generally accepted today in polymer chemistry) substantiation of the statistical method for the calculation of different classes of polymerization processes proceeding from their kinetic consideration. Expressions for the distribu­ tion of macromolecules in size and composition, arrived at as a result of such a consideration, are compared with the analogous expressions derived in the framework of the theory of stochastic processes. The coin­ cidence of corresponding distributions argues for the applicability of the chosen variant of the statistical approach. Under such a comparison, the dependencies of the probability parameters of the stochastic process on stoichiometric and kinetic parameters of a reaction system are simulta­ neously revealed. At the next stage, the possibility appears to calculate in the framework of the statistical approach necessary statistical character­ istics controlled by the chemical structure of polymer molecules. This general strategy has been successfully applied under mathematical mod­ eling of different polymers obtained by free-radical copolymerization (Kuchanov, 1992, 2000), polycondensation (Kuchanov, 2000; Kuchanov et al., 2004), and chemical modification of macromolecules (Kuchanov, 1996, 2000).

5.2 Kinetic method The kinetic method turns out to be especially efficient in the mathematical modeling of the processes of obtaining of polymers describable by the ideal kinetic model. In this case, the material balance equations for the concentrations of polymer molecules containing fixed numbers of mono­ meric units and active centers often can be integrated analytically using the method of generating functions (Kuchanov, 1978). Such a function is completely equivalent to the distribution of concentrations of molecules in size, composition, and functionality (SCF distribution), which can be obtained as coefficients of the series expansion of the generating function. This proves to be especially convenient for the calculation of the statistical moments of the SCF distribution, which are expressed through the deri­ vatives of the generating function at a single point when the values of all its arguments are unity. An infinite set of ordinary differential equations for the distribution of concentrations of polymer molecules in case of the ideal kinetic model is reduced to one equation for the generating function. It will be ordinary differential equation or the first-order partial differen­ tial equation, depending on whether linear or branched polymers are formed in the course of the process of interest. Along with the rate constants of elementary reactions, the coefficients of this equation are controlled by average concentrations of the active centers and low molecular weight reagents. They can be found from the solution of a set of

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differential equations, which is always closed under the applicability of the ideal kinetic model since the elementary reactions can be con­ sidered in its framework separately from polymeric reactions. In many cases of practical importance, a partial differential equation for the gen­ erating function is possible to integrate analytically using the method of characteristics. When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the prob­ ability of arbitrary sequences {Uk} in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively.

5.3 Extension of statistical and kinetic methods These important properties are not peculiar to polymers synthesized in reaction systems, which are describable by nonideal kinetic models. Never­ theless, for some of them, in which the deviations from ideality are due to the short-range effects, the modern approaches of the statistical chemistry permit formulating a rather general algorithm by means of which any characteristics of the chemical structure of such polymers can be calcu­ lated. Underlying to this algorithm is the Flory principle in its extended formulation that makes possible to write down kinetic equations for the reagents in closed form. Using these equations it is usually possible to substantiate rigorously the correctness of the recourse to a particular version of the statistical method in order to provide an exhaustive descrip­ tion of the chemical structure of polymer products. Noteworthy, the stochastic processes engaged in such a description are characterized by the states having some additional attributes in comparison with those peculiar to the states of Markovian or Gordonian polymers. For example, under kinetic modeling of “living” anionic copolymer­ ization in the framework of the terminal model, a macromolecule is associated with the realization of a certain stochastic process. Its states (
,) are monomeric units, each being characterized along with chemical type
and also by some label . This random quantity equals the moment when this monomeric unit entered in a polymer chain as a result of the addition of
-type monomer to the terminal active center. It has been

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rigorously shown (Kuchanov et al., 2002) that a stochastic process of conventional movement along macromolecules whose units are labeled in such a way is Markovian. Besides, the expressions for the transition probabilities between successive states of this process have also been derived there. This makes possible in principle to find any statistical characteristics of the chemical structure of macromolecules with labeled units. Once these characteristics are found, they should be averaged over the probability distribution of random quantity , which can be formally interpreted as “erasing” of labels. As a result, statistical characteristics of real copolymers with non-Markovian statistics of the succession of units in macromolecules can be found (Kuchanov et al., 2002). The above-described “labeling-erasing” procedure is in common use in statistical chemistry of polymers (Kuchanov, 2000). It gives a chance to obtain a number of important theoretical results under kinetic modeling of polymerization and polycondensation processes, where the deviation from their description in terms of the ideal kinetic model is due to the short-range effects. A different situation arises when in the kinetic model any long-range effects should be taken into account. As a rule, no results can be achieved here analytically. As for the equations for the statistical moments of SC distribution and for the fractions of the fragments of macromolecules, they are not closed as it takes place for the ideal model. Hence, finding of these statistical characteristics implies the necessity of a numerical solution of the material balance equations for concentrations of molecules with fixed numbers of monomeric units and active centers. Here the problem of the solution of an infinite set of equations arises. To escape this problem one is supposed to switch from discrete variables characterizing the size and composition of macromolecules to continuous variables. Under such a transition, the mathematical problem is reduced to the solution of a single integro-differential equation containing partial derivatives. Of considerable promise for mathematical modeling of macromolecu­ lar reaction is the Monte Carlo method (Lowry, 1970). It consists in modeling the dynamics of any particular process of polymers’ obtaining by means of statistical tests, which obey the probability laws correspond­ ing to the kinetic model chosen. Subsequent averaging over all computersimulated realizations of this stochastic process enables one, in principle, to find any statistical characteristics of synthesized polymers. The Monte Carlo method is particularly effective when polymers are not Markovian or Gordonian. This method makes possible a direct computer simulation of particular macromolecular reactions, avoiding thus the derivation and solution of corresponding kinetic equations which can be either too com­ plicated or even cannot be derived at all. The method of computer simula­ tions has already found its application for calculations of the statistical characteristics of the products of some processes of polymer synthesis and

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chemical modification (Lowry, 1970; Motoc and O’Driscoll, 1981; Plate et al., 1995). However, its potentialities in this area are still far from being exhausted.

6. SOME GENERAL THEORETICAL RESULTS In this section, some general results are reported of the theoretical con­ sideration of the main processes of polymer synthesis in the framework of different kinetic models. This information could be of assistance to an engineer-researcher in making a scientifically grounded choice of such a model.

6.1 Polycondensation Functional groups play here the role of active centers. If their number in each monomer is two, only linear macromolecules will be obtained as a result of the synthesis. In order to prepare branched and network poly­ mers, among monomers involved those having three and more functional groups should be present. The quantitative theory of polycondensation processes describable by the ideal kinetic model may be thought of as completed (Kuchanov et al., 2004). So, it has been found (Kuchanov, 1976) that the sequence distribution in the products of joint polycondensation of any set of bifunctional mono­ mers is described by a certain Markov chain. Its parameters are related in a known way with the reactivity of the functional groups and the initial stoichiometry of monomers. As for branched polymers, a branching process has been developed and rigorously kinetically substantiated (Korolev et al., 1981; Kuchanov, 1978) to describe the products of co-polycondensation of arbitrary mixture of monomers, each comprising any numbers of functional groups of arbitrary types. Every type of reproducing particles of this branching process is associated with a particular type of reacted functional groups. These results have been extended (Kuchanov, 1987) to arbitrary multistage processes in the course of which the reactive oligomers first form, cross-linking then to generate a network polymer. Here the statistical characteristics of both intermediate and final products turn out to be describable by the formulas of branching process with properly chosen probability parameters. Being universal, this fundamental property mani­ fests itself in polycondensation processes conducted in any number of steps with possible arbitrary alteration of temperature and/or catalyst’s concen­ tration during the cross-linking stage (Kuchanov et al., 2004, 1987). Many service properties of the products of cross-linking of reactive oligomers are essentially controlled by their distribution in functionality types (Entelis et al., 1989). This important statistical characteristic of

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forepolymers obtained in the first stage can be found for polycond­ ensation of any monomer mixture by known formulas (Korolev et al., 1981; Kuchanov, 1976). The statistical method allows a mathematical modeling of branched polycondensation describable by the ideal kinetic model not only before the gel point but also after it. A complete statistical description of sol molecules is performed by the same branching process as before the gelation point but with the values of the probability parameters renorma­ lized in a certain way (Korolev et al., 1981; Kuchanov, 1978). As for gel, not only local characteristics of its topological structure can be calculated but also global ones. Among these latter is the cyclic rank of the molecular graph of polymer network, whose value controls its elasticity properties (Kuchanov, 1987). The quantitative theory of branched polycondensation enables one, in particular, to predict the moment of gelation, to describe the evolution of compositions of sol and gel, as well as to calculate in each of them the conversions of functional groups of all types (Korolev et al., 1981). The above reasoning allows a conclusion that once a researcher has decided upon the particular ideal kinetic model of polycondensation, he or she will be able to readily calculate any statistical characteristics of its products. The only thing he or she is supposed to do is to find the solution of a set of several ordinary differential equations for the concentrations of functional groups, using then the expressions known from literature. Addressing now nonideal kinetic models, it should be emphasized that for those of them which allow for the short-range effects, the quantitative theory of polycondensation is developed reasonably well (Kuchanov et al., 2004). The most typical manifestation of such effects is the change in the reactivity of the second functional group in a bifunctional monomer caused by the entry in the reaction of the first group. This “substitution effect” is known to be peculiar to many aromatic monomers (Allen, 1988; Sokolov, 1979). It has been theoretically established (Kuchanov, 1979) that the participation of such monomers in linear co-polycondensation can result in the formation of non-Markovian copolymers. A criterion has been formulated (Kuchanov, 1979) for deciding whether the substitution effect in some monomers violates the Markovian statistics of the succes­ sion of units along polymer chains. An analogous criterion has been suggested (Kuchanov, 1979) for mathematical modeling of a branched polycondensation. It permits specifying the conditions under which the polymers formed will be Gordonian despite the involvement of monomers with kinetically dependent groups. Mathematical modeling in this case is performed in such a simple manner as that for the ideal model. Somewhat more complicated is the modeling of a branched polycon­ densation of monomers with kinetically dependent groups, the products of which represent Gordonian polymers (Kuchanov et al., 2004). Their

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chemical structure can be exhaustively described by means of the “labeling­ erasing” procedure based on the results of work by Kuchanov and Zharnikov (2003). Those authors, when writing down the material bal­ ance equations for polymer molecules in line with the extended Flory principle, have chosen the monads (i.e., monomeric units with adjacent functional groups) as kinetically independent elements. At such a level of detail, a polymer molecule is completely characterized by the numbers of constituting monads of different kinds. The analysis of the equation for the generating function of distribution of molecules in a number of monads brought Kuchanov and Zharnikov (2003) to the conclusion about the existence of a general branching process (Harris, 1963), which permits finding any characteristics of the products of a polycondensation of monomers with kinetically dependent groups. Reproducing particles of this process is associated with the reacted functional groups, each being supplied by a pair of labels, discrete and continuous ones. The first of them denotes the kind of a monad incorporating the functional group under consideration before its entry into a condensation reaction. The second label designates a moment at which this reaction occurred. Having performed these labels’ “erasing,” one can find, in particular, the dependence of statistical characteristics of the polymer network of a gel responsible for its elasticity properties on the conversion of functional groups (Kuchanov and Zharnikov, 2003). Further progress of a general theory of branched polycondensation of monomers with kinetically dependent groups is connected with allowance for the effect of monomer configuration on the change of their reactivity due to the entry of the neighboring groups into a reaction. An account of this factor, first realized in Kuchanov et al. (2006), calls for the extension of the traditional kinetic model of the “substitution effect.” In this publica­ tion, the validity of the statistical method for the complete description of the chemical structure of the products of branched polycondensation proceeding under the applicability of the above-mentioned model has been rigorously substantiated. The recourse to this method enables the allowance for the configurational effects when calculating any statistical characteristics of sol and gel (Kuchanov et al., 2006).

6.2 Conventional free-radical copolymerization The active centers in this process are free radicals, whose reaction with double bonds of monomers leads to the growth of a polymer chain. In the framework of the ideal kinetic model, the reactivity of a macroradical is exclusively governed by the type of its terminal unit. According to this model, the sequence distribution in macromolecules formed at any moment is described by the Markov chain with elements controlled by the instantaneous composition of the monomer mixture in the reactor as

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well as by the reactivity ratios whose values were tabulated for hundreds of particular monomeric pairs (Eastmond and Smith, 1976; Greenley, 1980). Since the above composition changes during the synthesis, the final products certainly will not be Markovian copolymers representing a mixture of such copolymers formed at different moments of this process. The quantitative theory of copolymerization of an arbitrary number of monomers, describable by the ideal model, is thoroughly elaborated (Kuchanov, 1992). It has been shown (Kuchanov, 1992; Kuchanov et al., 1989; Yakovlev and Kuchanov, 2000) how its application provides the possibility to predict by means of mathematical modeling the transpar­ ency and thermostability of industrial terpolymers. The allowance for the short-range effects has been carried out in two types of kinetic models (Kuchanov, 1992). In the first of them, the reactiv­ ity of a macroradical is presumed to be dependent on the types of n monomeric units preceding the terminal one. Here the mathematical formalism differs from that used in the case of the ideal model only in one point. The states of the Markov chain are associated in the framework of these models with monomeric units, each supplied by the label specify­ ing the type of sequence {Un} of units acting upon the reactivity of the active center. The second type of nonideal models takes into account the possible formation of donor–acceptor complexes between monomers. Essentially, along with individual entry of these latter into a polymer chain, the possibility arises for their addition to this chain as a binary complex. A theoretical analysis of copolymerization in the framework of this model revealed (Korolev and Kuchanov, 1982) that the statistics of the succession of units in macromolecules is not Markovian even at fixed monomer mixture composition in a reactor. Nevertheless, an approach based on the “labeling-erasing” procedure has been developed (Kuchanov et al., 1984), enabling the calculation of any statistical characteristics of such nonMarkovian copolymers. Under copolymerization of some monomers, a number of anomalies have been experimentally revealed (Semchikov et al., 1990a, 1990b, 1996) which cannot be explained in the framework of the above-discussed traditional kinetic models. Among such anomalies are the dependence of a copolymer composition on its molecular weight as well as a strongly pronounced intramolecular inhomogeneity of polymer chains. In order to provide an explanation for these anomalies, a new model taking into account the phenomenon of the preferential sorption of monomers by growing macroradicals has been put forward (Kuchanov and Russo, 1997). Evidently, the monomer mixture composition inside each of them differs, generally speaking, from its value in bulk of the reactor because of the physical interactions of monomers with monomeric units of a polymer chain. That is why the monomer mixture composition inside

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a macroradical will be controlled by its composition, i.e., the rate of the monomers’ addition to the active center will be predetermined not only by its neighboring units but also by all units of a polymer chain. The allow­ ance for such long-range effects in the kinetic model of copolymerization permits the explanation of the anomalies observed and, in particular, the intramolecular inhomogeneity of polymer products (Kuchanov and Pogodin, 2008; Kuchanov and Russo, 1997). It should be emphasized that performing mathematical modeling of a copolymerization in systems where the preferential sorption of monomers plays a decisive role, it is not enough to confine the consideration to the kinetic equations describing the growth of macroradicals. They should be complemented with thermodynamic expressions establishing the depen­ dence of monomers’ concentrations in the vicinity of the active center of a macroradical on its chemical size and composition, as well as on concen­ trations of monomers in bulk of the reactor. The joint solution of the equations of chemical kinetics and interphase equilibrium provides the possibility to calculate both the rate of polymerization and the character­ istics of the molecular structure of the copolymer. The results of such a calculation (Kuchanov and Pogodin, 2008; Kuchanov and Russo, 1997) testify to the efficiency of combining the approaches of statistical chemis­ try and thermodynamics of polymers to tackle the problems of utmost practical importance.

6.3 Chemical modification of polymers Potentialities of the statistical chemistry for the description of such pro­ cesses can be illustrated by consideration of polymer-analogous reactions (PARs) of a linear homopolymer (Plate et al., 1995) whose macromolecules comprise in each monomeric unit A a reactive functional group. Its reac­ tion with a low molecular weight compound results in transformation of unit A into unit B. Successive transformations of such a kind lead to the formation in a reaction system of heteropolymer macromolecules varying in composition and in pattern of arrangement of A and B units. As an example of PARs, we might mention the esterification of polymethacrylic acid and saponification of polyvinyl acetate. In the course of the first of these reactions, carboxyl groups transform into ester ones, whereas for the second reaction acetate groups transform into hydroxyl ones. If the rate constant k of the elementary reaction of transformation A ! B is supposed to be the same for all groups, the pattern of arrange­ ment of units in macromolecules will be perfectly random. However, such an ideal kinetic model is not appropriate for a vast majority of real polymers because of the necessity to take into consideration under mathematical modeling of PARs proceeding in their macromolecules the short-range and long-range effects. The easiest way to take account

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of the first of them is to resort to the “neighboring-group” (NG) model (Kuchanov, 1978; Plate et al., 1995), according to which the rate constant of an elementary reaction of an arbitrary functional group depends solely on the types of two adjacent monomeric units. This model is characterized by three rate constants k0,k1,k2 of elementary reaction of the groups in unit A whose nearest neighbors are, respectively, 0,1,2 units B. Kinetically inde­ pendent fragments here are triads of units. Surprisingly, even such sim­ plest model invokes for its solution a rather sophisticated mathematical apparatus (Kuchanov, 1996). Difficulties emerging thereby originate from the non-Markovian character of the distribution of monomeric units along macromolecules. However, these difficulties have been successfully over­ come, which made possible to solve rigorously in the framework of the NG model the problems of finding both the CD of heteropolymers formed in the course of PAR (Brun et al., 1982; Flory, 1953; Kuchanov and Brun, 1976, 1983) and the structure of their macromolecules (Brun and Kuchanov, 1977; Flory, 1953; Kuchanov and Aliev, 1997; Noa et al., 1973; Plate et al., 1974). A detailed description of the quantitative theory of PAR is pre­ sented in the review articles (Ewans, 1993; Kuchanov, 1996). The statistical characteristics of the products of PAR are significantly affected by the thermodynamic quality of the solvent with respect to the polymer. This influence is especially strongly pronounced in dilute solu­ tions, where every macromolecule can be considered individually. The above-outlined theory of PAR is relevant to the case of a thermody­ namically good solvent, when macromolecules are in coil-conformational state (Grossberg and Khokhlov, 1994). Here the concentration of a low molecular weight reagent z in the vicinity of reactive centers (i.e., func­ tional groups) coincides with the concentration of this reagent throughout the reactor. A completely different situation takes place when PAR proceeds in moderately poor solvent. In such systems, macromolecules are in a con­ formational state of globules inside which the density of units is not high enough to hamper the diffusion of reagent z into the globular nanoreactor. At the same time, this density appreciably exceeds the density of units in the polymer coil. This can be responsible for the strongly pronounced influence of preferential sorption of reagent z on the kinetics of PAR (Kuchanov et al., 2008). As the thermodynamic quality of the solvent deteriorates, the density of monomeric units in a globule increases, leading to hindering of the diffusion of low molecular weight reagent inside nanoreactors. This density can become so high in a poor solvent that PAR will proceed in the diffusion-controlled regime instead of a kineti­ cally controlled one. In line with theoretical analysis (Kuchanov and Khokhlov, 2003; Kuchanov et al., 2003), the chemical structure of macro­ molecules obtained in nanoreactors in these two regimes of PAR will substantially differ even under the applicability of the ideal kinetic

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model. Thus, the chemical structure of heteropolymers formed in a diffu­ sion-controlled regime is described by a rather sophisticated stochastic process of conventional movement along macromolecules, which resem­ bles in a manner that describing the chemical structure of some biological macromolecules (Kuchanov and Khokhlov, 2003).

7. EFFECT OF HYDRODYNAMIC STIRRING IN A REACTOR ON SOME SERVICE PROPERTIES OF POLYMER PRODUCTS 7.1 General discussion It has been experimentally established that the mechanical and physico­ chemical properties of the products of the synthesis or chemical modifica­ tion of polymers obtained by batch and continuous methods can markedly differ despite the similarity of temperature and hydrodynamic regimes. Essentially, the character of hydrodynamic stirring in a continuous reactor can substantially affect the service properties of polymer products pre­ pared. That is why designing continuous polymer processes an engineer is supposed to perform an in-depth examination of the dependence of poly­ mer product quality on the regime of stirring in a reactor. The reason is that the properties of commercial polymers can essentially differ from those of polymers synthesized on the laboratory scale. This section is devoted to the consideration of some general aspects of the above problem as applied to three main classes of the processes of obtaining of polymeric synthetic materials, i.e., polymerization, polycondensation, and chemical modification of polymers. The kinetic peculiarities of macromolecular reactions in open systems controlling the above-mentioned dependencies are still not sufficiently studied. It especially concerns two last classes of the processes. In the meantime, the importance of the investigation of the kinetics of the reactions in open systems for chemical technology has been outlined (see, for instance, Beisenberger and Sebastian, 1983; Dotson et al., 1996; Emanuel, 1979; Meyer and Keurentjes, 2005). As noted in the foregoing, any grade of synthetic polymer represents practically an infinite set of different individual compounds whose macro­ molecules can vary in the degree of polymerization (chemical size) l, composition, and chemical structure (Kuchanov, 1978, 2000). Therefore the performance parameters governing the quality of a polymer specimen are always determined as a result of certain averaging of some statistical characteristics of macromolecules over this set. The probability measure over which this averaging is performed depends on the preparation con­ ditions of given polymer specimen being controlled by kinetic, diffusion, hydrodynamic, and other factors exerting influence on macrokinetic con­ ditions in a reactor. Thus, in order to connect them with the quality of

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polymer obtained, one should do the following. First, it is necessary to reveal the correlation between the performance parameters characterizing the quality and corresponding statistical characteristics of a specimen, and, second, using kinetic calculations to express these characteristics through the parameters of a reaction system. In the present state of the art of polymer physics, an exhaustive solu­ tion of the first of these two problems has not been found so far, although some attempts in this direction have been undertaken. At the same time, an intensive development of production of polymers puts forward demands of the prediction of their properties. To cope with this task chemical engineers generally use in practice some simple semi-empirical correlations. In doing so, they resort to certain qualitative theoretical approaches to treat the available experimental data. According to the most reputable adherent of this method, van Krevelen, such semi-empiri­ cal correlations are highly effective and provide rather reliable results in most practically important cases (van Krevelen and te Nijenhuis, 2009). However, even in the framework of the above approach, considerable difficulties are encountered, because often there is no clear idea about which specific statistical characteristics of a polymer are responsible for a particular mechanical and physicochemical property. It especially con­ cerns copolymers because the number of their characteristics of such a kind is larger than that for homopolymers. Evidently, it would be pertinent to discuss the hierarchy of different statistical characteristics of a polymer proceeding from the degree of their influence on each of its properties. Belonging to the first hierarchy level are such characteristics as average degree of polymerization and (in case of copolymers) average composition of a polymer specimen. On the next hierarchy level, the subject of consideration is the effect on the properties of the manner of distribution of macromolecules in a specimen in their chemical size and (for copolymers) in composition at fixed average values of such quantities. It is precisely this factor that plays a key role in the prediction of the quality of a polymer when switching from batch to continuous process of its production. Indeed, by varying the conditions of polymerization (reagent concentrations, temperature, catalyst, and so on), it is possible to select the regimes under which the average character­ istics of the products will be identical in both processes. Nevertheless, the character of MWD in homopolymers, as well as the CD of macro­ molecules of copolymers in their composition can essentially differ for the above-discussed two methods, depending on the regime of hydro­ dynamic stirring in a reactor. Changing this regime, for example, by varying the number of steps of stirring by means of the apparatus sectioning or by switching to a cascade reactor, an engineer is able to control the quality of polymer products retaining unaltered their average characteristics.

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All commercial specimens of synthetic polymers are characterized by a certain degree of inhomogeneity of macromolecules in size and composi­ tion, which predetermines the quality of a polymer. That is why it is necessary to analyze the factors responsible for the inhomogeneity of a specimen appearing in the course of its synthesis. These factors are largely controlled by the type of chemical reactions, the process mode, the char­ acter of hydrodynamic stirring, and (in case of heterophase systems) also by macrokinetic parameters. The analysis of heterophase systems is a far more complex issue because no general regularities exist for their model­ ing, and, consequently, every particular case calls for a special theoretical consideration. Therefore, it is expedient to invoke a structure multilevel approach by which, at every hierarchy level, the factors are revealed responsible for the degree of molecular inhomogeneity of a polymer. Having realized a sequential estimation of the contribution of each level to the overall inhomogeneity of a specimen, one can specify by calcula­ tions the main factors, leaving apart those of less importance. An example of practical implementation of such an approach for the mathematical modeling of the commercial process of chlorination of polyethylene with allowance for the quality of the product obtained is the investigation reported by Brun and Kuchanov (1980). However, as the analysis of macrokinetic factors is beyond the scope of this chapter, the subsequent consideration of the continuous process will be focused mainly on homo­ geneous systems. The commonly known advantages of continuous commercial pro­ cesses over batch processes are basically peculiar to the polymer-obtaining technology as well, although some specific problems arise in the latter case. One of the most important of them is the effect of hydrodynamic stirring on the quality of polymers. In a plug flow reactor (PFR), where there is no stirring at all, the regularities of a periodic process with the same residence time  are realized. Another limiting hydrodynamic regime takes place in a contin­ uous stirred tank reactor (CSTR), characterized by exponential distribu­ tion function of the residence time. In any real apparatus, which is intermediate between these two extreme types of reactors, the degree of hydrodynamic stirring is normally characterized by the number of steps of ideal stirring, n, in a cell model or by the values of the effective Peclet number Pe
in the framework of the diffusion model. If the degree of stirring is low enough, both these models yield the same results at n = Pe
/2 >> 1 (Aris, 1961). It turns out that even slightly pronounced stirring in a reactor can be responsible for a dramatic change of polymer’s inhomogeneity. This should be necessarily taken into account when switching from periodic processes to continuous ones. The pioneer investigator of polymerization in continuous-flow systems was Denbigh (1947). He formulated (Denbigh, 1951) the conditions

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enabling one to distinguish two limiting types of systems differing in character of the effect of the regime of hydrodynamic stirring on the MWD of the polymer formed. Belonging to the first type are systems in which the characteristic time of the formation of a macromolecule tf being in active state is large enough as compared to its average residence time  in a reactor. The values of these two quantities are controlled by different factors, namely tf depending on kinetics while  being governed by hydro­ dynamic stirring, so that the ratio between them can be arbitrary. A typical example of the first type of systems is living anionic polymerization, characterized by fast initiation and sufficiently slow chain termination. For such a polymerization, inequality tf >>  holds. In systems of the second type, for example, in conventional radical polymerization, oppo­ site ratio between the rates of chain initiation and chain termination takes place, so that inequality tf <<  is true. These two types of systems are characterized by qualitatively different effects of hydrodynamic stirring in a reactor on the width of MWD of the polymer synthesized. In the first type of systems, on going from PFR to CSTR, the MWD widening occurs, unlike in the second type of systems, where the MWD narrowing happens. In this case, a very narrow Poisson distribution of the products of living anionic polymerization with the value of polydispersity coefficient K close to unity is replaced by the exponential Flory distribution with K = 2. Conversely, under the same change of the regime of hydrodynamic stir­ ring, the rather wide MWD of a polymer with K essentially larger than 2 formed at high conversions during radical polymerization switches to narrower distribution with the polydispersity coefficient in the interval between 1.5 and 2. Quantity K is found in this case by formula K = 0.5 (1 + )(3), where  and(1) are, respectively, fractions of radicals, terminating by disproportionation and recombination mechanisms. The reason for increase in the polydispersity of the products of living anionic polymerization conducted in the perfect stirring regime instead of the plug flow regime is the variation in the residence time of macromole­ cules in a reactor. The wider is the distribution of the residence time (DRT) of macromolecules, the larger is the dispersion of MWD of a polymer. Besides, since for living anionic polymerization, the contribution to poly­ dispersity connected with the random character of the monomer addition is negligible, the appearance of MWD of macromolecules obtained in a reactor will be exactly described by the function of DRT of this reactor.

7.2 Radical polymerization Quite different factors exert an influence on the polydispersity of the products of radical polymerization, where the formation of any macro­ molecule is so fast that it practically has no time for moving in space over a distance sufficient for a considerable change in monomer concentration M.

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So, it is conceivable that in analogous second type of systems, in each elementary reactor’s volume dV with particular value of M, the formation occurs of macroscopic number dN of polymer molecules with some local MWD. The resulting MWD of final products represents the superposition of such local distributions of macromolecules formed all over the reaction volume which leave the reactor simultaneously. If there is certain distribu­ tion of concentration of monomer M along PFR, macromolecules leaving the reactor at every instant are formed at different values of M at different points of the reactor. This can lead to an essential polydispersity of a product. In case of CSTR, the gradient of monomer concentration in the reactor is absent; therefore, all polymer molecules are formed under the same conditions. This circumstance enables the exclusion of one of the reasons of the appearance of polydispersity due to the inhomogeneity of the formation conditions. In this case coefficient K drops to value K = 0.5 (1 + )(3) which is minimal for given mechanism of the reaction. Sometimes, for technology reasons, a radical polymerization should be conducted in PFR. However, this option inevitably entails a pronounced polydispersity of the products, for avoiding which it is necessary to find the ways for increasing time tf of the formation of macromolecules. This can be realized by performing a polymerization, for example, in the presence of polyfunctional initiators whose molecules comprise several peroxide groups. A distinctive feature of the mechanism of polymerization in the presence of such polyinitiators is the appearance of macromolecules con­ taining peroxide groups in the main chain in the course of the process. Under the cleavage of these groups, macromolecules transform in macroradicals capable of further propagation to form subsequently new macro­ molecules as a result of mutual recombination. Under repeatedly occurring acts of cleavage and recombination, the formation of inactive macromole­ cule will be completed only when it contains no peroxide group. The corresponding characteristic time tf is prescribed here by the half-decay period of a peroxide group, rather than by the lifetime of a radical as in the case of a regular polymerization in the presence of monofunctional initia­ tors. As the scales of these times can differ by several orders, the conditions are conceivable when inequality tf << , indicative of the belonging of a system to the second type; however, it does not hold because of the large value of tf. Kinetic calculations show (Kuchanov, 2007) that in such systems a certain narrowing of MWD can be expected in CSTR when switching from monofunctional initiators to polyfunctional initiators. One more option for increasing time scale tf of the formation of macro­ molecules in the processes of radical polymerization is the employment of nontraditional initiators, such, as iniferters. The term “iniferter,” intro­ duced by Otsu and Yoshida (1982), is a result of the fusion of three words: initiator, transfer agent, and terminator. A special feature of an iniferter is its participation in each of the three reactions mentioned.

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What all iniferters have in common is that despite possible essential differences in structural formulas, their molecules contain labile intera­ tomic bonds, capable of cleavage in the course of synthesis resulting in the formation of a pair of primary radicals. At least one of them does not turn into polymer radical, although it does take part in the chain termination reaction. This reaction leads to the formation of a macromolecule with labile terminal bond, able to decay under the action of heat or light. As the consequence of such a decay, the macromolecule decomposes into two components. One of them is the active macroradical, whereas another one is the small radical which is stable enough to avoid the initiation of a new polymer chain. In its turn, the macroradical after its propagation recom­ bines with stable small radical to become eventually a polymer molecule of macroiniferter. The latter decomposes again into a pair of radicals dramatically differing in their reactivity, and so on. According to this kinetic scheme of stepwise polymerization, every polymer chain can suc­ cessively be in two states: alive (as a macroradical) and dormant (as a macromolecule). Because a polymer chain lives for a rather short time in comparison with the time of its inactive state, macromolecule growth is believed to occur by separate steps. At every such step, the length of polymer chain increases by just one length of elementary chain, whose number in macromolecules can be large enough. In this case, polymeriza­ tion products will be characterized by narrow MWD and high molecular weight. Polymerization with participation of iniferters represents just one of the conceivable ways of performing “living” radical polymerization (LRP). A comprehensive description of its kinetics and mechanism is presented in voluminous literature (see, for example, Davis and Matyjaszewski, 2002; Goto and Fukuda, 2004; Matyjaszewski, 2003,). A common trait of any LRP is the ability of a polymer chain for alternating its active and non­ active states. As far as the mechanism is concerned, LRP holds an inter­ mediate position between conventional free-radical polymerization and living anionic polymerization. In the first of them, macromolecules are incapable of activating in principle, contrary to the second one, where being constantly in active state they retain their ability to growth. As noted above, each of these two limit cases of LRP calls for a particular limit regime of hydrodynamic stirring in a reactor, the most appropriate for carrying out a continuous process of polymerization. So, the best-suited candidates for conducting conventional free-radical polymerization and living anionic polymerization are PFR and CSTR, respectively. What can be recommended to an engineer designing a continuous LRP process with regard to optimum choice of hydrodynamic stirring? To answer this question one is supposed to compare average time of a polymer chain being in dormant state tf with its average residence time  in a reactor. If the values of tf and  are comparable, LRP products will be

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macromolecules consisting, on average, of few elementary chains, and thus differing noticeably in chemical size and composition. With this in mind, when designing a continuous LRP process, preference should be given to reactors in which the regime of hydrodynamic stirring is as close as possible to the ideal mixing. The products of such polymerization can be expected to have rather low polydispersity. The opposite recommenda­ tion to an engineer choosing the type of continuous reactor can be given when inequality tf <<  holds. Under such conditions, PFR is preferable in which polymer specimens close to monodisperse ones will be formed. The explanation is that macromolecules in this specimen are characterized by large values of average number m of elementary chains involved. Upon turning number m to infinity and average length of the elementary chain to zero, we will arrive at this limit at living anionic polymerization.

7.3 Polycondensation Among typical systems of the first type are polycondensation processes. Here the lifetime tf of a macromolecule during which its formation occurs in the absence of impurities or deliberately added chain terminators is practically infinite. The reason is that at any growth step, a macromolecule remains active because it comprises reactive functional groups. As with any system belonging to the first type, there are two reasons responsible for the polydispersity of the products of continuous polycondensation. The first of them is due to the statistical nature of chemical transformations in the course of polycondensation, whereas the second one owes its origin to the disper­ sion of the residence time of molecules in the reactor. If the reactor operates in the plug flow regime, the second reason is excluded, so that the inhomo­ geneity of a polymer specimen obtained in such a reactor (in comparison with any other reactor at the same conversion) will be minimal. Obviously, hydrodynamic stirring in the apparatus providing residence time distribu­ tion (RTD) with nonzero dispersion is always the factor causing an additional increase of polydispersity of the products. The comparison of two polymer specimens obtained in PFR and CSTR at the same value of conversion p and number average degree of polymerization PN = (1p)1 shows that MWD of the second specimen turns out to be essentially wider than MWD of the first one. This distinction is especially strongly pronounced in the most important region of conversions close to unity, where PN >> 1. Here the value of polydispersity coefficient of the products prepared in CSTR is PN >> 1 times larger than K of the polymer with the same value of PN but synthesized in PFR. As typical values of PN for commercial polycondensa­ tion polymers are close to 100, the distinction in polydispersity of the specimens synthesized in PFR and CSTR is tremendous, exceeding con­ siderably the analogous distinction in the same characteristic of the poly­ merization products. In a search for ways to reduce the polydispersity of

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the products of continuous polycondensation, it is possible to increase the number of steps of hydrodynamic stirring in a system by apparatus sec­ tioning or by employment of a cascade consisting of n successive CSTRs. The problem of calculation of the dependence of the width of MWD on number n has been solved earlier (Kuchanov, 1981). The results of this work allow a general conclusion about the advisability of conducting a continuous polycondensation in hydrodynamic regime which is as close to plug flow regime as possible. This recommendation is essential because even a minor deviation from this regime substantially decreases the appa­ ratus capacity, increasing simultaneously undesirable polydispersity of the products of continuous polycondensation.

7.4 Effect of stirring on composition inhomogeneity Hydrodynamic stirring evidently exerts an influence on the inhomogene­ ity of macromolecules not only in chemical size but also in composition. Analysis of the factors affecting this inhomogeneity shows that the reg­ ularities outlined above for the consideration of polydispersity remain in force exclusively for the second type of systems. In these systems, the composition inhomogeneity of copolymer specimens (characterized by dispersion 2 of CD of macromolecules) synthesized in CSTR is markedly less than the inhomogeneity of a copolymer obtained in PFR. So, the products of radical copolymerization prepared in CSTR have the minimal possible value of 2 (as compared to the reactor of any other type), which is controlled only by the stochastic character of chemical reactions. Quan­ tity 2 in this case is inversely proportional to number average degree of polymerization PN, whereas proportionality coefficient D is of order 101 being governed by kinetic parameters and monomer mixture composition. Inasmuch as the typical value of PN for radical copolymerization is 103104, corresponding values of 2 are equal to 104105. Due to their smallness they are out-of-experimental accuracy. Therefore, it is possible to neglect in practice this contribution to the composition inho­ mogeneity in comparison with the contribution originated by the spatial inhomogeneity of monomer concentrations inside a reactor. The existence of gradients of these concentrations in PFR can lead to substantially larger values of 2 as compared to the dispersion of copolymers synthesized in CSTR, which affects some macroscopic properties of the copolymers formed. Because of poor compatibility of the majority of different poly­ mers, even relatively slight composition inhomogeneity of a copolymer induces phase separation of a reaction mixture accompanied by a dramatic deterioration of the service properties. A characteristic attribute of such a phenomenon is the appearance at certain monomers’ conversion of the opalescence of this mixture with subsequent complete loss of its transpar­ ency. The final product of this process is heterophase turbid copolymer,

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whose poor performance properties make it normally improper for prac­ tical applications. It is clear from the above how important for practice is the investiga­ tion of the conditions at which a reaction mixture undergoes in the course of copolymerization the phase transition to heterophase state accompa­ nied by the loss of its transparency. In this connection, the fundamental research by Slocombe (Slocombe, 1957), who studied about 43 component systems, deserves special attention. He analyzed the dependence of the transparency of terpolymers obtained at high conversions in a batch reactor on the initial monomer mixture composition. This analysis enabled Slocombe to establish some empirical rules which later have been theore­ tically substantiated (Brun and Kuchanov, 1977). Using as examples par­ ticular copolymers, the authors of this paper found quantitative correlations between the dispersion 2 of CD of a copolymer and its transparency. They showed that limit value of dispersion, 2cr , exists for each copolymer. If dispersion 2 of any copolymer specimen exceeds 2cr , this specimen loses its transparency due to the pronounced inhomogeneity of its macromolecules in composition. If the parameters of the kinetic model of copolymerization are known, dispersion 2 can be theoretically calculated for any model of hydrodynamic stirring. Therefore, the approach put forward in work (Brun and Kuchanov, 1977) enables an engineer to make the choice of the conditions of conducting industrial processes of obtaining transparent copolymers scientifically grounded.

7.5 Polymer-analogous reactions Unlike for radical copolymerization, where the ideal mixing regime is preferable for preparing compositionally homogeneous specimens, it is recommended to conduct the reactions of chemical modification of polymers by their polymer-analogous transformations in a regime which is as close as possible to the plug flow regime. Such reactions as chlorina­ tion of polyethylene and polyvinylchloride or the synthesis of polyvinyl alcohol by the hydrolysis of polyvinyl acetate pertain to the first type of reactions. This conclusion ensues from the fact that the period of the formation tf of polymer products in these reactions is rather prolonged, being predetermined by the time of complete transformation of all functional groups belonging to a particular macromolecule. The most practically important products of the processes of such a kind are copoly­ mers in which the above-mentioned transformation is realized only par­ tially. Theoretical analysis of the dependence of 2 on number n of steps of stirring showed (Brun and Kuchanov, 1980) that at low values of n the composition inhomogeneity of these copolymers is very pronounced, whereas at n >> 1 the dispersion decreases proportionally to n1 until n becomes comparable with the average degree of polymerization of

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macromolecules. This means that the number of steps (103104) should be very large in order for the degree of composition inhomogeneity of the products of polymer-analogous transformations in a continuous process to be comparable with that taking place in a periodic process.

7.6 Microsegregation As the overall kinetic order of the reactions of the synthesis and chemical modification of polymers is generally larger than unity, the products’ inhomogeneity in size and composition depends not only on the distribu­ tion of the residence time in a reactor but also on the degree of micromix­ ing in this apparatus. This factor has been theoretically investigated for homopolymerization and homo-polycondensation (Kuchanov, 1981; Tadmor and Biensenberger, 1966) as well as for radical copolymerization (O’Driscoll and Knorr, 1969; Szabo and Nauman, 1969) by comparing the results of calculation of the inhomogeneity in two limit cases, namely complete mixing at microlevel and complete segregation. The effect of micromixing in a reactor is most pronounced in the case of polycondensa­ tion products, where the transition from the first limit case to the second one is accompanied by an appreciable rise in the productive capacity of the reactor with concurrent decrease of the products’ polydispersity.

ACKNOWLEDGMENTS The author owes sincere gratitude to his wife, Natasha, for her help and patience. He is also very indebted to Prof. Yablonsky, a co-editor of this volume, for the kind suggestion to contribute a chapter to the present work.

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Principles of Statistical Chemistry as Applied to Kinetic Modeling of Polymers

199

Kuchanov, S. I., Korolev, S. V., and Panyukov, S. V. Adv. Chem. Phys. 72, 115–326 (1988). Kuchanov, S. I., Korolev, S. V., Zubov, V. P., and Kabanov, V. A.. Polymer 25, 100 (1984). Kuchanov, S. I., Orlova, Z. V., Kosheleva, A. F., and Gorelov, Yu. P.. Vysokomol. Soedin. Ser. A 31, 474 (1989). Kuchanov, S. I., and Panyukov, S. V., in “Comprehensive Polymer Science” (G. Allen Ed.), pp. 441–496, Chapter 13, 2nd Supplement. Pergamon Press, New York (1996). Kuchanov, S. I., and Panyukov, S. V.. J. Polym. Sci. B 36, 937 (1998). Kuchanov, S. I., and Pogodin, S. G.. J. Chem. Phys. 128, 244902 (2008). Kuchanov, S. I., Pogodin, S., ten Brinke, G., and Khokhlov, A.. Macromolecules 41, 2689 (2008). Kuchanov, S. I., and Russo, S.. Macromolecules 30, 4511 (1997). Kuchanov, S., Slot, H., and Stroeks, A. Progr. Polym. Sci. 29, 563–633 (2004). Kuchanov, S. I., Tarasevich, K. V., and Zharnikov, T. V.. J. Stat. Phys. 122, 875 (2006). Kuchanov, S., and Zharnikov, T.. J. Stat. Phys. 111, 1273 (2003). Kuchanov, S., Zharnikov, T., and Khokhlov, A. Eur. Phys. J. Ser. E 10, 93 (2003). Lowry, G. G., (Ed.), “Markov Chains and Monte Carlo Calculation in Polymer Science”. Marcel Dekker, New York (1970). Matyjaszewski K. (Ed.), “Controlled/Living Radical Polymerization”, ACS Symposium Series, Vol. 854 (2003) Meyer, T., and Keurentjes, J., (Eds.), “Handbook of Polymer Reaction Engineering”, Vol. I, II. Wiley, New York (2005). Motoc, I., and O’Driscoll, K. F.. Lect. Notes Chem. 27, 62 (1981). Noa, O. V., Toom, A. L., Vasilyev, N. B., Litmanovich, A. D., and Plate, N. A.. Vysokomol. Soedin. Ser. A 15, 877(1973). O’Driscoll, K. F., and Knorr, R.. Macromolecules 2, 507 (1969). Otsu, T., and Yoshida, M.. Makromol. Chem. Rapid Commun. 3, 127 (1982). Panyukov, S. V., and Kuchanov, S. I.. J. Phys. (Fr.) II 2, 1973 (1992). Plate, N. A., Litmanovich, A. D., Noa, O. V., Toom, A. L., and Vasilyev, N. B.. J. Polym. Sci. Polym. Chem. 12, 2165 (1974). Plate, N. A., Litmanovich, A. D., and Noah, O. V., “Macromolecular Reactions”. John Wiley and Sons, New York (1995). Rudin, A., “The Elements of Polymer Science and Engineering”. Academic Press, New York (1982). Seavey, K. C., and Liu, Y. A., “Step-Growth Polymerization Process Modeling and Product Design”. Wiley, New York (2008). Semchikov, Yu. D.. Macromol. Symp. 111, 317 (1996). Semchikov, Yu. D., Slavnitskaya, N. N., Smirnova, L. A., Sherstyanykh, V. I., Sveshnikova, T. G., and Borina, T. I.. Eur. Polym. J. 26, 889 (1990). Semchikov, Yu. D., Smirnova, L. A., Knyazeva, T. E., and Sherstyanykh, V. I.. Eur. Polym. J. 26, 883 (1990). Semchikov, Yu. D., Smirnova, L. A., Kopylova, N. A., and Izvolenskij, V. V.. Eur. Polym. J. 32, 1213 (1996). Slocombe, R. J.. J. Polym. Sci. 26, 9 (1957). Sokolov, L. B., “Fundamentals of the Synthesis of Polymers by Polycondensation Method”. Khimia, Moscow (1979). Szabo, T. T., and Nauman, E. B.. Am. Inst. Chem. Eng. J. 15, 575 (1969). Tadmor, Z., and Biensenberger, J. A.. Ind. Eng. Chem. Ser. Fundam. 5, 337 (1966). Tonelli, A. E., “NMR Spectroscopy and Polymer Microstructure”. VCH, Weinheim (1989). van Krevelen, D. W., and te Nijenhuis, K., “Properties of Polymers: Their Correlation with Chemical Structure”, 4th ed., Elsevier, Amsterdam (2009). Vasnev, V. A., Vinogradova, S. V., Markova, G. D., and Voitekunas, V. U.. Vysokomol. Soedin. Ser. A 39, 412 (1997). Yakovlev, A., and Kuchanov, S.. Macromol. Symp. 160, 35 (2000).

SUBJECT INDEX

Note: The letters ‘f’ and ‘t’ following locators refer to figures and tables respectively. “Active” or “end-functionalized” chains, 144 Applications of mesoscale field-based models interaction of two grafted monolayers by attractive chains creation of stable dispersion, aim, 155 stable vs. unstable dispersion of tethered chains, 156f interaction of two grafted monolayers by end-functionalized chains analysis of density profiles at different gallery heights, 158–159, 160f calculated free energy profiles, SCFT approaches, 157–158, 157f calculated nanocomposite phase diagram, 160f Flory–Huggins interaction parameters used, 156t practical implications of the model, 161 predictions of the “compressible model,” 161 interaction of two grafted monolayers with different segment sizes, 154, 155f interaction of two grafted monolayers with equal segment sizes density profiles of grafted/free polymers at separations, 148, 149f interpenetration of grafted monolayers/free polymer, 148 SCFT/iSAFT calculations, comparison, 148–154, 150f–153f of micro- and nanostructured materials, 132–133

structure of grafted polymer monolayers in a polymer melt, 147–148 comparison of density profiles, cases, 147f Bioengineering, 76 Branched polymers gelation, 174–175 calculation of cyclic rank of polymer network (Flory theory), 175 gel point, 175 mathematical modeling of branched polymerization, objectives, 175 stochastic branching process, 173–174 Galton–Watson process, 174 Gordonian polymers, algorithm for, 174 tree/molecular forest, molecular graph representation, 174 Canonical-ensemble statistical mechanics, 88 CG-MC. See Coarse-grained Monte Carlo (CG-MC) CG-MD. See Coarse-grained molecular dynamics (CG-MD) Chapman–Enskog method, 122 Chemical correlators, 172–173 Chemical equilibrium, 90 Chemical modification of polymers PAR outlined theory for good/poor solvent, 187–188 PARs, example esterification of polymethacrylic acid, 186 neighboring-group (NG) model, 187 saponification of polyvinyl acetate, 186

201

202

Subject Index

Classical equilibrium thermodynamics, 2, 6, 8, 78–79, 81, 88, 106, 116–117 Clausius inequality, 4 Coal burning, interrelated processes burning of coke, 63–64

burning of volatiles, 63

coal pyrolysis, 63

Coal pyrolysis, 63 Coarse-grained molecular dynamics (CG-MD), 134 Coarse-grained Monte Carlo (CG-MC), 134 “Coarse-graining” process, 110 Coke burning, 63–64 Complex fluids, 76, 95, 110–111, 127 Contact geometry, 76–78, 80, 120 Continuous stirred tank reactor (CSTR), 190 Conventional free-radical copolymerization copolymerization of some monomers, anomalies in, 185–186 mathematical modeling, considerations, 186 quantitative theory of copolymerization short-range effects, types of kinetic models, 185 Convex programming (CP), 19 CSTR. See Continuous stirred tank reactor (CSTR) Denbigh, 190–191 Density functional theory (DFT), 134 DFT. See Density functional theory (DFT), 134 Dissipative macroscopic systems, equilibrium thermodynamic modeling of analysis of equilibrium models (Euler and Lagrange), 3 MEIS geometrical interpretations, 33–38 of spatially inhomogeneous systems, 26–28 with variable flows, 20–26 with variable parameters, 17–20 variants of kinetic constraints formalization, 29–33 MEIS application, examples of formation of nitrogen oxides during coal combustion, 54–64 isomerization, 50–54

stationary flow distribution in hydraulic circuits, 64–66 MEIS vs. models of nonequilibrium

thermodynamics

areas of computational efficiency,

46–50 areas of effective applications, 39–46 “Model Engineering” (Gorban), 5 reversible and irreversible processes (Galileo), 3–4 substantiation for irreversible processes equilibrium and reversibility, interpretations, 8–11 equilibrium approximations, 16–17 experience of classics, 5–8 nonequilibrium thermodynamics, equilibrium interpretations of, 12–16 thermodynamics, emergence of principle of entropy (second law of thermodynamics), 3 thermodynamics of nonconservative systems, problems of analysis and development of MEISs, 70 analysis of computational problems in MEIS application, 70–71 reduction of models of irreversible motion to models of rest, 69–70 solution of specific theoretical/applied problems on MEIS, 71 Dissipative particle dynamics (DPD), 134 Distribution of the residence time (DRT), 191, 197 DPD. See Dissipative particle dynamics (DPD) DRT. See Distribution of the residence time (DRT) Equations of motion, 6 Equilibrium models, analysis of (Euler and Lagrange), 3 Equilibrium states, 91 Euler–Lagrange equations, 137 Extended Flory principle, 176–177, 184 Fenimore mechanism, 55–56 “Field-based” mesoscale theories. See Density functional theory (DFT)

Subject Index

Flory–Huggins interaction parameters, 156t Flory principle, 176, 180, 184 Fluid mechanics, 105–113, 117–118, 122–127 “Free” or “matrix” homopolymer chains, 144 Fuel nitrogen oxides, formation of, 54–55 Galileo, 3 Galton–Watson branching process, 174 Gel, 175 Gelation, 174–175 Gel point, 175, 183 Gibbs paradox, 77 Gibbs phase rule, 16, 19, 47, 67 Gordonian polymers, 174–175, 178, 180, 183 Grafted polymer monolayers interaction by attractive chains creation of stable dispersion, aim, 155 stable vs. unstable dispersion of tethered chains, 156f interaction by end-functionalized chains analysis of density profiles at different gallery heights, 158–159, 160f calculated free energy profiles, SCFT approaches, 157–158, 157f calculated nanocomposite phase diagram, 160f Flory–Huggins interaction parameters used, 156t practical implications of the model, 161 predictions of the “compressible model,” 161 interaction of, with different segment sizes, 154, 155f interaction of, with equal segment sizes density profiles of grafted/free polymers at separations, 148, 149f interpenetration of grafted monolayers/free polymer, 148 SCFT/iSAFT calculations, comparison, 148–154, 150f–153f structure in a polymer melt, 147–148 comparison of density profiles, cases, 147f Henry law, 16 H-theorem, 3, 6 Hydraulic circuit theory, 24 Hydrodynamic fields, 106

203

Hydrodynamic stirring effects on properties of polymers effect on composition inhomogeneity, 195–196 general considerations advantages of continuous commercial processes over batch processes, 190 control of inhomogeneity of polymer, factors, 190 conventional radical polymerization systems, 191 degree of hydrodynamic stirring in PFR/CSTR, 190 living anionic polymerization systems, 191 statistical characteristics of a polymer, hierarchy, 189

microsegregation, 197

polycondensation, 194–195

polymer-analogous reactions,

196–197

radical polymerization, 191–194

Ideal kinetic model, 175–176 Irreversible processes, equilibrium thermodynamic modeling of equilibrium and reversibility, interpretations, 8–11 Boltzmann trajectories of motion, 9 dynamics of a system with periodic agitation (Gorban), 10, 10f equilibrium approximations, 11 equilibrium, main feature in mechanics, 8–9 “far from equilibrium,” meaning (Gorban), 11 equilibrium approximations, 16–17 “damnation of dimension,” 17 experience of classics, 5–8 classical equilibrium thermodynamics, computational tool used, 8 equilibrium and reversibility, analysis of interrelations, 6 equilibrium trajectories study and mathematical relations (Gibbs), 6–7 law of Fick, 8 partial equilibria notion, irreversible process of light diffusion, 8

204

Subject Index

Irreversible processes, equilibrium thermodynamic modeling of (Continued) principle of entropy increase (Boltzmann), 6

theory of electric circuits (Kirchhoff), 7

nonequilibrium thermodynamics, equilibrium interpretations of, 12–16 equilibrium interpretation of Prigogine theorem, situations (entropy equations), 12–14 Onsager reciprocal relations, 14–16 ISAFT model classical DFT, tool application in modeling interfacial properties of LJ fluid, 135–136

Helmholtz free energy as function of

density distribution, basis, 136

prediction of microscopic structure/

thermodynamics/phase behavior, 135

extension to grafted polymer chains,

140–141

homogeneous systems

PRISM, application, 136

Wertheim’s TPT1, development of

SAFT equation, 136

modeling of polyatomic molecules

application in heterogeneous polymer

systems, features, 140

density profile, expression, 139

ideal gas free energy functional, 138

linear polymer chain formation of m

segments from m associating

spheres, 137–140, 138f

open system in canonical ensemble, free

energy computation, 137–140

quantum DFT, 135

Isolated systems, study of (Gorban), 10, 10f

Kinetic block of model, thermodynamic approaches constraint on process rate determined only by one reaction, 31–32 thermodynamic analysis of kinetic

equations

constraints used, 30–31

unity of thermodynamics and kinetics constraints used, 29–30

Kinetic models of macromolecular reactions ideal kinetic model

Flory principle, assumptions, 176

process of radical polymerization,

175–176 models allowing for the deviations from ideality extended Flory principle, 176–177 polymer nature of reagents, long-range effects, 177

substitution effects, short-range effects,

176

“Kink” mechanism, 161

Kirchhoff theorem of minimum heat

production, 23

“Labeling-erasing” procedure, 181, 184, 185

Lagrange equilibrium equation, 6

Law of Fick, 8

Least action principle (PLA), 7, 16

Le Chatelier–Brown principle, 16

Legendre transformation, 76, 78–81, 83–85,

89, 92

Lennard–Jones (LJ) fluid, 136

“Living” radical polymerization (LRP),

193–194 LJ fluid. See Lennard–Jones (LJ) fluid LRP. See “Living” radical polymerization (LRP) Mass action laws, 16, 77, 99, 101, 175

Maximum entropy principle, 76, 81

‘Mechanics,’ 8

Mechanisms of NO formation, 54–56

MEIS. See Model of extreme intermediate

states (MEIS) MEIS application, examples of formation of nitrogen oxides during coal

combustion

advantages of MEIS-based modeling, 64

coal burning, interrelated processes

(kinetic models), 63–64

formulation of inequality by kinetic

equations, 59

fuel nitrogen oxides, formation of,

54–55

kinetic constraints formulations in

slow/fast subsystem, 57

Subject Index

NO formation from dinitrogen oxide, 56 prompt nitrogen oxides, formation of (Fenimore mechanism), 55–56 rate of nitrogen oxide formation, equation, 58 theoretical/experimental NO emissions at coal combustion, calculations, 61– 62, 62f thermal nitrogen oxides, formation of (Zeldovich mechanism), 55 isomerization, 50–54 computational methods/accuracy, 52–54

constraint used, 50–51

graphical interpretation of

isomerization process, 51f interpretation of studied problem, advantage, 52 kinetic equations for isomerization process, curves of, 53f “physico-economic” self-organization problem, analysis, 52 study of multistage processes, difficulties, 53 stationary flow distribution in hydraulic circuits, 64–66 final equilibrium model, form, 65 isothermal flow of incompressible fluid in three-loop circuit, example, 64–66 Prigogine theorem, aaplication, 66 results of flow distribution calculation, 66t scheme of the hydraulic circuit, 65f MEISs isomerization, 50–54 MEIS vs. models of nonequilibrium thermodynamics areas of computational efficiency, 46–50 areas of effective applications, 39–46 Mesoscale approaches, 134 Mesoscale field-based models, applications in polymer melts applications interaction of two grafted monolayers in presence of attractive chains, 154–155 interaction of two grafted monolayers in presence of end-functionalized chains, 156–161

205

interaction of two grafted monolayers with different segment sizes, 154 interaction of two grafted monolayers with equal segment sizes, 148–154 of micro- and nanostructured materials, 132–133 structure of grafted polymer monolayers in a polymer melt, 147–148 modeling of polymeric systems

mesoscale approaches, 134

problems, 133

short-range structure, role in

applications, 133 theory extension of iSAFT model to grafted polymer chains, 140–141 iSAFT model, 135–140 self-consistent field theory, 141–146 Microcanonical-ensemble statistical mechanics, 88 Microsegregation, 197 “Model Engineering,” 5, 29, 39, 40, 68, 70 Model of extreme intermediate states (MEIS), 2 geometrical interpretations, 33–38, 35f hexane isomerization reaction, analysis, 36f, 37 idea of tree in formalization of macroscopic kinetic constraints, 38 notion of thermodynamic tree (Gorban), 36–38, 36f polyhedron of material balance, 34, 36f use of tree notion in constructing algorithms, 38 of spatially inhomogeneous systems, 26–28 equations, 26–28 graph of spatially inhomogeneous system, 27f indication of harmful substance distribution in vertical air column, 28 macroscopic kinetics constraints inclusion, difficulties, 28 material balances in model, 28 parametric and flow MEIS features, 28 with variable flows, 20–26

construction of flow models of

hydraulic systems, 24–26

206

Subject Index

Model of extreme intermediate states (MEIS) (Continued) “equilibrium” derivation, hydraulic circuit theory, 24 flow modifications, groups, 20 interpretation of flows as coordinates of states, 20 nonstationary flow distribution, equations, 23–24 stationary flow distribution in closed circuit, equations, 20–22 thermodynamic model of passive circuit, 22–23 with variable parameters, 17–20

convex programming (CP), 19

list of stages in model, need for

indication, 19 model equations, assumptions, 18–19 variants of kinetic constraints formalization, 29–33 Boltzmann assumption, basis, 29 kinetic block of model, thermodynamic approaches, 29–32 MEIS modifications, difficulties, 32–33 “Model Engineering,” 29 optimal description of constraints on macroscopic kinetics, issues, 33 Monads, 184 Multiscale equilibrium thermodynamics classical equilibrium thermodynamics, 78–79 mesoscopic equilibrium thermodynamics contact geometry, applications, 80–81 example: equilibrium kinetic theory (ideal gas), 81–84 example: equilibrium kinetic theory (van der Waals gas), 84–86 example: Gibbs equilibrium statistical mechanics, 86–89 example: multicomponent isothermal systems, 89–91 example: multicomponent nonisothermal systems, 91 fundamental thermodynamic relation, 79–80 Gibbs and Gibbs–Legendre manifolds, 81, 82f Multiscale nonequilibrium thermodynamics

combination of scales example: direct molecular simulations, 111–116 single scale realizations example: a simple illustration, 96–98 example: chemically reacting isothermal systems, 98–101 example: complex fluids, 110–111 example: fluid mechanics, 105–109 example: kinetic theory of chemically reacting systems, 101–105 example: particle dynamics, 109–110 Multiscale thermodynamics in chemical engineering Gibbs formulation of classical thermodynamics, 76 macroscopic/microstructure behavior of multicomponent systems, 76–77 multiscale equilibrium thermodynamics classical equilibrium thermodynamics, 78–79 mesoscopic equilibrium thermodynamics, 79–91 multiscale nonequilibrium

thermodynamics

combination of scales, 111–116

single scale realizations, 95–111

multiscale nonequilibrium thermodynamics of driven systems example: a simple illustration, 120–122 example: Chapman–Enskog reduction of kinetic theory to fluid mechanics, 122–127 Nano-engineering, 76 Neighboring-group (NG) model, 187 Nonconservative systems, 6, 9, 66, 69–71 Nonequilibrium thermodynamics, 4 Nonideal kinetic models, 180, 183 Opalescence phenomenon, 8 PARs. See Polymer-analogous reactions (PARs) “Particle-based” mesoscale simulations. See Dissipative particle dynamics (DPD) Paul Flory, 167, 178

Subject Index

PFR. See Plug flow reactor (PFR) PLA. See Least action principle (PLA) Plug flow reactor (PFR), 190 Polycondensation, 182–184, 194–195 choice of ideal kinetic model, 182–183 cross-linking of reactive oligomers, 182–183 extension of “substitution effect,” 184 Gordonian polymers (branching process) “labeling-erasing” procedure, 183–184 monads, kinetically independent elements, 184

nonideal kinetic models, 183

statistical description of sol/gel

molecules, 183 Polymer adsorption, 132 Polymer-analogous reactions (PARs), 186, 196–197 Polymer–clay nanocomposites, 134 dispersion in clay platelets, stabilization of, 135

equilibrium morphology of, 135

synthesis of, 134–135

Polymer properties, hydrodynamic stirring effects on effect on composition inhomogeneity, 195–196 general considerations advantages of continuous commercial processes over batch processes, 190 control of inhomogeneity of polymer, factors, 190 conventional radical polymerization systems, 191 degree of hydrodynamic stirring in PFR/CSTR, 190 living anionic polymerization systems, 191 statistical characteristics of a polymer, hierarchy, 189 microsegregation, 197 polycondensation, 194–195 polymer-analogous reactions, 196–197 radical polymerization, 191–194 Polymer reference interaction site model (PRISM), 136 Polymers, chemical modification of

207

PAR outlined theory for good/poor

solvent, 187–188

PARs, example esterification of polymethacrylic acid, 186 neighboring-group (NG) model, 187 saponification of polyvinyl acetate, 186 Polymers, kinetic modeling of choice of model, considerations, 166–167 chemical modification of polymers, 167 polydispersity of products for synthesis, 167 description of polymers, peculiarities chemical correlators, 172–173 microstructure parameters, 171–172 quantitative description of macromolecules, problems, 168–169 statistical approach, 169–171 general theoretical results chemical modification of polymers, 186–188 conventional free-radical copolymerization, 184–186 polycondensation, 182–184 hydrodynamic stirring effects on properties of polymers effect of stirring on composition inhomogeneity, 195–196 microsegregation, 197 polycondensation, 194–195 polymer-analogous reactions, 196–197 radical polymerization, 191–194 kinetic models of macromolecular reactions ideal kinetic model, 175–176 models allowing for the deviations from ideality, 176–177 methods of calculations extension of statistical and kinetic methods, 180–182 kinetic method, 179–180 statistical method, 178–179 specificity of branched polymers gelation, 174–175 stochastic branching process, 173–174 Polymers, peculiar features in chemical correlators, 172–173 microstructure parameters, 171–172 statistical approach

208

Subject Index

Polymers, peculiar features in (Continued) isomerisms, types, 171 Markovian copolymers, features, 170 mathematical modeling for nonMarkovian copolymers, 170 microstructure of copolymer molecules, characteristics of second group, 171 SCD function, characteristics of first group, 170 Principle of entropy, 3, 6 Principle of the least energy dissipation (Rayleigh), 7 Principles of statistical chemistry applied to kinetic modeling of polymers choice of model, considerations, 166–167 chemical modification of polymers, 167 polydispersity of products for synthesis, 167 description of polymers, peculiarities chemical correlators, 172–173 microstructure parameters, 171–172 quantitative description of macromolecules, problems, 168–169 statistical approach, 169–171 general theoretical results chemical modification of polymers, 186–188 conventional free-radical copolymerization, 184–186 polycondensation, 182–184 hydrodynamic stirring effects on properties of polymers effect of stirring on composition inhomogeneity, 195–196 microsegregation, 197 polycondensation, 194–195 polymer-analogous reactions, 196–197 radical polymerization, 191–194 kinetic models of macromolecular reactions ideal kinetic model, 175–176 models, allowing for the deviation from ideality, 176 methods of calculations extension of statistical and kinetic methods, 180–182 kinetic method, 179–180 statistical method, 178–179

specificity of branched polymers gelation, 174–175 stochastic branching process, 173–174 PRISM. See Polymer reference interaction site model (PRISM) Prompt nitrogen oxides, formation of, 55–56 Quantum theory of radiation (Beiträge), 16 Quasiparticles/”ghost” particles, 116 Radiation thermodynamics, 7–8 Radical polymerization, 175–176, 191–194 The Raoult law, 16 SCD. See Size–composition distribution (SCD) SCFT. See Self-consistent field theory (SCFT) Self-consistent field theory (SCFT), 134, 141–146 applications, 141 de Gennes–Edwards description of polymer molecule, 141 exfoliated/intercalated/immiscible morphologies, theories, 142 lattice model, schematic depiction, 143f free/active/grafted chain polymers, evaluation of propogators, 144–145 free energy/density profile, expressions, 143–145 nanocomposite phase diagrams, generation, 142 Single scale realizations in multiscale nonequilibrium thermodynamics example: a simple illustration, 96–98 example: chemically reacting isothermal systems, 98–101 example: complex fluids, 110–111 example: fluid mechanics, 105–109 example: kinetic theory of chemically reacting systems, 101–105 exchange-of-identity collisions, 105 inelastic collisions, 105 multicomponent systems with binary chemical reactions, 105 spatially nonlocal collisions, 105

Subject Index

example: particle dynamics, 109–110

Size–composition distribution (SCD), 170

Spatially inhomogeneous systems, 17, 26–28,

67, 70, 71

Stoichiometric coefficients, 90

Synergetics, 4, 12, 39, 46, 66, 67, 69

Theory of dynamic systems, 4, 12, 39,

67, 69, 167

Theory of electric circuits (Kirchhoff), 7

Thermal nitrogen oxides, formation of, 55

Thermodynamic Lyapunov functions, 3

209

Thermodynamic perturbation theory (TPT1),

136

‘Thermodynamics,’ 3

TPT1. See Thermodynamic perturbation theory (TPT1) WAXS. See Wide-angle X-ray scattering (WAXS) Wertheim’s thermodynamic perturbation

theory, 136

Wide-angle X-ray scattering (WAXS), 142

Zeldovich mechanism, 55

CONTENTS OF VOLUMES IN THIS SERIAL

Volume 1 (1956) J. W. Westwater, Boiling of Liquids A. B. Metzner, Non-Newtonian Technology: Fluid Mechanics, Mixing, and Heat Transfer R. Byron Bird, Theory of Diffusion J. B. Opfell and B. H. Sage, Turbulence in Thermal and Material Transport Robert E. Treybal, Mechanically Aided Liquid Extraction Robert W. Schrage, The Automatic Computer in the Control and Planning of Manufacturing Operations Ernest J. Henley and Nathaniel F. Barr, Ionizing Radiation Applied to Chemical Processes and to Food and Drug Processing Volume 2 (1958) J. W. Westwater, Boiling of Liquids Ernest F. Johnson, Automatic Process Control Bernard Manowitz, Treatment and Disposal of Wastes in Nuclear Chemical Technology George A. Sofer and Harold C. Weingartner, High Vacuum Technology Theodore Vermeulen, Separation by Adsorption Methods Sherman S. Weidenbaum, Mixing of Solids Volume 3 (1962) C. S. Grove, Jr., Robert V. Jelinek, and Herbert M. Schoen, Crystallization from Solution F. Alan Ferguson and Russell C. Phillips, High Temperature Technology Daniel Hyman, Mixing and Agitation John Beck, Design of Packed Catalytic Reactors Douglass J. Wilde, Optimization Methods Volume 4 (1964) J. T. Davies, Mass-Transfer and Inierfacial Phenomena R. C. Kintner, Drop Phenomena Affecting Liquid Extraction Octave Levenspiel and Kenneth B. Bischoff, Patterns of Flow in Chemical Process Vessels Donald S. Scott, Properties of Concurrent Gas–Liquid Flow D. N. Hanson and G. F. Somerville, A General Program for Computing Multistage Vapor–Liquid Processes

211

212

Contents of Volumes in this Serial

Volume 5 (1964) J. F. Wehner, Flame Processes—Theoretical and Experimental J. H. Sinfelt, Bifunctional Catalysts S. G. Bankoff, Heat Conduction or Diffusion with Change of Phase George D. Fulford, The Flow of Lktuids in Thin Films K. Rietema, Segregation in Liquid–Liquid Dispersions and its Effects on Chemical Reactions Volume 6 (1966) S. G. Bankoff, Diffusion-Controlled Bubble Growth John C. Berg, Andreas Acrivos, and Michel Boudart, Evaporation Convection H. M. Tsuchiya, A. G. Fredrickson, and R. Aris, Dynamics of Microbial Cell Populations Samuel Sideman, Direct Contact Heat Transfer between Immiscible Liquids Howard Brenner, Hydrodynamic Resistance of Particles at Small Reynolds Numbers Volume 7 (1968) Robert S. Brown, Ralph Anderson, and Larry J. Shannon, Ignition and Combustion of Solid Rocket Propellants Knud Østergaard, Gas–Liquid–Particle Operations in Chemical Reaction Engineering J. M. Prausnilz, Thermodynamics of Fluid–Phase Equilibria at High Pressures Robert V. Macbeth, The Burn-Out Phenomenon in Forced-Convection Boiling William Resnick and Benjamin Gal-Or, Gas–Liquid Dispersions Volume 8 (1970) C. E. Lapple, Electrostatic Phenomena with Particulates J. R. Kittrell, Mathematical Modeling of Chemical Reactions W. P. Ledet and D. M. Himmelblau, Decomposition Procedures foe the Solving of Large Scale Systems R. Kumar and N. R. Kuloor, The Formation of Bubbles and Drops Volume 9 (1974) Renato G. Bautista, Hydrometallurgy Kishan B. Mathur and Norman Epstein, Dynamics of Spouted Beds W. C. Reynolds, Recent Advances in the Computation of Turbulent Flows R. E. Peck and D. T. Wasan, Drying of Solid Particles and Sheets Volume 10 (1978) G. E. O’Connor and T. W. F. Russell, Heat Transfer in Tubular Fluid–Fluid Systems P. C. Kapur, Balling and Granulation Richard S. H. Mah and Mordechai Shacham, Pipeline Network Design and Synthesis J. Robert Selman and Charles W. Tobias, Mass-Transfer Measurements by the Limiting-Current Technique

Contents of Volumes in this Serial

213

Volume 11 (1981) Jean-Claude Charpentier, Mass-Transfer Rates in Gas–Liquid Absorbers and Reactors Dee H. Barker and C. R. Mitra, The Indian Chemical Industry—Its Development and Needs Lawrence L. Tavlarides and Michael Stamatoudis, The Analysis of Interphase Reactions and Mass Transfer in Liquid–Liquid Dispersions Terukatsu Miyauchi, Shintaro Furusaki, Shigeharu Morooka, and Yoneichi Ikeda, Transport Phenomena and Reaction in Fluidized Catalyst Beds Volume 12 (1983) C. D. Prater, J, Wei, V. W. Weekman, Jr., and B. Gross, A Reaction Engineering Case History: Coke Burning in Thermofor Catalytic Cracking Regenerators Costel D. Denson, Stripping Operations in Polymer Processing Robert C. Reid, Rapid Phase Transitions from Liquid to Vapor John H. Seinfeld, Atmospheric Diffusion Theory Volume 13 (1987) Edward G. Jefferson, Future Opportunities in Chemical Engineering Eli Ruckenstein, Analysis of Transport Phenomena Using Scaling and Physical Models Rohit Khanna and John H. Seinfeld, Mathematical Modeling of Packed Bed Reactors: Numerical Solutions and Control Model Development Michael P. Ramage, Kenneth R. Graziano, Paul H. Schipper, Frederick J. Krambeck, and Byung C. Choi, KINPTR (Mobil’s Kinetic Reforming Model): A Review of Mobil’s Industrial Process Modeling Philosophy Volume 14 (1988) Richard D. Colberg and Manfred Morari, Analysis and Synthesis of Resilient Heat Exchange Networks Richard J. Quann, Robert A. Ware, Chi-Wen Hung, and James Wei, Catalytic Hydrometallation of Petroleum Kent David, The Safety Matrix: People Applying Technology to Yield Safe Chemical Plants and Products Volume 15 (1990) Pierre M. Adler, Ali Nadim, and Howard Brenner, Rheological Models of Suspenions Stanley M. Englund, Opportunities in the Design of Inherently Safer Chemical Plants H. J. Ploehn and W. B. Russel, Interations between Colloidal Particles and Soluble Polymers Volume 16 (1991) Perspectives in Chemical Engineering: Research and Education Clark K. Colton, Editor Historical Perspective and Overview L. E. Scriven, On the Emergence and Evolution of Chemical Engineering Ralph Landau, Academic—industrial Interaction in the Early Development of Chemical Engineering

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Contents of Volumes in this Serial

James Wei, Future Directions of Chemical Engineering Fluid Mechanics and Transport L. G. Leal, Challenges and Opportunities in Fluid Mechanics and Transport Phenomena William B. Russel, Fluid Mechanics and Transport Research in Chemical Engineering J. R. A. Pearson, Fluid Mechanics and Transport Phenomena Thermodynamics Keith E. Gubbins, Thermodynamics J. M. Prausnitz, Chemical Engineering Thermodynamics: Continuity and Expanding Frontiers H. Ted Davis, Future Opportunities in Thermodynamics Kinetics, Catalysis, and Reactor Engineering Alexis T. Bell, Reflections on the Current Status and Future Directions of Chemical Reaction Engineering James R. Katzer and S. S. Wong, Frontiers in Chemical Reaction Engineering L. Louis Hegedus, Catalyst Design Environmental Protection and Energy John H. Seinfeld, Environmental Chemical Engineering T. W. F. Russell, Energy and Environmental Concerns Janos M. Beer, Jack B. Howard, John P. Longwell, and Adel F. Sarofim, The Role of Chemical Engineering in Fuel Manufacture and Use of Fuels Polymers Matthew Tirrell, Polymer Science in Chemical Engineering Richard A. Register and Stuart L. Cooper, Chemical Engineers in Polymer Science: The Need for an Interdisciplinary Approach Microelectronic and Optical Material Larry F. Thompson, Chemical Engineering Research Opportunities in Electronic and Optical Materials Research Klavs F. Jensen, Chemical Engineering in the Processing of Electronic and Optical Materials: A Discussion Bioengineering James E. Bailey, Bioprocess Engineering Arthur E. Humphrey, Some Unsolved Problems of Biotechnology Channing Robertson, Chemical Engineering: Its Role in the Medical and Health Sciences Process Engineering Arthur W. Westerberg, Process Engineering Manfred Morari, Process Control Theory: Reflections on the Past Decade and Goals for the Next James M. Douglas, The Paradigm After Next George Stephanopoulos, Symbolic Computing and Artificial Intelligence in Chemical Engineering: A New Challenge The Identity of Our Profession Morton M. Denn, The Identity of Our Profession Volume 17 (1991) Y. T. Shah, Design Parameters for Mechanically Agitated Reactors Mooson Kwauk, Particulate Fluidization: An Overview Volume 18 (1992) E. James Davis, Microchemical Engineering: The Physics and Chemistry of the Microparticle Selim M. Senkan, Detailed Chemical Kinetic Modeling: Chemical Reaction Engineering of the Future Lorenz T. Biegler, Optimization Strategies for Complex Process Models

Contents of Volumes in this Serial

215

Volume 19 (1994) Robert Langer, Polymer Systems for Controlled Release of Macromolecules, Immobilized Enzyme Medical Bioreactors, and Tissue Engineering J. J. Linderman, P. A. Mahama, K. E. Forsten, and D. A. Lauffenburger, Diffusion and Probability in Receptor Binding and Signaling Rakesh K. Jain, Transport Phenomena in Tumors R. Krishna, A Systems Approach to Multiphase Reactor Selection David T. Allen, Pollution Prevention: Engineering Design at Macro-, Meso-, and Microscales John H. Seinfeld, Jean M. Andino, Frank M. Bowman, Hali J. L. Forstner, and Spyros Pandis, Tropospheric Chemistry Volume 20 (1994) Arthur M. Squires, Origins of the Fast Fluid Bed Yu Zhiqing, Application Collocation Youchu Li, Hydrodynamics Li Jinghai, Modeling Yu Zhiqing and Jin Yong, Heat and Mass Transfer Mooson Kwauk, Powder Assessment Li Hongzhong, Hardware Development Youchu Li and Xuyi Zhang, Circulating Fluidized Bed Combustion Chen Junwu, Cao Hanchang, and Liu Taiji, Catalyst Regeneration in Fluid Catalytic Cracking Volume 21 (1995) Christopher J. Nagel, Chonghum Han, and George Stephanopoulos, Modeling Languages: Declarative and Imperative Descriptions of Chemical Reactions and Processing Systems Chonghun Han, George Stephanopoulos, and James M. Douglas, Automation in Design: The Conceptual Synthesis of Chemical Processing Schemes Michael L. Mavrovouniotis, Symbolic and Quantitative Reasoning: Design of Reaction Pathways through Recursive Satisfaction of Constraints Christopher Nagel and George Stephanopoulos, Inductive and Deductive Reasoning: The Case of Identifying Potential Hazards in Chemical Processes Keven G. Joback and George Stephanopoulos, Searching Spaces of Discrete Soloutions: The Design of Molecules Processing Desired Physical Properties Volume 22 (1995) Chonghun Han, Ramachandran Lakshmanan, Bhavik Bakshi, and George Stephanopoulos, Nonmonotonic Reasoning: The Synthesis of Operating Procedures in Chemical Plants Pedro M. Saraiva, Inductive and Analogical Learning: Data-Driven Improvement of Process Operations Alexandros Koulouris, Bhavik R. Bakshi and George Stephanopoulos, Empirical Learning through Neural Networks: The Wave-Net Solution Bhavik R. Bakshi and George Stephanopoulos, Reasoning in Time: Modeling, Analysis, and Pattern Recognition of Temporal Process Trends Matthew J. Realff, Intelligence in Numerical Computing: Improving Batch Scheduling Algorithms through Explanation-Based Learning

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Contents of Volumes in this Serial

Volume 23 (1996) Jeffrey J. Siirola, Industrial Applications of Chemical Process Synthesis Arthur W. Westerberg and Oliver Wahnschafft, The Synthesis of Distillation-Based Separation Systems Ignacio E. Grossmann, Mixed-Integer Optimization Techniques for Algorithmic Process Synthesis Subash Balakrishna and Lorenz T. Biegler, Chemical Reactor Network Targeting and Integration: An Optimization Approach Steve Walsh and John Perkins, Operability and Control inn Process Synthesis and Design Volume 24 (1998) Raffaella Ocone and Gianni Astarita, Kinetics and Thermodynamics in Multicomponent Mixtures Arvind Varma, Alexander S. Rogachev, Alexandra S. Mukasyan, and Stephen Hwang, Combustion Synthesis of Advanced Materials: Principles and Applications J. A. M. Kuipers and W. P. Mo, van Swaaij, Computional Fluid Dynamics Applied to Chemical Reaction Engineering Ronald E. Schmitt, Howard Klee, Debora M. Sparks, and Mahesh K. Podar, Using Relative Risk Analysis to Set Priorities for Pollution Prevention at a Petroleum Refinery Volume 25 (1999) J. F. Davis, M. J. Piovoso, K. A. Hoo, and B. R. Bakshi, Process Data Analysis and Interpretation J. M. Ottino, P. DeRoussel, S., Hansen, and D. V. Khakhar, Mixing and Dispersion of Viscous Liquids and Powdered Solids Peter L. Silverston, Li Chengyue, Yuan Wei-Kang, Application of Periodic Operation to Sulfur Dioxide Oxidation Volume 26 (2001) J. B. Joshi, N. S. Deshpande, M. Dinkar, and D. V. Phanikumar, Hydrodynamic Stability of Multiphase Reactors Michael Nikolaou, Model Predictive Controllers: A Critical Synthesis of Theory and Industrial Needs Volume 27 (2001) William R. Moser, Josef Find, Sean C. Emerson, and Ivo M, Krausz, Engineered Synthesis of Nanostructure Materials and Catalysts Bruce C. Gates, Supported Nanostructured Catalysts: Metal Complexes and Metal Clusters Ralph T. Yang, Nanostructured Absorbents Thomas J. Webster, Nanophase Ceramics: The Future Orthopedic and Dental Implant Material Yu-Ming Lin, Mildred S. Dresselhaus, and Jackie Y. Ying, Fabrication, Structure, and Transport Properties of Nanowires Volume 28 (2001) Qiliang Yan and Juan J. DePablo, Hyper-Parallel Tempering Monte Carlo and Its Applications Pablo G. Debenedetti, Frank H. Stillinger, Thomas M. Truskett, and Catherine P. Lewis, Theory of Supercooled Liquids and Glasses: Energy Landscape and Statistical Geometry Perspectives

Contents of Volumes in this Serial

217

Michael W. Deem, A Statistical Mechanical Approach to Combinatorial Chemistry Venkat Ganesan and Glenn H. Fredrickson, Fluctuation Effects in Microemulsion Reaction Media David B. Graves and Cameron F. Abrams, Molecular Dynamics Simulations of Ion–Surface Interactions with Applications to Plasma Processing Christian M. Lastoskie and Keith E, Gubbins, Characterization of Porous Materials Using Molecular Theory and Simulation Dimitrios Maroudas, Modeling of Radical-Surface Interactions in the Plasma-Enhanced Chemical Vapor Deposition of Silicon Thin Films Sanat Kumar, M. Antonio Floriano, and Athanassiors Z. Panagiotopoulos, Nanostructured Formation and Phase Separation in Surfactant Solutions Stanley I. Sandler, Amadeu K. Sum, and Shiang-Tai Lin, Some Chemical Engineering Applications of Quantum Chemical Calculations Bernhardt L. Trout, Car-Parrinello Methods in Chemical Engineering: Their Scope and potential R. A. van Santen and X. Rozanska, Theory of Zeolite Catalysis Zhen-Gang Wang, Morphology, Fluctuation, Metastability and Kinetics in Ordered Block Copolymers Volume 29 (2004) Michael V. Sefton, The New Biomaterials Kristi S. Anseth and Kristyn S. Masters, Cell–Material Interactions Surya K. Mallapragada and Jennifer B. Recknor, Polymeric Biomaterias for Nerve Regeneration Anthony M. Lowman, Thomas D. Dziubla, Petr Bures, and Nicholas A. Peppas, Structural and Dynamic Response of Neutral and Intelligent Networks in Biomedical Environments F. Kurtis Kasper and Antonios G. Mikos, Biomaterials and Gene Therapy Balaji Narasimhan and Matt J. Kipper, Surface-Erodible Biomaterials for Drug Delivery Volume 30 (2005) Dionisio Vlachos, A Review of Multiscale Analysis: Examples from System Biology, Materials Engineering, and Other Fluids-Surface Interacting Systems Lynn F. Gladden, M.D. Mantle and A.J. Sederman, Quantifying Physics and Chemistry at Multiple Length- Scales using Magnetic Resonance Techniques Juraj Kosek, Frantisek Steeˇpa´nek, and Milosˇ Marek, Modelling of Transport and Transformation Processes in Porous and Multiphase Bodies Vemuri Balakotaiah and Saikat Chakraborty, Spatially Averaged Multiscale Models for Chemical Reactors Volume 31 (2006) Yang Ge and Liang-Shih Fan, 3-D Direct Numerical Simulation of Gas–Liquid and Gas–Liquid– Solid Flow Systems Using the Level-Set and Immersed-Boundary Methods M.A. van der Hoef, M. Ye, M. van Sint Annaland, A.T. Andrews IV, S. Sundaresan, and J.A.M. Kuipers, Multiscale Modeling of Gas-Fluidized Beds Harry E.A. Van den Akker, The Details of Turbulent Mixing Process and their Simulation Rodney O. Fox, CFD Models for Analysis and Design of Chemical Reactors Anthony G. Dixon, Michiel Nijemeisland, and E. Hugh Stitt, Packed Tubular Reactor Modeling and Catalyst Design Using Computational Fluid Dynamics

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Volume 32 (2007) William H. Green, Jr., Predictive Kinetics: A New Approach for the 21st Century Mario Dente, Giulia Bozzano, Tiziano Faravelli, Alessandro Marongiu, Sauro Pierucci and Eliseo Ranzi, Kinetic Modelling of Pyrolysis Processes in Gas and Condensed Phase Mikhail Sinev, Vladimir Arutyunov and Andrey Romanets, Kinetic Models of C1–C4 Alkane Oxidation as Applied to Processing of Hydrocarbon Gases: Principles, Approaches and Developments Pierre Galtier, Kinetic Methods in Petroleum Process Engineering Volume 33 (2007) Shinichi Matsumoto and Hirofumi Shinjoh, Dynamic Behavior and Characterization of Automobile Catalysts Mehrdad Ahmadinejad, Maya R. Desai, Timothy C. Watling and Andrew P.E. York, Simulation of Automotive Emission Control Systems Anke Gu¨thenke, Daniel Chatterjee, Michel Weibel, Bernd Krutzsch, Petr Kocˇ´ı, Milosˇ Marek, Isabella Nova and Enrico Tronconi, Current Status of Modeling Lean Exhaust Gas Aftertreatment Catalysts Athanasios G. Konstandopoulos, Margaritis Kostoglou, Nickolas Vlachos and Evdoxia Kladopoulou, Advances in the Science and Technology of Diesel Particulate Filter Simulation Volume 34 (2008) C.J. van Duijn, Andro Mikelic´, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Pe´clet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited Liqiu Wang, Mingtian Xu, and Xiaohao Wei, Multiscale Theorems Volume 35 (2009) Rudy J. Koopmans and Anton P.J. Middelberg, Engineering Materials from the Bottom Up – Overview Robert P.W. Davies, Amalia Aggeli, Neville Boden, Tom C.B. McLeish, Irena A. Nyrkova, and Alexander N. Semenov, Mechanisms and Principles of 1 D Self-Assembly of Peptides into � -Sheet Tapes Paul van der Schoot, Nucleation and Co-Operativity in Supramolecular Polymers Michael J. McPherson, Kier James, Stuart Kyle, Stephen Parsons, and Jessica Riley, Recombinant Production of Self-Assembling Peptides Boxun Leng, Lei Huang, and Zhengzhong Shao, Inspiration from Natural Silks and Their Proteins Sally L. Gras, Surface- and Solution-Based Assembly of Amyloid Fibrils for Biomedical and Nanotechnology Applications Conan J. Fee, Hybrid Systems Engineering: Polymer-Peptide Conjugates

Contents of Volumes in this Serial

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Volume 36 (2009) Vincenzo Augugliaro, Sedat Yurdakal, Vittorio Loddo, Giovanni Palmisano, and Leonardo Palmisano, Determination of Photoadsorption Capacity of Polychrystalline TiO2 Catalyst in Irradiated Slurry Marta I. Litter, Treatment of Chromium, Mercury, Lead, Uranium, and Arsenic in Water by Heterogeneous Photocatalysis Aaron Ortiz-Gomez, Benito Serrano-Rosales, Jesus Moreira-del-Rio, and Hugo de-Lasa, Miner­ alization of Phenol in an Improved Photocatalytic Process Assisted with Ferric Ions: Reaction Network and Kinetic Modeling R.M. Navarro, F. del Valle, J.A. Villoria de la Mano, M.C. Alvarez-Galva´n, and J.L.G. Fierro, Photocatalytic Water Splitting Under Visible Light: Concept and Catalysts Development Ajay K. Ray, Photocatalytic Reactor Configurations for Water Purification: Experimentation and Modeling Camilo A. Arancibia-Bulnes, Antonio E. Jime´nez, and Claudio A. Estrada, Development and Modeling of Solar Photocatalytic Reactors Orlando M. Alfano and Alberto E. Cassano, Scaling-Up of Photoreactors: Applications to Advanced Oxidation Processes Yaron Paz, Photocatalytic Treatment of Air: From Basic Aspects to Reactors Volume 37 (2009) S. Roberto Gonzalez A., Yuichi Murai, and Yasushi Takeda, Ultrasound-Based Gas–Liquid Interface Detection in Gas–Liquid Two-Phase Flows Z. Zhang, J. D. Stenson, and C. R. Thomas, Micromanipulation in Mechanical Characterisation of Single Particles Feng-Chen Li and Koichi Hishida, Particle Image Velocimetry Techniques and Its Applications in Multiphase Systems J. P. K. Seville, A. Ingram, X. Fan, and D. J. Parker, Positron Emission Imaging in Chemical Engineering Fei Wang, Qussai Marashdeh, Liang-Shih Fan, and Richard A. Williams, Electrical Capacitance, Electrical Resistance, and Positron Emission Tomography Techniques and Their Applications in Multi-Phase Flow Systems Alfred Leipertz and Roland Sommer, Time-Resolved Laser-Induced Incandescence Volume 38 (2009) Arata Aota and Takehiko Kitamori, Microunit Operations and Continuous Flow Chemical Processing Anıl Ag˘ıral and Han J.G.E. Gardeniers, Microreactors with Electrical Fields Charlotte Wiles and Paul Watts, High-Throughput Organic Synthesis in Microreactors S. Krishnadasan, A. Yashina, A.J. deMello and J.C. deMello, Microfluidic Reactors for Nano­ material Synthesis Volume 39 (2010) B.M. Kaganovich, A.V. Keiko and V.A. Shamansky, Equilibrium Thermodynamic Modeling of Dissipative Macroscopic Systems Miroslav Grmela, Multiscale Equilibrium and Nonequilibrium Thermodynamics in Chemical Engineering

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Prasanna K. Jog, Valeriy V. Ginzburg, Rakesh Srivastava, Jeffrey D. Weinhold, Shekhar Jain, and Walter G. Chapman, Application of Mesoscale Field-Based Models to Predict Stability of Particle Dispersions in Polymer Melts Semion Kuchanov, Principles of Statistical Chemistry as Applied to Kinetic Modeling of PolymerObtaining Processes