Colloids and Interfaces in Life Sciences

Colloids and Interfaces in Life Sciences Willem Norde Wageningen University Wageningen and University of Groningen Gron...

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Colloids and Interfaces in Life Sciences

Willem Norde Wageningen University Wageningen and University of Groningen Groningen, The Netherlands

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved. MARCEL

MARCEL DEKKER, INC. D E K K E R

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Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide speciﬁc advice or recommendations for any speciﬁc situation. Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identiﬁcation and explanation without intent to infringe. Cover ﬁgure reprinted with permission from IPF, Dresden, Germany. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-0996-9 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc., 270 Madison Avenue, New York, NY 10016, U.S.A. tel: 212-696-9000; fax: 212-685-4540 Distribution and Customer Service Marcel Dekker, Inc., Cimarron Road, Monticello, New York 12701, U.S.A. tel: 800-228-1160; fax: 845-796-1772 Eastern Hemisphere Distribution Marcel Dekker AG, Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright # 2003 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microﬁlming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8

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Preface

In spite of the revived interest from researchers in both pure and applied sciences, colloids and interfaces are still poorly represented among the various adjacent subjects taught at our educational institutions. Colloids and Interfaces in Life Sciences gives a concise treatment of physical-chemical principles determining interrelated colloidal and interfacial phenomena. These phenomena are generic; their occurrence in colloidal systems as blood, cell plasma, food products, waste water, soil systems, and other forms can be traced back to the same laws of nature. Although, of course, all interfacial and colloidal phenomena are ultimately dictated by interactions and movements of molecules, it is, for many purposes, not necessary to search for a molecular interpretation. There are laws of general validity that apply to macroscopic systems; examples are the laws of thermodynamics and hydrodynamics. Such laws are applied without considering the existence of molecules. On the other hand, theories based on models on a molecular level may be used. Colloid and interface science uses both approaches, as they are considered to be complementary. The interactions are often described in terms of forces and energies. The presentation in this book focuses on physical-chemical concepts that form the basis of understanding colloidal and interfacial phenomena, rather than on experimental methods and techniques. Because colloidal systems in life sciences are more often than not aqueous solutions and gels of biopolymers, and self-assembled amphiphilic structures, emphasis is placed on these reversible, soft colloids.

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Preface

The book starts with an introductory chapter giving a bird’s eye view of the history of colloid science and in which the relevance of colloids and interfaces for life sciences is indicated. Colloids are in the ﬁrst place characterized by their dimensions. Several properties—e.g., the interfacial area per unit mass of dispersed material and hence the capacity as adsorbent, particle–particle interaction forces and rheological behavior—are directly related to particle sizes. Chapter 2 addresses shapes and, to an even greater extent, sizes and size distribution of particles. Chapter 3 refers to thermodynamic principles, in particular applied to interfaces. It provides the essential thermodynamic background for the analyses and interpretations of phenomena presented in subsequent chapters. Water is pre-eminently the medium for biological systems on earth. Because of its unique and extraordinary properties, water plays a striking role in interfacial and assembly processes. Chapter 4 deals with molecular and macroscopic characteristics of water determining its role as a solvent and dispersion medium. As interfacial properties are often decisive for the behavior of colloidal systems, the most relevant interfacial properties, i.e., interfacial tension, curvature, monolayer formation, wetting, and the electrical double layer at charged interfaces, are treated in Chapters 5 through 9, respectively. In Chapter 10 electrokinetic phenomena are discussed. Electrokinetic phenomena are relatively easily accessible by experiments and they are usually studied to derive information on the electric charge and potential at interfaces. Chapter 11 explains why and how amphiphilic molecules assemble spontaneously to form different types of supramolecular structures. Polymer molecules, which by their mere size belong to the colloid family, are described in Chapter 12. Special attention is paid to polymer–solvent interaction and its inﬂuence on the structure adopted by polymer molecules. Proteins are a special class of biopolymers. Because of their central role in life sciences a full chapter, Chapter 13, is dedicated to describing their threedimensional structure and structure stability in an aqueous environment. It is shown that the compact structures of globular protein molecules is the result of intramolecular self-assembly. Chapter 14 continues with the most elementary theories of adsorption of low-molecular-weight components. The mechanisms underlying the adsorption of polymers, including proteins, are more intricate. Polymer adsorption, with emphasis on globular proteins, is discussed in Chapter 15. The main forces that rule colloidal stability, that is, the tendency of particles to aggregate, are the topic of Chapter 16. Colloid stability appears to be the net outcome of a subtle interplay among dispersion, osmotic, and steric forces. Evaluation of these forces provides the clue for manipulating the stability of colloidal systems. In Chapter 17 we discuss rheological properties, in particular viscosity and elasticity, of colloidal systems. These properties are at the basis of quality

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Preface

v

characteristics as strength, pliancy, ﬂuidity, texture, and other mechanical properties of various materials and products. In addition to bulk rheology, the rheological features of interfaces are brieﬂy discussed. Interfacial rheological behavior is crucial for the existence of deformable dispersed particles in emulsions and foams. Emulsions and foams, notably their formation and stabilization, are considered in more detail in Chapter 18. Chapter 19 is about physiochemical aspects of the most ubiquitous interface in living systems, the biological membrane. We conclude with Chapter 20, which deals with bioadhesion, the accumulation of biological cells at interfaces. Bioadhesion may lead to adverse effects—for instance, fouling of surfaces—but in other applications it is desired—for instance, immobilization of cells in bioreactors. Thus, the goal of Colloids and Interfaces in Life Sciences is to make the reader understand colloidal and interfacial phenomena, their mutual relations and connections, and their relevance in diverse areas of the life sciences. The book is written for upper-level undergraduate and graduate students in materials science, biotechnology, biomedical sciences, food science, environmental technology, and molecular biology. The level of the text is introductory, yet it supports a basic knowledge of physical chemistry and mathematical calculus. More advanced and thorough treatises of colloids and interfaces can be found in various other books. A few, more or less classical, comprehensive books are mentioned at the end of Chapter 1. In addition, more speciﬁc, topical texts are suggested for further reading at the end of each of the other chapters. I could accomplish this volume only thanks to the cooperation of and interaction with others. During the more than 30 years I worked in the scientiﬁcally fertile environment of the Laboratory of Physical Chemistry and Colloid Science of Wageningen University I was involved in fundamental research with an open eye for applications, especially in the ﬁelds of food and soil science as well as environmental and biotechnology. I greatly beneﬁted from collaboration and discussions with colleagues, in particular Hans Lyklema, Martien Cohen Stuart, Gerard Fleer, Frans Leermakers, Arie de Keizer, Mieke Kleijn, Luuk Koopal, and Pieter Walstra. Joining the Department of Biomedical Engineering of the University of Groningen gave me the opportunity to teach and apply colloid and interface science in a biomedical context. I am grateful to my colleagues in Groningen, speciﬁcally Henk Busscher and Henry van der Mei, for introducing me smoothly into the realm of biomaterials. I also thank my students who, over the years, gave their unprejudiced criticism and comments that kept me alert in teaching. Finally, I acknowledge Mrs. Ina Heidema-Kol for her accurate word processing and secretarial services and Mr. Gert Buurman for his indispensable help with the artwork.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

vi

Preface

I hope and expect that Colloids and Interfaces in Life Sciences will inspire students and scientists to future research and be valuable to anyone who wants to appreciate the fascinating roles colloids and interfaces play in life sciences. Willem Norde

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Contents

Preface 1

Introduction 1.1 1.2 1.3 1.4

2

Colloidal Particles: Shapes and Size Distributions 2.1 2.2 2.3 2.4

3

The Colloidal Domain Interfaces Are Closely Related to Colloids Colloid and Interface Science in a Historical Perspective Classiﬁcation of Colloidal Systems Suggestions for Further Reading

Shapes Particle Size Distributions Average Molar Mass Speciﬁc Surface Area Exercises Suggestions for Further Reading

Some Thermodynamic Principles and Relations, with Special Attention to Interfaces 3.1 3.2 3.3

Energy, Work, and Heat. The First Law of Thermodynamics The Second Law of Thermodynamics. Entropy Reversible Processes. Deﬁnition of Intensive Variables

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viii

Contents

3.4 3.5

Introduction of Other Functions of State. Maxwell Relations Molar Properties and Partial Molar Properties. Dependence of the Chemical Potential on Temperature, Pressure, and Composition of the System Criteria for Equilibrium. Osmotic Pressure Phase Equilibria, Partitioning, Solubilization, and Chemical Equilibrium Entropy of Mixing The Excess Nature of Interfacial Thermodynamic Quantities. The Gibbs Dividing Plane The Gibbs–Duhem Equation The Gibbs Adsorption Equation Some Applications of the Gibbs Adsorption Equation Exercises Suggestions for Further Reading

3.6 3.7 3.8 3.9 3.10 3.11 3.12

4

Water 4.1 4.2 4.3

5

Interfacial Tension 5.1 5.2 5.3 5.4 5.5 5.6

6

Phenomenological Aspects of Water Molecular Properties of Water Water as a Solvent Exercises Suggestions for Further Reading

Interfacial Tension: Phenomenological Aspects Interfacial Tension as a Force. Mechanical Deﬁnition of the Interfacial Tension Interfacial Tension as an Interfacial (Gibbs) Energy. Thermodynamic Deﬁnition of the Interfacial Tension Operational Restrictions of the Interfacial Tension Interfacial Tension and the Works of Cohesion and Adhesion Molecular Interpretation of the Interfacial Tension Exercises Suggestions for Further Reading

Curvature and Capillarity 6.1 6.2 6.3

Capillary Pressure. The Young–Laplace Equation Some Consequences of Capillary Pressure Curvature and Chemical Potential. Kelvin’s Law and Ostwald’s Law

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Contents

6.4

7

7.5 7.6

The Interfacial Pressure Gibbs and Langmuir Monolayers. Equations of State Formation of Monolayers Pressure-Area Isotherms of Langmuir Monolayers. Two-Dimensional Phases Transfer of Monolayers to Solid Surfaces. Langmuir– Blodgett and Langmuir–Schaefer Films Self-Assembled Monolayers Exercises Suggestions for Further Reading

Wetting of Solid Surfaces 8.1 8.2 8.3 8.4 8.5 8.6 8.7

9

Curvature and Nucleation Exercises Suggestions for Further Reading

Monolayers at Fluid Interfaces 7.1 7.2 7.3 7.4

8

ix

Contact Angle. Equation of Young and Dupre´ Some Complications in the Establishment of the Contact Angle: Hysteresis, Surface Heterogeneity, and Roughness Wetting and Adhesion. Determination of Surface Polarity Approximation of the Surface Tension of a Solid. The Critical Surface Tension of Wetting Wetting by Solutions Containing Surfactants Capillary Penetration Some Practical Applications and Implications of Wetting: Impregnation, Flotation, Pickering Stabilization, Cleansing Exercises Suggestions for Further Reading

Electrochemistry of Interfaces 9.1 9.2 9.3 9.4 9.5

Electric Charge Electric Potential The Gibbs Energy of an Electrical Double Layer Models for the Electrical Double Layer Donnan Effect; Donnan Equilibrium; Colloidal Osmotic Pressure; Membrane Potential Exercises Suggestions for Further Reading

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x

10

Contents

Electrokinetic Phenomena 10.1 10.2 10.3 10.4 10.5

11

Self-Assembled Structures 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9

12

Self-Assembly as Phase Separation Different Types of Self-Assembled Structures Aggregation as a ‘‘Start-Stop’’ Process. Size and Shape of Self-Assembled Structures Mass Action Model for Micellization Factors that Inﬂuence the Critical Micelle Concentration Bilayer Structures Reverse Micelles Microemulsions Self-Assembled Structures in Applications Exercises Suggestions for Further Reading

Polymers 12.1 12.2 12.3 12.4 12.5 12.6 12.7

13

The Plane of Shear. The Zeta-Potential Derivation of the Zeta-Potential from Electrokinetic Phenomena Some Complications in Deriving the Zeta-Potential: Surface Conduction; Visco-Electric Effect Interpretation of the Zeta-Potential Applications of Electrokinetic Phenomena Exercises Suggestions for Further Reading

Polymers in Solution Conformations of Dissolved Polymer Molecules Coil-Like Polymer Conformations Semidilute and Concentrated Polymer Solutions Polyelectrolytes Phase Separations in Polymer Solutions: Coacervation, Complex-Coacervation, and Polymer-Induced Micellization Polymer Gels Exercises Suggestions for Further Reading

Proteins 13.1 13.2

The Amino Acids in Proteins The Three-Dimensional Structure of Protein Molecules in Aqueous Solution

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Contents

13.3 13.4 13.5 13.6

14

Adsorbent–Adsorbate Interactions Adsorption Kinetics Adsorption Equilibrium: A Thermodynamic Approach Binding of Ligands Applications of Adsorption Exercises Suggestions for Further Reading

Adsorption of (Bio)Polymers, with Special Emphasis on Globular Proteins 15.1 15.2 15.3 15.4 15.5 15.6 15.7

16

Noncovalent Interactions that Determine the Structure of a Protein Molecule in Water Stability of Protein Structure in Aqueous Solution Thermodynamic Analysis of Protein Structure Stability Reversibility of Protein Denaturation. Aggregation of Unfolded Protein Molecules Exercises Suggestions for Further Reading

Adsorption 14.1 14.2 14.3 14.4 14.5

15

xi

Adsorption Kinetics Morphology of the Interface Relaxation of the Adsorbed Molecule Adsorption Afﬁnity; Adsorption Isotherm Driving Forces for Adsorption of Globular Proteins Reversibility of Protein Adsorption Process: Desorption and Exchange Competitive Protein Adsorption Exercises Suggestions for Further Reading

Stability of Lyophobic Colloids Against Aggregation 16.1 16.2 16.3 16.4 16.5

Forces Operating Between Colloidal Particles DLVO Theory of Colloid Stability The Inﬂuence of Polymers on Colloid Stability Aggregation Kinetics Morphology of Colloidal Aggregates Exercises Suggestions for Further Reading

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xii

17

Contents

Rheology, with Special Attention to Dispersions and Interfaces 17.1 17.2 17.3 17.4

18

Emulsions and Foams 18.1 18.2 18.3 18.4

19

Phenomenological Aspects Emulsiﬁcation and Foaming Emulsion and Foam Stability Modulation of the Coarseness and Stability of Emulsions and Foams Exercises Suggestions for Further Reading

Physicochemical Properties of Biological Membranes 19.1 19.2 19.3 19.4

20

Rheological Properties Classiﬁcation of Materials Based on Their Rheological Behavior Viscosity of Diluted Liquid Dispersions Interfacial Rheology Exercises Suggestions for Further Reading

Structure and Dynamics of Biomembranes Electrochemical Properties of Biomembranes Transport in Biological Membranes The Transmembrane Potential Exercises Suggestions for Further Reading

Bioadhesion 20.1 20.2 20.3 20.4

A Qualitative Description of Bioﬁlm Formation Biological Surfaces Physicochemical Models for Cell Deposition and Adhesion General Thermodynamic Analysis of Particle Adhesion Exercises Suggestions for Further Reading

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Stem Cell Applications

The picture shows a stem cell from bone marrow adhered to a solid substratum. Stem cells may be applied to surfaces of materials for medical prostheses and artiﬁcial organs to improve biocompatibility. Adhesion involves close contact between surfaces and is therefore largely inﬂuenced by interfacial properties of both the cell and the substratum. Adhesion in biological systems discussed may be based on principles and concepts from colloid and interface science. (Figure courtesy of IPF, Dresden, Germany.)

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

1 Introduction

Life sciences deal directly or indirectly with biological systems. Disciplines such as biology, physics, chemistry, pharmacology, materials science, engineering, and so on are combined with the aim of improving the quality of agricultural, environmental, biotechnological, and biomedical processes and products as well as that of foodstuffs and pharmaceuticals. In these areas colloids and interfaces are omnipresent. Living systems are heterogeneous, made up largely of proteins, polysaccharides, and other polyelectrolytes, and self-assembled amphiphilic molecules, all contained in an aqueous medium. Many processes controlling life occur at interfaces. For example, the biological membrane itself is a selfassembled structure of mainly phospholipids and (glyco-)proteins. Glycoproteins on the cell wall participate in cellular aggregation and cellular growth; proteins of the reticuloendothelial system are likely to be involved in phagocytosis; the cytochrome enzyme system for oxidative phosphorylation is bound to the mitochondrion membrane and membrane proteins of chloroplasts have been shown to mediate in energy transfer processes during photosynthesis. Aggregation of protein molecules may be held responsible for the malfunctioning of cellular processes. For instance, ‘‘conformational diseases’’ such as BSE, Creutzfeldt–Jacob, and Alzheimer’s seem to be related to the aggregation of prion-like or amyloid proteins. Living or biological systems are often brought into contact with surfaces of synthetic materials, for instance, in biomedical applications (cardiovascular and other implants, hemodialysis, teeth and dental restoratives, voice prostheses, contact lenses, drug targeting, controlled release systems, and medical diagnostics), in biocatalysis (immobilization of enzymes or complete biological cells in bioreactors), food processing (e.g., heat exchangers and separation membranes), and offshore activities (ship hulls, desalination units).

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2

Chapter 1

As a rule, certain components of biological ﬂuids tend to become accommodated at the surface. The adsorption at the surface may induce structural rearrangements in the biomolecules with a concomitant change in their biological functioning. Almost all agricultural and industrial products, especially foodstuffs, contain colloidal structures that determine their rheological properties and textures. Manufacturers must control these structures in order to provide their products with the desired quality. In environmental science and technology we often deal with colloidal systems. For instance, clay and other soil constituents are of a colloidal nature and their state of aggregation affects the soil’s fertility. Sludge in waste water is colloidal material and efﬁcient waste water puriﬁcation requires knowledge of colloid science. All these examples, and many more, illustrate that colloids and interfaces have a great impact on our daily lives.

1.1 THE COLLOIDAL DOMAIN Colloids have been deﬁned classically as systems involving characteristic length scales ranging from a few to a few thousand nanometers. This dimensional range has attracted relatively little scientiﬁc interest. The two far ends of our perception of space, the subatomic elementary particles on one side and the universe with its galaxies at the other, have been much more appealing to scientists. Figure 1.1 shows the colloidal domain on a logarithmic length scale. It is situated between the microworld of atoms and molecules and the macroworld of biological and technological systems involving organisms and products. Colloidal dimensions may therefore be classiﬁed as mesoscopic. As said, for years colloids have suffered from stepmotherly scientiﬁc attention but during recent decades scientiﬁc interest has greatly increased, often under the cloak of ‘‘mesoscopic physics’’ and ‘‘nanotechnology.’’ Colloidal particles, having intermediate dimensions, possess characteristics pertaining to both the molecular and the macroscopic worlds. At the lower limit, particles of, say, a few nanometers dispersed in thermal motion

gravitational motion

meter 10–12 10–10 10–8 10–6 molecular sciences quantum physics

colloid sciences

10–4 10–2

100

engineering materials science medical sciences (micro) biology

102

104

106

108

geology environmental sciences

Figure 1.1 Length scales of various scientiﬁc disciplines.

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1010

1012

cosmology

Introduction

3

a liquid medium behave at ﬁrst sight as do molecular solutions. However, such colloidal solutions show strongly reduced colligative properties, such as osmotic pressure and freeze point suppression, as compared to solutions of regular-sized molecules of the same mass concentration. The upper limit for the particle size in a colloidal system is marked by the point where thermal (Brownian) motion, which tends to keep the particles in a dispersed state, is superseded by the gravitational force, which tends to segregate and settle the particles from solution. Thus, colloidal behavior is based on a delicate balance of the intrinsic thermal motion and external forces that act upon and between the particles. External forces are usually stronger for larger particles. For instance, the gravitational force between two bodies is proportional to the respective masses and the electric force is proportional to the total charge on the interacting bodies. Therefore, particles of very small (molecular) dimensions are barely inﬂuenced by external forces and their dynamics are essentially determined by thermal motion. At a given temperature, the energy of thermal motion has a ﬁxed value, independent of particle size. Hence, for the larger particles thermal motion becomes irrelevant; its effect is negligible in comparison with the effects of external forces. The colloidal, or mesoscopic, size is characterized by its sensitivity to both thermal motion and external forces. According to this approach colloidal systems encompass much more than dispersed particles. Systems that have colloidal dimensions in only one or two directions belong to the colloidal domain as well. In view of its intermediate position on the dimension scale, knowledge about matter on a mesoscopic level is a requirement for understanding macroscopic phenomenological behavior and characteristics in terms of molecular properties and interactions. In many, if not most, biological and technological systems molecules are clustered together thereby forming intriguing structures of mesoscopic dimensions. Examples are found in living cells, food products, pharmaceuticals, soil, and many biotechnological and biomedical appliances (see Figure 1.2). Thus from this point of view the colloidal domain takes an intermediate position as well: it is the interdisciplinary ﬁeld where chemistry, physics, biology, and engineering meet.

meter 10–9

10–8

micelles

10–7

biopolymers micro-emulsions biomembranes

virusses phages liposomes

10–6

10–5

emulsions bacteria (blood) cells

foam

Figure 1.2 Biological systems belonging to the colloidal domain.

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4

Chapter 1

1.2 INTERFACES ARE CLOSELY RELATED TO COLLOIDS A colloidal system is heterogeneous: one phase is ﬁnely dispersed in another, continuous, phase. Because of the characteristic dimensions of the dispersed phase, colloidal systems expose a large interfacial area. For instance, one liter of a colloidal dispersion in which the volume fraction of colloidal particles of 10 nm in diameter is only 1% contains an interfacial area of 6000 m2. This is about the size of two soccer ﬁelds! Interfaces are the seat of excessive Gibbs energy and this gives rise to various interfacial phenomena, such as interfacial tension, wetting, adsorption, and adhesion. The resulting interfacial properties govern the interactions between colloidal particles and therewith the macroscopic behavior and characteristics of a colloidal system, such as its rheological and optical properties and its stability against aggregation.

1.3 COLLOID AND INTERFACE SCIENCE IN A HISTORICAL PERSPECTIVE Empirical knowledge of colloidal and interfacial phenomena was already used in medieval and premedieval times. In ancient civilizations around the world colloidal pigments were used to produce paintings and writings. Colloidal systems were made use of in papermaking, pottery, cheese making, beer brewing, and various other crafts and arts. The lubricating effect of covering surfaces with a greasy substance was known and applied to facilitate transport of giant stones used for the construction of monumental buildings. Seafarers poured oil on the water to damp the waves. The scientiﬁc approach of interfacial phenomena started in the second half of the eighteenth century with Franklin’s reports (1765) on the amount of oil needed to cover the surface of Clapham Pond in England. Later, in the nineteenth century, Lord Rayleigh pursued these experiments, and Pockels and Langmuir did the ﬁrst quantitative studies on the properties of monolayers of surface active substances in liquid–air interfaces. Colloidal dispersions were ﬁrst described by Selmi (1845). He called these dispersions ‘‘pseudosolutions’’ and explained the anomalous colligative properties by assuming that the ‘‘dissolved’’ entities or particles were much larger than regular-sized molecules, so that at a given concentration (in mass per unit volume) the particle concentration was extremely low. In 1861 the name ‘‘colloids’’ (from the Greek kolla, which means glue) was assigned to the particles in Selmi’s pseudosolution. By choosing this name, Graham intended to emphasize the low rate of diffusion indicating a particle size of, at least, a few nanometers in diameter. After discovering and explaining some typical phenomena of colloidal dispersions during the last half of the nineteenth century, the distinction between

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Introduction

5

hydrophobic and hydrophilic (if water is not the continuous medium: lyophobic and lyophilic) colloids was clearly formulated at the beginning of the twentieth century (Perrin; Ostwald). In hydrophobic colloidal systems the water molecules have a higher afﬁnity for one another than for the colloidal particles. Consequently, the particles stick together on each encounter, unless they repel each other. It was discovered by Schulze (1883) that addition of electrolyte destabilizes hydrophobic colloidal dispersions, and Hardy (1900) showed that the destabilization is accompanied by a reduction in the electrophoretic mobility of the particles. From this it was inferred that colloidal stability is maintained by electrostatic repulsion between charged particles. The behavior of hydrophilic colloids is quite different. Hydrophilic colloids often, but not always, are electrically charged as well. Addition of electrolyte lowers the electrophoretic mobility, but it usually does not lead to destabilization of the colloidal dispersion. The obvious explanation is that the particles interact favorably with water molecules. The particles are surrounded with a layer of hydration water that keeps the particles apart. Then, between 1920 and 1950 insights into the mechanisms determining colloidal stability changed drastically, both for hydrophobic and hydrophilic colloids. Lowering of the electrostatic repulsion between the particles resulting from the addition of electrolyte was no longer ascribed to discharging the particles, but to compression of the electrical double layer adjacent to the charged particle surface (Kruyt, 1934). It was further made clear by De Boer (1936) and Hamaker (1937) that, in addition to electrostatic interaction, dispersive (London–Van der Waals) interaction makes an important contribution to the overall interaction between particles of colloidal dimensions. These reﬁnements became the basis for Derjaguin and Landau (1941) and Verwey and Overbeek (1948) to formulate a quantitative theory for the stability of hydrophobic colloids, known as the DLVO-theory. With respect to hydrophilic colloids, the concept of hydration water combined with the usually observed extraordinary high viscosity of the dispersion led to the suggestion that hydrophilic colloids contain many volumes of water per unit volume of dry material (Kruyt and Bungenberg de Jong, 1920s). It was gradually agreed upon that hydrophilic colloidal dispersions were just solutions of giant molecules having molar masses of several thousands to over a million Daltons, rather than agglomerates of smaller, ‘‘normal-sized’’ molecules. Thus, hydrophilic colloidal particles are water-soluble macromolecules, often polymers. These may be synthetic polymers, but also various natural ones (i.e., biopolymers, such as proteins, polysaccharides, and nucleic acids). Most of these polymers are made up of very long chains, the backbone, to which relatively short side groups may be attached. Depending on the mutual interactions within and between the polymers, and between the polymer and the water molecules, the macromolecular chain adopts a more or less ﬂexible structure. The high viscosity often observed

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6

Chapter 1

for polymer solutions can be easily explained by the enclosure of a relatively large amount of water by the coil-like open structure of the polymer molecule (a ratio of 99% water and 1% macromolecular material is quite normal), whereas only a small fraction of that water is bound to the macromolecule to prevent it from aggregation. The recent advancements in macromolecular chemistry (Flory, 1950–1980) and physics (de Gennes, 1970–2000) have greatly contributed to our understanding of hydrophilic colloids. The strongly swollen polymer molecules already overlap each other at a rather low concentration. This results in a complex network of entangled chains that may form intermolecular crosslinks. Such systems are referred to as gels. They are more or less rigid, although they usually consist of more than 90% solvent that is entrapped in the network. A well-known example is gelatin-gel. Thus, gels are more often than not built from hydrophilic colloids, but hydrophobic gels exist as well. Hydrophobic gels may be formed when hydrophobic colloidal particles aggregate into loose open structures. Another group of colloids that deserve special attention is the so-called ‘‘association colloids.’’ Association colloids are formed by amphiphilic molecules. A part of each amphiphilic molecule interacts favorably with solvent, whereas another part is disliked by the solvent molecules. In an aqueous environment, beyond a certain rather sharply deﬁned concentration these molecules aggregate spontaneously to form supramolecular structures of colloidal dimensions. These structures may attain all kinds of forms—spheres, cylinders, sheets, and so on—or combinations thereof. Pioneering studies on such systems were done during the ﬁrst half of the twentieth century by McBain, Hartley, and Harkins among others. Association colloids, or, as they are alternatively called, self-assembled structures, are more complex than the other colloidal dispersions. The reason is that a balance of different interactions determines the association pattern. Their occurrence, size, and shape may respond to changes in the surrounding conditions. In living systems self-assembled amphiphilic molecules are found in organized structures that often play roles of vital importance. The biological membrane is a pronounced example. Currently, self-assembled structures, often referred to as ‘‘soft condensed matter’’ are a popular area of research for colloid scientists, biophysicists, and molecular biologists.

1.4 CLASSIFICATION OF COLLOIDAL SYSTEMS We have already pointed out the difference between hydrophobic and hydrophilic colloids. Closely related to this distinction is the division into reversible and irreversible colloids. Reversible colloids are thermodynamically stable, which means that they are, at constant temperature and pressure, in their state of minimum Gibbs energy.

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Introduction

7

Table 1.1 Classiﬁcation of irreversible colloidal dispersions Dispersed material Medium

Solid (S)

Solid (S)

Solid suspension (bone, wood, various composite materials) Sol, Suspension (blood, polymer latex, paint, ink)

Liquid (L)

Gas (G)

Liquid (L) Solid emulsion (opals, pearls)

Emulsion (milk, rubber, crude oil, shampoo, mayonnaise) Aerosol ( fog, sprays)

Aerosol (smokes, dust)

Gas(G) Solid foam (loofah, bread, pumice, styrofoam) Foam (detergent foam, beer foam)

Table 1.2 Examples of reversible and irreversible colloids Reversible colloids Solution of (bio)polymers

Hydrophilic gels Association colloids

Examples Various body ﬂuids, such as blood, digestive juices, lachrymal ﬂuid; fruit juices; waste water Gelatin-gel; Sephadex and other matrices for, e.g., gel permeation chromatography Detergents; microemulsions; vesicles and liposomes; biological membranes

Irreversible colloids Hydrophobic sols and suspensions Hydrophobic gels Emulsions

Foams

Aerosols

Examples Fine dispersions of metals, metal oxides and halogenides in soils and ground water, etc.; blood; paint and ink Silica-gel; iron oxide (Fe2O3)-gel Milk and other dairy products; sauces; globules of alimentary fats in the duodenum; crude oil Beer foam, froth in a bioreactor, (shaving)soap, whipped cream, foam concrete Smokes; dust; clouds and fog; sprays

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8

Chapter 1

It implies that the dispersion is formed spontaneously upon mixing the components. Solutions of macromolecules (hydrophilic colloids) and association colloids belong to the reversible colloids. Irreversible colloids are thermodynamically unstable. The material of which the particles are composed is not soluble in the surrounding medium. Hence, irreversible colloids are not spontaneously formed upon mixing; special tricks are required to prepare (and maintain) them. The colloidal state of irreversible colloids is only metastable: it requires an activation energy to transfer the system to the thermodynamically stable state, which is the state where the particles are segregated to minimize the contact area between them and the surrounding medium. The activation energy may be so high that the dispersion is colloidally stable for prolonged periods of time, say, days or years. Clearly, hydrophobic colloids are irreversible colloids. Because irreversible colloids are a ﬁne dispersion of one phase (the dispersed phase) in another, continuous, phase (the medium), this group of colloids may be classiﬁed on the basis of its constituting phases, as is done in Table 1.1. Some examples of reversible and irreversible colloids are summarized in Table 1.2, but many more could be given.

SUGGESTIONS FOR FURTHER READING A. W. Adamson, A. P. Gast. Physical Chemistry of Surfaces, 6th edition, New York: John Wiley, 1997. D. F. Evans, H. Wennerstro¨m. The Colloidal Domain, Berlin: VCH Publishers, 1994. P. C. Hiemenz, R. Rajagopalan. Principles of Colloid and Surface Chemistry, 3rd edition, New York: Marcel Dekker, 1997. R. J. Hunter. Foundations of Colloid Science, 2nd edition, New York: Oxford University Press, 2001. J. N. Israelachvili. Intermolecular and Surface Forces, 2nd edition, London: Academic, 1992. H. R. Kruyt (ed.). Colloid Science I Irreversible Systems; II Reversible Systems, Amsterdam: Elsevier, 1949, 1952. J. Lyklema. Fundamentals of Interface and Colloid Science: I Fundamental; II Solid– Liquid Interfaces; III Fluid–Liquid Interfaces, London: Academic, 1992; 1995; 2000. C. J. van Oss. Interfacial Forces in Aqueous Media, New York: Marcel Dekker, 1994.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Von Willebrand Disease

Blood platelets of patients suffering from the Von Willebrand disease are difﬁcult to activate. They do adhere to damaged blood vessel walls, but without changing their shapes (top picture). Healthy platelets adhering to damaged arterial walls form irregular shapes with all kinds of protrusions allowing embracements of erythrocytes to form a blood clot (bottom picture). (Figure courtesy of School of Medicine, Wayne State University, Detroit, MI, U.S.A.)

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

2 Colloidal Particles: Shapes and Size Distribution

Cytoplasm, blood, and other bioﬂuids, milk, margarine, fruit juices, mayonnaise, bread, beer foam and whipped cream, pesticides, soils, pharmaceutical and cosmetic creams and lotions, paper, paint and ink, detergents, lubricants, and so on and so forth, are colloidal systems. The shape, size, and size distribution of the colloidal particles strongly inﬂuence several macroscopic properties of those systems, such as aggregation, sedimentation, rheological behavior, and optical properties. These characteristics are discussed in the following chapters. For instance, sensory properties such as texture and optical appearance are of utmost importance for the quality of food products. In living systems, structures and shapes are strongly related to biological functioning. Examples are the inﬂuence of the (local) curvature on the barrier properties of biological membranes, the change of shape of blood platelets in the blood clotting process, and the dependence of enzymatic activity on the three-dimensional structure of protein molecules.

2.1 SHAPES The colloidal domain contains particles having all kinds of shapes. The geometrically simplest shape is the sphere. The size and shape of a sphere is described by only one parameter: its radius. Spherical particles are found, for example, in emulsions, (synthetic) latexes, vesicles, liposomes, some bacterial cells, and some globular proteins. Ellipsoidally shaped particles are found among

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12

Chapter 2

various globular proteins and bacterial cells. Long cylinders are observed for, for instance, collagen and (double stranded) DNA. Erythrocytes have disc-like shapes and biological cells such as leucocytes and ﬁbroblasts are readily deformable and may adapt their shape in response to the environment. Various minerals, such as bentonites and zeolytes, form rigid sheets and the shape of biological membranes may be approached as ﬂexible lamellae. Thread-like structures are observed for some inorganic colloids, such as the rigid ’’needles‘‘ of vanadium pentoxide (V2O5) particles, and for long polymer chains which in general are more or less ﬂexible. Many inorganic hydrophobic colloids are polycrystalline particles having irregular shapes and, as a ﬁrst approximation, such particles are often treated as having a regular shape, for example, a sphere, cylinder, and the like. Solid particles are essentially nondeformable. Most reversible colloids have a more or less ﬂexible three-dimensional structure; as a result of thermal motion their shape may vary in time. Examples are the (random) coil structure of a dissolved polymer molecule and the undulation of the lamellar structure of phospholipid bilayers. A few colloidal systems with different forms of the above-mentioned structures are depicted in Figure 2.1.

2.2 PARTICLE SIZE DISTRIBUTIONS Various properties of a colloidal system, such as its stability and rheological and optical characteristics, are directly related to the size of the particles. For many colloidal systems the particles vary in size; such systems are called heterodisperse or polydisperse. Almost all synthetic colloids are heterodisperse. In a few cases the particles in a colloidal system are all of the same size. Such systems are referred to as homodisperse or monodisperse. Examples are solutions of one type of protein, or synthetic colloids that are prepared under specially controlled conditions. Of course, heterodisperse systems may be fractionated to yield more or less homodisperse fractions. In a heterodisperse system there is a particle size or, for that matter, particle mass distribution around an average value. We now further elaborate on the particle mass distributions within a heterodisperse population. Let M be the molar mass (kg mole1); then M ¼ NAvm, where NAv is Avogadro’s number and m the mass per particle. A heterodisperse system may be considered as a series of fractions i (i ¼ 1, 2, 3, . . .), each fraction containing particles of molar mass Mi (or particle mass mi). Each fraction is approximated as being homodisperse, so that Mi is well deﬁned. For each fraction the number of particles ni is related to the total mass wi, according to ni mi ¼ ni Mi =NAv ¼ wi :

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ð2:1Þ

Figure 2.1 Different shapes of biocolloidal particles: (a) erythrocytes from human blood, responsible for transporting oxygen from the lungs to all cells as well as removing carbon dioxide as a product of cell oxidation; (b) Staphylococcus aureus, a pathogenic bacterium causing infections of the skin; (c) trehalose particles, used as a protecting agent in freeze-drying sensitive (biological) materials (courtesy of Ytkemiska Institutet AB, Stockholm, Sweden); (d) hydroxyapatite crystals, the matrix material of dental enamel and dentine (courtesy of Osaka Dental University, Japan); (e) tobacco mosaic virus (courtesy of Department of Plant Sciences, Wageningen University, The Netherlands); (f) fat globules in (creamed) milk (courtesy of Nizo Food Research, Ede, The Netherlands). (The scales of the pictures do not correspond to each other.)

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14

Chapter 2

The values for wiNAv=N are, according to (2.1) calculated as niMi=N, and those for wi=W by dividing wiNAv=N by WNAv=N, where N is the total number of particles and W their total mass. The values in the last two columns of Table 2.1 give the cumulative number distribution Nk and the cumulative mass distribution Wk, which are obtained by adding up the numbers, respectively, masses of the fractions 1, 2, 3, . . . , k: Nk ¼

k X

ni

and

Wk ¼

i¼1

k X

wi :

ð2:2Þ

i¼1

Thus Nk is the total number of particles below a certain mass and Wk is the corresponding total mass. The data collected in Table 2.1 may be graphically represented: ni(Mi) or for that matter ni(Mi)=N gives a number histogram and wi(Mi) a mass histogram. Similarly, Nk(Mk) and Wk(Mk) give cumulative number and mass histograms. These histograms are shown in Figure 2.2. As compared to the number distributions, the mass distributions are shifted towards higher values of i, and, hence of Mi, because wi and Wk are derived from ni and Nk by multiplying with Mi resp., Mk , and the value for the molar mass increases with increasing fraction number i (see Table 2.1). Thus in the number distribution each particle contributes equally, whereas in the mass distribution the particles contribute proportionally to their mass. In the foregoing we have considered the heterodisperse system as composed of a number of distinct homodisperse fractions. However, as a rule, the distributions are continuous. This is approximated by taking Mi þ 1 7 Mi ( dM) inﬁnitesimally small; n(M) is the number of particles having a molar mass between M and M þ dM. The function n(M) is called the (differential) Table 2.1 Mass distribution in a heterodisperse system i 1 2 3 4 5 6 7 8 9 10

ni

Mi

ni=N

wiNAv=N

wi=W

Nk=N

Wk

40 60 90 110 150 200 200 100 40 10

10 20 30 40 50 60 70 80 90 100

0.04 0.06 0.09 0.11 0.15 0.20 0.20 0.10 0.04 0.01

0.4 1.2 2.7 4.4 7.5 12.0 14.0 8.0 3.6 1.0

0.007 0.021 0.049 0.080 0.137 0.219 0.255 0.146 0.066 0.018

0.040 0.100 0.190 0.300 0.450 0.650 0.850 0.950 0.990 1.000

0.007 0.028 0.077 0.157 0.294 0.515 0.770 0.916 0.982 1.000

1.00

54.8

1.000

N ¼ 1000

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Colloidal Particles: Shapes and Size Distribution

15

Figure 2.2 Histograms for the number and mass distributions of particles in a heterodisperse system. The histograms correspond to the data in Table 2.1. The curves in (a) and (b) represent the differential distribution functions and those in (c) and (d) cumulative distribution functions. The shaded area in Figure 2.2(b) gives the cumulative mass of all particles with M < 50 kg mol1 corresponding to the dot in Figure 2.2(d).

number distribution function. Note that n(M)dM is a number, and, because M is expressed in kg mole1, n(M) has the dimension mole kg1. In a similar way, w(M)dM is the mass of the particles having a molar mass between M and M þ dM; w(M) is the (differential) mass distribution function, having the dimension mole. Analogous to (2.1) nðM ÞM =NAv ¼ wðM Þ:

ð2:3Þ

The cumulative distribution functions N(M) and W(M) are ðM N ðM Þ ¼

ðM nðM ÞdM

and

0

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W ðM Þ ¼

wðM ÞdM : 0

ð2:4Þ

16

Chapter 2

The curves in Figure 2.2 are approximations of the differential distribution curves, pertaining to the system given in Table 2.1. Clearly, according to (2.4), N(M) and W(M) can be obtained by integrating the differential distribution curves. For instance, the cumulative mass of all particles with M < 50 is given by the shaded area in Figure 2.2(b). It corresponds to one point on the curve in Figure 2.2(d). Conversely, n(M) and w(M) may be derived by differentiation of N(M) and W(M). It goes without saying that completely analogous reasoning leads to the establishment of distributions of other variables, such as the particle size. To determine number and mass distributions, the heterodisperse systems are usually fractionated (e.g., by size exclusion chromatography or by sedimentation). In some cases the distribution functions can be derived from the theory of particle preparation.

2.3 AVERAGE MOLAR MASS If a heterodisperse system is subjected to a measurement to determine the molar mass (or the particle mass), an average value for that mass is obtained. However, for one and the same system different averages may result. The kind of average depends on the type of measurement. When the measured quantity is proportional to the number of particles (as is the case for colligative properties such as osmotic pressure, boiling point elevation, freeze point depression, etc.) a number average mass is obtained, but when it scales with the particle mass (as in light scattering) a mass average mass is derived. In addition to these two, other types of averages are distinguished. The number average particle mass mn is deﬁned as the total P mass of P the particles in the system divided by the number of particles, mn inimi= i ni . Hence, for the number average molar mass Mn, P P P C i ni Mi =NAv i ci M i Mn ¼ P ¼ P ¼P i i : i ni =NAv i ci i Ci =Mi

ð2:5Þ

Note that ci is the molar concentration (mole m3) and Ci is the mass concentration (kg m3) of i; Ci and ci are related through Ci ¼ ciMi. Applying (2.5) to the system of Table 2.1 yields Mn ¼ 54.8, derived by dividing P P P i wi NAv =N ð¼ i ni Mi =N Þ by i ni =N. P P The mass average molar mass Mw is deﬁned as i wi Mi = i wi , so that, combined with (2.1), P P P n M 2 =N c M2 i Ci Mi : Mw ¼ Pi i i Av ¼ Pi i i ¼ P n M =N c M Ci Av i i i i i i

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ð2:6Þ

Colloidal Particles: Shapes and Size Distribution

17

The Z-average molar mass (Z refers to the German word for centrifuge, i.e., Zentrifuge) is derived from equilibrium sedimentation. It is deﬁned as P P P n M 3 =N c M3 C M2 ð2:7Þ Mz ¼ Pi i i2 Av ¼ Pi i i2 ¼ Pi i i : i C i Mi i ni Mi =NAv i ci M i In the calculation of Mn all particles are counted once. In Mw, and, even more so in Mz, the particles having the higher molar mass are more strongly represented. For continuous distributions, Eqs. (2.5) through (2.7) are transformed into ð1 ð1 MnðM ÞdM = nðM ÞdM ; ð2:8Þ Mn ¼ 0

ð1 Mw ¼

0

ð1 M nðM ÞdM = MnðM ÞdM;

ð2:9Þ

2

0 ð1

Mz ¼

0 ð1

M 3 nðM ÞdM = 0

M 2 nðM ÞdM :

ð2:10Þ

0

An average molar mass for macromolecules is often inferred from viscosity measurements. Chapter 17 on rheology discusses the so-called intrinsic viscosity of a macromolecular solution which scales with Ma, with 0.5 < a < 0.8 depending on the macromolecule–solvent interaction. The viscosity average molar mass is deﬁned as P 1=a Ci Mia i P : ð2:11Þ Mv ¼ i Ci It follows that Mn Mv Mw. Obviously, a homodisperse system has only one average molar mass Mn ¼ Mw ¼ Mz ¼ Mv ¼ : For heterodisperse systems the various kinds of averages differ with higher degrees of heterodispersity. Therefore the ratio Mw=Mn is commonly taken as a measure of the heterodispersity of the system. For the system in Table 2.1 values of 54.8, 62.5, and 67.7 are calculated for Mn, Mw, and Mz, respectively, so that Mn : Mw : Mz ¼ 1 : 1.14 : 1.24. This is considered to be a rather narrow distribution. In comparison, the most likely distribution of the molar mass of a polymer formed by a polycondensation process is Mn : Mw : Mz ¼ 1 : 2 : 3.

2.4 SPECIFIC SURFACE AREA The speciﬁc surface area of a dispersion is deﬁned as the surface area per unit mass of dispersed material. The surface-to-volume ratio is larger for smaller

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18

Chapter 2

particles and, therefore, the speciﬁc surface area is larger for a more ﬁnely dispersed system. For a system containing n homodisperse spheres of radius a and density r, the speciﬁc surface area S is calculated as S¼

n4pa2 3 : ¼ nð4=3Þpa3 r ra

ð2:12Þ

In the case of a heterodisperse system S is calculated by summing up the surface areas and the masses of each (homodisperse) fraction, so that P n a2 3 S ¼ Pi i i3 : ð2:13Þ r i ni ai Conversely, if the speciﬁc area of a heterodisperse system is (experimentally) known, an average radius a may be derived according to a ¼ 3=rS. Here, a is the volume=surface average radius, deﬁned as P n a3 a ¼ Pi i i2 ð2:14Þ i ni ai

EXERCISES 2.1

A micrograph of an o=w emulsion (oil droplets dispersed in water) shows 28 oil droplets. The droplets are classiﬁed in ﬁve size-fractions, as indicated in the ﬁgure (ni is the number of particles in fraction i and di the droplet diameter). The density of the oil is 0.9 g cm3 and that of water 1.0 g cm3

(a)

Give expressions for the number average diameter dn and the mass average diameter dw. Calculate the speciﬁc interfacial area (m2 g1) of the oil droplets.

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Colloidal Particles: Shapes and Size Distribution

(b)

2.2

19

In the fractions with di ¼ 4 mm and di ¼ 5 mm ﬁve droplets are counted. Which percentage of the total interfacial area is represented by these two fractions and which percentage of the total volume?

Consider a colloidal system with a continuous molar mass distribution between M1 and M2 ¼ xM1 (with x 1). Each molar mass is represented by the same number of particles and there are no particles with a molar mass outside the interval between M1 and M2. (a)

Show graphs for the number distribution function n(M) and the mass distribution function w(M) (b) Express the number average molar mass Mn and the mass average molar mass Mw in terms of M1 and M2 (c) Derive Mw 4 1 þ x þ x1 : ¼ Mn 3 2 þ x þ x1 For the given molar mass distribution, what is the maximum variation in Mw=Mn? Give a graph for the number distribution function n(M) in case of the maximum value for Mw=Mn. (d) The most probable distribution resulting from a polycondensation process is characterized by Mw=Mn ¼ 2. Is this distribution included in the one given above? Give a qualitative indication of the number distribution for which Mw=Mn ¼ 2.

SUGGESTIONS FOR FURTHER READING T. Allen. Particle Size Measurement, Boca Raton, FL: Chapman and Hall, 1997. H. G. Barth (ed.). Modern Methods of Particle Size Analysis, New York: WileyInterscience, 1984. S. Hyde, S. Andersson, K. Larsson, Z. Blum, S. Lnadh, S. Lidin, B. W. Ninham. The Language of Shape, Amsterdam: Elsevier, 1997. E. Kissa. Dispersions. Characterization, testing and measurement, in Surfactant Science Series 84, New York: Marcel Dekker, 1999.

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The Thermodynamic Route

Thermodynamics is a powerful intellectual method for solving problems. The problem is taken from the ‘‘real physical world’’ into the ‘‘world of the mind’’ by introducing abstract concepts such as energy, entropy, chemical potential, and so on. Then, in the abstract world of our minds, we use mathematics to work on the problem. The solution will consequently present itself in terms of these abstract notions. By way of example, the thermodynamic answer to the question of the equilibrium distribution of a compound partitioning between two phases is equal values for the chemical potential of that component in the two phases. Now, to bring back the thermodynamic solution into the real world we have to call upon model assumptions, such as, for instance, ideal behavior of a solution or a gas. Thus, by taking a thermodynamic detour, a large collection of mathematical relations between (experimentally) observable variables may be derived.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

3 Some Thermodynamic Principles and Relations, with Special Attention to Interfaces

In the following chapters thermodynamics is frequently applied to derive relations between macroscopic parameters. In writing this book it is assumed that the reader is familiar with the basics of thermodynamics of reversible processes. Nevertheless, this chapter is included as a reminder. It presents a concise summary of thermodynamic principles that are relevant in view of the topics discussed in forthcoming chapters, and special attention is paid to heterogeneous systems that contain phase boundaries.

3.1 ENERGY, WORK, AND HEAT. THE FIRST LAW OF THERMODYNAMICS Generally, when a system passes through a process it exchanges energy U with its environment (¼ rest of the universe). The energy change in the system DU may result from performing work w on the system or letting the system perform work, and from exchanging heat q between the system and the environment DU ¼ q þ w:

ð3:1Þ

The heat and the work supplied to a system are withdrawn from the environment, such that, according to the ﬁrst law of thermodynamics, DUsystem þ DUenvironment ¼ 0:

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ð3:2Þ

22

Chapter 3

The First Law of thermodynamics states that the energy content of the universe (or any other isolated system) is constant. In other words, energy can neither be created nor annihilated. It implies the impossibility of designing a perpetuum mobilae, a machine that performs work without the input of energy from the environment. The First Law also implies that for a system passing from initial state 1 to ﬁnal state 2 the energy change D1!2 U does not depend on the path taken to go from 1 to 2. A direct consequence of that conclusion is that U is a function of state: when the macroscopic state of a system is fully speciﬁed with respect to composition, temperature, pressure, and so on (the so-called state variables), its energy is ﬁxed. This is not the case for the exchanged heat and work. These quantities do depend on the path of the process. For an inﬁnitesimal small change of the energy of the system dU ¼ dq þ dw:

ð3:3Þ

(Note that the symbol d is used for the differential of a state function or state variable whereas d just indicates an inﬁnitesimal small amount of a quantity that is dependent on the path taken.) For w and, hence, dw, various types of work may be considered, such as mechanical work resulting from compression or expansion of the system, electrical work, interfacial work associated with expanding or reducing the interfacial area between two phases, and chemical work due to the exchange of Ðmatter between system and environment. All types of work are expressed as X dY , where X and Y are state variables. X is an intensive property (independent of the extension of the system) and Y the corresponding extensive property (it scales with the extension of the system). Examples of such combinations of intensive and extensive properties are pressure p and volume V , interfacial tension g and interfacial area A, electric potential c and electric charge Q, the chemical potential mi of component i, and the number of moles ni of component i. As a rule, X varies with Y but for an inﬁnitesimal small change of Y , X is approximately constant. Hence, we may write X dU ¼ dq pdV þ gdA þ cdQ þ m dni : ð3:4Þ i i The terms of type X dY in Eq. (3.4) representP mechanical (volume), interfacial, electric, and chemical works, respectively. i implies summation over all components in the system. It is obvious that for homogeneous systems the gdA term is not relevant.

3.2 THE SECOND LAW OF THERMODYNAMICS. ENTROPY According to the First Law of thermodynamics the energy content of the universe is constant. It follows that any change in the energy of a system is accompanied

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Some Thermodynamic Principles and Relations

23

by an equal, but opposite, change in the energy of the environment. At ﬁrst sight, this law of energy conservation seems to present good news: if the total amount of energy is kept constant why then should we be frugal in using it? The bad news is that all processes always go in a certain direction, a direction in which the energy that is available for performing work continuously decreases. This direction is described by the Second Law of thermodynamics. An evident example is that heat ﬂows from a region of high temperature to a region of low temperature and never in the opposite direction. Another example is that gas molecules in a container do not accumulate in a part of the available volume; on the contrary, they distribute themselves homogeneously over the entire volume. A similar phenomenon is observed after injecting a dye in a solvent. Less visible, but just as real, is the dispersion of, for example, exhaust gases and propellants in the atmosphere and of (waste) products in the soil, surface water, and oceans. A third example showing the direction of processes is the impossibility of allowing a system to perform work by extracting heat from its environment that has the same temperature as the system. If this were possible, a ball lying on the ﬂoor could lift itself, rivers could ﬂow uphill, and a car could drive without using fuel, and all this could occur because heat is taken from the environment. In common experience it is just the opposite: the directed coherent motion of the molecules of a falling ball is, upon hitting the ﬂoor, transformed into heat, that is, nondirected incoherent movement of molecules. The same applies to the friction between the directionally moving river and car and their respective environments. These examples are manifestations of one principle: the natural tendency of ‘‘things’’ (in our examples heat, matter, and coherence) to disperse. This is related to the tendency of storing the constant amount of energy of the universe in as many ways as possible. This is the quintessence of the Second Law of thermodynamics. Entropy, S, is the central notion in the Second Law. The entropy of a system is a measure of the number of ways the energy can be stored in that system. In view of the foregoing, any spontaneous process goes along with an entropy increase in the universe ð¼ system þ environmentÞ; that is, DS > 0. If as a result of a process the entropy of a system decreases, the entropy of the environment must increase in order to satisfy the requirement DS > 0. Based on statistical mechanics, the entropy of a system, at constant U and V can be expressed by Boltzmann’s law Su;v ¼ kB ln O;

ð3:5Þ

where O is the number of states accessible to the system and kB is Boltzmann’s constant. For a given state O is ﬁxed and, hence, so is S. It follows that S is a function of state. It furthermore follows that S is an extensive property: for a system comprising two subsystems (a and b) O ¼ Oa Ob and therefore, because of (3.5), S ¼ Sa þ Sb . The entropy change in a system undergoing

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24

Chapter 3

a process ð1 ! 2Þ is thermodynamically formulated in terms of the heat dq taken up by that system and the temperature T at which the heat uptake occurs: ð2 DS

dq : 1 T

ð3:6Þ

Because the temperature may change during the heat transfer (3.6) is written in differential form.

3.3 REVERSIBLE PROCESSES. DEFINITION OF INTENSIVE VARIABLES In contrast to the entropy, heat is not a function of state. For the heat change it matters whether a process 1 ! 2 is carried out reversibly or irreversibly. For a reversible process, that is, a process in which the system is always fully relaxed, ð2 DS ¼

dqrev : 1 T

ð3:7Þ

Inﬁnitesimal small changes imply inﬁnitesimal small deviations from equilibrium and, therefore, reversibility. The term dq in (3.4) may then be replaced by T dS, which gives X dU ¼ T dS pdV þ gdA þ cdQ þ m dni ; ð3:8Þ i i where all terms of the right-hand side are now of the form X dY . Equation (3.8) allows the intensive variables X to be expressed as differential quotients, such as, for instance, @U ; g¼ @A S;V ;Q;n0 s

ð3:9Þ

i

where the subscripts indicate the properties that are kept constant. In other words, the interfacial tension equals the energy increment of the system resulting from the reversible extension of the interface by one unit area under the conditions of constant entropy, volume, electric charge, and composition. The required conditions make this deﬁnition very impractical, if not inoperational. If the interface is extended it is very difﬁcult to keep variables such as entropy and volume constant. The other intensive variables may be expressed similarly as the change in energy per unit extensive property, under the appropriate conditions.

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Some Thermodynamic Principles and Relations

25

3.4 INTRODUCTION OF OTHER FUNCTIONS OF STATE. MAXWELL RELATIONS At equilibrium, implying that the intensive variables are constant throughout the system, (3.8) may be integrated, which yields X U ¼ TS pV þ gA þ cQ þ mn: ð3:10Þ i i i To avoid impractical conditions when expressing intensive variables as differential quotients as, for example, in (3.9), auxiliary functions are introduced. These are the enthalphy H, deﬁned as H U þ pV ;

ð3:11Þ

the Helmholtz energy F U TS;

ð3:12Þ

and the Gibbs energy G U þ pV TS ¼ H TS ¼ F þ pV :

ð3:13Þ

Since U is a function of state, and p, V , T , and S are state variables, H, F, and G are also functions of state. The corresponding differentials are X m dni ; ð3:14Þ dH ¼ T dS þ V dp þ gdA þ cdQ þ i i X dF ¼ SdT pdV þ gdA þ cdQ þ m dni ; ð3:15Þ i i X dG ¼ SdT þ V dp þ gdA þ cdQ þ m dni : ð3:16Þ i i Expressing g, c, or mi as a differential quotient requires constancy of S and V, S and p, T and V, and T and p, when using the differentials dU, dH, dF, and dG, respectively. In most cases the conditions of constant T and V or constant T and p are most practical. It is noted that for heating (or cooling) a system at constant p, the heat exchange between the system and its environment is equal to the enthalpy exchange. Hence, for the heat capacity, at constant p, dq dH Cp ¼ : ð3:17Þ dT p dT p In general, for a function of state f that is completely determined by variables x and y, df ¼ Adx þ Bdy. Cross-differentiation in df gives ð@A=@yÞx ¼ ð@B=@xÞy , known as a Maxwell relation. Similarly, cross-differentiation in dU, dH, dF, and

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26

Chapter 3

dG yields a wide variety of Maxwell relations between differential quotients. For instance, by cross-differentiation in dG we ﬁnd @g @S ¼ : ð3:18Þ @T p;A;Q;n0 s @A T ;p;Q;n0 s i

i

3.5 MOLAR PROPERTIES AND PARTIAL MOLAR PROPERTIES. DEPENDENCE OF THE CHEMICAL POTENTIAL ON TEMPERATURE, PRESSURE, AND COMPOSITION OF THE SYSTEM Molar properties, indicated by a lowercase symbol, are deﬁned as an extensive property Y per mole of material: y Y =n. Since they are expressed per mole, molar quantities are intensive. For a single component system Y is a function of T ; p; . . . ; n. Many extensive quantities vary linearly with n, but for some (e.g., the entropy) the variation with n is not proportional. In the latter case y is still a function of n. In a two-, three- or multi-component system (i.e., a mixture), the contribution of each component to the functions of state, say, the energy of the system cannot be assigned unambiguously. This is because the energy of the system is not simply the sum of the energies of the constituting components but includes the interaction energies between the components as well. It is impossible to specify which part of the total interaction energy belongs to component i. For that reason partial molar quantities yi are introduced. They are deﬁned as the change in the extensive quantity Y pertaining to the whole system due to the addition of one mole of ni under otherwise constant conditions. Because by adding component i the composition of the mixture and, hence, the interactions between the components are affected, yi is deﬁned as the differential quotient @Y yi : ð3:19Þ @ni T;p;...;nj6¼1 The partial molar quantities are P operational; that is, they can be measured. Now YT ;p;...;n0i s can be obtained as i ni yi . A partial molar quantity often encountered is the partial molar Gibbs energy, @G gi : @ni T;p;...;nj6¼i

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Some Thermodynamic Principles and Relations

27

According to (3.16), @G gi ¼ mi ; @ni T;p;...;nj6¼i

ð3:20Þ

that is, at constant T ; p; . . . ; nj6¼i , the chemical potential of component i in a mixture equals its partial molar Gibbs energy. By cross-differentiation in (3.16) the temperature- and pressure-dependence of mi can be derived as @mi @S ¼ ¼ si ð3:21Þ @ni T;p;...;nj6¼i @T p;...;n0 s i

with mi ¼ gi hi Tsi ;

ð3:22Þ

it can be deduced that @ðmi =T Þ h ¼ 2i : T @T 0 p;...;n s

ð3:23Þ

i

The pressure-dependence of mi is also obtained from (3.16): @mi @V ¼ ni : @p T ;...;n0 s @ni T ;p;...;nj6¼i

ð3:24Þ

i

For an ideal gas ni ¼

ð3:25Þ

in which < is the universal gas constant. Combining (3.24) and (3.25) gives, after integration, an expression for mi ð pi Þ in an ideal gas mi ¼ myi þ
ð3:26Þ pi ; myi

mi ðpi ¼ 1Þ, its where my is an integration constant that is independent of value depending on the units in which pi is expressed. Similarly, without giving the derivation here, it is mentioned that for the chemical potential of component i in an ideal solution mi ¼ moi þ
ð3:27Þ

where ci is the concentration of i in the solution. In more general terms, for an ideal mixture mi ¼ moi þ
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ð3:28Þ

P i

ni .

28

Chapter 3

Note that the value of moi is the one obtained for mi by extrapolating to Xi ¼ 1 assuming ideality of the mixture. This value deviates from the real value of mi for pure i, because in the case of pure i the ‘‘mixture’’ is as far as possible from ideal. As said, moi ; moi , and myi are deﬁned per unit Xi , ci , and pi, respectively, and their values are therefore independent of the conﬁgurations of i in the mixture. They do depend on the interactions between i and the other components and therefore on the types of substances in the mixture. Because Xi , ci , and pi are expressed in different units, the values for moi ; moi , and myi differ. The
which, because of (3.21), gives si ¼ soi < ln Xi :

ð3:30Þ

The partial molar entropy of i is composed of a part soi , which is independent of the conﬁgurations of i in the mixture but dependent on the interactions of i with the other component(s), and a part < ln Xi , which takes into account the possible conﬁgurations of i. It follows that the
3.6 CRITERIA FOR EQUILIBRIUM. OSMOTIC PRESSURE A system in equilibrium is fully relaxed; its state does not change in time. Of course, ﬂuctuations occur on a molecular level but these are random and do not result in changes in the macroscopic properties of the system. For instance, for the liquid–gas phase equilibrium of a component i molecules of i are continuously transferred between the two phases but there is no net transport of i from the one phase to the other. The requirements for a system to attain equilibrium depend on its interactions with the environment. As we have seen in Section 3.2, the Second Law of thermodynamics tells us that a process proceeds irreversibly if dS > dq=T and that equilibrium is reached when dS ¼ dq=T . The Second Law may be combined with expressions for U, H, F, or G to ﬁnd the appropriate criterion for equilibrium. Consider homogeneous closed systems, that is, systems that consist of one phase and that do not exchange matter or electric charge with the environment.

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Some Thermodynamic Principles and Relations

29

Equation (3.4) applied to such systems gives dU ¼ dq pdV . A special case is an isolated system that has no interaction whatsoever with its environment, implying that dq ¼ 0 and pdV ¼ 0. Thus, off-equilibrium, when T dS > dq, dS > 0 and at equilibrium, when T dS ¼ dq; dS ¼ 0. It follows that the entropy of an isolated system (U , V constant) in equilibrium is at a maximum. Systems are usually not isolated, but interact with their environment and they do so at certain speciﬁed conditions. In most cases these conditions are constant p and T or constant V and T . For instance, for heat exchange at constant V and T combining the expressions for dS and dF yields that off-equilibrium dF < 0 and at equilibrium dF ¼ 0. Hence, for a closed system at constant V and T to be in equilibrium, its Helmholtz energy F is at a minimum. Similarly, a closed system, at constant p and T, is at equilibrium when the Gibbs energy G is minimal. Let us now return to an isolated system (U , V , n0i s constant) which we divide in two parts a and b. This is schematically shown in Figure 3.1. Note that the isolated system ða þ bÞ may be understood as a nonisolated system a interacting with its environment b. The state variables U a ; V a , and nai can be independently chosen. However, because of the condition of constant U , V , b dnai ¼ dnbi . For a and b we n0i s: dU a ¼ dU b ; dva ¼ dV P a , and P can write a a a a a a dU ¼ T dS p dV þ i mi dni and dU b ¼ T b dS b pb dV b þ i mbi dnbi . For the isolated system ða þ bÞ in equilibrium we derived dS ¼ 0, so that

ðdSÞU ;V ;n0i s ¼

a 1 1 p pb a dU dV a þ Ta Tb Ta Tb ! X ma mb i i b dnai ¼ 0: i Ta T

U V

α α

α

β

U

ð3:31Þ

β

β

V

β

n iα

ni

T

T

Figure 3.1 Two phases, a and b, in an isolated system.

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30

Chapter 3

In general, U a ; V a , and nai can be varied independently and Eq. (3.31) therefore holds only if each of the terms on the right-hand-side is zero. Hence, Ta ¼ Tb T;

pa ¼ pb p;

mai ¼ mbi mi ;

ð3:32Þ

showing that for an isolated system in equilibrium each of the intensive variables T ; p, and mi has a constant value throughout the system. It may be clear that in the case where electrical work is involved (3.32) should be supplemented with ca ¼ cb and for heterogeneous systems ga ¼ gb . We stress again that the criteria for equilibrium as expressed in (3.32) apply only if the corresponding terms in (3.31) are independent of each other. Sometimes, however, these terms are not independent but coupled. An example of such coupling is the transport of ions. In an ion matter and charge are linked to each other and, hence, transporting ions involves both chemical and electrical work, represented by the terms mi dni , and cdQ. These terms are coupled since dQ ¼ zi Fdni (where zi is the valency of ion i and F is Faraday’s constant). Then the reasoning followed above leads to mai þ zi Fca ¼ mbi þ zi Fcb as the criterion for equilibrium with respect to ionic species i. Furthermore, in a system containing two (or more) components equilibrium may apply to one of the components but not to the other. Such a situation results when a solution is separated from the solvent by a boundary that is freely permeable for the solvent but that cannot be crossed by molecules of the solute. We show in forthcoming chapters that such semipermeable boundaries are often encountered in colloidal systems and near interfaces. Therefore, by way of example, we derive here how equilibrium between two solutions, a and b, containing different concentrations of solute, gives rise to a pressure inequality pa 6¼ pb. The difference pb pa is called the osmotic pressure. Consider a solvent a separated by a semipermeable membrane from a solution b, as schematically depicted in Figure 3.2. The membrane is permeable for the solvent (component 1) but impermeable for the solute (component 2). Phases a and b are in thermal equilibrium, that is, T a ¼ T b , and in equilibrium with respect to the distribution of solvent molecules: ma1 ¼ mb1 . Because X1a 6¼ X1b ; pa 6¼ pb in order to satisfy ma1 ¼ mb1. Hence, the inﬂuence on m1 by the lower value of X1 in phase b must be compensated by a pressure difference pb pa > 0. In mathematical terms: dm1 ðp; X Þ ¼ ð@m1 =@pÞX dp þ ð@m1 =@X ÞdX ¼ 0. With (3.24) and (3.28) [deﬁning X X2 and hence ð1 X Þ X1 ] we obtain

dp ¼

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ð3:33Þ

Some Thermodynamic Principles and Relations

pα X1=1

α

pβ X1

X2=0

X2

T

T

31

β

Figure 3.2 Semipermeable membrane separating a solution ðbÞ from its solvent ðaÞ.

For dilute solutions, that is, X 1 so that lnð1 X Þ may be approximated by X , this results in dp ¼

ð3:34Þ

Integration yields the osmotic pressure p, p ¼ pb pa ¼

ð3:35Þ

because in dilute solutions the molar concentration of component 2, c, can be approximated by X =n1.

3.7 PHASE EQUILIBRIA, PARTITIONING, SOLUBILIZATION, AND CHEMICAL EQUILIBRIUM 3.7.1 Phase Equilibria Matter exists in different phases, that is, states of aggregation. These are solid (S), liquid (L), and gas (G). The transition between the phases is indicated in a phase diagram. By way of example, Figure 3.3 shows the phase diagram for water. The curves in the diagram represent p; T -combinations at which the phases are in equilibrium. For instance, curve AB gives the vapor pressure for water, AD is the melting curve, and AC gives the vapor pressure of ice. For a component i each point of the curve representing equilibrium between two phases I and II is characterized by, mIi ð p; T Þ ¼ mIIi ð p; T Þ. For an inﬁnitesimal

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Chapter 3

p ×10 – 5 / P a

32

D

S

B

L

1

G

A C 273

373 T/K

Figure 3.3 Schematic representation of a phase diagram for water.

small trajectory along the curve: dmI ð p; T Þ ¼ dmII ð p; T Þ. As for both phases (3.21) and (3.24) apply, sIi dT þ nIi dp ¼ sIIi dT þ nIIi dp

ð3:36Þ

and, hence, dp sIIi sIi ¼ : dT nIIi nIi

ð3:37Þ

At equilibrium I ! II is reversible and therefore qrevðI!IIÞ dp ¼ : dT T ðnIIi nIi Þ

ð3:38Þ

Equation (3.38) is known as the Clapeyron equation. When applied to vaporization (i.e., when one of the phases, say, phase II, is the gas phase and, moreover, if the gas phase behaves ideally nIIi ¼
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð3:39Þ

Some Thermodynamic Principles and Relations

33

3.7.2 Partitioning and Solubilization For a component i partitioning between two phases a and b equilibrium requires mai ¼ mbi . If a and b are liquids and i is dissolved in a and b, we ﬁnd after applying (3.27) the partitioning coefﬁcient kð cbi =cai Þ as ln k ¼

o o Da!b go m;a m;b i i ¼
ð3:40Þ

and, using (3.23), for the inﬂuence of the temperature on the partitioning d ln k Da!b ho ¼ :
ð3:41Þ

Analogously, for the solubilization of i, ln ci ¼

moi mi D go ¼ dissolution ;
ð3:42Þ

where refers to pure i, and for the temperature-dependence of the solubility of i d ln ci Ddissolution ho ¼ : dT
ð3:43Þ

The same types of equations may be derived for evaporation, micellization, and so on.

3.7.3 Chemical Equilibrium For a chemical reaction aA þ bB p and T is

pP þ qQ the equilibrium condition at constant

amA þ bmB ¼ pmP þ qmQ :

ð3:44Þ

With (3.27) for components in solution this leads to the following expression for the equilibrium constant Kð c pP cqQ =caA cbB Þ,
ð3:45Þ

in which Dr Go ð¼ amoA þ bmoB pmoP qmoQ Þ is the standard Gibbs energy of the reaction. For gaseous components mi is usually expressed in partial pressures, using (3.26). The shift in the equilibrium composition due to temperature variation is given by d ln K Dr Ho ¼ :
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ð3:46Þ

34

Chapter 3

3.8 ENTROPY OF MIXING As a result of the increased number of conﬁgurational possibilities the entropy of a mixture is higher than the sum of the entropies of the components in the unmixed state, under otherwise identical conditions. For the sake of simplicity, let us consider an ideal mixture of two components, 1 and 2. The entropy of mixing, Dm S, is Dm S ¼ Sm S1 S2 :

ð3:47Þ

Sm is the entropy of the mixture containing N1 molecules of 1 and N2 molecules of 2, and S1 and S2 are the entropies of N1 molecules of pure 1 and N2 molecules of pure 2, respectively. For an ideal system the changes in energy and volume upon mixing are zero; that is, U and V are constant. At constant U and V the molecular degrees of freedom as translation, rotation, and vibration are the same for the mixed and the unmixed state and, hence, so is the entropy associated with these degrees of freedom. Therefore, to derive the entropy of mixing, we only have to compare the conﬁgurational possibilities Oconfig: and apply (3.5) after and before mixing, respectively. In the unmixed state (i.e., for the pure components), there is only one way to distribute N1 indistinguishable molecules over N1 sites, which means that ¼ 0. Similarly, ln Oconfig: ¼ 0 for pure 2. For the mixture of 1 and 2 ln Oconfig: 1 2 Oconfig: 1þ2 ¼

ðN1 þ N2 Þ! : ðN1 Þ!ðN2 Þ!

ð3:48Þ

Combining (3.47), (3.5), and (3.48) we ﬁnd for the entropy of mixing Dm S ¼ kB ln

ðN1 þ N2 Þ! ; ðN1 Þ!ðN2 Þ!

ð3:49Þ

which can be simpliﬁed using the Stirling approximation ln N ! ¼ N ln N N ;

ð3:50Þ

which applies for large values of N. Thus, Dm S ¼ kB ðN1 ln X1 þ N2 ln X2 Þ

ð3:51Þ

with X1

N1 N1 þ N2

and

X2

N2 N1 þ N2

being the mole fractions of 1 and 2 in the mixture. Because 0 < X < 1, ln X < 0 and, hence, Dm S > 0.

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Some Thermodynamic Principles and Relations

35

It follows (with N ¼ NAv: n and < ¼ NAv: kB, where NAv is Avogadro’s constant) that for one mole of mixture Dm s ¼ <ðX1 ln X1 þ X2 ln X2 Þ:

ð3:52Þ

Because for ideal mixtures Dm U ¼ 0 and Dm H ¼ 0, the Helmoltz energy and the Gibbs energy of mixing are determined by the entropy Dm g ¼ Dm f ¼
ð3:53Þ

3.9 THE EXCESS NATURE OF INTERFACIAL THERMODYNAMIC QUANTITIES. THE GIBBS DIVIDING PLANE Consider a two-phase system containing the homogeneous phases a and b divided by a ﬂat interface. Each phase is thermodynamically characterized by its variables of state. The crossing over from a to b takes place over a region of ﬁnite thickness. In this transition region the intensive thermodynamic quantities change gradually from their values in the bulk of a to those in the bulk of b. The problem is that it is impossible to determine the contribution of each phase to the properties of the interfacial region, or, stated otherwise, there is no way of telling where the one phase ends and the other phase begins. Some convention is required to locate the interface. Here, we adopt the Gibbs convention. According to the Gibbs convention the interface is an inﬁnitely thin layer, that is, a plane, the position of which is determined by letting the interfacial excess of the major component be zero. This plane is called the Gibbs dividing plane. By way of example, for a liquid–vapor interface this is schematically depicted in Figure 3.4: a is the liquid phase and b the gas phase. Let the major component in the system be water; then the density change of the water molecules across the interfacial region is given by the full line. The location of the Gibbs dividing plane, indicated by the dashed line in Figure 3.4, is such that the deﬁcit of water molecules in phase a (which is thought to extend up to the dividing plane) equals the excess of water molecules in phase b (also extending up to the dividing plane), implying that the net interfacial excess of water nswater ¼ 0. (The superscript s is used to indicate quantities pertaining to the interface.) The deﬁcit of water in a and the excess in b are indicated by the hatched areas. The interfacial excesses of all components other than water are now assumed to be located at the Gibbs dividing plane. It implies that all interfacial excesses, that is, all adsorbed amounts, are counted with respect to water, the density of which is taken constant, up to the dividing plane, in both the liquid and the gas phases.

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36

Chapter 3

ρ

ρ liquid

ρ vapor

position Figure 3.4 Gibbs dividing plane between a liquid and a vapor. r is the density of the phase indicated.

Having established the position of the Gibbs dividing plane, the volumes of a and b, V a , and V b, are ﬁxed and because the dividing plane is volumeless the total volume V of the system is given by V ¼ V a þ V b:

ð3:54Þ

After having ﬁxed V a and V b, any other extensive property Y of the system is assumed to be made up of its contributions from a, b and the interface s, Y ¼ Y a þ Y b þ Y s:

ð3:55Þ

It shows that Y s is an excess quantity. For an (inﬁnitesimally small) change in Y, dY ¼ dY a þ dY b þ dY s :

ð3:56Þ

For instance, a change in the energy dU is composed of X ma dnai dU a ¼ T a dS a pa dV a þ i i

ð3:57Þ

and dU b ¼ T b dS b pb dV b þ

X i

mbi dnbi

ð3:58Þ

and dU s ¼ T s dS s þ gdA þ

X i

msi dnsi :

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ð3:59Þ

Some Thermodynamic Principles and Relations

37

In equilibrium, that is, T a ¼ T b ¼ T s T , pa ¼ pb p, and mai ¼ mbi mi , Eqs. (3.57) through (3.59) can be integrated, to yield X U a ¼ TS a pV a þ m na ; i i i X U b ¼ TS b pV b þ m nb ; i i i X U s ¼ TS s þ gA þ m ns : i i i

¼ msi ð3:60Þ ð3:61Þ ð3:62Þ

In equilibrium, the interfacial region which, according to the Gibbs convention, is assumed to be contracted in a volumeless dividing plane, may be considered as an independent phase. Now, as we did for bulk phases [see (3.11) through (3.13)], auxiliary functions of state may be deﬁned for the interfacial phase. Thus, the interfacial enthalpy, H s is deﬁned as H s U s gA ¼ TS s þ

X i

mi nsi :

ð3:63Þ

Note that for the interfacial phase gA is equivalent to pV for the bulk phase. It follows from (3.59) and (3.63) that X dH s ¼ T dS s Adg þ m dnsi : ð3:64Þ i i In analogy to the bulk phase, the interfacial Helmholtz energy F s and the interfacial Gibbs energy Gs are deﬁned as X F s U s TS s ¼ gA þ m ns ð3:65Þ i i i and Gs U s gA TS s ¼ H s TS s ¼ F s gA ¼

X i

mi nsi

ð3:66Þ

with the differentials dF s ¼ S s dT þ gdA þ

X i

mi dnsi

ð3:67Þ

mi dnsi :

ð3:68Þ

and dGs ¼ S s dT Adg þ

X i

Equation (3.67) is useful for describing processes under conditions of constant T and A, and (3.68) for processes at constant T and g.

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38

Chapter 3

As a consequence of the deﬁnitions given above, U ¼ U a þ U b þ U s;

ð3:69Þ

H ¼ H a þ H b þ H s þ gA;

ð3:70Þ

F ¼ F a þ F b þ F s;

ð3:71Þ

G ¼ G þ G þ G þ gA:

ð3:72Þ

a

b

s

Notice that the expressions for H and G are not of the type Y ¼ Y a þ Y b þ Y s , but that they contain an extra gA term. This could be avoided by choosing another set of deﬁnitions for the interfacial functions of state. However, the deﬁnitions as given in (3.63), (3.65), and (3.66) have the advantage that they are consistent with the deﬁnitions for the corresponding quantities in the bulk phases (3.11) through (3.13). It is expedient to express the interfacial thermodynamic functions of state per unit interfacial area (indicated by the subscript a). X Uas ¼ TSas þ g þ mG; ð3:73Þ i i i X Has ¼ TSas þ mG; ð3:74Þ i i i X Fas ¼ g þ mG; ð3:75Þ i i i X Gas ¼ mG: ð3:76Þ i i i In (3.73) through (3.76) Gi ¼ nsi =A. Note that in these equations the interfacial excess thermodynamic quantities and Gi depend on the choice of the dividing plane, whereas g is independent of it. Fas presents the simplest integral expression for the interfacial tension g, namely, X g ¼ Fas mG; ð3:77Þ i i i which for a single component system ðGi ¼ 0Þ gives g ¼ Fas :

ð3:78Þ

Equation (3.77) shows that g is affected by Gi and, hence, by the composition ðXi0 s or, for that matter, n0i s) of the system. The dependency of g on the composition could also be inferred from (3.16), @G ð3:79Þ g¼ @A T;p;...;n0 s i

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Some Thermodynamic Principles and Relations

39

but the advantage of (3.77) is that this dependency is explicitly given in operational, that is, measurable, parameters without any restrictive conditions.

3.10 THE GIBBS–DUHEM RELATION The thermodynamic state of a system is deﬁned by its state variables ðT ; p; . . . ; m0i sÞ. Because the state variables are not independent of each other, speciﬁcation of only a limited number, given by the Gibbs–Duhem relation, is sufﬁcient to determine the thermodynamic state. The Gibbs–Duhem relation is obtained by equating the differential of a function of state [as deﬁned by (3.10) through (3.13)] to the corresponding differential expressions [(3.8) and (3.14) through (3.16)]. For instance, for the energy U of a nonelectric (no cQ term), P homogeneous P (no gA term) system,P dU ¼ T dS þ SdT pdV V dp þ i mi dni þ i ni dmi ¼ T dS pdV þ i mi dni , which results in X SdT V dp þ n dmi ¼ 0; ð3:80Þ i i or, expressed per mole, X sdT vdp þ X dmi ¼ 0: i i

ð3:81Þ

Equation (3.80) or (3.81) is the Gibbs–Duhem relation. It indicates the number of variables that can be independently varied for a system. At constant T and p, for a homogeneous system containing k components the chemical potentials of ðk 1Þ components can be chosen at will.

3.11 THE GIBBS ADSORPTION EQUATION The Gibbs adsorption equation may be regarded as a two-dimensional Gibbs– Duhem relation. The derivation is analogous to that of the Gibbs–Duhem relation, but now using functions of state for the interface instead of for the bulk phase. Thus, equating the differential of U s , given by (3.62), to dU s , given by (3.59), yields X Adg þ S s dT þ ns dmi ¼ 0 ð3:82Þ i i or dg ¼ Sas dT

X i

Gi dmi ;

ð3:83Þ

which is the well-known Gibbs adsorption equation. It is noted that (3.83) would also result from the integral and differential expressions for functions of state other than U s that is, H s , F s , and Gs . It is further noted that the summation in (3.82) and (3.83) includes all components except the one (usually the solvent) for

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40

Chapter 3

which G is set to zero by the Gibbs convention on positioning the interface. Under isothermal conditions, which is often the case in practice, dg ¼

X i

Gi dmi ;

ð3:84Þ

showing the interdependence between g, Gi , and mi or [through (3.28)] Xi . If the relation between two of the three variables is known, the third one can be calculated using the Gibbs equation. For instance, the adsorption isotherm gives the relation between Gi and Xi from which the variation of g can be derived. For ﬂuid interfaces g can be readily measured, but for solid interfaces g is not directly experimentally accessible. However, from the isothermal adsorption of i at a solid interface, keeping components j 6¼ i constant, Dg may be evaluated as Dg ¼

ð Xi Xi ¼0

Gi dmi :

ð3:85Þ

For an ideal gas or solution, for which mi ðXi Þ is given by (3.28), (3.85) may be transformed into Dg ¼
ð Xi Xi ¼0

Gi d ln Xi ;

ð3:86Þ

which implies that Dg can be derived as the area under the curve of a Gi versus ln Xi plot. For ﬂuid interfaces the Gibbs equation is often used to establish adsorbed amounts of i from the experimentally determined dependency of g on Xi . This approach is especially useful when little surface area is available so that Gi cannot be established analytically. In this way, the functionality Gi ðXi Þ, the adsorption isotherm, can be derived. In the case of competitive adsorption, that is, the simultaneous adsorption of two or more components from a mixture, it follows from cross-differentiation in (3.83) that

@Gj @mk

@Gk ¼ @mj T;mi6¼k

! :

ð3:87Þ

T;mi6¼j

Hence, the interdependency of the adsorption isotherms Gi ðXi Þ or, for that matter, Gi ðmi Þ should satisfy (3.87) to be thermodynamically consistent. As the Gibbs adsorption equation is derived from thermodynamics it does not contain model parameters and is therefore generally applicable to all types of interfaces.

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Some Thermodynamic Principles and Relations

41

3.12 SOME APPLICATIONS OF THE GIBBS ADSORPTION EQUATION The examples presented in this section anticipate discussions in the following chapters.

3.12.1 Adsorption of (Ionic) Surfactants Assume a surface in contact with a solution of an ionic surfactant in water containing low molecular weight electrolytes. Let, by way of example, the surfactant be NaDS (sodium dodecyl sulphate) and the low molecular weight electrolyte NaCl. Then, at constant temperature, dg ¼ GNaDS dmNaDS GNaCl dmNaCl :

ð3:88Þ

The chemical potentials of the strong electrolytes NaDS and NaCl in aqueous solution may be written as the sum of their ionic components, so that dg ¼ GNaDS ðdmNaþ þ dmDS Þ GNaCl ðdmNaþ þ dmCl Þ;

ð3:89Þ

which, for ideal solutions of the ionic components [See (3.27), and realizing that d ln ci ¼ dci =ci ], gives dcNaþ dcDS dcNaþ dcCl dg ¼
ð3:90Þ

When the NaCl concentration is low such that cNaDS cNaCl , cNaþ cDS ¼ cNaDS , the ﬁrst term on the right-hand side of (3.90) approaches
2dcNaDS cNaDS

¼ 2
and, because of the low GNaCl, the second term approaches zero. Thus, at low ionic strength a factor 2 enters the Gibbs adsorption equation for a strong monovalent ionic surfactant dg ¼ 2
ð3:91Þ

If cNaCl cNaDS , dcNaþ =cNaþ is negligible upon increase of cNaDS and the ﬁrst term on the right-hand side of (3.90) becomes
dcDS ¼
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42

Chapter 3

Because dcNaþ =cNaþ 0 and, at constant cNaCl ; ðdcCl =cCl Þ ¼ 0, the second term of (3.90) is zero. It follows that for the adsorption of an ionic surfactant from a solution of high ionic strength dg ¼
ð3:92Þ

It is thus demonstrated that the factor 2 in the Gibbs adsorption equation for a monovalent strong ionic surfactant vanishes when changing the ionic strength of the solution from low to high. In other words, at high ionic strength the ionic surfactant tends to behave as a nonionic surfactant. At intermediate ionic strengths dg ¼ p
ð3:93Þ

where, in the case of a monovalent surfactant and monovalent electrolyte, p¼

csurfactant : csurfactant þ celectrolyte

Equation (3.93) does not take into account the effect of possible coadsorption of the low molecular weight electrolyte on g.

3.12.2 Adsorption of (Bio)Polymers Adsorption of polymers, including proteins, polysaccharides, and so on, is usually irreversible with respect to changes in the concentrations of these components in solution. It implies that, within the time of the experiment, the ascending branch of the adsorption isotherm Gi ðci Þ does not coincide with the descending branch, or, in other words, that the adsorption isotherm contains a hysteresis loop. This situation is schematically depicted later in Figure 15.13. If the Gibbs equation is applied to this adsorption isotherm the conclusion would be that g decreases less on the way up than it increases on the way down. Hence, the problem is that, for a given ci , no unique relation between Gi and g is found. Moreover, computation of Gi from dg=d ln ci usually leads to erroneous results. This is due to the fact that an adsorbed polymer molecule is attached at the surface via several segments whereas other segments of the adsorbed molecule reach out in the solution (see Figure 15.11). The interfacial tension is related to the density proﬁle of the adsorbed polymer in such a way that segments in the interface contribute more strongly to the reduction of g than those farther out. The problem with respect to applying the Gibbs equation to polymer adsorption is the lack of detailed knowledge about the segment density proﬁle (normal to the interface) in the adsorbed layer and, even more so, about the

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Some Thermodynamic Principles and Relations

43

contribution of the segments to the interfacial tension as a function of their location within the adsorbed layer. Another complication is that polymers are often heterodisperse. The larger molecules adsorb more strongly (more attached segments per molecule) but the smaller ones adsorb faster (due to their higher molar concentration and higher diffusivity). As a result, there is a gradual change in G that may not be unambiguously reﬂected in a change in g. On the other hand, even after having reached a steady-state situation with respect to G; g may further decrease because of relaxation of the adsorbed molecule during which it places more segments in the interface. This phenomenon is discussed in more detail in Chapter 15.

3.12.3 Adsorption of Uncharged Compounds at a Charged Interface Assume a (solid) surface, for example, the outer wall of a bacterial cell, of which the charge is determined by dissociation or association of protons in an aqueous medium containing NaCl as the only low molecular weight electrolyte. The charge density s0 at the surface of the solid is given by s0 ¼ FðGHCl GNaOH Þ;

ð3:94Þ

where F is Faraday’s constant. s0 can be readily determined by titrating the surface with HCl or NaOH; more details are discussed in Chapter 9. Figure 9.2(d) presents such a titration curve for the bacterium Bacillus brevis. Now, let A be the uncharged compound that adsorbs at the (bacterial) surface. Thermodynamic analysis of the adsorption, at constant T , starts with (3.84), dg ¼ GHCl dmHCl GNaOH dmNaOH GNaCl dmNaCl GA dmA ;

ð3:95Þ

in which, because of HCl þ NaOH ! NaCl þ H2 O; dmHCl þ dmNaOH ¼ dmNaCl þ dmP H2 O . From the Gibbs–Duhem relation (3.81), at constant P T and p, dmH2 O ¼ i6¼H2 O Xi dmi =XH2 O , which for dilute solutions, that is, i6¼H2 O Xi XH2 O , is negligible. Then dmHCl þ dmNaOH ¼ dmNaCl allows elimination of one variable. If we select dmNaOH , and combine (3.94) and (3.95), we obtain dg ¼

s0 dm ðGNaCl þ GNaOH ÞdmNaCl GA dmA : F HCl

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ð3:96Þ

44

Chapter 3

Cross-differentiation between the ﬁrst and the third term, at constant cNaCl or mNaCl gives 1 @s0 @GA ¼ ; @mHCl mA F @mA mHCl which, using (3.27), gives 1 @s0 @GA ¼ 0:434 : @pH cA F @ ln cA pH

ð3:97Þ

ð3:98Þ

With (3.98) the variation of GA with pH, at constant cA , can be computed from the experimentally accessible variation of s0 with cA , at constant pH. As the relation between s0 and pH is also known, the dependence of GA on s0 can be established, even if this cannot be directly measured.

EXERCISES 3.1

Does the surface tension of a liquid increase or decrease with increasing pressure, at constant temperature?

3.2

ÐConsider the isothermal volume change of an ideal gas. Show that ðdqrev =T Þ is a function of state, whereas qrev is no`t.

3.3

3.4

b a að1Þ V a Þ, where Gið1Þ is the adsorpShow that AGð1Þ i AGi ¼ ðci ci ÞðV tion (¼ interfacial excess per unit interfacial area) of component i relative to component 1ðG1 0Þ at the a=b interface. A is the total interfacial area and ci the concentration of i in the volume V of the phase indicated. Derive ! cai cbi ð1Þ Gi ¼ Gi G1 : ca1 cb1

Derive ! @nj @g ¼ @A p;T ;m0 s @mj i

p;T;mi6¼j

Do we need a convention to replace the left-hand side by Gj ? 3.5

An electrocapillary device allows the measurement of the interfacial tension g of a mercury=aqueous electrolyte solution as a function of the electric potential c across the interface A. Equation (3.16) may be applied to this

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Some Thermodynamic Principles and Relations

45

system. Q is the total electric charge at the interface; the interfacial charge density s0 is deﬁned as s0 ð@Q=@AÞp;T;n0i s;c . For constant differential capacity ds0 =dc of the interface, prove that the functionality gðcÞ is parabolic and that the maximum value for g is reached at zero interfacial charge.

SUGGESTIONS FOR FURTHER READING E. A. Guggenheim. Thermodynamics, 5th edition, Amsterdam: North Holland, 1976. J. Lyklema. Interfacial thermodynamics with special reference to biological systems, in Physical Chemistry of Biological Interfaces, A. Baszkin and W. Norde (eds.), New York: Marcel Dekker, 2000, Chapter 1.

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The Constructive Power of Water

Water was present on earth long before the evolution of life. It is an essential constituent of living cells and this suggests that life on our planet has been conditioned by the properties of water. At ﬁrst sight it may seem illogical, yet it is true: inorganic liquid water is indispensable for the structuring, maintenance, and functioning of organic biological systems. The picture shows a water-ﬂea (Daphnia). As in any other organism, the high water content of the ﬂea helps to shape the animal, its membranes, and its protein molecules. (Figure courtesy of P. Frappell, La Trobe University, Melbourne, Australia.)

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4 Water

Water is one of the most abundant chemical components on Earth. The hydrosphere consists of more than 90% water; 50–90% of the gas in the atmosphere is water vapor; the lithosphere (the Earth’s crust) contains about 15% water and living organisms on our planet typically contain 50–90% water. Because of its ubiquity water may be considered as very normal and common. Yet, from a physical–chemical point of view, water is a unique extraordinary substance. By virtue of its unique properties water is the medium, par excellence, in which life has evolved and is sustained. To be more precise, in living organisms water fulﬁlls the following functions. — It is essential for the formation and maintenance of the structures of cell walls, membranes, proteins, and the like. — It is the environment for biochemical pathways; often it participates as a reagent in biochemical reactions. — It is the medium in which components are transported. — It regulates the temperature of warm-blooded organisms. Two questions arise immediately: which properties make water so special, and how can these properties be explained and understood on a molecular level?

4.1 PHENOMENOLOGICAL ASPECTS OF WATER In Table 4.1 the most relevant (at least for our goal) physicochemical characteristics of water are summarized. To demonstrate the uniqueness of water, some of its properties are compared with those of other compounds. In Table 4.2 this is done for the

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48

Chapter 4 Table 4.1 Some physicochemical properties of water Chemical composition Molar mass Freezing temperature (at 1atm) Boiling temperature (at 1atm) Expansion during liquid ðLÞ ! solid (S) transition: rH2 OðLÞ ð0 C; 1 atmÞ rH2 OðSÞ ð0 C; 1 atmÞ Heat of evaporation at 100 C and 1atm Heat of melting at 0 C and 1atm Heat capacity ðCp Cv Þ of H2 O ðLÞ Viscosity ð20 C; 1 atmÞ Surface tension at interface with air ð20 CÞ Speciﬁc electrical conductivity ð20 CÞ Dielectric constant ð20 CÞ Dipole moment Relaxation time Excellent solvent for most salts: e.g., NaCl Good solvent for polar components: e.g., ethanol n-butanol Poor solvent for apolar components

H2O 18 g mol1 0 C 100 C 1 kg dm3 0:9 kg dm3 40:6 kJ mol1 6:02 kJ mol1 75:25 J K1 mol1 1:002103 N m2 s 72 mN m1 4104 O1 m1 80 1:84 De 1013 1011 s 360 g dm3 all proportions 80 g dm3

Table 4.2 Temperatures and heats for the melting and evaporation of various compounds Compound Water Methane Ethane Propane Methanol Ethanol Propanol Ethylether Carbon dioxide

H2O CH4 C2H6 C3H8 CH3OH C2H5OH C3H7OH C4H10O CO2

T

Molar mass (g mole1)

Tm ð CÞ

qmm ðJ g1 Þ

T

Tev ð CÞ

qevev ðJ g1 Þ

18 16 30 44 32 46 60 74 44

0 184 172 190 98 114 127 116

333 58 95 71 99 109 86 98

100 162 88 42 64 78 97 34 78

2255 511 531 433 1100 838 691 358 393

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Water

49 T

melting and evaporation temperatures (Tm and Tev ) and the heats of melting qmm T and evaporation qevev at Tm and Tev . When judging the values given in Table 4.2 one should realize that, as a rule, Tm and Tev increase with increasing polarity of the compound, and that Tev increases with increasing molar mass. This is clearly reﬂected in the values for compounds within a homologous series, for example, the alkanes and the alcohols. It is evident that the values of Tm , Tev , qm , and qev for water are exceptionally high! The values of Tm and Tev for the alcohols are also relatively high. In the melting process ðS ! LÞ and to a much larger extent in the evaporation process ðL ! GÞ physical bonds between molecules are disrupted. Hence, the values for Tm and, even more so, Tev are an indication of the resistance against thermal motion of the individual molecules or, more precisely stated, of the strength of attractive interaction between the molecules of the heated substance. Our conclusion is that there is an extraordinary strong internal coherence between the molecules in H2 O (S) and also in H2 O ðLÞ. Hence, a lot of heat (¼ energy) is needed to disrupt bonds between the H2 O molecules. This is reﬂected in the high values for the heats of melting and evaporation. The strong internal coherence in water is also reﬂected in its anomalous high value for the heat capacity (at constant pressure) Cp ; see Table 4.3. The heat capacity equals the amount of heat required to raise the temperature of the system by 1 C. For a fair comparison between the different compounds Cp is expressed per unit mass. The high value for H2 O ðLÞ implies that, despite its liquid state, water is strongly internally structured: it requires a large amount of heat to disrupt that structure in order to increase the motion of the individual molecules, that is, to increase the temperature. Because of its large heat capacity the temperature of water is relatively insensitive to heat changes. One of the consequences is the tempering effect of oceans on the climate. Furthermore, it is important for the role water plays in the heat regulation of warm-blooded organisms. Let us now proceed to examine the viscosity. Roughly, it may be stated that the viscosity of a substance is a measure of its ﬂuidity, that is, the ease by which it ﬂows upon imposing a stress. The more ﬂuid a substance, the lower its viscosity. Table 4.3 Heat capacities of various compounds at 20 C and 1 atmosphere Heat capacity (J K1 g1)

Compound Water Ethanol Carbon tetrachloride Chloroform Ether

H2O (L) C2H5OH (L) CCl4 (L) CHCl3 (L) C4H10O (L)

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4.18 2.49 0.98 0.90 2.31

50

Chapter 4

Table 4.4

Viscosities of some liquids at 20 C and 1 atmosphere Viscosity (N m2 s) 103)

Compound Water Ethanol Carbon tetrachloride Chloroform Ethyl ether Olive oil

H2O (L) C2H5OH (L) CCl4 (L) CHCl3 (L) C4H10O (L)

1.00 1.20 0.97 0.58 0.23 84.0

A detailed discussion on viscosity (and other rheological properties) is given in Chapter 17. Table 4.4 compares the viscosity of H2 O ðLÞ with that of some other liquid compounds. Here we see that water does not have an anomalous viscosity. In view of the foregoing conclusions concerning the strong cohesive interaction in H2 O ðLÞ this is a rather surprising observation. For, if H2 O molecules strongly attract each other, one might expect that their individual mobilities are severely restricted, which would result in a high viscosity. If the viscosity of water were much higher it would take much more energy to circulate the blood in our body. Viscosity depends on temperature and pressure. It decreases with increasing temperature. For essentially all liquids the viscosity increases with increasing pressure, but not so for water. For H2 O ðLÞ the viscosity is lower at higher pressure and this is advantageous for the energy household of deep sea animals. The ability of water to solvate (hydrate) different components is an instructive property. Charged species (ions) and polar components readily dissolve in water, whereas the solubility of apolar components is very poor. Apolar (e.g., lipid-like) particles or droplets may be ﬁnely dispersed in water by coating their exterior with a layer of polar molecules. This principle is used by nature and also in various technical applications. Examples from nature are the emulsiﬁcation of fat in the duodenum to allow enzymatic digestion, the formation and stabilization of biological membranes, and the folding of globular proteins in an aqueous environment. These phenomena are discussed in following chapters. Electrically charged molecules, that is, ions, exist in water as single entities. This is important for ion transport in organisms, especially for the role of ions in conveying nerve signals. Why do oppositely charged ions not associate in water, as they do in most other liquids? The answer to this question is connected with the exceptionally high dielectric permittivity of water. (See Table 4.5.) The dielectric constant represents the capability to screen the electrostatic interaction between two charges at a given separation distance. In water of 25 C that interaction is 78.5 smaller than in vacuum. Therefore, in water the electrostatic attraction between oppositely charged ions may be outweighed by favorable hydration of the individual ions (see Section 4.3.1).

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Water

51 Table 4.5 Dielectric constants of some liquids relative to that of vacuum, at 25 C Compound Water Ethanol Carbon tetrachloride Chloroform Ethyl ether

Relative dielectric constant H2O (L) C2H5OH (L) CCl4 (L) CHCl3 (L) C4H10O (L)

78.5 24.3 4.8 2.2 4.3

Finally, we consider the relaxation time. The relaxation time of a system is the time it takes for the system to relax back to its ‘‘ground’’ state after releasing a preimposed stress. Solid substances have very long, but not inﬁnitely long, relaxation times, on the order of years up to geological time scales. The speed of relaxation of most liquids goes beyond our imagination with relaxation times typically ranging between nano- and picoseconds. As for the viscosity, it would be expected that because of the internal coherence the relaxation time of water is longer than for other nonassociating liquids. However, just as for the viscosity, that is not the case: H2 O ðLÞ behaves quite normally showing relaxation times between 1013 and 1011 seconds, depending on the type of perturbation. Summarizing, it is concluded that the uniqueness of (liquid) water is characterized by the strong association of its molecules which gives rise to an extraordinarily high boiling temperature and heat capacity. In spite of that association liquid water molecules have ‘‘normal’’ mobilities as is reﬂected in the viscosity and the relaxation time. Furthermore, water is a good solvent for ions and polar components, whereas apolar substances are poorly soluble in it, if soluble at all. The combination of these properties makes water life’s own essence; that is, the evolution of life on Earth has been strongly directed by the physical–chemical properties of water. The discussion given above includes the answer to the question of which properties make water so special. In order to answer the question of how to understand and explain the properties of water on a molecular level, we have to consider the architecture of a H2 O molecule in more detail.

4.2 MOLECULAR PROPERTIES OF WATER A model for the molecular architecture of H2 O is schematically shown in Figure 4.1. The H2 O molecule is composed of a relatively large O-atom to which two very small H-atoms are chemically linked. Hence, the H2 O molecule is nearly spherical, having a radius of almost 0:14 nm. Because of the positions of the electron-donating H-atoms relative to the electron-accepting O-atom, the

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52

Chapter 4

H +q

–q 105 o 3'

O –q

+q H Figure 4.1 Model of a water molecule showing the positive charges ðþqÞ on the hydrogen atoms and the negative charges ðqÞ on the oxygen-atom. The pole-center-pole angles are 105 30 .

charge in the (overall electrically neutral) H2 O molecule is not evenly distributed. Positive charges are centered on each H-atom and two compensating negative charges on the opposite side of the O-atom, representing the two unshared electron pairs. Hence, the H2 O molecule is a quadrupole. The quadrupole may be approximated as a dipole in which an excess of positive charge dþ is separated over a distance x from an equal excess of negative charge d . The dipolar moment m d x

ð4:1Þ

of H2 O equals 1.84 Debye-units (De), which is very high considering the small size of the H2 O molecule. The charge distribution causes the H2 O molecules to attract each other: a H-atom is shared between two O-atoms, forming a so-called hydrogen bond. Thus, because of the quadrupolar character, each O-atom tends to be surrounded by four H-atoms. As a consequence of the electron-donating and -accepting characters of H and O, respectively, the H O separation between two H2 O molecules is less than it would be otherwise, resulting in a relatively large value

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Water

53

for the energy of stabilization of the hydrogen bond, that is, 16:7 kJ per mol of H-bonds ( 7 kB T , at room temperature, per single H-bond). When the four poles of the H2 O-quadrupole are connected a tetraeder-like geometry is obtained. See Figure 4.2. (The pole-center-pole angles are 105 30 , whereas in a perfect tetraeder these are 109 50 .) Tetraeders can be easily stacked to obtain a regular three-dimensional structure in which, in the case of H2 O, all possible hydrogen bonds are realized. It results in a very open crystal structure, including a large fraction of interstitial voids. This is the case in ice, H2 O (S), as depicted in Figure 4.3. The packing density in ice is about 55%. In ice the separation between nearest-neighbor O-atoms is 0:276 nm, whereas the distance to a second nearest O-atom is 1.63 times that value, and so on. In a random packing the distance to a second nearest neighbor would be twice as large as to the nearest one. The fact that the hydrogen bonds are aligned in the regular network adds to the stability of these bonds and, hence, to the internal cohesion of the structure. When a pressure is exerted on the regular crystal structure, the bond angles may become distorted thus deviating from 105 30 . As a result, the regular threedimensional structure is (partly) disrupted: ice melts under pressure. The structure

+

– +

–

– – +

+

Figure 4.2 Dipolar interaction between H2 O molecules.

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54 Figure 4.3 Three-dimensional structure of ice, H2 O (S). Each oxygen-atom is at the center of four oxygen-atoms at a separation distance of 0:276 nm. Chapter 4

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Water

55

of H2 O ðLÞ is less open than that of H2 O (S); at the melting point ð0 C; 1 atmÞ the density of H2 O ðLÞ exceeds that of H2 O (S) by approximately 10%. At the melting point in the liquid state an average of approximately three (out of the possible four) hydrogen bonds are still intact. Hence, although in the liquid state the individual H2 O molecules are highly mobile they are still strongly associated. The associative structures are dynamic; the H2 O molecules readily rotate and hop from one orientation to the other. As the temperature increases more hydrogen bonds are disrupted resulting in a further breakdown of the regular structure, so that the density increases, reaching a maximum at 4 C. Then, beyond 4 C, the density gradually decreases due to an overcompensating effect of thermal expansion. As mentioned, breakdown of the regular structure leads to a decreased water volume. Stated in the reversed way: a reduced volume (for instance, by exerting a pressure) leads to a breakdown of structure which manifests itself by, for example, a lower viscosity. This is the reason why, in contradiction to essentially all other liquids, water has a lower viscosity at higher pressure, at least in the temperature range between 0 and 30 C and at pressures below 1000 atm. Thus the unique and extraordinary features of water can be reduced to its molecular architecture. The uneven distribution of charge over the molecule and the tetraeder-like geometry of the H2 O-quadrupole result in a strong association between the molecules and, at the same time, a very open network structure. In liquid water hydrogen bonds are continuously formed and disrupted, where, on the average, most of the hydrogen bond potentialities are realized without compromising the mobility of the individual H2 O molecules. The resulting quality of water as a solvent is discussed in more detail in the following section.

4.3 WATER AS A SOLVENT In the simple case that the solubility of solute B in solvent A is determined by molecular contacts only, the effective A–B interaction, w (expressed per mole B), may be written in terms of the contributions, W, from AB, AA, and BB contacts, as follows, 1 1 w ¼ NAv z WAB WAA WBB ; 2 2

ð4:2Þ

where NAv is Avogadro’s number and z is the number of contacting neighbors. It follows from the discussion in the foregoing section that for water WAA has an unusually high value.

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56

Chapter 4

4.3.1 Electrolytes Because of their strong dipolar character H2 O molecules strongly orient in an electric ﬁeld. If, furthermore, the electric ﬁeld is inhomogeneous, as it is around an ion, the H2 O dipole will be strongly attracted to the ion in the electrostatically preferred orientation. Ions are therefore strongly hydrated, allowing salts to dissociate in water. The value for WAB is large and in the same range as those for WAA and WBB0 (B and B0 being the cation and anion of the salt, resp.). Dissociation of the salt largely increases the conﬁgurational entropy of the system. The net effect of these factors determines whether salts are soluble in water, which is the case for most salts. As a rule, the hydration layer around a dissolved ion extends over several molecular H2 O layers, the extension being primarily determined by the electric ﬁeld strength around the ion. This ﬁeld strength, in turn, is a function of the valency and the radius of the ion. The preferred direction of the H2 O dipoles in the hydration layer leads to a reduction of the entropy. The structured hydration layer is connected to the differently structured bulk water via a transition layer in which the H2 O molecules have to switch over from one structure to the other. As a result, the transition layer is relatively chaotic having a higher entropy. See Figure 4.4. Now, depending on the extension of the inner hydration layer relative to that of the transition region, ions per saldo promote or break water structure (the reference state being the structure of bulk water). Small and=or multivalent ions, such as Liþ ; Naþ ; Mgþ ; Ca2þ ; Sr2þ ; Al3þ , and so on, are structure promoters whereas large and=or monovalent ions, for example, Csþ ; Br ; I ; ClO4 ; SO4 2 , are structure breakers. It follows from the foregoing that the overall hydration entropy may be the most straightforward criterion to distinguish between structure promoting and breaking. However, the partial molar volume

structured hydration water transition layer between hydration layer and bulk water Figure 4.4 Effects of ion hydration on water structure.

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Water

57

of the ion and the change in the viscosity may serve that purpose as well. Structure promoting ions have a negative partial molar volume and cause an increase in viscosity, whereas structure breakers have the opposite effects.

4.3.2 Noncharged Components Polar groups interact favorably with H2 O molecules (large WAB ), mainly through electrostatic interactions. These interactions (partly) compensate for the strong cohesion between the water molecules ðW AA Þ and, possibly, between the molecules of the solute ðW BB Þ. Together with the entropy of mixing it renders polar components readily soluble in water. Examples of such soluble components are acetone, the lower alcohols (methanol, ethanol, propanol), sugars, and so on. For apolar components, such as the hydrocarbons, that have not much to offer in terms of electrostatic interactions (i.e., WAB small), the cohesion between H2 O molecules makes the solubility low, unless the component ﬁts well in the cavities of the H2 O clusters. This mechanism is at the basis of hydrophobic interaction, that is, the tendency of apolar compounds to aggregate in an aqueous environment. Although this explanation seems simple and straightforward, contradictory views on the phenomenon of hydrophobic interaction are given in the literature. Before discussing this further, some experimental data are presented. Figure 4.5

alkanes

∆tr go / kJ mole–1

40

alkenes

30

alkadienes

20 4

5

6

7

8 nC

Figure 4.5 Change in the standard Gibbs energy for the transfer of hydrocarbons from their own medium to water, as a function of the number of C-atoms, nC , in the chain. T ¼ 25 C. (From Tanford, 1973.)

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58

Chapter 4

shows the change in the standard chemical potential (¼ standard molar Gibbs o ) for the transfer of hydrocarbons from their own medium to water. energy Dtr gHC This change may be established by studying the equilibrium between pure hydrocarbon (HC) and hydrocarbon dissolved in water (mole fraction HC in water, XHC ). At equilibrium, and for an ideal solution of HC in water, mHC ð mHC Þ ¼ mHCðaqÞ ¼ moHCðaqÞ þ RT ln XHC ;

ð4:3Þ

where mHC is the chemical potential of HC, the superscript * refers to the pure component, and ‘‘o’’ to the solution standard condition (XHC ¼ 1, but still a function of p and T ). It follows from the change in the standard chemical potential for the transfer from bulk HC to water that o ð¼ Dtr moHC Þ ¼ moHCðaqÞ mHC ¼ RT ln XHC ; Dtr gHC

ð4:4Þ

o where Dtr refers to the change due to the transfer and gHC is the standard molar o Gibbs energy of HC. For not too short hydrocarbon chains Dtr gHC depends linearly on the number of C-atoms in the hydrocarbon chain nC , so that o ¼ A þ BnC ; Dtr gHC

ð4:5Þ

where A and B are empirical constants. It follows from Figure 4.5 that B assumes a CH2 increment, value of 3:8 kJ ð¼ 1:5
C2H6 C3H8 C4H10 C6H6 C6H5CH3 C6H5C2H5

Dtr g o kJ mol1

Dtr ho kJ mol1

16 20 25 19 22 25

10 7.1 3.3 þ2.5 þ2.5 þ1.7

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JK

Dtr s 1

o 1

mol

88 93 96 54 67 79

TDtr so kJ mol1

Dtr Cp J K1 mol1

þ26 þ27 þ28 þ16 þ20 þ23

150 — 272 451 451 451

Water

59

unfavorable entropy change: T Dtr so Dtr ho . The large positive value for Dtr Cp implies that Dtr ho and Dtr so are strongly temperature-dependent: @Dtr ho @Dtr so Dtr Cp ¼ ¼ : ð4:6Þ @T p @ ln T p For benzene these temperature dependencies are shown in Figure 4.6. Clearly, the positive and relatively temperature-independent value for Dtr g o ð¼ Dtr ho T Dtr so Þ is the result of a strong enthalpy–entropy compensation. A large positive value for Dtr Cp and, hence, enthalpy–entropy compensation, is a major characteristic of hydration of apolar hydrophobic solutes. Now, having presented some experimental data, let us return to the physicochemical explanation of the characteristics of hydrophobic hydration. At high temperatures, say, close to 100 C; H2 O ðLÞ has lost most of its internal

40

kJ mole–1

30 ∆tr G 20

10 ∆tr H

–T∆tr S

0

–10 – 20 250

300

350

400

450

T/K Figure 4.6 Changes in the Gibbs energy, enthalpy, and entropy for the transfer of benzene from its own medium to water, as a function of temperature. (From Privalov and Gill, Adv. Protein Chem. 39:191, 1988.)

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60

Chapter 4

structure and at that condition Dtr go is essentially of enthalpic nature. Such a result would be expected if Dtr go were solely determined by the formed AB contacts and, hence, the broken AA and BB contacts, so that Eq. (4.2) applies. The deviating result at lower temperatures suggests some structural reorganization among the H2 O molecules when the apolar solute is introduced, leading to a lower entropy. The reason for the entropy reduction may be that at the apolar solute the H2 O molecules are not able to form hydrogen bonds in all four directions, thus restricting their rotational freedom. This reasoning should not give rise to the idea (which is often encountered in the literature and textbooks) that it is the restructuring of the water in the hydration layer around the apolar solute which makes the solubility low. In fact, the opposite is true: according to Le Chatelier’s principle any restructuring of the water upon introducing an apolar solvent ðDtr go > 0Þ should have a lowering effect on Dtr go . This could well be the reason for the increasing solubility of hydrocarbons in water as the temperature is decreased below, say, 20 C. See Figure 4.6.

EXERCISES 4.1

Comment on the following statements. (a) The melting temperature of ice increases with increasing pressure. (b) Hydrophobic bonding results from preferred interaction between apolar compounds rather than interaction between apolar compounds and water. (c) The solubility of apolar substances in water increases with increasing temperature. (d) The strong enthalpy–entropy compensation in the temperaturedependency of the solubility of apolar compounds in water is related to the relatively large change in the heat capacity upon dissolution.

4.2

1 gram of a compound i partitions between 1 dm3 oil and 1 dm3 water. At 27 C the concentration of i in the aqueous phase is 0:10 g dm3 and at 57 C this concentration is reduced to 0:08 g dm3 . (a)

Is the transfer of i from oil to water an exothermic or an endothermic process? (b) What are the values for the enthalpy and entropy effects associated with the transfer? (c) Explain the notion ‘‘hydrophobic interaction.’’ What is the main driving force for hydrophobic interaction?

4.3

From the solubilities of a homologous series of alkanes in water it is derived that at 25 C the Gibbs energy effect of dehydration of 1 nm2 of the hydrocarbon chain amounts to 10:4 kJ mol1 . This effect is practically

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61

completely ascribed to entropy increase of the water. Estimate the relative increase in the number of conﬁgurations of water upon dehydration of the hydrocarbon chain of the alkanes. The radius of a water molecule is 0:14 nm. 4.4

(a)

Give an expression (in terms of mole fraction X) for the solubility of alkanes in water as a function of the number n of C-atoms in the hydrocarbon chain. The increment in the Gibbs energy of dissolution at 20 C equals 3:5 kJ per mole per C-atom in the hydrocarbon chain. Calculate XC8 =XC12 . (b) The ﬁgure below gives for octane the ratio of the mole fractions in water and tetra as a function of temperature. Can you infer from this ﬁgure that hydration of octane involves a reduction of entropy in the system? What are your conclusions with respect to the change in heat capacity? Give a physical explanation.

SUGGESTIONS FOR FURTHER READING W. Blokzijl, J. B. F. N. Engberts. Hydrophobic effects. Opinions and facts, in Angew. Chem. Int. Ed. Engl. 32: 1545–1579, 1993. F. Franks (ed.). Water: A Comprehensive Treatise. I: The Physics and Physical Chemistry of Water; II: Water in Crystalline Hydrates; Aqueous Solutions of Simple Nonelectrolytes; III: Aqueous Solutions of Simple Electrolytes; IV: Aqueous Solutions of Amphiphiles and Macromolecules; V: Water in Disperse Systems: VI: Recent Advances; VII: Water and Aqueous Solutions at Subzero Temperatures, New York: Plenum, 1972–1982. C. Tanford. The Hydrophobic Effect, New York: John Wiley, 1973.

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Honeybees at Work

Honeybees are smart! The hexagonal symmetrical structure of the honeycomb represents a minimum surface area for a given number of cells, like the foam shown in Figure 5.3. The bees act highly ergonomically: they perform a minimum amount of work to create the surface area of the thin-walled cells. (Figure courtesy of Bijenhuis, Wageningen, The Netherlands.)

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5 Interfacial Tension

Interfaces are the boundaries between immiscible phases. They may be formed between solid (S) and liquid (L), solid (S) and gas (G), liquid (L) and gas (G), and between a polar liquid (L) and an apolar liquid (L). In the case of a boundary with a gas (G) or with vacuum (-) the interface is usually called the ‘‘surface.’’ To indicate the physical exterior of a body, irrespective of its surroundings, the notion of ‘‘surface’’ may be used as well. It is generally experienced that a relaxing interface tends to minimize its area. This tendency results from a contractile force in the interface, which is referred to as interfacial tension. As a consequence, work has to be added to enlarge the interfacial area. The existence of the interfacial tension is at the basis of phenomena such as wetting and adhesion, phenomena that ubiquitously occur in biological systems and biotechnological applications. These, and related phenomena, are discussed in the following chapters.

5.1 INTERFACIAL TENSION: PHENOMENOLOGICAL ASPECTS For ﬂuid interfaces, that is, L=G and L=L interfaces, the action of the interfacial tension manifests itself in the shape the interface adopts. For instance, in the absence of external forces, a ﬁxed amount of liquid assumes a spherical shape being the geometry with the lowest surface-to-volume ratio. Examples are emulsion droplets, water droplets in clouds and fog, soap bubbles, and glassy spherules in lunar dust (that must have been ﬂuid before solidiﬁcation). Another illustration of the contractile force in an interface is the minimum interfacial area

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Chapter 5

Figure 5.1 Crossing soap ﬁlms (a) are unstable. The triangular pattern shown in (b) is spontaneously formed.

of a soap ﬁlm. This can be elegantly demonstrated in a simple experiment: cover the bottom of a glass dish with a 0.5–1 cm thick layer of mercury and on top of this another 0.5 cm thick layer of concentrated surfactant solution. By moving a glass rod over the bottom across the dish a vertical soap ﬁlm in the mercury is formed that persists for some time (a few minutes). Making another ﬁlm, crossing the ﬁrst one, never leads to the situation shown in Figure 5.1(a) but always to the one depicted in Figure 5.1(b). The reason is that in the case of Figure 5.1(b) the total path length is less than the total path length in Figure 5.1(a). In fact, from mathematical analysis it follows that the path has a minimum length when all the angles in Figure 5.1(b) are 120 . In three dimensions the same phenomenon is observed in soap ﬁlms in a cubic or other three-dimensional frame. See Figure 5.2. For the same reason, the gas bubbles in foams tend to pack in a perfect hexagonal symmetry. In practice, however, the foam structure is usually less homogeneous. Polydispersity leads to somewhat curved

Figure 5.2 Stable three-dimensional conﬁgurations for soap ﬁlms.

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Figure 5.3 Cross section through a layer of foam showing the tendency of hexagonal packing.

lamellae, and variations in the interfacial tensions of adjacent lamellae cause angles to deviate from 120 . A picture of foam structure is shown in Figure 5.3.

5.2 INTERFACIAL TENSION AS A FORCE. MECHANICAL DEFINITION OF THE INTERFACIAL TENSION Consider a ﬂat, deformable, interface A, as shown in Figure 5.4. Somewhere along its perimeter a length ‘ is taken. It requires a force f to pull this length ‘ outwards over a distance D x, that is, to enlarge the interfacial area with DA ¼ ‘ Dx, to compensate the contractile force acting in the interface. The force f is

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Chapter 5

A ∆x

f Figure 5.4

Interfacial tension is a force per unit length.

proportional to ‘ and, if the change in the interfacial area occurs reversibly (which implies that at any time equilibrium is maintained), f ¼ g‘, or f g¼ : ‘

ð5:1Þ

Equation (5.1) deﬁnes the interfacial tension g as a force per unit length. It follows that the work w involved in enlarging the interfacial area is given by ð ð w ¼ g‘dx ¼ gdA ð5:2Þ which, for the case where g is invariant with A, can be written as gDA. Thus the interfacial tension is mechanically measurable as the work or, for that matter, the force required to reversibly enlarge the interfacial area (by an inﬁnitesimally small amount). As a rule, ﬂuid interfaces are isotropic. The value of g is therefore independent of the direction in which the interface is expanded. In other words, it is a scalar property. The usual mechanistic explanation of the interfacial tension is as follows. In the bulk of an isotropic phase a molecule is equally attracted by its neighbors from all directions, so that the net force acting on that central molecule averages to zero. At an interface the molecules are in an asymmetric force ﬁeld. They are attracted to the phase having the strongest cohesive attraction force. As a result, the molecules tend to leave the interface, thereby reducing the interfacial area. In other words, the molecules in the interface have a higher potential energy. From this reasoning it follows that the interfacial tension is higher the more the internal cohesions within the two adjoining phases differ. There is, however, a snag in this mechanistic approach. An interface cannot be at rest if there is a net force acting on the molecules in the interfacial region. There must exist another, compensat-

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ing, force to ensure equilibrium. For a more adequate discussion of this issue we depart from the mechanistic explanation and choose a thermodynamic approach.

5.3 INTERFACIAL TENSION AS AN INTERFACIAL (GIBBS) ENERGY. THERMODYNAMIC DEFINITION OF THE INTERFACIAL TENSION According to basic thermodynamics (see Chapter 3) for an inﬁnitesimally small (i.e., reversible) change in a heterogeneous system X dU ¼ T dS pdV þ gdA þ m dni ; ð5:3Þ i i where U is the internal energy of the system, T the temperature (in K), S the entropy, p the pressure, V the volume, g the interfacial tension, A the interfacial area, mi the chemical potential of component i, and ni the number of moles of i in the system. The term T dS equals the heat absorbed by the system from its surroundings and the other terms on the right-hand side of Eq. (5.3) represent work performed on the system. Under conditions of constant S, V and ni0 s @U g¼ : ð5:4Þ @A S;V ;ni0 s The interfacial tension is thermodynamically deﬁned as the increment in energy when reversibly extending the interface by one unit area, at constant entropy, volume, and composition of the system. Note that g deﬁned in this way has the dimension energy=area ðJ m2 Þ which is equivalent to force=length ðN m1 Þ. Keeping S and V constant during variation of the interfacial area is highly impractical. As a rule, the density in the interfacial region differs from that in the bulk phases and, consequently, the same applies to the entropy and the volume. In practice usually p and T are ﬁxed, so that it is more convenient to deﬁne g from X m dni ; ð5:5Þ dG ¼ SdT þ V dp þ gdA þ i i where G is the Gibbs energy of the system which gives @G ; g¼ @A T;p;ni0 s

ð5:6Þ

that is, the interfacial tension equals the increment in Gibbs energy when reversibly extending the interfacial area by one unit, at constant temperature, pressure, and composition of the system. It implies that the interfacial tension is experimentally accessible as the isothermal isobaric work required to reversibly extend the interfacial area by one unit, at constant composition.

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Chapter 5

Because G U þ pV TS ¼ H TS [cf. Eq. (3.13)] in which the enthalpy H U þ pV [cf. Eq. (3.11)], @G @U @V @S g¼ ¼ þp T @A T;p;ni0 s @A T;p;ni0 s @A T ;p;ni0 s @A T ;p;ni0 s @H @S ¼ T : ð5:7Þ @A T ;p;ni0 s @A T;p;ni0 s Relative to the other terms p

@V @A T ;p;ni0 s

is very small, so that @U @H

: @A T ;p;ni0 s @A T ;p;ni0 s Equation (5.7) reveals the error made when stated that, at T , p, and ni0 s constant, the interfacial tension is equivalent to the (differential) energy per unit interfacial area. That error is given by @V @S @S T

T : p @A T ;p;ni0 s @A T ;p;ni0 s @A T;p;ni0 s Hence, interpreting the interfacial tension in terms of an interfacial energy implies neglect of entropy contributions. The value of ð@S=@AÞT ;p;ni0 s is not experimentally accessible, but it can be readily obtained as the temperature-dependence of g. Cross-differentiation in (5.5) gives @S @g ¼ : ð5:8Þ @A T;p;ni0 s @T p;A;ni0 s Inserting this in (5.7) yields @H @g g¼ þT : @A T ;p;ni0 s @T p;A;ni0 s

ð5:9Þ

For essentially all interfaces g decreases with increasing temperature which means that in the interfacial region the entropy is higher than in the liquid bulk phase. In equilibrium the excess energy in the interface is just compensated by the excess entropy. Values of thermodynamic properties of the L=G interfaces for a variety of compounds are listed in Table 5.1. Over a rather wide temperature range, say, at least some tens of degrees Celsius (or Kelvin), @g=@T is essentially constant. The

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Table 5.1 Characteristic thermodynamic properties of L=G interfaces in single component systems, at 20 C and 1 bar

Water n-Hexane Ethyl ether Chloroform Benzene Mercury

g (mJ m 2)

@ g= @ T ¼ @ S=@ A (mJ m 2 K 1)

@ H= @ A @U=@ A (mJ m 2)

72.8 18.4 17.0 28.5 29.0 484

0.152 0.105 0.116 0.135 0.099 0.220

117.3 49.2 51.0 68.3 58.0 548

values in Table 5.1 clearly indicate that the interfacial entropy largely, and often dominantly, contributes to the interfacial tension. In the case of L=G and S=G interfaces the excess interfacial entropy can easily be understood in terms of decreasing packing density when passing the interfacial region (having a thickness of a few molecular layers) towards the G-phase. As a consequence, the molar conﬁguration entropy in the interfacial region is higher than in the bulk phases. L=L interfaces have an excess entropy as well; it is caused by the mixing of the two liquids in the interfacial region (cf. Section 3.8).

5.4 OPERATIONAL RESTRICTIONS OF THE INTERFACIAL TENSION 5.4.1 Interfacial Tension of Solids According to Eq. (5.6) the interfacial tension can be assessed as the isothermal isobaric work required to reversibly expand the interface under constant composition of the system. Reversibility is easily satisﬁed with ﬂuid interfaces. However, it is, with a few exceptions, impossible to extend a solid (S=L or S=G) interface reversibly. Furthermore, (crystalline) solids are usually not isotropic and the formation of new interfacial area by cleavage occurs along preferred cracks. Various procedures have been proposed to indirectly assess the interfacial tension of solids. None of them is unambiguous and we refrain here from discussing them. In most applications it is the replacement of S=G by S=L, or vice versa, that is relevant, rather than the formation of new S or S=G. Hence, the quantities (gSG gS ) or (gSL gSG ) are of interest. These are operational quantities because exposure of a solid to a gas or liquid and replacement of a gas by a liquid at a solid easily reach equilibrium within the time scale of the measurement. This matter is discussed in more detail in Chapter 8.

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Chapter 5

5.4.2 Constant Composition The requirement of constant composition poses a problem for solutions and other mixtures. When the interfacial area is extended adsorption will occur at the newly formed interface. To relate the interfacial tension to the excess interfacial Gibbs (or, for that matter, Helmholtz) energy, adsorbed amounts and concomitant changes in the composition of the bulk phases have to be established, as has been discussed in Section 3.9 [Eq. (3.75)]. Further elaboration on the relation between interfacial tension, adsorbed amounts, and composition of the bulk phase can be found in Chapters 7, 11, and 14.

5.4.3 Dynamic Interfacial Tension So far we have treated the interfacial tension as an equilibrium property which can be determined in a system that is relaxed during the time of the measurement. However, if the interface is off-equilibrium, that is, during the relaxation process towards the equilibrium state, the interfacial tension is time-dependent. Such a nonequilibrium, time-dependent interfacial tension is referred to as dynamic interfacial tension. Interpretation of dynamic interfacial tensions is usually in terms of surface rearrangements, transport of surface-active compounds to or from the interface, conformational and orientational changes of adsorbed molecules, and so on. In a single-component system the time-dependency of the interfacial tension is determined by the time needed for the molecules in the interfacial region to attain their equilibrium distribution. Except for solids (as discussed above) this is a fast process typically on the order of milliseconds, so that essentially all measuring procedures yield the equilibrium interfacial tension. On the other hand, for solutions containing surface active compound(s), adsorption and desorption processes usually determine the rate of relaxation of the interface. Depending on the system and the conditions the time scale may be much longer, say, on the order of seconds up to hours. We return to this in Chapter 17.

5.5 INTERFACIAL TENSION AND THE WORKS OF COHESION AND ADHESION When deﬁning cohesion and adhesion we consider two phases a and b without specifying their physical state. The interface between the two phases is indicated by a=b and their individual surfaces (interfaces with their own vapor or with vacuum) are indicated by either a or b. The processes of cohesion and adhesion are schematically depicted in Figures 5.5(a) and (b). Cohesion involves the merging of two volumes of a (or b) into one volume in an environment of its vapor.

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Figure 5.5 Schematic representations of (a) cohesion and (b) adhesion.

Then, in view of Eq. (5.6) the Gibbs energy of cohesion in phase a is deﬁned as the reverse of the isothermal isobaric work per unit cross-sectional area to reversibly separate two volumes of a, Daa Ga 2ga :

ð5:10Þ

Adhesion involves the merging of a and b, that is, the formation of an a=b interface at the expense of surfaces of a and b. Correspondingly, the Gibbs energy of adhesion may be expressed as Dab Ga gab ga gb :

ð5:11Þ

It should be realized that the illustrations in Figure 5.5 are schematic. In other words, the volumes and areas depicted are meant to be portions of macroscopic samples. The shapes of the affected materials are irrelevant. Values of Daa Ga and Dab Ga for various liquids are collected in Table 5.2. By comparing values of Daa Ga and Dab Ga information may be inferred on the interfacial behavior of molecules. As the value of DG is a measure of the afﬁnity between the compounds involved, it is concluded that the afﬁnity between molecules of the saturated hydrocarbons (e.g., heptane and octane) among Table 5.2 Gibbs energy of cohesion Daa Ga and Gibbs energy of adhesion DabGa for various compoundsa a Water Heptane Octane Octanol Octanoic acid ( ¼ Caprylic acid) a

b

DaaGa (mJ m 2)

DabGa (mJ m 2)

Water Water Water Water Water

146 41 44 55 55

— 42 44 92 95

T ¼ 20 C.

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Chapter 5

one another is similar to the afﬁnity between these hydrocarbons and water. The apolar hydrocarbons interact only through dispersion interactions and these are for water about the same as for the hydrocarbons. Polar compounds, such as octanol and octanoic acid interact more strongly with water than with themselves. Apparently, these compounds orient themselves in the interface with their polar part towards the water to form hydrogen bonds with the water molecules, which gives, in addition to the dispersion contribution, a polar contribution to the Gibbs energy of adhesion. The high value of Daa Ga for water reﬂects the strong internal cohesion in this liquid, as has been amply discussed in Chapter 4.

5.6 MOLECULAR INTERPRETATION OF THE INTERFACIAL TENSION 5.6.1 Nearest-Neighbor Interactions Differences between interactions between molecules located in an interface and in a bulk phase manifest themselves as an interfacial tension. In a bulk phase each molecule is equally attracted in all directions by its neighbors, whereas at an interface the molecules are in an asymmetric force ﬁeld. For a liquid (L) or a solid (S) phase adjoining their vapor (G), or vacuum, there are hardly any neighbors, or no neighbors at all, in the gas phase or the vacuum. As a result, the molecules are apparently pulled towards the L or S bulk phase. Let the nearest-neighbor interaction between a pair of molecules of type A be eAA ; then the interaction expressed per molecule is eAA =2. Assuming that each molecule is surrounded by z nearest neighbors, the Gibbs energy of the total nearest-neighbor interaction in the bulk phase a can be written as a ¼ Gnn

za e : 2 AA

ð5:12Þ

Similarly at the surface s of L or S, s Gnn ¼

zs e : 2 AA

ð5:13Þ

s [Note that, in contradiction to Gs in (3.66) and equations derived thereof Gnn is not an excess quantity. The excess Gibbs energy due to nearest neighbor s a Gnn Þ]. interactions is ðGnn Because eAA < 0 and za > zs, moving a molecule from the bulk to the surface, that is, extending the surface area, increases the Gibbs energy of the system. In other words, work must be performed on the system to create new

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s a interfacial area. This work ðGnn Gnn Þ, expressed per unit surface area, equals ga . Hence,

ga ¼

ðzs za ÞeAA ; 2ao

ð5:14Þ

where ao is the area per molecule at the interface. eAA may comprise various types of interaction. For nonpolar compounds that interact by dispersion (d) forces only, eAA ¼ edAA ¼

bAA ; r6

ð5:15Þ

where b is a constant for the dispersion interaction between two molecules at separation distance r. In the case of a cubic surrounding lattice ao can be approximated by r2, so that combining Eqs. (5.14) and (5.15) gives ga;d ¼

bAA : 2r8

ð5:16Þ

(In a cubic lattice each molecule in the bulk of a is surrounded by six nearest neighbors, whereas at the surface of a the number of nearest neighbors is ﬁve.) For more complex substances, such as polar compounds, eAA may be approximated as X ex ; ð5:17Þ eAA ¼ edAA þ x AA P where x stands for the summation of all kinds of nondispersive interactions, such as stacking between p-electrons, hydrogen bonding, ion pairing, and so on. Dispersion interactions are always effective; therefore, the term edAA is excluded from the summation term. Based on Eqs. (5.14) and (5.17) the following additivity rule has been proposed for the surface tension, X ga;x ; ð5:18Þ ga ¼ ga;d þ x in which ga;x represents the contribution of the interaction of type x to ga . For an L=L or an L=S interface a molecule in the interface has nearest neighbors in both adjoining phases a and b. When a molecule A is moved from the bulk of phase a to the a=b interface it loses interactions with other A molecules but it gains interactions with B molecules in phase b. Similarly, B molecules lose interactions with B and gain interactions with A. Assuming that the coordination number z is the same in a and b and that the molecules A and B are of similar size, the Gibbs energy to extend the interface with one unit area is zs zab 1 1 eAA þ eBB eAB ; gab ¼ ð5:19Þ 2 2 ao

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Chapter 5

which for a cubic lattice with edge r in both phases reduces to gab ¼

eAA þ eBB 2eAB : 2r2

Combined with (5.14), applied to a cubic lattice, 2e : gab ¼ ga þ gb AB 2r2

ð5:20Þ

ð5:21Þ

To further evaluate gab , the relation between eAB , eAA , and eBB is required. This matter is discussed in Section 5.6.2. Considering interactions between the nearest neighbors only seems rather restrictive. In particular for dispersion interactions, which are in good approximation additive, the inﬂuence of the numerous less-near neighbors may contribute signiﬁcantly. A model including dispersion interactions with more distant neighbors leads, for L=G or S=G interfaces, to ga;d ¼

pbAA : 6r8

ð5:22Þ

This result differs only slightly from the expression for ga;d derived from the nearest neighbor model, that is, (5.16). In view of the many extra assumptions to be made and, hence, the additional uncertainties introduced, the approximation is not really improved by including interaction with neighboring molecules beyond the nearest ones. Finally, the main shortcomings of the molecular interpretation of the interfacial tension should be mentioned. First, in deriving (5.16) it is assumed that the molecular interactions within one phase, and the intermolecular distance throughout the system, are everywhere the same. However, as has been discussed in Section 5.3, for interfaces with gas (or vacuum) the interfacial layer is less dense than the bulk of the liquid or solid phase. The separation distance between the molecules at the surface is larger than in the bulk of L or S, and, hence, the values for eAA are less negative. Moreover, the less dense packing allows a larger translational freedom for the molecules at the surface. The resulting excess surface entropy may be regarded as a mixing entropy of molecules and ‘‘holes.’’ Similarly, for L=S and L=L interfaces the mixing entropy of A and B in the interfacial region is ignored. Second, entropy effects arising from preferred orientations of molecules in the interfacial region are not taken into account.

5.6.2 Relations Between the Interfacial Tension cab and the Surface Tensions ca and cb As discussed in Section 5.3, ga and gb equal the reversible (isothermic and isobaric) work of extending the surfaces of the condensed phases a and b by one

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unit area. In turn, these works are directly related to the attractive interactions between A–A and B–B molecules, respectively. In the interfacial region between the condensed phases a and b, the molecules A and B attract each other across the interface. This A–B attraction partially overcomes the A–A and B–B attractions that oppose A (resp., B) to move to the interface. It is thus inferred that gab < ðga þ gb Þ. In a ﬁrst approximation it may be assumed that interaction between A and B, eAB , across the interface is dominated by dispersion forces. It is agreed that eAB may be calculated as the geometric average of edAA and edBB . Thus, eAB ¼ ðedAA edBB Þ1=2 :

ð5:23Þ

The sign indicates attractive interaction. Substitution of Eq. (5.23) in (5.21), and using (5.14) for a cubic lattice, yields gab ¼ ga þ gb 2ðga;d gb;d Þ1=2 :

ð5:24Þ

This result is known as the Fowkes’ approximation of the interfacial tension. More often than not, ga;d and gb;d are unknown. Only for completely nonpolar compounds, such as hydrocarbons, that interact through dispersion forces only gd ¼ g. Hence, for an interface at which at least one of the adjoining phases is completely nonpolar, (5.24) can be applied to ﬁnd gd of the other phase. For example, the measured values, at 20 C, for the interfacial tension of the octane=water interface and for the surface tensions of water and octane are 50:8 mJ m2 , 72:8 mJ m2 , and 21:8 mJ m2, respectively. Applying (5.24) gives gwater;d ¼ 22:0 mJ m2 . Thus, for water dispersion interactions contribute only about 30% to its surface tension. This result illustrates the relatively strong polar interactions between water molecules (cf. Chapter 4). If other than dispersion interactions are effective across the a=b interface as well, Eq. (5.24) gives an overestimation of gab . In view of this, the following modiﬁcation of (5.24) is made, þ eAB ; gab ¼ ga þ gb 2 ðga;d gb;d Þ1=2 þ r2

ð5:25Þ

in which eþ AB represents the polar (electron donor–electron acceptor) interactions. These interactions may be further elaborated, as that, for example, by Van Oss:

eþ AB ¼ 2fðga;þ gb; Þ1=2 þ ðga; gb;þ Þ1=2 g; r2

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ð5:26Þ

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where gþ and g represent the electron donor and electron acceptor properties of the components involved. In his theory Van Oss indicates how gþ and g may be derived from wetting studies and he presents gþ and g values for a large variety of biological and synthetic compounds. A more general relation among gab , ga , and gb may be obtained by expressing the differences between ðga þ gb Þ and gab as ðga þ gb Þ gab ¼ 2Fðga gb Þ1=2 ;

ð5:27Þ

in which F, the so-called Girifalco and Good parameter, is deﬁned as F

eAB : ðeAA eBB Þ1=2

ð5:28Þ

Substitution of Eq. (5.28) in (5.21) and using (5.15) for a cubic lattice, gives Girifalco and Good’s approximation of the interfacial tension gab ¼ ga þ gb 2Fðga gb Þ1=2 :

ð5:29Þ

Values of F may be derived from the works or, for that matter, the Gibbs energies of cohesion and adhesion (see Section 5.5), because F may also be expressed as F¼

Dab Ga aa

ðD Ga Dbb Ga Þ0:5

:

ð5:30Þ

The value for F approaches a maximum of unity if the interactions between the molecules within phase a are similar to those within phase b and, hence, similar to those between molecules of a and b. Table 5.3 lists a few approximate values of F for liquid=water interfaces, as obtained by applying Eq. (5.30) to experimental values for the interfacial and

Table 5.3 Values for the Girifalco and Good parameter for water=liquid interfaces Liquid Alkanes Aromatic hydrocarbons Polar liquids H-bonding compounds

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F 0.55 0.7 0.6–0.9 1

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Table 5.4 Interfacial tensions of water (a)=liquid(b) interfaces calculated using the models of Fowkes and Girifalco and Good, and derived from direct measurementa Measured Liquid Water Hexane Octane Octanol Benzene Aniline a

Fowkes

Girifalco and Good

gb

gab

gab,d

gab

72.8 18.4 21.8 27.5 28.9 42.9

— 51.1 50.8 10.7 35.0 5.8

— 51.1 51.0 52.7 51.5 53.7

— 51.0 50.8 10.9 37.5 3.9

T ¼ 20 C.

surface tensions. Alternatively, F may be evaluated theoretically. It is noted that Fowkes’ equation for the interfacial tension, (5.24), is a special case of Girifalco and Good’s approximation, namely, for the condition that the attraction within and between the phases across the interface are governed by dispersion forces. In Table 5.4 values for the interfacial tensions of various water=liquid interfaces as calculated by the models of Fowkes and Girifalco and Good are compared with measured values. It is clear that Fowkes’ model gives good results for aliphatic hydrocarbons that interact with water by dispersion forces only. As soon as polar interactions (p-electrons, dipoles, hydrogen bonding, etc.) play a role as well, gab deviates strongly from gab;d . It is further noted that, because of the relative insensitivity of dispersion interactions to the type of liquid, the constancy of the calculated values for gab;d adds to their plausibility.

EXERCISES 5.1

Comment on the following statements. (a)

As a rule, the surface tension of pure liquids increases with increasing temperature. (b) Low molecular mass ions tend to be strongly hydrated. As a result, the surface tension of an aqueous solution of NaCl is higher than that of pure water.

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Chapter 5

(c)

The interfacial tension of water=(higher) alcohol interfaces can be calculated using Fowkes’ approximation. (d) The molar Gibbs energy near the surface of a liquid is, at constant temperature and pressure, higher than in the bulk of the liquid. (e) The Gibbs energy of adhesion between an alkane and water is more negative than the Gibbs energy of cohesion of the alkane.

5.2

Give an expression in terms of interfacial tensions for the Gibbs energy of adhesion between phases a and b across a medium d, DabðdÞ Ga , and across a vacuum Dab Ga . For the case of an apolar medium d derive that DabðdÞ Ga Dab Ga ¼ ð2gd Þ1=2 fðgd Þ1=2 ðga;d Þ1=2 ðgb;d Þ1=2 g:

5.3

(a)

What is the relation between the interfacial tension and changes in the enthalpy and entropy due to expanding an interface at constant temperature and pressure? Derive the relation between the differential interfacial entropy and the temperature-dependence of the interfacial tension. The temperature-dependence of the surface tension of water in the temperature range 10–40 C and at 1 bar is as follows. Temperature ( C) Surface tension (mN m 1)

10 74.2

15 73.5

20 72.8

25 72.0

30 71.2

40 69.6

(b)

Calculate the differential surface enthalpy and differential surface entropy at 25 C and 1 bar. (c) Calculate the work required to reversibly expand the surface at 25 C and 1 bar. Calculate the heat effect due to this expansion. (d) The pressure is increased from 1 bar to 10 bar by supplying an inert gas. Does this lead to an increase or a decrease of the interfacial tension?

5.4

A molecular interpretation of the interfacial tension requires a model. Assume that in the solid (S) and in the liquid (L) phases at either side of an S=L interface molecules are arranged in a cubic lattice and that the molecules in both S and L have a radius of 0.2 nm. Furthermore, assume that the Gibbs energy of interaction between molecules is solely determined by nearest-neighbor interactions e. Derive an expression for the excess Gibbs energy of the S=L interface in terms of nearest-neighbor interactions.

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SUGGESTIONS FOR FURTHER READING I. Benjamin, Molecular structure and dynamics at liquid–liquid interfaces, Ann. Rev. Phys. Chem. 48: 407–457, 1997. F. M. Fowkes. Attractive forces at interfaces, in Chemistry and Physics of Interfaces, D. E. Gushee (ed.), Washington, DC: ACS, 1965, Chapter 1. A. W. Neumann, J. K. Spelt (eds.), Applied surface thermodynamics, in Surfactant Science Series 63, New York: Marcel Dekker, 1996. J. F. Padday, Surface tension. Part I: The theory of surface tension, in Surface and Colloid Science, Vol. I, E. Matijevic (ed.), New York: Wiley-Interscience, 1969, pp. 39–251. C. J. van Oss, Polar or Lewis acid-base interactions, in Interfacial Forces in Aqueous Media, New York: Marcel Dekker, 1994.

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Smog

A small strongly curved droplet of liquid evaporates more readily than a larger one. This is the reason why water droplets in fog are not so small. For the same reason fog is more easily developed in areas where the atmosphere already contains other particles: water condenses around those particles. We see this phenomenon in industrial and urban areas where smoke provides the nuclei for ‘‘smog’’ formation. (Figure courtesy of Schiphol-Aerophoto, The Netherlands.)

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6 Curvature and Capillarity

The colloidal domain comprises a wide variety of curved interfaces. For instance, in biotechnological and biomedical systems we encounter foams, emulsions, and other dispersions of (deformable) mesoscopic particles such as vesicles, liposomes, and various biological cells. Furthermore, there are numerous physical phenomena where curved interfaces play a crucial role. Capillary rise and capillary depression, illustrated in Figure 6.1, are well-known examples. If the inside of a capillary is wetted by the liquid capillary rise is observed whereas nonwetting results in capillary depression. The angle a between the liquid and the solid, the so-called contact angle, is a measure of the wettability of the solid: the

Figure 6.1 Capillary effects due to different wettabilities of the inner wall of a capillary, reﬂected by the contact angle of wetting a: (a) capillary rise; (b) no effect; (c) capillary depression.

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Chapter 6

better the wetting, the sharper the contact angle. Wetting phenomena are treated in detail in Chapter 8. Quantitative interpretation of capillary rise and depression requires the introduction and discussion of the notion of ‘‘capillary pressure,’’ that is, the pressure difference across a curved interface due to the interfacial tension in that interface. Curvature further plays an important role in phase transitions. The formation of a new phase starts with nuclei, very small particles, droplets, or bubbles. The strong curvature retards the growth of the nuclei and formation of the new phase therefore occurs only after superheating, supercooling, or supersaturation.

6.1 CAPILLARY PRESSURE. THE YOUNG–LAPLACE EQUATION Consider the formation of an air bubble in a liquid medium, for instance, the blowing of a soap bubble. To blow a bubble an excess pressure is applied. This excess pressure is called the capillary pressure or the Laplace pressure pL . The excess pressure inside the bubble is balanced by a stress in the liquid=air interface. The question at issue is how the capillary pressure is related to the interfacial tension and the size of the bubble. Figure 6.2 illustrates a cross section of a bubble with radius R. When the bubble is inﬁnitesimally expanded to a radius R þ dR the corresponding change of the volume is 4 dV ¼ p ðR þ dRÞ3 R3 4p R2 dR; 3 and of the area (for one L=G phase boundary of the bubble), dA ¼ 4p ðR þ dRÞ2 R2 8p RdR:

ð6:1Þ

ð6:2Þ

At equilibrium the net change in Helmholtz energy is zero; that is, at constant temperature and composition, dF ¼ pdV þ gdA ¼ 0:

ð6:3Þ

Substituting Eqs. (6.1) and (6.2) in (6.3) gives the relation between the excess pressure, the interfacial tension, and the radius of the bubble: pL ¼

2g : R

ð6:4Þ

For a cylindrical geometry of radius R and length l, similar reasoning results in pL ¼

g : R

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ð6:5Þ

Curvature and Capillarity

83

p

L

p + ∆ pL

G

R

dR

Figure 6.2 Cross section of an expanding gas bubble in a liquid medium.

The relevance of (6.4) and (6.5) is that they enable us to formulate balances between body forces and interfacial forces. Consider, for instance, the rise of a liquid in a vertically positioned capillary. In the case where a ¼ 0 and the meniscus inside the capillary is perfectly spherical with radius R, (6.4) can be applied to calculate the underpressure in the liquid just below the meniscus. This underpressure causes the liquid to rise until pL is just balanced by the hydrostatic pressure hrg, where h is the capillary rise, r the density of the liquid, and g the gravitational constant. In equilibrium h¼

2g : Rgr

ð6:6Þ

Thus, in capillaries having, say, a radius of 105 m, as may exist in capillary blood vessels, (super)absorbents, soils and sediments, and other textiles, powders,

porous materials, water g 72 mN m1 would rise about 1:50 m. Equations (6.4) and (6.5) are special forms (i.e., for the geometries of a sphere and a cylinder, resp.) of the Young and Laplace equation for an arbitrary curved interface where R1 and R2 are the two principal radii of curvature characterizing the spatial curvature of an interface 1 1 þ pL ¼ g : ð6:7Þ R1 R2 Equation (6.7) [and (6.4) and (6.5)] shows that constancy of the pressure throughout an equilibrated system does not hold when curved interfaces are

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84

Chapter 6

present. The pressure difference across the curved interface is larger for the more strongly curved interfaces, that is, in the case of spheres, for the smaller particles. It should be realized that when the amount of material at the concave side of the interface becomes smaller, the system may eventually no longer be considered macroscopic in the sense that notions such as interfacial tension and pressure lose their physical meaning and hence the applicability of (6.7) breaks down.

6.1.1 Radii of Curvature The spatial curvature at a given point on an arbitrarily curved interface can be characterized by the two principal radii of curvature. For the one-dimensional situation depicted in Figure 6.3 the curvature at point C along the line AB is given by the radius RC of the circle whose circumference merges as much as possible with the section of the line around C. RC is called the radius of curvature and 1=RC is the curvature at point C. The same reasoning leads to 1=RD as the curvature at point D. The overall curvature may include convex and concave sections and different signs are assigned to the respective radii of curvature. The curvature at the concave (¼ hollow) side of the interface is taken positive and at the convex side negative. (This is consistent with an excess pressure inside a soap bubble.) In the case of a two-dimensional geometry, that is, an interface, two radii of curvature are required. Consider point E in the spatially curved interface ABCD shown in Figure 6.4(a). To ﬁnd the curvature in E the normal NN0 to the interface is constructed. This normal is in a plane PQRS that intersects the interface through FEG. An inﬁnite number of planes containing NN0 are possible. In the plane PQRS a circle tangent at the curve in E is constructed as explained for the one-dimensional geometry; see Figure 6.4(b). The corresponding radius of curvature is R0E . A second radius of

B C RC D

RD A

Figure 6.3 Radii of curvature along a curved line.

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Curvature and Capillarity

85

Figure 6.4 Radii of curvature for a curved interface.

curvature at E, R00E is found by the same procedure in a plane containing NN0 but normal-to-plane PQRS. R0E and R00E fully determine the curvature at E. As the choice of orientation of plane PQRS was arbitrary, an inﬁnite number of pairs of radii of curvature R0E and R00E exist. To standardize the characterization of the curvature the plane PQRS is chosen such that R0E is maximal and, hence, R00E is minimal. The two radii thus obtained are called the principal radii of curvature and denoted R1 and R2 . The curvature is now characterized by ð1=R1 þ 1=R2 Þ. R1 and R2 may take any value. It could, for instance, be that R1 > 0 and R2 < 0. The general mathematical expressions for 1=R1, and 1=R2 are rather complicated and are not given here. For geometries of revolution, which are often encountered in practice, the expressions can be simpliﬁed. Let the axis of revolution coincide with the z-axis; then any cross-section by a plane, containing the z-axis, is identical. Without giving the derivation, the following expressions for 1=R1 and 1=R2 are given. " 2 #3=2 1 d2 z dz ¼ 1þ R1 dx2 dx

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ð6:8Þ

86

Chapter 6

and " 2 #1=2 1 1 dz dz 1þ ¼ ; R2 x dx dx

ð6:9Þ

where x is the axis in the zx-plane, perpendicular to the z-axis. Figure 6.5 shows some well-known relatively simple bodies of revolution of which R1 and R2 are indicated.

6.2 SOME CONSEQUENCES OF CAPILLARY PRESSURE As a consequence of the pressure difference across a curved interface, such an interface will resist deformation by exerting an external force. This resistance is larger the larger the pressure difference is. Therefore, smaller drops or bubbles are less easily deformable than larger ones. This phenomenon is relevant for the preparation and stability of emulsions and foams (see Chapter 18). The capillary pressure along the perimeter of an irregularly shaped droplet of liquid varies according to the variation of the curvature. For instance, in the liquid droplet resting on a solid support, as depicted in Figure 6.6, pA > pB . Due to this pressure gradient liquid moves from A to B until pA ¼ pB . It implies equalizing the curvature along the perimeter. Anticipating the discussion in Chapter 18, we mention here that the drainage of foam (i.e., the reduction in time of the liquid fraction in foam) is strongly affected by capillary pressure differences. Figure 6.7 shows a crosssection through the contacting lamellae in foam. According to the Young–Laplace equation pA < pexternal and pB ¼ pexternal, so that pA < pB . As a result, liquid is sucked into the A-area, called the Plateau-border. The liquid then further ﬂows down the borders due to gravity. Another interesting phenomenon related to capillary pressure differences is the rupture into small droplets of a slowly ﬂowing thin jet of liquid. This is called capillary instability and can be explained as follows. A cylindrical jet of liquid is subject to a sinusoid-like perturbation, as illustrated in Figure 6.8. The principal radii of curvature in A and B follow from Figure 6.5: in A Rf ¼ R D and R2 ¼ ‘2 =4p2 D, and in B R1 ¼ R þ D and R2 ¼ ‘2 =4p2 D. The capillary pressures at A and B are pA ¼ g

1 4p2 D 2 RD ‘

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ð6:10Þ

Curvature and Capillarity

87

y R1

sphere 2a

z

R2

R1 = R2 = a

x y

R2 =

cylinder

R1

z

R1 = a

2a

R2 =

x y A

ellipsoid

b B

x

z A :

R1 = a R2 = a2 / b

B :

R1 = R2 = b2 / a

a

y A sinusoid

a

z B

x

A :

R1 = a R2 = 2 / 4π2 a

B :

R1 = R2 =

Figure 6.5 Radii of curvature for some well-deﬁned geometries.

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88

Chapter 6

pB

B

pA

A Figure 6.6 Droplet with varying capillary pressure along its irregularly shaped perimeter.

and pB ¼ g

1 4p2 D þ 2 ; RþD ‘

ð6:11Þ

so that pA pB ¼ pA pB ¼ 2gD

1 4p2 ; ‘2 R2 D2

ð6:12Þ

B pB pexternal

Plateau-border A pA

Figure 6.7 Cross section through contacting foam lamellae. The pressure in the curved Plateau-border is reduced.

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Curvature and Capillarity

89

R

∆ A

∆ B

Figure 6.8 Capillary instability induced by a sinusoid-like perturbation. Due to different capillary pressures at A and B the cylindrical jet of liquid may break up into droplets.

which reduces for D R to 1 4p2 pA pB ¼ 2gD 2 2 : ‘ R

ð6:13Þ

If pB > pA liquid in the jet ﬂows from B to A and deformation of the jet will be restored. This occurs when ‘ < 2pR. If ‘ > 2pR pA > pB and the ensuing ﬂow of liquid from A to B will cause rupture of the jet. Applications of the phenomenon of capillary instability are found, for example, in sprayers that produce aerosols, in emulsifying devices, and in spray-drying.

6.3 CURVATURE AND CHEMICAL POTENTIAL. KELVIN’S LAW AND OSTWALD’S LAW Consider a curved interface between two phases a and b. In equilibrium, the chemical potential mi for any component i is the same in a and b: mai ¼ mbi .

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90

Chapter 6

Because of the curvature the pressure at the concave side of the interface, say, phase a, is higher than for a ﬂat interface, which, in turn, increases mai . The equilibrium condition mai ¼ mbi then requires an increased value for mbi as well. For a small droplet of radius R of a pure liquid L surrounded by its saturated vapor G mL ðRÞ > mL ðR ¼ 1Þ. Then, also mG ðRÞ > mG ðR ¼ 1Þ. At constant temperature T mG ðRÞ can only exceed mG ðR ¼ 1Þ by increasing the vapor pressure pG . The conclusion is that the saturated vapor pressure of a liquid droplet is higher than the saturated vapor pressure mass of the liquid that has a ﬂat surface. of a large

The difference is given by @mG =@p T . Cross-differentiation between Vdp and mdn in dG ¼ SdT þ V dp þ mdn gives @m @V ¼ n; @p T;n @n T;p

ð6:14Þ

where G is the Gibbs energy, S the entropy, n the number of moles of the compound, V the volume of the system, and n the molar volume of the compound. Because dmG ¼ dmL , written as G L @m @m G dp ¼ dpL ; @pG T @pL T

ð6:15Þ

it follows, after applying (6.14), that nG dpG ¼ n L dpL :

ð6:16Þ

For ideal behavior of the vapor nG ¼
ð6:17Þ

Since nL is essentially independent of pL , the inﬂuence of curvature is given by ð pðRÞ d ln p ¼ n

G

pðR¼1Þ

L

ð p

dpL :

ð6:18Þ

o

In good approximation dpL ¼ dð pL pG Þ, which is the change in capillary pressure. Ð p Hence, o dpL equals the capillary pressure, due to the curvature of the surface, given by (6.4).

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Curvature and Capillarity

91

Integration of Eq. (6.18) then yields

pðRÞ 2gnL : ¼ R pðR ¼ 1Þ

ð6:19Þ

Similar reasoning for a curved surface with principal radii of curvature R1 and R2 gives us

pðRÞ 1 1 ¼ gnL þ pðR ¼ 1Þ R1 R2

ð6:20Þ

or, since nL ¼ M =rL , where M is the molar mass of the compound and rL its density in the liquid phase,

pðRÞ gM 1 1 ¼ L þ : pðR ¼ 1Þ r R1 R2

ð6:21Þ

Equations (6.20) and (6.21) are known as Kelvin’s law. For a convex surface, that is, R > 0, as is the case for liquid droplets in vapor, it follows that the saturated vapor pressure is higher than for a ﬂat surface. The droplets tend to evaporate and the evaporation accelerates as the droplets become smaller. If the dispersion is heterodisperse, containing a variety of droplet sizes, the larger droplets grow at the expense of the smaller ones. This process is called isothermal distillation. For a concave interface, R < 0, as for a gas bubble in a liquid environment, the saturated vapor pressure in the bubble decreases with an increase in the curvature. As a consequence, the larger bubbles tend to grow and the smaller ones disappear. For the same reason, a gas having a pressure below its saturation value (for R ¼ 1) may condense into the liquid state upon entering a narrow pore. This phenomenon is called capillary condensation. It is instructive to inspect the Kelvin equation somewhat more closely. For the water=air interface at 20 C gM =rL ¼ 1:31 106 J m mol1 and
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92

Chapter 6 Table 6.1 Saturated vapor pressures and Laplace pressures for aqueous dispersed systems Liquid dispersed in vapor

Vapor dispersed in liquid

R(m)

p(R)=p(R ¼ 1 )

DpL(bar)

p(R)=p(R ¼ 1 )

DpL(bar)

106 107 108 109

1.001 1.018 1.114 2.94

1.44 14.37 143.7 1437

0.999 0.989 0.898 0.339

1.44 14.37 143.7 1437

surrounding liquid L medium (phase b). As for ideal solutions ð@m=@cÞT ¼
ð6:22Þ

in which M ; r, and R1;2 apply to the dispersed compound. For salts having the general formula Mm Zn which in the dissolved state is dissociated in its ionic components M and Z, (6.22) has to be modiﬁed into ðm þ nÞ
cðRÞ gM 1 1 ¼ : þ cðR ¼ 1Þ r R1 R2

ð6:23Þ

Equation (6.22) or (6.23), now called the Ostwald equation, shows that the solubility increases with decreasing particle size. Hence, as with the L in G and G in L dispersions, heterodisperse L in L (emulsions) and S in L (sols or suspensions) systems coarsen because the smaller particles are more soluble than the bigger ones. This process is called Ostwald ripening. The rate of this ripening process depends on the solubility, the size of the particles, and the diffusivity through the medium. It increases strongly with increasing temperature, because of a higher diffusion rate. This matter is further treated in Chapter 18, Section 18.3. In contrast to liquid droplets, solid particles usually have an irregular shape with planes having different interfacial tensions. In such cases Ostwald ripening manifests itself as a change towards more perfect crystals. This phenomenon, often referred to as aging, is applied in the preparation of sols and suspensions to obtain particles having better-deﬁned interfacial characteristics. It was mentioned in Section 5.4 that the interfacial tension of a solid cannot be determined by direct measurement. The Ostwald equation gives, in principle, access to the interfacial tension of a S=L interface. However, although the effect of particle size on the

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Curvature and Capillarity

93

solubility may be well measurable, evaluation of gSL is not unambiguous because the surface tension of a solid particle is usually not uniform and the presence of more strongly curved areas (e.g., edges and protuberances) has an increasing effect on the solubility. Finally, based on the Ostwald equation, the stability of a liquid in a liquid emulsion is compared with that of a foam. As a rule, the density in the gas phase is much lower than in the liquid phase and it then follows from Eqs. (6.21) and (6.22) that droplets of a liquid in another liquid can be much smaller than air bubbles in a liquid. Emulsion droplets have dimensions typically in the range of some tenths of mms up to a few mms, whereas air bubbles in foams are usually in the mm range.

6.4 CURVATURE AND NUCLEATION The formation of a new phase begins with nucleation, the formation of small particles, and therefore the Kelvin (or Ostwald) effect plays an important role. The Kelvin effect implies that the ratio cðRÞ=cðR ¼ 1Þ or pðRÞ=pðR ¼ 1Þ increases rapidly as R decreases to very small values, as can be seen in Table 6.1. In view of this, phenomena such as supersaturation, supercooling, and superheating are involved in the onset of phase separation, that is, the formation of nuclei of the new phase. By way of example the condensation of vapor into liquid is considered. Molecules from the vapor must associate to form a nucleus onto which more molecules condense. The newly formed liquid droplet has to pass through a stage where it has a very strong curvature and therefore a high saturated vapor pressure. Such a droplet will only grow if the pressure is at least equal to the saturated vapor pressure of the droplet. In other words, relative to the saturated vapor pressure of a larger amount of liquid the vapor must be supersaturated for growth of the liquid droplets into bulk liquid to occur. If the pressure is insufﬁciently supersaturated the nucleus will not grow but spontaneously disintegrate. The Gibbs energy of forming a liquid droplet of radius R is

G ¼ n
pðRÞ þ 4pR2 g; pðR ¼ 1Þ

ð6:24Þ

where n is the number of moles in a droplet. The ﬁrst term on the right-hand side of (6.24) originates from the difference in chemical potential between a phase

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94

Chapter 6

under curvature and a bulk phase, and the second term represents the work to create the interfacial area of the droplet. With n ¼ 4pR3 r=3M (6.24) becomes G ¼

4pR3 r
ð6:25Þ

When pðRÞ > pðR ¼ 1Þ the ﬁrst term on the right-hand side of (6.25) makes G < 0 for sufﬁciently large values of R; if R is smaller, the second term dominates over the ﬁrst one, resulting in G > 0. The shape of the curve representing the variation of G with R is qualitatively shown in Figure 6.9, for pðRÞ > pðR ¼ 1Þ. The function GðRÞ contains a maximum at a certain value for R, the so-called critical radius Rc . Because, at constant p and T, any spontaneous process is characterized by a decrease in the Gibbs energy, introducing a liquid droplet of radius R into a vapor of pressure p leads to further growth of the droplet if R > Rc but if R < Rc the droplet vaporizes. Evaporation and crystallization of a liquid as well as precipitation from a solution can be treated in the same way as described for vapor condensation. For instance, for the precipitation process this results in G ¼

4pR3 r
ð6:26Þ

∆nucleationG

∆G (R)

Rc

R

Figure 6.9 Gibbs energy of formation of a droplet as a function of the droplet radius.

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Curvature and Capillarity

95

where the density r and the molar mass M refer to the solute and where g is the interfacial tension of the precipitate–solution interface. In all these cases of phase separation the thermodynamic driving force for forming the new bulk phase must balance the excess Gibbs energy in the interfacial area of the nuclei. Taking the very onset ðR ! 0Þ of nucleation as the reference state, the maximum in GðRÞ may be considered as the activation Gibbs energy nucleation G of the nucleation process. A drawback of this approach is that at the level of the ﬁrst associated molecules the notion of interfacial tension loses its meaning, and, hence, application of (6.25) or (6.26) becomes questionable. Combination of a molecular model for the aggregation of molecules into nuclei with the macroscopic thermodynamic approach allows evaluation of the rate of the nucleation process: Because of the resistance against homogeneous nucleation (i.e., the formation of nuclei made of one single compound), in practice phase separation often begins on small ‘‘foreign’’ particles, such as dust and the like, that are present in any system that is not rigorously puriﬁed. In the case of condensation from the vapor phase, molecules adsorb from the vapor on the (dust) particle and from there on further growth of the nucleus takes place until a macroscopic liquid phase has been formed. Nucleation around such predispersed particles is called heterogeneous nucleation. It may be clear that, compared to homogeneous nucleation, for heterogeneous nucleation relatively little supercooling, superheating, or supersaturation sufﬁces for phase separation to occur. Examples of heterogeneous nucleation are the use of seed crystals to initiate the formation of a new phase, bubble trails in carbonated drinks, the formation of smog in air-polluted areas, and the dispersion of condensation seeds to initiate rain from supersaturated clouds.

EXERCISES 6.1

Comment on the following statements. The pressure difference between an emulsion droplet (radius 5 mm and interfacial tension 30 mN m1 ) and its surrounding is 104 N m2 . (b) In two connecting capillary tubings of identical material but with different radii a well-wetting liquid moves from the wider to the narrower side. (c) Crystallization from a supersaturated solution gives more pure crystals when the degree of saturation is higher. (d) Ostwalds’s law [Eq. (6.22)] predicts that from a freeze-dried sample the larger particles dissolve more quickly than the smaller ones. (e) Zero curvature of an interface implies planar geometry.

(a)

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96

6.2

Chapter 6

(a)

Calculate the capillary rise of water in a glass capillary with an inner radius of 0:3 mm, for the cases where —the glass is completely wetted by the water; —the glass–water contact angle is 30 . The surface tension of water is 72 mN m1 , the density 1:0 g cm3, and the gravitational constant 9:8 m s2 .

(b)

Calculate the pressure in —a spherical air bubble immersed in an aqueous surfactant solution; —a spherical soap bubble drawn from the same surfactant solution, ﬂoating in the air. The radius of both bubbles is 1 mm, the surrounding pressure 105 nM2 ð¼ 1 barÞ, and the surface tension of the surfactant solution is 40 mN m1 .

6.3

Show that Kelvin’s equation, Eq. (6.21), can be derived from Eq. (6.25) for the condensation of a vapor into a liquid.

6.4

In a moist environment solid particles tend to stick to each other due to capillary condensation of vapor.

(a) What is the attractive driving force between the particles? (b) The picture shows a liquid water ﬁlm between two solid particles (S, L, and G denote the solid, liquid, and gas phases, resp.). Derive an expression for Kelvin’s law using the variables x and y. What is the value for the relative humidity when x ¼ 0:2 mm and y ¼ 20 nm? The molar volume of liquid water is 18 cm3 mol1 . (c) Show that the force f by which two particles attract each other can be approximated as f ¼g

x2 ; 2y

where g is the surface tension of the liquid.

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Curvature and Capillarity

(d)

97

The reversible mechanical work w to disrupt the particle–particle bond is given by 1 w ¼ pgx2 : 2 Derive this equation.

SUGGESTIONS FOR FURTHER READING S. Hyde, S. Andersson, K. Larsson, Z. Blum, S. Lnadh, S. Lidin, B. W. Ninham. The Language of Shape, New York: Elsevier, 1997. H. M. Princen. The equilibrium shapes of interfaces, drops and bubbles. Rigid and deformable particles at interfaces, in Surface and Colloid Science, Vol. 2, E. Matijevic (ed.), New York: Wiley-Interscience, 1969, pp. 1–84. J. W. Rowlinson, B. Widom. Molecular Theory of Capillarity, New York: Clarendon, 1982.

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Breath of Life

The ﬁrst inhalation of a newborn baby requires an extraordinary effort: the lungs become inﬂated and a large ﬂuid–air interface is created. This is opposed by the interfacial tension of the alveoli. To overcome this difﬁculty a special lung surfactant is released from alveolar cells and is spread to form a monolayer at the surface of the alveoli. It reduces the interfacial tension and provides the alveoli with the proper interfacial rheological characteristics that allow easy breathing. (Figure courtesy of Isoned b.v., Tiel, The Netherlands.)

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7 Monolayers at Fluid Interfaces

The notion of ‘‘monolayer’’ usually refers to a layer of amphiphilic molecules at a ﬂuid interface, being either a liquid=liquid or a liquid=gas interface. Even when the layer is incomplete or when it is more than one molecular layer thick it is still called a monolayer. The term monolayer may also be used in the case of adsorption at solid surfaces to distinguish it from a bilayer or multilayer. Amphiphilic molecules contain a polar and an apolar part. As a result, such molecules have an ambiguous (amphi) afﬁnity (philos) for water. The apolar parts behave hydrophobically: the water molecules tend to escape from contact with these parts. The polar parts are hydrophilic. They interact favorably with water. The consequence of the amphiphilic character is that the molecules are preferably located at interfaces with water, where the polar parts are exposed to the aqueous phase and the apolar parts to the nonaqueous phase. Low-molecular-weight amphiphilic compounds are often called surfactants. Well-known examples of surfactant are the classical soaps (single chain fatty acids), phospholipids, cholesterol, bile acids, long surfactant, and so on. In Figure 7.1 the chemical structures showing the polar and apolar parts of some of these surfactants are given. Monolayers may also be formed by polymers, polyelectrolytes, and proteins that contain polar and apolar parts. This chapter deals primarily with monolayers of surfactants at ﬂuid interfaces. Monolayers demonstrate well how interactions on a molecular level affect macroscopic phenomena. For instance, a monolayer of oil on the surface of an ocean damps the waves, and precious water supplies in ponds and lakes may be conserved by a monolayer that retards evaporation of the water underneath. Studying monolayers provides information on the orientation and association of the amphiphiles at interfaces. This information may also be useful for

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100

Chapter 7 octadecanoic acid (stearic acid)

COO –

O O

O O O

O

P

O

+

N (CH3)3

O 1, 2 distearoyl phosphatidylcholine (lecithin)

HO cholesterol

Figure 7.1 Structure formulae for some well-known surfactants.

understanding self-assembled structures of such amphiphilic compounds (Chapter 11) as well as the role they play in the formation and stabilization of emulsions and foams (Chapter 18). With respect to the preparation of monolayers two principally different ways are distinguished: by adsorption and by spreading. Adsorbed monolayers are formed by letting the surfactant molecules adsorb from either one of the adjoining phases. The molecules in the adsorbed monolayer are in chemical equilibrium with the ones dissolved in the phase from which they adsorb. The amount adsorbed in the interface is related to its mole fraction or, for that matter, concentration in solution according to Gibbs’ law, Eq. (3.86), showing that the interfacial tension decreases with increasing adsorbed amount. When an adsorbed monolayer is compressed molecules will desorb from the interface in order to let it relax back to the equilibrium situation given by the Gibbs adsorption equation. Therefore, adsorbed monolayers are also called ‘‘Gibbs monolayers’’ or ‘‘soluble monolayers.’’ Spread monolayers are obtained with molecules that, at least on the time scale of the experiment, do not or barely dissolve in the adjoining phases. Such monolayers are called ‘‘insoluble monolayers’’ or ‘‘Langmuir monolayers.’’ They are, as distinct from the Gibbs monolayers, not in chemical equilibrium with the adjoining phases. It means that there is no exchange of molecules to and from the monolayer and, hence, Gibbs law is not applicable to Langmuir monolayers. When Langmuir monolayers are compressed the molecules are conﬁned in a smaller area resulting in a further decrease of the interfacial tension.

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7.1 THE INTERFACIAL PRESSURE The difference between the interfacial tension g* of the bare interface and the interfacial tension g of the interface containing the monolayer is called the interfacial pressure p. Hence, p g g:

ð7:1Þ

The interfacial pressure is a two-dimensional pressure, expressed in force per unit length. For ﬂuid interfaces p is readily obtained because both g* and g can be directly measured. The interfacial pressure at solid interfaces is deﬁned according to (7.1) as well. However, in this case p cannot be measured, but may be accessible through Eq. (3.85). Although there is a formal analogy with the three-dimensional pressure of a gas in a given volume, the origin of the interfacial pressure is different. The pressure a gas exerts on the walls of the container in which it is conﬁned is due to the gas molecules colliding against the wall, whereas the interfacial pressure is the difference in contractile force in a bare interface and a monolayer, respectively. The three-dimensional analogue of the interfacial pressure is the osmotic pressure difference between a solution and a solvent. See Section 3.6. Interfacial equations of state, relating the interfacial pressure p to the adsorbed amount, the available interfacial area A, and the temperature T, can be formulated analogously to the three-dimensional equivalent relating the pressure, number of molecules, volume, and temperature.

7.2 GIBBS AND LANGMUIR MONOLAYERS. EQUATIONS OF STATE The Gibbs monolayer is continuously in equilibrium with the adjacent solution. Hence, for an ideal solution (where ci is proportional with xi) at constant temperature, combining the Gibbs adsorption equation (3.86) with (7.1) yields ðc p ¼
GðcÞd ln c;

ð7:2Þ

0

which shows that p increases with increasing surfactant concentration in solution, c. To derive the interfacial equation of state of the monolayer, relating p, G (or the interfacial area per molecule), and T, we need to know the functionality G(c). In Chapter 14 equations for G(c) at constant T, so-called adsorption isotherm

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102

Chapter 7

equations, are derived and discussed. Thus, anticipating (14.25) for an ideally ‘‘diluted’’ monolayer the following equation of state is derived, pam ¼ kB T ;

ð7:3Þ

in which am is the interfacial area available per surfactant molecule and kB is Boltzmann’s constant. Similarly, more complicated equations of state are obtained using expressions for G(c) for nonideally behaving adsorbed layers. As mentioned in Section 7.1, Langmuir monolayers are not in chemical equilibrium with the solution and, as a consequence, the Gibbs adsorption equation is not applicable to such monolayers. However, relations among p, G, and T are completely determined by the number of molecules, and the interactions among them, in the monolayer, irrespective of the way the monolayer has been formed. Equations of state are therefore identical for Gibbs and Langmuir monolayers.

7.3 FORMATION OF MONOLAYERS From the foregoing it follows that Gibbs and Langmuir monolayers are prepared in different ways. The procedure to obtain a Gibbs monolayer is, in principle, simple. A solution containing the surfactant is brought in contact with another, immiscible, phase which could be a gas (usually air) or a liquid. Adsorbed monolayers may thus be formed in colloidal systems, that is, on emulsion droplets (L=L interface) or foam lamellae (L=G interface). The large interfacial area in emulsions and foams allows analytical determination of the adsorbed amount G as a function of the concentration in solution. Because of the mesoscopic dimensions of the interfaces in such dispersed systems, the interfacial tension g cannot be measured. However, this quantity and, hence, the interfacial pressure may be derived from G(c) using Gibbs’ adsorption equation, (7.2). On the other hand, with macroscopic interfaces the adsorbed amount is often difﬁcult to assess accurately, but the interfacial tension is readily measurable. Then G can be obtained from g(c), again by applying Gibbs’ adsorption equation. When comparing the characteristics of an adsorbed monolayer on dispersed particles with those of a monolayer at a ﬂat macroscopic surface, one should be aware of ambiguities and pitfalls: as a consequence of the intricate rupture and recoalescence processes involved in the formation of emulsion droplets and foam bubbles, the composition and orientation of the surfactant molecules at the interface may not be in complete equilibrium and may deviate from the situation for a monolayer at a quiescent macroscopic interface. We return to this in Chapter 18. Langmuir monolayers are prepared by depositing, in one way or another, the amphiphilic molecules in the interface. The usual deposition procedure is spreading. Spreading starts with the application of a drop of liquid or of a solid

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crystal of the insoluble substance at the interface. Then, spreading requires the breakdown of internal coherence in the liquid or the solid in favor of adherence between this material and the supporting phase. However, for most amphiphiles Langmuir monolayers are prepared indirectly by dissolving the material to be spread in a volatile solvent, the spreading solvent, that spreads over the interface and then disappears quickly by evaporation (L=G interfaces) or by dissolution in one of the liquid phases (L=L interfaces). Figure 7.2 shows a cross section of a drop of liquid (g) at an interface between the phases a and b. Consider point A, where the three interfaces a=b, a=g, and b=g meet. The drop ﬂattens if gab > gag cos yag þ gbg cos ybg . During ﬂattening the angles y become sharper and their cosines approach unity. Hence, full spreading occurs if S abðgÞ gab gag gbg > 0:

ð7:4Þ

S abðgÞ is called the spreading coefﬁcient for the spreading of g at the ab interface. Applying (5.10) and (5.11) it can be easily veriﬁed that S abðgÞ equals the difference between the Gibbs energies of cohesion in g and of adhesion between a and b, ðgg G þ ab GÞ, and the Gibbs energies of adhesion between a and g and b and g, ðag G þ bg GÞ. By way of example, the data given in Table 5.4 predict that, at 20 C, benzene spreads over an air=water interface ðS ¼ 72:8 28:9 35:0 ¼ 8:9 mJ m2 Þ. However, benzene and water are not completely immiscible and after some time mutual saturation is reached. The surface tension of water saturated with benzene is dropped to 62.2 mN m 2 and that of benzene saturated with water to 28.8 mN m 2. It follows that after mutual saturation S < 0. This is the reason why a drop of benzene deposited on a surface of pure water initially spreads but subsequently contracts into a lens. This example demonstrates that care should be taken when predicting whether deposition of a liquid results in spreading.

γ αγ

γ

θ αγ

γ αβ

A

α

θ βγ γ βγ

β

Figure 7.2 Cross section of a sessile drop of liquid (g) at an interface between a and b. The contact angles are indicated by y.

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104

Chapter 7

When a surfactant is added to a spreading solvent gag and gbg are reduced leading to a more positive value for S abðgÞ and, hence, to a stronger spreading tendency. The spreading procedure allows exact control of the number N s of surfactant molecules to be placed in the interface. This makes quantitative interpretation of monolayer studies possible. In such studies it is usual to ﬁrst deposit an amount of surfactant that is far less than the amount that can be accommodated in a close-packed monolayer and, thereafter, to compress the interfacial area to reach close-packing of surfactant molecules. Thus, in contrast to Gibbs monolayers (in which the interfacial pressure is not affected by the available interfacial area), Langmuir monolayers are eminently suitable to determine pressure-area isotherms. These are described and discussed in Section 7.4.

7.4 PRESSURE-AREA ISOTHERMS OF LANGMUIR MONOLAYERS. TWO-DIMENSIONAL PHASES Langmuir monolayers are usually studied in a so-called Langmuir trough. The basic features of this apparatus are shown in Figure 7.3. It usually consists of a rectangular container carrying the ﬂuid phase(s). The trough is equipped with a

B B'

γ*

T

γ

γ*

T

B'

B

R R

Figure 7.3 Schematic representation of a Langmuir trough. The barrier BB0 , is movable over the slidings R to vary the enclosed interfacial area. The corresponding force is measured by a torsion wire T that is connected to the barrier. As an alternative, the interfacial tensions at both sides of the barrier may be measured independently.

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Monolayers at Fluid Interfaces

105

barrier that is positioned on the surface or, in the case of a L=L interface, in the interface. The area enclosed by the barrier can be varied in a controlled way. Spreading molecules at one side of the barrier results in a difference between the interfacial tensions at the two sides: g g ¼ p. This difference exerts a force p l on the barrier of length l in the direction of the clean interface. Compressing the spread molecules over the available area A by sliding the barrier over the interface leads to a higher value of p. The pðAÞ functionality at constant temperature, the pðAÞ-isotherm, gives for a known number of spread surfactant molecules information on the orientation of these molecules in the interface as well as on their lateral interaction. It goes without saying that for a quantitative interpretation of the pðAÞ-isotherm extreme care must be taken to avoid contaminants from adsorbing at the interface. A schematic example of a pðAÞisotherm of an insoluble monolayer is displayed in Figure 7.4. At very low values of p, say, 0.5 mN m 1, the monolayer exhibits gaseous (G) behavior. At these conditions the molecules in the monolayer are so far apart from each other that they do not interact. Because of their amphiphilic character the surfactant molecules have different interactions with the two phases at either side of the interface so that they adopt a preferential orientation. In the G-state the isotherm obeys the relation pA ¼ N s kB T ;

ð7:5Þ

collapse π

S LC LC + LE LE LE + G

G am

Figure 7.4 Schematic example of a pressure-area isotherm of a spread monolayer showing different two-dimensional phases: gas (G), liquid expanded (LE), liquid condensed (LC), and solid (S).

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106

Chapter 7

which is identical to (7.3), with A=N s am . At room temperature kB T ¼ 4:11 1021 N m, so that p ¼ 0:5 mN m1 corresponds to am ¼ 8:2 nm2 . For a fatty acid or a phospholipid with a hydrocarbon chain length of, say, 20 C-atoms 8.2 nm2 exceeds by far the area occupied by a molecule that lies ﬂat in the interface. This should of course result for noninteracting gaseous behavior. In this context, it is noted that at the water–gas interface the surfactant molecules are indeed expected to be oriented ﬂat in the interface because of dispersion interactions with the aqueous phase. At liquid–liquid interfaces the apolar part of the surfactant has dispersion interaction with the apolar liquid as well, so that the apolar parts probably protrude into the apolar liquid in order to minimize unfavorable hydrophilic hydration by the water phase. When the monolayer is compressed, the packing density of the molecules in the monolayer gradually increases and the monolayer undergoes a phase transition from the gaseous (G) state into the liquid expanded (LE) state. This phase transition is characterized by considerable compression at constant p, indicating a ﬁrst-order transition during which the G and LE states coexist. In the LE state p further increases with decreasing A until another transition occurs bringing the monolayer into the liquid condensed (LC) state. For most systems p slightly increases during the LE–LC transition, but some experiments using ultrapure components have shown this transition to occur at constant p as well. Extrapolation of p from the LE part of the isotherm to p ¼ 0 gives, for a single chain, unbranched, surfactant a value of am in the range of 0.5 nm2. This is still much larger than the cross-sectional area of the surfactant molecule, but smaller than the area the molecule needs to ﬂoat freely in the interface. It indicates that in the LE state the molecules are interacting with each other. In the LC state the monolayer is still compressible but p rises steeply with decreasing A. The p-values reached in this state are in the range of a few to a few tens of mN m1. Essentially all the apolar hydrocarbon chains are pushed out of the interface and they strongly interact with each other. It is remarkable that in the two-dimensional monolayer two distinct L-phases (LE and LC) exist, whereas this is not the case in the three-dimensional situation. The coexistence of LE and LC phases can be made visible, for instance, by using a Brewster angle microscope. Examples are shown in Figure 7.5. It requires only a small reduction of A to convert the monolayer from the LC into the solid (S) state. In the S-state the compressibility of the monolayer is essentially zero, which implies that the surfactant molecules are closely packed as in a two-dimensional crystal. The cross-sectional area of the surfactant molecule may be estimated as the value of am corresponding to the S-state. In the LE-state and, even more so, in the LC- and S-states interactions between the surfactant molecules may induce a certain structure in the monolayer that does not readily

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Monolayers at Fluid Interfaces

107

Figure 7.5 Brewster angle microscopy of monolayers showing the coexistence of LC (light) and LE (dark) phases. (From M. A. Cohen Stuart et al. Langmuir 12: 2863, 1996.)

break down upon expansion. This may give rise to hysteresis, that is, to pðAÞisotherms that deviate between compression and expansion. Further compression of the S-state leads to collapse of the monolayer. The molecules are forced to leave the interface. Because of their insolubility in the adjoining phases they will form a (disorganized) bilayer or multilayer. The slope of the pðAÞ-isotherm is a measure of its isothermal compressibility kT : @ ln A kT ¼ : @p T

ð7:6Þ

Thus, a monolayer in the G-state is highly compressible, but in the more condensed states it is not. The orientation of the surfactant molecules corresponding to the various stages in the pðAÞ-isotherm are schematically depicted in Figure 7.6.

7.4.1 Inﬂuence of the Temperature on the pðAÞ-Isotherm Figure 7.7 shows the general trend for the inﬂuence of temperature on the pðAÞisotherm. As the temperature increases, the pressure at the onset of the LE ! LC transition increases and the corresponding value of am decreases. Hence, when the temperature is raised, the monolayer may change from a condensed into an expanded state. Furthermore, the region over which LE and LC coexist decreases with increasing temperature.

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108

Chapter 7

G

G + LE

LE

LE + LC

LC

S

collapse

Figure 7.6 Orientations of amphiphilic molecules in a spread monolayer corresponding to the various phases that may be encountered along the pressure-area isotherm.

In general, a phase transition I ! II, in a monolayer may thermodynamically be treated analogously to that in a three-dimensional system (cf. Section 3.7.1). Thus the Clapeyron equation, (3.38), relating the variation of p with T, reads dp qI!II ¼ ; ð7:7Þ dT N s T I!II am where qI!II is the heat involved in the transition I ! II and I!II am the difference in the area per surfactant molecule in states II and I, respectively. If the transition is reversible, that is, in the absence of hysteresis, the change in the entropy I!II S can be derived as q I!II S ¼ I!II ð7:8Þ T

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Monolayers at Fluid Interfaces

109

π

decreasing T

πc

am, LC Figure 7.7

am, LE

am

Interfacial pressure-area isotherms as a function of temperature.

and that of the enthalpy I!II H, as I!II H ¼ qI!II

ð am;II

p dam :

ð7:9Þ

am;I

By way of example, Table 7.1 presents values of qLC!LE and LC!LE S for the LC ! LE transition in monolayers of myristic acid (CH3(CH2)15COOH) at the air=water interface. The main conclusions from these data are that heat is required to disrupt energetically favorable interactions between the surfactant

Table 7.1 Heat and entropy changes involved in the LC ! LE transition in monolayers of myristic acid at the air–water interface. T( C)

qLC ! LE (kJ mole1)

DLC ! LES (J K1 mole1)

7.2 9.1 12.1 14.1 17.0 18.0 22.3 26.2

34.3 30.1 24.7 21.3 18.4 18.0 16.3 14.0

121.2 108.7 86.6 74.2 62.7 62.7 54.3 50.2

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Chapter 7

molecules when releasing them from the condensed state. This is accompanied by an increased entropy of the monolayer. Furthermore, the temperature-dependence of q, at constant p, @qLC!LE ; LC!LE Cp ¼ @T p is highly negative. This feature also indicates the rupture of interactions between the surfactant molecules when passing from the condensed to the expanded state. Pressure-area isotherms at different temperatures also allow the evaluation of the isobaric expansion coefﬁcient ap , @ ln A ; ð7:10Þ ap @T p Like the isothermal compressibility kT ; ap is large for the G-state, but it is strongly reduced for the LC- and, even more so, for the S-state.

7.5 TRANSFER OF MONOLAYERS TO SOLID SURFACES. LANGMUIR–BLODGETT AND LANGMUIR–SCHAEFER FILMS Monolayers ﬂoating in a liquid=gas (usually air) interface may be transferred to a solid support by the so-called Langmuir–Blodgett (LB) technique. The procedure is to move a solid plate (various times) vertically through the monolayer while keeping the interfacial pressure constant. See Figure 7.8. If the surface of the solid substrate is hydrophilic, as is the case for glass, quartz, and most other (metal) oxides, the ﬁrst downward stroke through the water=air interface (Figure 7.8) does not result in monolayer transfer. The hydrophilic surface is wetted upon touching the liquid phase, but because the apolar parts of the amphiphilic surfactant molecules do not favorably interact with the hydrophilic solid surface these molecules are not transferred. In the subsequent upward stroke the meniscus slides over the surface thereby transferring the surfactant molecules from the liquid interface with their polar head groups down onto the solid support. As the apolar parts are directed outward, the solid surface is now hydrophobized. In the following downward stroke the meniscus is turned down and a second monolayer is deposited with its apolar parts contacting the apolar parts of the molecules deposited in the ﬁrst layer. Now, the surface is hydrophilic again so that in the next upward stroke surfactant molecules deposit head-down, and so on. If the surface of the solid support is hydrophobic, for example, for most synthetic polymers, deposition starts during the ﬁrst immersion in the polar liquid phase (cf. stroke 3 for the hydrophilic support).

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Monolayers at Fluid Interfaces

111

gas

liquid

support

Figure 7.8 Formation of a Langmuir–Blodgett ﬁlm (Y-type) by monolayer transfer from a liquid=gas interface onto a solid support.

In this way (multi)layers, referred to as Langmuir–Blodgett ﬁlms or, for short, LB-ﬁlms, are formed with alternating head–head and tail–tail orientations of the surfactant molecules. This type of transfer in which a monolayer is deposited during each passage of the substrate through the interface is called Y-transfer. It follows that with hydrophilic supports at the end of each completed cycle ﬁlms containing an odd number of layers are deposited, whereas for hydrophobic surfaces an even number of layers are transferred. Formation of Y-type ﬁlms is not always possible. Other types of ﬁlms are formed when the monolayer is transferred only during a downstroke (X-type) or only during an upstroke (Z-type). X-type of transfer is sometimes observed with hydrophobic supports. Deposition occurs during the ﬁrst immersion but there is no transfer on lifting the solid. Then, in the next downstroke a second layer is deposited on the ﬁrst layer through tail–head interaction, and so on. Z-type ﬁlms may be formed on hydrophilic surfaces by deposition during upward strokes. The structures of the ﬁlms formed by X- and Z-type deposition are shown in Figure 7.9. Obviously, for the formation of X- and Z-type ﬁlms tail–head interaction is more favorable than head–head and tail–tail interactions. This is rather uncommon and it requires special molecular structures of the amphiphiles. When X- and Z-type layers are formed they often turn out to be unstable and to reorient into Y-type layers. For

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112

Chapter 7

(a)

(b)

Figure 7.9 (a) X-Type and (b) Z-Type transfer of a monolayer onto a solid support.

polymeric amphiphiles such a reorientation process may occur very slowly, implying a kinetically stable X- or Z-type ﬁlm. As a rule, the liquid subphase from which the monolayer is transferred is water or, more often, an aqueous electrolyte solution. Indeed, when ionic surfactant molecules are involved, low-molecular-weight ions play an important role in stabilizing the interaction between the charged head groups in the Y-type ﬁlm. In particular, divalent (cat)ions serve to stabilize ﬁlms of monovalent (an)ionic surfactants. For a controlled transfer from the liquid to the solid surface the surface pressure should be kept constant. Hence, the LB-ﬁlm preparation takes place in a Langmuir trough (see Figure 7.3) equipped with an automated feedback system that moves the barrier during the removal of the monolayer from the liquid interface. Transfer is usually most successful and reproducible from condensed monolayers. The down- and up-strokes should occur at a rate slow enough to let the monolayer relax and the liquid subphase to drain. Rates are usually in the range of 1 mm s 1. When the structure of the monolayer is maintained during transfer the transfer ratio, that is, the ratio of the area of the monolayer removed from the liquid interface to the area of the solid support covered by the transferred monolayer, approaches unity. In this sense, the transfer ratio is usually taken as a measure of the quality of the deposition. It should be realized, however, that a transfer ratio substantially deviating from unity does not necessarily mean unsatisfactory deposition. If the deviation is consistent it indicates a reproducible change in molecular organization in the transferred monolayer.

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Monolayers at Fluid Interfaces

113

Another procedure for transferring monolayers is the Langmuir–Schaefer (LS) method, in which the solid support is moved downward until it horizontally touches the monolayer at the liquid interface. After contacting for a few or a few tens of seconds the solid substrate is withdrawn or pushed down into the liquid subphase. The LS-method is far less popular than the LB-method but it may be preferred for use with monolayers that are rather rigid. Solid supports made of glass or quartz (hydrophobized or not) are commonly used when studying LB- or LS-ﬁlms. Their optical transparency allows optical and spectroscopic investigation of the transferred layer(s). When interpreting the data it should be realized that the equilibrium structure of the transferred monolayer may not be identical to that of the parent monolayer in the liquid interface. Interaction of the amphiphiles with the solid substrate (or with the predeposited layers) is probably different from that with the liquid interface and, furthermore, at the solid surface the amphiphiles do not experience the same, constant, interfacial pressure that keeps the parent monolayer in the original state. Due to all of this, structural rearrangements may occur during or after deposition. During the aging process of the transferred ﬁlm phase separation, terrace and crack formation may take place. Where the ﬁlms kept in air are stable over several weeks or months, the aging in water may occur within minutes or hours. In addition to this thermodynamic instability the deposited ﬁlms may lack mechanical stability. When a liquid is ﬂown against or along the ﬁlm it may tear off from the solid support. Both the thermodynamic and the mechanical stability may be improved by, after deposition, covalently linking the amphiphilic molecules to each other, for example, by polymerization. Langmuir–Blodgett (and Langmuir–Schaefer) ﬁlms have, in principle, great potentialities in various applications, such as in optical switches, biosensors, highly selective membranes, and biocompatible materials, among others. However, for the time being, lack of long-term stability often has frustrated the practical usefulness of these novel applications. As alternatives for the LB- and LS-techniques there may be other ways of forming layers of amphiphilic molecules on solid supports. Among these is the self-assembly technique in which ordered monolayers (or multilayers) are formed by physical or chemical adsorption. Such self-assembled monolayers are brieﬂy discussed in Section 7.6.

7.6 SELF-ASSEMBLED MONOLAYERS Self-assembled monolayers are spontaneously formed upon immersing an appropriate solid substrate in a solution of molecules that adsorb tenaciously in a preferential orientation on that solid surface. Well-known examples are ﬁlms consisting of alternating layers of anionic and cationic polymers, layers of alkane

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Chapter 7

thiols on gold surfaces and of organosilicium (silane), and vinyl-terminated compounds on silicon and hydroxylized surfaces of, for example, glass, silica, titanium oxide, and other metal oxides.

7.6.1 Alternating Polyelectrolyte Layers Polyelectrolytes, that is, electrically charged polymer molecules, adsorb strongly on surfaces that are oppositely charged. The adsorbing polyelectrolyte may overcompensate the charge on the solid support and, hence, cause charge inversion. This creates a favorable support for adsorption of a polyelectrolyte having a charge sign opposite to that of the polyelectrolyte adsorbed in the ﬁrst layer. (See Figure 7.10). Because exact matching of charges by the adsorbing polyelectrolytes would be coincidental, low-molecular-weight ions are incorporated in the deposited layer. The process of alternating adsorption of a polyanion and a polycation may be repeated several times yielding a layer of which the thickness can be varied accordingly.

7.6.2 Alkane Thiols on Gold Surfaces When an ultraclean (hydrophilic) gold, Au, substrate is immersed in a, say, 103 M solution of alkane thiols, R–SH (where R is the alkyl chain), in an organic solvent a monolayer of the R–SH molecules at the Au surface is formed within a few minutes up to a few hours. The R–SH molecules become chemically linked to the Au surface by virtue of commensurate ﬁtting of the –SH group in the Au (111) crystal lattice. The resulting value for am is 0.22 nm2. Considering that the molecular cross-section of a hydrocarbon chain is 0.18 nm2, it is inferred that a close-packed condensed monolayer is obtained.

Figure 7.10 Alternating adsorbed layers of oppositely changed polyelectrolytes.

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Monolayers at Fluid Interfaces

115

7.6.3 Organosilicium Compounds on Hydroxylated Surfaces Alkylsilicium compounds of the types RSiX3, R1R2SiX2, and R1R2R3SiX, where R is an alkyl group and X a halogen atom (usually Cl) or an alk-oxy group (usually methoxy or ethoxy), react with hydroxylized surfaces, according to

Silica surfaces contain about 5–8 silanol groups per nm2. For an average of two bonds per alkyl sylicium molecule, am is in the range of 0.25–0.40 nm2. Both for the alkane thiols and the silanes, the alkyl chain length may vary between, say, 1 and 18 C-atoms. Furthermore, the alkyl chain may be functionalized with, for example, amine groups which allows further chemical reaction with different compounds. In this way surfaces may be modiﬁed, tailormade for their speciﬁc purposes.

7.6.4 Vinyl-Terminated Compounds on Hydrogen-Terminated Silicon and on Silica Surfaces Vinyl groups react with hydrogen-terminated or hydroxyl-terminated groups on silicium surfaces, according to

and vinyl-terminated compound

The R-group may be an oligomer or polymer of which the degree of polymerization and therewith the chain length can be well controlled. This provides a way to apply uniform molecular brushes on such silicon or silica surfaces (see also Section 16.3).

EXERCISES 7.1

Comment on the following statements. (a)

The Gibbs adsorption equation applies to soluble and insoluble monolayers. (b) The equation of state for a gaseous monolayer, Eq. (7.3), is a twodimensional analogue for the ideal gas law ( pv ¼ RT).

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Chapter 7

(c)

Addition of a surfactant to a solvent promotes spreading of that solvent at an interface. (d) Lateral interactions between surfactant molecules in a monolayer may lead to hysteresis in the pressure-area isotherm. (e) The compressibility of a monolayer decreases with increasing temperature. 7.2

The following interfacial tensions g, at 20 C, are given.

Interface Water=air Water=hexane Water=octyl alcohol Hexane=air Octyl alcohol=air

g(mN m 1) 72 50 9 18 28

(a) Does hexane spread at an air=water interface? (b) Does water spread at an air=hexane interface? (c) Does octyl alcohol spread at an air=water interface? (d) Does hexane spread at an octyl alcohol=water interface?

7.3

(a)

The ﬁgure shows the equilibrium surface tension g of aqueous butanol solutions as a function of the butanol concentration c in water, at 25 C. Derive the pressure-area isotherm p(am) for the soluble butanol monolayer.

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Monolayers at Fluid Interfaces

(b)

7.4

117

The ﬁgure shows pressure-area isotherms for an insoluble monolayer of a phospholipid at an air=water interface. The ﬂat regions in the isotherms reﬂect the transition between the liquid-expanded (LE) and the liquid-condensed (LC) states of the monolayer. Calculate the heat, enthalpy, entropy, and Gibbs energy for the transition of one mole of the phospholipid from the LE into the LC state. Compare the isobaric expansion coefﬁcients and the isothermal compressibilities between the LE and the LC states.

30 mg of a protein spread at an air–water interface form an insoluble monolayer. The interfacial pressure p is measured at different areas A available for the protein. The measurements are performed at 25 C. p(mN m1) A(cm2)

0.8 250

1.4 215

2.0 200

2.6 192

Is the monolayer in the gaseous state? Explain the dependency of p on A. (b) Calculate the molecular area of the protein in the monolayer. Calculate the molar mass of the protein. (a)

SUGGESTIONS FOR FURTHER READING K. S. Birdi. Lipid and Biopolymer Monolayers at Liquid Interfaces, New York: Plenum Press, 1989. G. L. Gaines. Insoluble Monolayers at Liquid–Gas Interfaces, New York: Interscience, 1966. V. M. Kaganer, H. Mo¨hwald, P. Dutta. Structure and Phase Transitions in Langmuir Monolayer. Rev. Mod. Phys. 71: 779–819, 1999. F. MacRitchie. Chemistry at Interfaces, San Diego: Academic, 1990. G. Roberts (ed.). Langmuir–Blodgett Films, New York: Plenum, 1990.

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The Lotus Effect

Engineers and product designers learn from nature. They learn, for instance, from the lotus ﬂower. The leaves of the lotus are known for their purity and cleanliness. They owe their reputation to wax crystals that form a rocky landscape on the leaves. Because of the small contact area the crystals hamper ﬁrm attachment of dust, soils, spores, and so on. Moreover, the wax makes the leaves extremely hydrophobic. Water droplets from dew or rain just roll over the leaf’s surface, rinsing off any impurity. The lotus effect may be copied in technical applications. Thus, ‘‘self-cleaning’’ paints have been developed and bricks and tiles may be designed that prevent growth of moss and mold.

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8 Wetting of Solid Surfaces

Solid materials are often characterized by their wetting behavior. Wetting of solid surfaces by liquids is a very common and important phenomenon in nature as well as in technological applications. Examples are the poor wetting by water of the feathers of (water)birds, the wetting of plant leaves by solutions of pesticides, the treatment of a textile to make it waterproof, the inﬂuence of detergents on the wettability of dishes, the wettability of contact lenses by lachrymal ﬂuid, and so on. Just as at ﬂuid interfaces, the spreading tendency of a liquid at a solid=gas interface is given by Eq. (7.4). If a is the solid, b the gas, and g the liquid, the spreading coefﬁcient, deﬁned in (7.4), is now written as S SGðLÞ gSG gSL gLG ;

ð8:1Þ

which equals the difference between the Gibbs energy of cohesion LL G in the liquid and the Gibbs energy of adhesion SL G between the liquid and the solid (assuming that there is no cohesion and adhesion for the gas phase).

8.1 CONTACT ANGLE. EQUATION OF YOUNG AND DUPRE In systems where LL G SL G < 0 the solid is not completely wetted by the liquid but a sessile drop of L is formed on S. Such a drop on a ﬂat, smooth, and horizontally positioned solid surface is depicted in Figure 8.1. The shape of the

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Chapter 8

γ LG

γ SG

α γ SL

G

L S

Figure 8.1 Interfacial tensions determine the shape of a sessile drop of liquid at an interface.

drop, or, more precisely, the contact angle a with the solid is determined by the balance of interfacial tensions. In equilibrium gSG gSL gLG cos a ¼ 0

ð8:2Þ

gSG gSL ; gLG

ð8:3Þ

or cos a ¼

which is known as the equation of Young and Dupre. Note that this equation contains two terms gSG and gSL that cannot be established independently. Hence, the Young and Dupre equation is only useful in combination with another equation to eliminate one of the two unknowns. Below, in Section 8.3, Eqs. (8.3) and (5.11) are combined to analyze the polarity of a solid surface. Generally, G contains the vapor of L (assuming nonvolatile S). Molecules of L adsorb from the vapor on S causing an interfacial pressure pSG , deﬁned as pSG gS gSG ;

ð8:4Þ

where gS is the tension of the solid–vacuum interface. pSG may be obtained by determining the adsorption of molecules of L from the vapor phase on S and applying Eqs. (3.86) or (3.85). Substituting (8.4) in (8.3) gives cos a ¼

gS gSL pSG : gLG

ð8:5Þ

For practical purposes, if a exceeds, say, 15 pSG 0. Most apolar solid surfaces in contact with a polar liquid and its vapor belong to this category. Polar solid surfaces are usually well wetted by polar liquids but also by apolar liquids, so that pSG is not negligible. Also, if L is a solution in which the dissolved molecules (e.g., surfactant molecules) are nonvolatile but have a certain afﬁnity for the SG interface an interfacial pressure pSG builds up. The (surfactant) molecules move from the liquid phase onto the SG interface to form a molecular ﬁlm. This usually

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121

is a slow process taking several minutes or even hours and hence causes a timedependent value of pSG and, consequently, a.

8.2 SOME COMPLICATIONS IN THE ESTABLISHMENT OF THE CONTACT ANGLE: HYSTERESIS, SURFACE HETEROGENEITY AND ROUGHNESS The determination of the contact angle of a liquid drop on a (solid) surface is not always straightforward and unambiguous. A phenomenon often observed is hysteresis, that is, the advancing and receding contact angles have different values. An example of contact angle hysteresis is illustrated in Figure 8.2. It shows that the advancing contact angle is larger than the receding one, aa > ar . Hysteresis reﬂects that the system is not in equilibrium: it has, at constant T and p, not reached the minimum value of the Gibbs energy. Instead, the system is in a so-called metastable state, where the Gibbs energy is in a local minimum that is separated from other local minima by a Gibbs energy barrier. The situation is schematically depicted in Figure 8.3. Various reasons may cause a metastable state: one is nonequilibrium adsorption at the SG interface; another one is heterogeneity of the solid surface with respect to wetting afﬁnity; and a third one is surface roughness. Hysteresis at a heterogeneous surface results from the existence of energy barriers between patches of different wettabilities. The front of the perimeter of the liquid drop moving over the solid substrate tends to stop at such an energy barrier. Then, the advancing contact angle is mainly determined by the parts of the surface with the lower wettability (higher value for a) and the receding contact angle by the parts with the higher wettability (lower value for a). The

αr

G αa

Figure 8.2

L S

Advancing ðaa Þ and receding ðar Þ contact angles.

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122

Chapter 8

G

α Figure 8.3 Gibbs energies for (meta)stable states of wetting.

contact angle ac for a surface having two types of areas with intrinsic contact angles a1 and a2 is given by cos ac ¼ f1 cos a1 þ f2 cos a2 ;

ð8:6Þ

where fi is the fraction of the surface having a contact angle ai . Equation (8.6) is known as the Cassie equation and the contact angle ac as the Cassie angle. For not too complex heterogeneous surfaces it has been shown that cos ac may be approximated by cos ac ¼ 0:5ðcos aa þ cos ar Þ:

ð8:7Þ

When the surface is not smooth the observed apparent contact angle differs from the local contact angle a0 . The inﬂuence of surface roughness on the apparent contact angle is shown in Figure 8.4, where two drop proﬁles are displayed that have the same a0 . With respect to the horizontal, two very different contact angles are observed, identiﬁed as aa and ar . The two drop shapes in Figure 8.3 differ in interfacial area and in the position of the center of gravity and, hence, the two drops have different energies. The change from one shape to the other involves

αo

αo

αa

αr

Figure 8.4 Inﬂuence of surface roughness on the apparent contact angle.

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Wetting of Solid Surfaces

123

distortion of the drop shape, which accounts for the energy barrier between the two drop shapes. Surface roughness usually gives rise to a large number of metastable states and a relation between the apparent contact angles aa and ar and a0 on the other hand cannot be determined unequivocally. As a rule, ar is more sensitive to surface roughness than aa . Hence aa is more reliable as a parameter characterizing the surface wettability. Altogether, it may be clear that as long as wetting is not free of hysteresis one should take extreme care in further interpretation and analysis of the contact angle. As an alternative for a sessile drop of liquid, a captive gas (air) bubble may be used to probe the wettability of a solid surface. See Figure 8.5. Analysis and interpretation of the contact angle of the captive bubble is, mutatis mutandis, analogous to those for the sessile drop.

8.3 WETTING AND ADHESION. DETERMINATION OF SURFACE POLARITY It may be obvious to relate the equation of Young and Dupre, (8.3), to the Gibbs energy of adhesion between S and L, SL Ga . It follows from Eq. (5.11) (with a ¼ S and b ¼ L) that SL Ga ¼ gSL gS gL ;

ð8:8Þ

which, combined with (8.4) gives SL Ga ¼ gSL gSG gL pSG

ð8:9Þ

ðg ¼ g , because G consists of L’s vapor). Now, combining (8.9) and (8.3) yields LG

L

SL Ga ¼ gL ð1 þ cos aÞ pSG :

ð8:10Þ

Thus, by measuring a and determining pSG (from adsorption on the solid from the vapor phase), SL Ga can be calculated for a liquid of known gL that wets the

S γ SL

γ SG

G γ

L

LG

Figure 8.5 Captive gas bubble at a solid–liquid interface.

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124

Chapter 8

surface with a ﬁnite contact angle. SL Ga comprises an apolar (¼ dispersion) and a polar contribution. Assuming these contributions to be additive, SL Ga ¼ SL;apolar Ga þ SL;polar Ga :

ð8:11Þ

In Chapter 5, Section 5.6, it has been demonstrated that Fowkes’ approximation of the interfacial tension between a and b, or, for that matter, S and L, expressed in (5.24) applies well for apolar components. Then, combining Eqs. (5.11) and (5.24) results in SL;apolar Ga ¼ 2ðgS;d gL;d Þ1=2

ð8:12Þ

which, combined with (8.8), (8.9), and (8.11) yields gSL ¼ gS þ gL 2ðgS;d gL;d Þ1=2 SL;polar Ga ;

ð8:13Þ

and (8.13) with (8.5), cos a þ

pSG ðgL;d Þ1=2 SL;polar Ga þ 1 ¼ 2ðgS;d Þ1=2 : L g gL gL

ð8:14Þ

Using a series of apolar liquids (for which SL;polar Ga ¼ 0 of known gL ð¼ gL;d ), a plot of pSG cos a þ L þ 1 g

versus

ðgL;d Þ1=2 =gL

gives a straight line of which the slope equals 2ðgS;d Þ1=2 . See Figure 8.6. For a liquid that also has a polar interaction with the solid SL;polar Ga < 0. The experimental datapoint will deviate from the straight line obtained with the apolar liquids. The deviation equals SL;polar Ga =gL , so that by using a polar liquid of known gL SL;polar Ga can be derived. Now, because SL Ga is known from (8.10), SL;apolar Ga follows by applying (8.11) Thus, by taking water as the polar liquid the hydrophobic character of the solid surface may be determined as SL;apolar Ga =SL;polar Ga . Because in the plot shown in Figure 8.6 the line for the apolar liquids passes the origin, in principle the use of only one test liquid sufﬁces. For many practical purposes one is interested in the differences in wettability of different surfaces by a given liquid. For a series of surfaces, for which a 15 and, hence, pSG is negligible, it follows from Eq. (8.10) that the value of a can be taken as a measure of the wettability. Thus, with water as the wetting liquid the contact angle unambiguously reﬂects the surface hydrophobicity.

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cos α +

π SG +1 γL

Wetting of Solid Surfaces

125

x ∆SL,polar G γL

x x

x

(γ L,d )1/2 / γ L Figure 8.6 Plot for the determination of the polarity of a solid surface.

8.4 APPROXIMATION OF THE SURFACE TENSION OF A SOLID. THE CRITICAL SURFACE TENSION OF WETTING A rather popular parameter to characterize the wetting of a solid surface is the so-called critical surface tension of wetting, gc . By deﬁnition, gc of a solid equals the surface tension of the (hypothetical) liquid that just wets the surface with 0 contact angle. The lower the value of gc the poorer the wettability of the solid.

8.4.1 Zisman Method The critical surface tension of wetting may be evaluated from a ‘‘Zisman-plot’’ in which for a homologous series of liquids cos a is plotted against gLG ð¼ gL Þ. This usually gives a straight line. Figure 8.7 shows plots for a Teﬂon surface wetted by different series of homologous liquids. Extrapolation to cos a ¼ 1 yields gc . The value of gc may be compared with that of gS by applying (8.5), gc ¼ gS gSL pSG ;

ð8:15Þ

showing that gc < gS by a difference of ðgSL þ pSG Þ. The value of gSL strongly depends on the polarity contrast between S and L and the value of pSG is not negligible when cos a approaches unity ða ! 0 Þ. Only when using a series of apolar liquids, for which SL Ga is solely determined by dispersion interaction, on poorly wettable surfaces ðpSG 0Þ, gc is exclusively determined by the solid surface: gc ¼ gS;d . This follows from (8.10)

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126

Chapter 8

1.0

x x x x xx x

cos α 0.8

xx x

0.6 0.4 0.2 0.1

0

10

20

30

40 γ

L

/ mN

50 m– 1

Figure 8.7 Zisman plots for a Teﬂon surface from which critical surface tensions of wetting are derived. T ¼ 20 C. . alkyl benzenes; n-alkanes; dialkyl esters; u siloxanes; n polar liquids. (Data from W. A. Zisman. Adv. Chem. Ser. 43: 1, 1964.)

and (8.12) with gc ¼ gL ðcos a ¼ 1Þ. In all other cases gc -values should be interpreted with caution. For instance, the use of gc as a characteristic parameter for a (bio)material in an aqueous environment is questionable.

8.4.2 Wu Method Combining Young and Dupre’s law (8.3) with Girifalco and Good’s approximation (5.29) allows for a further analysis of gc . Thus,

gS cos a ¼ 2F LG g

1=2 1

pSG : gLG

ð8:16Þ

When using pure liquids and for a > 15 pSG may be neglected. Then, based on the deﬁnition of gc gc ¼ F2 gS :

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ð8:17Þ

Wetting of Solid Surfaces

127

Because, as a rule F < 1 (see Chapter 5, Table 5.3) it follows again that gc < gS . Substitution of gS according to (8.17) in (8.16) results in 0:5 gc 1 ð8:18Þ cos a ¼ 2 LG g or gc ¼

ð1 þ cos aÞ2 LG g ; 4

ð8:19Þ

γ c / mN m– 1

showing that the experimentally obtained value of gc depends on the test liquid used and on the contact angle of that liquid with the solid surface. Hence, applying a series of test liquids (with a 15 ) yields a collection of gc values that, as a function of gLG ð¼ gL Þ are given in Figure 8.8 for polymethyl metacrylate and for parafﬁn wax. According to (8.17) the maximum in gc ðgL Þ is reached when F attains a maximum value. The theoretical maximum for F is unity (see Section 5.6.2, Eq. (5.30)) and is obtained when the adhesion force between L and S are similar to the cohesion forces in L and in S. Hence, the maximum value for gc ðgL Þ may be a good approximation of gS . The approximation of gS by the Wu method is often used to characterize low-energy surfaces, in particular, surfaces of polymers.

50 40 30 20 10 0 10

20

30

40

50

60

70 80 γ L / mN m– 1

90

Figure 8.8 Critical surface tensions of wetting gc of the solids poly(methyl methacrylate) (s) and parafﬁn wax (u) using test liquids of various surface tensions gL . According to Wu the maximum of gc ðgL Þ corresponds to the surface tension of the solid. (From S. Wu. Polymer Interface and Adhesion, New York: Marcel Dekker, 1982, Ch. 5.)

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Chapter 8

8.5 WETTING BY SOLUTIONS CONTAINING SURFACTANTS If the test liquid is not a pure liquid but a solution containing a compound that adsorbs at the LG and=or SL interface, the contact angle is affected by the adsorbed amount G, which, in turn, is related to the concentration of that surface active compound in solution. Hence, a pure liquid that wets the surface poorly ðcos a < 1Þ may change into a good wetting liquid by adding a surface active compound, and vice versa. According to Gibbs’ law, (3.86), adsorption leads to a lowering of the interfacial tension. Reduction of gSL and gLG, at constant gSG (no surfactant adsorption at the SG interface), implies an increase in cos a; that is, improved wetting, as can be inferred from Eq. (8.3).

8.6 CAPILLARY PENETRATION In Section 6.1 we discussed the rise of a liquid in a capillary of which the inner wall is well wetted. In vertical position the rise is compensated by the hydrostatic pressure, as expressed in Eq. (6.6). For a horizontally positioned capillary the liquid is sucked into the capillary over an inﬁnite distance. Based on the same principle, a liquid in a capillary is spontaneously displaced by another liquid if the latter liquid wets the inner surface better. This phenomenon is illustrated in Figure 8.9. At the left-hand side the capillary is supplied with a liquid L1 and at the right-hand side with a liquid L2. The velocity v of the moving meniscus is given by Washburn’s equation n¼

gR cos a ; 4ðZ1 ‘1 þ Z2 ‘2 Þ

ð8:20Þ

where g ¼ gL1=L2 , R equals the radius of the capillary, Z1 and Z2 the viscosities of L1 and L2, and ‘1 and ‘2 the lengths over which the respective liquids ﬁll the

1

2

α

L1

L2

Figure 8.9 Displacement of a liquid by another liquid in a capillary.

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Wetting of Solid Surfaces

129

capillary. The driving force for the displacement is the capillary pressure (see Section 6.1) pL ¼

2g ; r

ð8:21Þ

where r is the radius of curvature of the meniscus ðr ¼ R= cos aÞ. The velocity of the displacement is related to the volume ﬂux Q per unit cross-sectional area, A, n¼

Q Q ¼ : A pR2

ð8:22Þ

When air is taken for L2, for which Z 0, Q is given by Poiseuille’s equation Q¼

pR4 pL ; 8Z‘

ð8:23Þ

in which Z Z1 and ‘ ‘1 . Combination of Eqs. (8.21) through (8.23) yields (8.20) for Z2 ‘2 ¼ 0. It is inferred from (8.20) that, for a given capillary, the velocity of penetration of the one liquid into the capillary, thereby displacing the other liquid (or air) is enhanced by lower values of Z, a high value of g, and a small value of aðcos a ! 1Þ. The last two conditions, high g and small a, are usually not compatible. General experience has shown that addition of a surfactant to the liquid leads to an increased velocity by which the liquid displaces air. The surfactant adsorbs at the LG and SL interfaces. The observation of increased penetration velocity indicates a dominant effect of the reduction in gSL . As gLG cos a ¼ gSG gSL (and gSG essentially being unaffected because pSG 0), a reduction in gSL corresponds to a higher value for gLG cos a, which, according to (8.20) implies a proportional increase of the penetration velocity.

8.7 SOME PRACTICAL APPLICATIONS AND IMPLICATIONS OF WETTING: IMPREGNATION, FLOTATION, PICKERING STABILIZATION, CLEANSING 8.7.1 Impregnation Textiles and other fabrics may be impregnated to render them ‘‘waterproof,’’ that is, impenetrable for water. In such applications the material should still be penetrable for air. These requirements may be achieved by applying a (mono)layer of hydrophobic coating to the tissue ﬁbers. As a result, the tissue is poorly wetted by water causing capillary suppression of the liquid water in the capillaries within the fabric. Conversely, small values for the contact angle ða > 0Þ lead to

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Chapter 8

penetration of the liquid between the ﬁbers. These two situations are depicted in Figure 8.10. According to the equation of Laplace and Young, (6.7), in situation (a) the capillary can withstand a certain pressure of the water column; in situation (b) water is sucked into the interﬁber space. The value of the Laplace (over)pressure at the concave side of the meniscus, which is given by the interfacial tension between the water and the tissue material and by the radii of curvature of the meniscus, determines the hydrostatic pressure the tissue can withstand before being penetrated by the water.

8.7.2 Flotation Flotation is a technique to separate ﬁnely dispersed solid particles of different surface wettabilities. The principle of ﬂotation is illustrated in Figure 8.11. Air bubbles are blown through a slurry of solid particles dispersed in a liquid. In the case of an aqueous liquid the more hydrophobic particles attach to the air bubbles that pass through the liquid and arrive in the froth, leaving the hydrophilic particles behind in the bulk liquid. If the wettability between the different kinds of particles does not sufﬁciently differ to allow such a separation, additives may be supplied that selectively adsorb to enhance the wettability contrast. Such additives are called collectors. Further improvement may be obtained by adding frothing agents that stabilize the froth layer containing the removed particles. Thus, ﬂotation is a complex process in which both kinetic and equilibrium aspects play important roles. The separation performance depends on the prob-

(a)

(b)

G G α

L

α

L Figure 8.10 (a) Capillary suppression and (b) rise in a capillary determining penetration of a fabric by a liquid.

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Wetting of Solid Surfaces

131

air foam layer air bubble

aqueous phase hydrophilic particle hydrophobic particle Figure 8.11 Flotation.

ability of bubble–particle encounter, the attachment of the particle to the bubble, and the stability of the bubble–particle complex. The attachment probability involves all three interfaces, SG, SL, and LG. In equilibrium, at constant p and T, the change in Gibbs energy per unit interfacial area, Ga , upon attachment of a particle (S) on a bubble (G) is given by Ga ¼ gSG ðgSL þ gLG Þ;

ð8:24Þ

which combined with (8.3) yields Ga ¼ gLG ðcos a 1Þ:

ð8:25Þ

It follows from (8.25) that attachment is favorable for cos a < 1; that is, a > 0 . In practice, in order to become effectively attached, the solid particles should be wetted with a contact angle of say, > 20 . Clearly, the larger the value of a the more favorable the attachment and the more stable the bubble–particle complex is. As mentioned before, particles having a well-wettable surface may be rendered poorly wettable by adsorbing collector molecules. The inﬂuence of the mole fraction Xc (or, for that matter, concentration) of the collector in solution on the particle–bubble interaction and, hence, on the separation efﬁciency, may be represented by dGa ¼
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ð8:26Þ

132

Chapter 8

which is obtained by differentiation of (8.24) to ln Xc and subsequent application of Gibbs’ adsorption equation (3.86). G is the adsorbed amount (in moles per unit area) at the interfaces indicated. It is inferred that for enhanced separation the collector molecules must preferably adsorb at the SG interface; that is, dGa =d ln Xc < 0 for GSG > GSL þ GLG. Frothing agents that stabilize the foam do so by lowering gSG and=or gLG. The ﬂotation technique was ﬁrst developed to separate minerals from the gangue in the processing of crushed ores. Later, it was also used for separations in other systems, for instance, for the removal of micro-organisms from waste and surface waters.

8.7.3 Pickering Stabilization Emulsions and foams may be stabilized against coalescence by powders. This is referred to as Pickering stabilization (after Pickering who discovered the stabilization of petroleum in water emulsions by clay particles). The stabilization results from the fact that the solid particles are not preferentially wetted by either of the ﬂuid phases L1 and L2 causing them to be located in the L1=L2 interface. This situation is illustrated in Figure 8.12. In the case where a ! 0 or a ! 180 the solid particle would be completely wetted by L1 or L2 and therefore be immersed in the corresponding liquid phase. Release of the particles from the L1=L2 interface would increase the interfacial Gibbs energy and hence promote rupture of the liquid ﬁlm between the emulsion droplets (resp., air bubbles in foam). See Figure 8.13. It appears that optimal stabilization is obtained when the continuous phase (L1 in Figure 8.13) wets the particle somewhat better than the dispersed phase does. Then the particles reach out farther into

α

L1 L2

S

α Figure 8.12

Nonpreferable wetting of a solid particle in a liquid–liquid interface.

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Wetting of Solid Surfaces

133

powder particles

L2

L2

L1

Figure 8.13 Pickering stabilization of a liquid–liquid interface.

the continuous phase and it is concluded that mechanical effects inﬂuence the stability as well. If the surface of the particles is electrically charged the emulsion or the foam is not only stabilized against coalescence but also against aggregation. Addition of a (small) amount of surfactant that adsorbs on the solid particles often causes preferential wetting thereby undoing the stabilization. Pickering stabilization is applied, for example, in emulsions and foams in foodstuffs, pharmaceutical products, and cosmetics. It may also play an adverse role in biological waste water puriﬁcation where bacteria stabilize undesired foaming.

8.7.4 Cleansing Cleansing of materials (substrates) implies the removal of soil and stains. A wide variety of stains and substrates are encountered. For instance, textiles may be of natural origin such as cotton, wool, or natural silk, or made of synthetic ﬁbers, such as nylon, polyester, or polyacryl. Cotton is cellulosis which has an intermediate hydrophobicity; wool and natural silk are proteins, both rather hydrophobic and in most cases negatively charged. Synthetic ﬁbers are usually polymers of which the backbone is characterized by a series of repeating units such as peptide units (in nylon), ester bonds (in polyesters), and cyan groups (in polyacryl). Stains may be even more diverse ranging from very hydrophobic (soot, grease, and oil), amphiphilic (proteinaceous stains), to hydrophilic (clay, sand,

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Chapter 8

cement). The stains may further be negatively or positively charged. Because of the diversity of both the substrates and the staining agents, a wide variety of interactions (dispersion interaction, hydrogen bonds, Coulomb interaction, and hydrophobic interaction) may play a role in the adhesion of the staining materials to the substrate. As a rule, a cleansing procedure comprises the steps: (a) detachment of the stain from the substrate, and (b) prevention of readhesion of the stain onto the substrate. Detachment of the stain F from the substrate S in an (aqueous) medium L occurs spontaneously if (at constant p and T) Ga ¼ gSL þ gFL gSF < 0:

ð8:27Þ

A negative value for Ga may be achieved by supplying a surfactant to L that adsorbs at the FL and SL interfaces, thereby lowering gSL and gFL . Removal of liquid-like stains from the voids between the ﬁbers of the substrate may be further enhanced by capillary penetration, as described by Eq. (8.20). Removal of polymeric stains is relatively difﬁcult because a polymer molecule attaches via several segments to the substrate (see Chapter 15). The most effective way to remove such substances is by using displacers, usually low molecular weight surfactants, that themselves adsorb with a high afﬁnity to the substrate. Moreover, binding of ionic surfactants to the polymer molecule renders them better soluble in water and by varying the pH, electrostatic repulsion induces removal of the charged polymer (polyelectrolyte) from a charged substrate. In addition to the above-mentioned physical effects stains may be released by chemical degradation such as oxidation and enzymatic hydrolysis, for example, lipolysis and proteolysis. Readhesion or readsorption is usually prevented by solubilization of the detached staining agents in surfactant micelles (see Chapter 11).

EXERCISES 8.1

Comment on the following statements. (a)

A high value for the critical surface tension of wetting implies that the surface is well wettable. (b) Improved wetting of the inner walls of pores causes faster capillary penetration. (c) The stability of an emulsion stabilized by the Pickering mechanism is increased by adding surfactant to the continuous phase. (d) Liquids having a higher surface tension show a stronger tendency to wet solid surfaces.

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Wetting of Solid Surfaces

(e)

8.2

135

The Gibbs energy of adhesion SL Ga between a solid and a liquid is related to the critical interfacial tension of wetting gc of the solid as SL Ga ¼ 2gc .

Two different brands of detergent, Fatex and Defa, are proposed to remove droplets of an oily residue from glassware. The following interfacial tensions are known. Interface

g(mN m1)

Fatex–oil Defa–oil Glassware–oil Fatex–glassware Defa–glassware

16 12 35 18 20

Which detergent would you recommend? Explain your choice by making drawings and calculations. 8.3

Consider the following system where two containers are connected by a narrow capillary.

L1 and L2 are two immiscible liquids that wet the capillary differently, resulting in a curved meniscus of the L1=L2 interface. (a)

Does the L1=L2 interface move and, if so, in which direction? What is the driving force for the interface to move and when does the motion stop? What is the shape of the meniscus then? Changes in hydrostatic pressure may be neglected.

The viscosity Z of both L1 and L2 is 102 N m2 s and the interfacial tension gL1=L2 is 40 mN m1 . L1 wets the capillary preferentially with a contact angle a ¼ 30 . (b)

Calculate the ﬂux per unit cross-sectional area in the capillary of 5 cm length and 1 mm diameter.

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136

Chapter 8

(c) 8.4

Is the observed contact angle the advancing or the receding contact angle?

A sperical protein molecule is located at a water=membrane interface. The ﬁgure shows the geometry of the system. Both the water and the membrane behave as liquids.

(a) Derive Young’s law at point A. (b) Derive an expression for the Gibbs energy of the system as depicted, GðhÞ, taking the protein totally immersed in the membrane phase as the reference state. Prove that GðhÞ ¼ 2pahðgPW gPN gMW Þ þ ph2 gMW . Discuss the results for h ¼ 0 and h ¼ 2a from a physical point of view. (c) Derive that h ¼ að1 cos aÞ. (d) Prove that in equilibrium G ¼ pa2 gMW ð1 cos aÞ2 . (e) Make a plot for GðhÞ for gMW ¼ 40 mN m1 and gPW gPM ¼ 20 mN m1 . What is the value of a? Indicate in the plot GðhÞ for h ¼ 0; h ¼ r; h ¼ 2r, and h at equilibrium. (f) Calculate the value of GðhÞ in equilibrium, given the interfacial tensions as in (e) and a ¼ 3 nm. Do you expect removal of the protein molecule from the MW interface due to thermal motion?

SUGGESTIONS FOR FURTHER READING J. C. Berg (ed.). Wettability, in Surfactant Science Series 49, New York: Marcel Dekker, 1993. R. J. Good. Contact angles and the surface free energy of solids, in Surface & Colloid Science, Vol. 11, E. Matijevic and R. R. Stromberg (eds.), New York: Plenum, 1979, Chapter 1.

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Wetting of Solid Surfaces

137

J. F. Padday (ed.). Wetting, Spreading and Adhesion, London: Academic, 1978. M. A. Schrader, G. I. Loeb (eds.). Modern Approaches to Wettability, New York: Plenum, 1992.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Bacterial Cell Walls

In an aqueous environment bacteria are charged, usually negatively charged. The electric characteristics of bacteria are strongly inﬂuenced by their cell wall structure. By way of example, the photograph shows a ‘‘bald’’ and a ‘‘hairy,’’ ﬁbrillated variant of Streptococcus salivarius. The electric potential at the surface of the smooth variant is much higher than at the outer surface of the ﬁbrillated one. The difference is due to the penetration and mobility of solvent molecules and ions in the ﬁbril-coat. Knowledge of the cell wall structure, in particular its softness with respect to solvent and ion penetration is a necessity for the assessment of electrostatic interaction in bacterial adhesion and aggregation. (Figure courtesy of Department of Biomedical Engineering, University of Groningen, The Netherlands.)

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

9 Electrochemistry of Interfaces

Water has a high dielectric permittivity and it is a good solvent for most ions (cf. Chapter 4). As a result, in an aqueous environment most electrolytes are dissociated in their individual ions. Surfaces of (solid) materials often contain ionizable groups which in contact with water are electrically charged. Together with the wettability, the presence of electric charge largely determines the stability of colloidal dispersions (Chapter 16) and interfacial processes such as adsorption and adhesion (Chapters 14, 15, and 20). The basic variables characterizing charged interfaces are the electric charge and potential. From these, properties such as capacitance and electric ﬁeld strength are derived. The following questions are at issue. (1) What is the origin of the charges and what is the interfacial charge density; (2) how is the charge distributed; and (3) what can be said about the ensuing potential proﬁle?

9.1 ELECTRIC CHARGE There are mainly two mechanisms by which surfaces in equilibrium with an aqueous solution can acquire electric charge. These are dissociation or association of surface groups, and binding of an excess of anions or cations in the crystal lattice of a poorly soluble salt. Most metals and some nonmetallic elements spontaneously generate an oxide surface layer when exposed to air or aqueous media. Examples are Fe2 O3 ; TiO2 ; SiO2 , and so on. These oxide layers react with Hþ . Hence, depending on the pH, these surfaces are positively or negatively charged. Biological interfaces, the surfaces of, for example, proteins, biomembranes, and biological cells in contact with an aqueous solution, often carry carboxyl,

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140

Chapter 9

amino, phosphate, sulphonate, and imidazole groups. All these groups associate= dissociate with protons. As both cationic and anionic groups are usually present, most biological surfaces are amphoteric. Charged surfaces of insoluble salts are found, for example, among silver halides, notably Agþ Cl ; Agþ Br, and Agþ I . The charge is due to excessive presence of either of the two constituting ions at the crystal surface. For synthetic materials all kinds of ionic groups may be introduced intentionally by controlling the synthesis conditions. The ions that cause the surface charge are referred to as charge determining ions (cdi). In our examples they are Hþ (or OH ) for the oxides and the biological surfaces, and Agþ (or Cl ; Br ; I ) for the silver halides. The charge density at the respective surfaces is determined by the concentration of cdi in solution, ccdi (or, for the sake of convenience, usually expressed as log ccdi ). The value for log ccdi (i.e., the pH if Hþ and OH are cdi, or pAg in the case of the silver halides) at which the net charge on the surface is zero is called the point of zero charge (pzc). Under equilibrium conditions the charge at the surface is neutralized by the charge in the adjacent phase(s). The surface charge together with the neutralizing charge is referred to as the electrical double layer. For our purpose, the adjoining phase to be considered is an aqueous solution. For instance, if the surface is charged by dissociation of protons, these protons and other ions (the counterions) that have a charge sign opposite to that of the surface tend to remain in the vicinity of the charged surface because they are electrically attracted to it. How far they go in the solution depends on the balance of that attraction and thermal motion. The same applies, mutatis mutandis, for the expulsion of ions (the co-ions) that have the same charge sign as the surface. The compromise between the minimum energy situation in which all counterions are as near as possible to the charged surface and the co-ions expelled away, and the maximum entropy situation where all ions are homogeneously distributed in solution, results in an ion distribution of minimum Gibbs energy. According to these principles the ion distribution as a function of the distance x from the interface is, qualitatively, as indicated in Figure 9.1. Here, the surface charge is taken negative so that the electric potential in the interfacial region is negative as well. The electrical double layer therefore contains an excess of cations and a deﬁcit of anions which are represented by the areas A and B in Figure 9.1. Because of the requirement of electroneutrality for the whole electrical double layer, the sum of A and B just compensates the charge at the surface. To establish the exact ion distribution, ci ðxÞ, the potential proﬁle cðxÞ across the electrical double layer should be known. We return to this in Section 9.2. The charge Q0 at the surface ðx ¼ 0Þ is commonly established by titration with charge determining ions. After deﬁning a reference point, for which the pzc is usually taken, an absolute value can be assigned to the surface charge. It may

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Electrochemistry of Interfaces

141

c+ , c–

c+ A

c+,

= c –,

B c– x Figure 9.1 Distribution of counterions and co-ions in a diffuse electrical double layer.

be convenient to express the charge as charge density s, that is, the charge per unit interfacial area. Thus, for the surface charge density s0 ; s0 ¼

X

z FGcdi ; cdi cdi

ð9:1Þ

where Gcdi is the surface excess (in moles per unit surface area) of the charge determining ions and zcdi the corresponding valencies. F is Faraday’s constant. Examples of titration curves are presented in Figure 9.2. From Eq. (9.1) it follows that for (de-)protonated surfaces s0 ¼ FðGHþ GOH Þ:

ð9:2Þ

The surface charge may be analyzed in terms of the contributions of classes of ionizable groups that are titrated over distinct pH regions. This allows determination of the number of titratable groups in each class; such an analysis is elaborated upon below. The Gibbs energy of dissociation diss G of a proton from any particular group can be split in a chemical (chem) and an electrical (el) term diss G ¼ diss Gchem þ diss Gel :

ð9:3Þ

The chemical term contains the intrinsic contribution of the dissociation and the electrical term the additional Gibbs energy to remove the proton from the charged

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Chapter 9

AgI KNO3

2

6 0

α – Fe2O3 0.001 M KCl

10

8

10

4

σo / µC cm–2

σo / µC cm–2

142

12 pAg

–2 (a)

(b)

4

0

10 o

–6

6 2

10 – 4 M 10 – 2

–4

8

4

6

8

10

–2

12 pH

–4 RNase 0.05 M KNO3

12 8

σo / µC cm–2

Qo × 10– 5 / C mol–1

16

(c)

4 0

400 1 M

0

0.01 M 0.001 M

–200 4

6

–4

8

10

(d)

12

pH

–400 0

–8

Bacillus brevis KNO3

0.1 M

200

2

4

6

8

10 pH

12

Figure 9.2 Proton titration curves for surfaces of (a) silver iodide, (b) iron oxide (hematite), (c) the protein ribonuclease, and (d) the cell wall of the bacterium Bacillus brevis. (From (a) B.H. Bijsterbosch, J. Lyklema, Adv. Colloid Interface Sci. 9: 147, 1978; (b) A. Breeuwsma, J. Lyklema. J. Colloid Interface Sci. 43: 437, 1973; (c) W. Norde, J. Lyklema. J. Colloid Interface Sci. 66: 266, 277, 1978; (d) A. van der Wal et al. Colloid Surfaces B. Biointerfaces 9: 81, 1997.)

site to inﬁnity (where the electric ﬁeld is zero). In the simple case where all titratable groups belong to one class Q0 ¼ azFN ;

ð9:4Þ

where Q0 is the total charge of the sample, N the number of titratable groups, and a the degree of dissociation. For the dissociation constant Kdiss the following expressions apply. 1a a

ð9:5Þ

diss Go ¼
ð9:6Þ

pKdiss ¼ pH þ log and

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Electrochemistry of Interfaces

143

Combining Eqs. (9.3), (9.5), and (9.6) yields a relation between the pH and a: pH ¼ 0:434

o diss Gchem þ diss Gelo 1a : log
ð9:7Þ

It should be realized that diss Gelo is a function of a; at a high degree of dissociation, when the surface has attained a high negative charge density and, consequently, a high negative potential, it requires more electrical work to remove a proton from the surface. diss Gelo is often expressed as diss Gelo Q ¼ 2W ¼ 2W azN : F
ð9:8Þ

W is the so-called electrostatic interaction factor and it depends primarily on the dielectric constant and the ionic strength of the medium. In differential form (9.8), in combination with (9.7), can be written as dpH 1 ¼ 0:868WzN þ 0:434 da að1 aÞ

ð9:9Þ

or dpH 0:434 W 0:868 : ¼ dQ0 zNFað1 aÞ F

ð9:10Þ

It is inferred that at a ¼ 0:5 dpH=dQ0 reaches a minimum, which corresponds to an inﬂection point in the titration curve Q0 (or s0 ) versus pH. Most (biological) surfaces contain more than one class of titratable groups; that is, different classes j of groups are titrated in distinct pH regions. In that case (9.10) has to be modiﬁed into dpH 0:434 W 0:868 ¼ P dQ0 F j Nj zj aj ð1 aj Þ F

ð9:11Þ

and the differential titration curve, that is, a plot of dpH=dQ0 versus Q0 (or vs. pH) then displays more than one minimum. By way of example, in Figure 9.3 the integral and differential titration curves for a hypothetical protein are given. The maxima in the differential titration curve are identiﬁed as the equivalence points of the titrations of the different classes. It follows that the separation between two maxima is determined by the number of groups titrated within one class. The minima indicate the pKdiss values [because, according to (9.5) pKdiss ¼ pH at a ¼ 0:5]. Adsorption of ions other than cdi can also lead to charging the surface, provided that the adsorption is speciﬁc. The notion ‘‘speciﬁc’’ implies that the adsorption forces are partly nonelectric in nature so that co-ions can overcome the repelling electric potential at the surface and, by their adsorption, even increase the surface potential. (Bio)surfactants, for example, phospholipids, of

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Chapter 9

10 pH 9 8 7

0.8

–

∆pH × 10 5 / mol C–1 ∆Qo

144

6

0.6

III 5

0.4

II

I

4

0.2 10

5

0

Qo ×

10 – 5

–5

/ C mol–1

Figure 9.3 Integral (!) and differential ( ) proton titration curves for a hypothetical protein in aqueous solution, indicating titration regions for different classes of groups: I carboxyl groups; II a-amino and imidazole groups; III e-amino, phenolic OH, and guanidyl groups.

which the hydrophobic part of the molecule has a high nonelectric afﬁnity for the surface show strong speciﬁc adsorption. In practice, the net charge at an interface results from both ionization of surface groups and speciﬁc ion adsorption. This situation is schematically depicted in Figure 9.4. How can we establish the charge density due to speciﬁcally adsorbed ions? The most common approach is to compare the titration charge with the electrokinetic charge. Anticipating Chapter 10 it is mentioned here that in electrokinetic phenomena (i.e., electrophoresis, electro-osmosis, streaming potential, and streaming current) a liquid moves tangentially with respect to a (solid) surface; hence, a moving and a stationary phase can be distinguished. Let us consider electrophoresis: the motion of a charged (colloidal) particle in an electric ﬁeld. For hydrodynamic reasons it is not only the particle itself that moves, but the moving entity, the electrokinetic particle, also includes a layer of solvent (water) around the particle: the so-called hydrodynamic slip layer. If the slip layer does not extend too far in the solution, the boundary of the electrokinetic particle is located somewhere in the electrical double layer. See Figure 9.4. The same

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Electrochemistry of Interfaces

145

+ +

– – – +

+

–

–

+ +

+

– +

–

– –

+

– +

+

+

–

Figure 9.4 Schematic representation of an electrical double layer. Speciﬁcally absorbed ions have lost (a part of) their hydration water and are in direct contact with the charged surface. At the right-hand side of the dotted line ions (þ =) are diffusely distributed in solution.

applies for the ﬂow of liquid along a stationary charged surface. It may be understood that the measured quantity, for example, the electrophoretic mobility or the streaming potential, pertains to the electrokinetic boundary. Thus the total charge within the hydrodynamic slip layer, that is, the electrokinetic charge Qek or expressed per unit interfacial area sek , can in principle be derived. If speciﬁc adsorption occurs the ensuing charge can be approximated by ðQek Q0 Þ. Figure 9.5 shows Q0 and Qek for the protein ribonuclease A. From the observation that at the pzc Qek < 0 it is inferred that anions have a stronger tendency for speciﬁc adsorption than cations. A problem arises if the surfaces are not smooth, but ‘‘hairy.’’ At various biological surfaces, such as bacterial cell walls, polymers or oligomeric molecules extend from the surface into the solution and may cause substantial modiﬁcation of the tangential ﬂow pattern; the slip plane might be much farther out from the surface and if nonspeciﬁcally adsorbed counterions inﬁltrate within the surfacepolymer layer sek is proportionally lowered and, at any rate, ðsek s0 Þ no longer represents the charge density of speciﬁcally adsorbed ions. Finally, it is mentioned that s0 may reach values as high as several tens of mC cm2 . Such high charge densities would give rise to high potentials causing very strong attraction of counterions which, as a result, become located im-

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146

Chapter 9

Q × 10 – 5 / C mol –1

12 8

Qo

4 0

Qek 4

–4

6

8 pH

10

–8 Figure 9.5 Net proton charge Q0 and electrokinetic charge Qek of the protein ribonuclease in aqueous solution. T ¼ 25 C; ionic strength 0.05 M KNO3. (Adapted from W. Norde, J. Lyklema. J. Colloid Interface Sci. 66: 266, 277, 1978.)

mediately adjacent to the surface. This phenomenon is called counterion condensation. Consequently, sek usually does not exceed a few mC cm2.

9.2 ELECTRIC POTENTIAL The potential cðrÞ at a place r is deﬁned as the reversible electrical work, at constant p and T, to transport a unit charge from inﬁnity to r, @G cðrÞ ¼ : ð9:12Þ @Q p;T This deﬁnition is theoretically adequate, but may give rise to confusion when measurements are to be interpreted. In electrolyte solutions and in the solution side of electrical double layers ions are the carriers of the charge. However, in ions charge is always linked to matter so that transport of the charge can only be achieved by simultaneously transporting matter. Consequently, the total work of ion transport includes an electrical and a chemical contribution. dG ¼ zi Fcdni þ mi dni ;

ð9:13Þ

where mi is the chemical potential of ion i. There is no unambiguous way to separate the total work into its electrical and chemical contributions. Hence, under such conditions, deﬁning electric potentials is a problem. Only when we are sure that no chemical work is involved is the potential deﬁned. This is the case if the ion is transported in a homogeneous environment, for instance, in a solution of

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Electrochemistry of Interfaces

147

invariant solvent structure. This situation is approximated in the outer part of the electrical double layer, that is, outside the hydrodynamic slip plane. Anticipating the forthcoming discussion in Section 9.4 it is noted here that the theory of diffuse double layers is based on the assumption that the transfer of ions from the bulk to the outer, solution-side part of the electrical double layer involves only an electrical contribution. In the inner part of the electrical double layer, adjacent to the (solid) surface, the structure of the hydration water differs from that in the bulk solution, and at the surface the environment is even more different. Hence, the surface potential c0 cannot be established as the work of charging the surface. For some substances, such as oxides and other inorganic materials, there may be a way out of this problem, namely, if an electrode can be made of the solid phase. In that case an electrical cell can be set up consisting of that electrode and a reference electrode (e.g., a calomel electrode) in a solution containing the charge determining ions (cdi), and then the potential difference between the two electrodes can be measured. Examples of such electrodes are the glass electrode and the Ag=AgI electrode which respond according to Nernst’s law (cf. Chapter 14, Section 14.3) c0 ¼

ð9:14Þ

pzc c0 0. However, most materials, for example, polymeric subwhere at ccdi stances, proteins, and bacterial cell walls have too low conductivity to be suitable as electrode material. Then, the only way to obtain c0 is by adopting a model for the charge distribution in the electrical double layer. The potential decay across the electrical double layer cðxÞ is related to the space charge density rðxÞ in that layer, which, in turn, is also a function of the distance x from the charged surface. The relation is given by Poisson’s law, which for a ﬂat geometry reads

d2 cðxÞ rðxÞ ¼ : dx2 ee0

ð9:15Þ

Hence, rðxÞ must be known to derive cðxÞ. In Section 9.4 the most current models for charge distributions in the electrical double layer are presented.

9.3 THE GIBBS ENERGY OF AN ELECTRICAL DOUBLE LAYER In Section 9.1 we have seen that surfaces attain an electric charge by dissociation or association with protons, by uptake or release of ions in=from crystals of inorganic salts, or by speciﬁc adsorption of ions other than cdi. These processes, governed by chemical nonelectric forces, occur spontaneously and, hence,

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148

Chapter 9

Gs ðchemÞ < 0. However, charging also includes an electric effect because charged groups of the same charge sign are brought in close proximity at the surface. In other words, charged groups are formed at a location of adverse potential, implying that Gs ðelÞ > 0. The charging process is accompanied by the formation of the electrical double layer and therefore the total change in Gibbs energy, Gs (total), resulting from the formation of the electrical double layer comprises a chemical and an electric term Gs ðtotalÞ ¼ Gs ðchemÞ þ Gs ðelÞ:

ð9:16Þ

As the electrical double layer is formed spontaneously Gs ðtotalÞ < 0. The electric term Gs (el) equals, at constant p and T, the reversible work of charging the surface against its potential ð s0 c00 ds00 ; ð9:17Þ Gs ðelÞ ¼ 0

where s00 and c00 are the variable charge density and potential at the surface ðx ¼ 0Þ during the charging process. Obviously, the integration runs from s00 ¼ 0 to s00 ¼ s0 (the ﬁnal surface charge density). Because the charging process is carried out reversibly, the electrical double layer is continuously in equilibrium implying that rearrangements in the distribution of charges in the solution side of the double layer should not be accounted for in Gs (el). To be more speciﬁc, the decrease in electric energy resulting from the transport of a counterion from bulk solution to a position in the double layer is just compensated by the entropy loss. The chemical contribution Gs (chem) may be approximated as follows. Chemical interactions are of much shorter range than electric ones. At not too high surface charge densities and assuming a more or less even distribution, lateral interactions between the surface charge groups are neglected. At that condition dGs ðchemÞ=dGcdi is essentially independent of Gcdi , the surface excess of charge determining ions. Charging the surface by adsorbing cdi continues until the decrement in Gs (chem) just equals the increment in Gs (el). Hence, for the adsorption of the last charge determining ion dGs ðchemÞ ¼ dGs ðelÞ ¼ c0 ds0 :

ð9:18Þ

Because Gs (chem) is independent of Gcdi , integration yields Gs ðchemÞ ¼ c0 s0

ð9:19Þ

and for Gs (total) Gs ðtotalÞ ¼

ð s0

c00 ds00 c0 s0 :

0

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ð9:20Þ

Electrochemistry of Interfaces

149

The two terms on the right-hand side of (9.20) may be taken in one term ð c0 s00 dc00 : ð9:21Þ Gs ðtotalÞ ¼ 0

Because s00 and c00 have the same signs Gs ðtotalÞ < 0, as it should be for the spontaneous formation of the electrical double layer upon exposing the surface to the (aqueous) solution. To evaluate the integrals in (9.17) and (9.21) the relation between c00 and s00 should be known. This relation may be obtained from s0 ðccdi Þ represented by the titration curves (see Figure 9.2), provided that Nernst’s law, (9.14), applies. If not, a model for the electrical double layer is required to derive c00 ðs00 Þ.

9.4 MODELS FOR THE ELECTRICAL DOUBLE LAYER 9.4.1 The Molecular Condenser The molecular condenser is the simplest imaginable double layer. It is depicted in Figure 9.6. The surface solution boundary is set at x ¼ 0, containing all the surface charge. All counterions are at x ¼ d. In reality, such a situation could arise if the electrostatic attraction of these ions to the surface were so strong that they

ψ

+ + + + + + + + +

– ψo

0

d

x

Figure 9.6 Charge distribution and potential proﬁle in a molecular condenser.

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150

Chapter 9

ruled out thermal motion and the counterions approached the surface at the shortest distance possible. That distance is determined by the radius of the ions, assuming them to be either hydrated or not; that is, in Figure 9.6, d would equal the sum of the radii of surface and counterion, with or without hydration water in between. The functionality cðxÞ is obtained by integrating Eq. (9.15) from x ¼ 0 to x ¼ d, which gives dcðxÞ s ¼ 0; dx ee0

ð9:22Þ

which implies that, at constant s0 ; cðxÞ drops linearly with x. It follows from the model that cðxÞ ¼

c0 x þ c0 d

ð9:23Þ

so that s0 ee0 ; ¼ c0 d

ð9:24Þ

being the equation for the capacitance of a ﬂat plate condenser with plate distance d. This explains the name of the model. Double layers resembling a molecular condenser are seldom met in practice, but as a limiting case they deserve attention.

9.4.2 The Diffuse Double Layer Figure 9.7 gives a representation of the diffuse double layer. This model is also known as the Gouy–Chapman layer (named after the persons who ﬁrst developed the model). The underlying picture is that the surface is located at x ¼ 0 and that the counterions are not only attracted by the surface but are also subject to thermal motion. The former force tends to accumulate all counterions at the distance of closest approach to the surface (as in the molecular condenser), whereas the latter tries to spread all counterions homogeneously in the solution. The co-ions are subjected to the same counteracting tendencies. See Figure 9.1. The resulting countercharge distribution is given by the Boltzmann equation ci ðxÞ ¼ ci;1 exp½zi FcðxÞ=
ð9:25Þ

with ci the concentration of ion i and ci;1 the concentration of i in the bulk of the solution far away from the surface.

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Electrochemistry of Interfaces

151

ψ

ψo

– e

+

+ +

+

+ +

+

– ψo

+

+

+

+

+ +

+

+ +

+

1/κ

x

Figure 9.7 A diffuse electrical double layer according to Gouy and Chapman.

The space charge density rðxÞ at the solution side in the electrical double layer follows from the excess of counterions and the deﬁcit of co-ions, X z c ðxÞ: ð9:26Þ rðxÞ ¼ F i i i Both the counterions and the co-ions are considered to be point charges that have no volume, so that a diffuse distribution is obtained up to the boundary with the solid surface, x ¼ 0. Combining Eqs. (9.15), (9.25), and (9.26) yields, after integration, for symmetrical electrolytes (i.e., zþ ¼ jz j z), zFcðxÞ zFc0 tanh exp½kx; ð9:27Þ ¼ tanh 4
ð9:28Þ

In (9.27) and (9.28) k is the so-called reciprocal Debye length; it is related to the ionic strength as (for symmetrical electrolytes) k2 ¼

2F 2 ci z2i : ee0
ð9:29Þ

In water at 25 C, (9.29) gives k2 ¼ 10cz2i ; in which c is expressed in mol dm3 and k in nm1.

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ð9:30Þ

152

Chapter 9

According to (9.28) the Debye-length ðk1 Þ equals the distance over which c reduces from c0 to c0 =e. By convention, that distance is referred to as the thickness of the electrical double layer. It follows from (9.29) that cðxÞ decays more steeply, or, in other words, that the thickness of the double layer decreases as the ionic strength of the solution increases. The diffuse charge density in the solution side of the double layer sd is obtained by adding up all countercharge ð1 rðxÞdx: ð9:31Þ sd ¼ x¼0

Taking rðxÞ from (9.15) and substituting in (9.31) we obtain, after integration dc dc : ð9:32Þ sd ¼ ee0 dx x¼1 dx x¼0 Because ðdc=dxÞx¼1 ¼ 0, dc : sd ¼ ee0 dx x¼0

ð9:33Þ

ðdc=dxÞx¼0 may be derived from (9.27) or (9.28). For relatively low potentials, where (9.28) is applicable, dc ¼ kc0 ð9:34Þ dx x¼0 and, hence sd ¼ ee0 kc0 :

ð9:35Þ

Because of the overall electroneutrality of the electrical double layer s0 ¼ sd ¼ ee0 kc0 :

ð9:36Þ

For the capacitance (per unit interfacial area) of the double layer, deﬁned as s0 =c0 , it follows for low potentials s0 ¼ ee0 k: ð9:37Þ c0 According to (9.14) the value for c0 of a Nernstian surface varies linearly with ln ccdi the proportionality constant being
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Electrochemistry of Interfaces

153

is reﬂected in the increased steepness of the titration curve at higher ionic strength, as is observed in Figure 9.2. Obviously, the pzc, where by deﬁnition s0 ¼ 0 and hence c0 ¼ 0, is invariant with the ionic strength. Based on this, the pzc is often established as the common intersection point of experimentally determined titration curves at varying ionic strengths. The diffuse double layer model of Gouy and Chapman works reasonably well for systems of relatively low surface potential ( few tens of mV) and low electrolyte concentration ð102 MÞ. At higher surface potential and ionic strength the outer part of the double layer may still obey this model, but the inner part close to the surface tends toward the molecular condenser. Therefore, these two pictures are integrated in the Gouy–Chapman–Stern model.

9.4.3 The Gouy–Chapman–Stern Model Stern modiﬁed the diffuse model ( Gouy–Chapman model) by assigning a ﬁnite volume to the counter- and co-ions, which, moreover, may adsorb speciﬁcally. The result is a double layer consisting of an inner part resembling the molecular condenser and an outer part obeying the diffuse model. See Figure 9.8. In the absence of speciﬁcally adsorbed ions there is a chargefree layer, the so-called Stern layer, adjacent to the surface that extends to x ¼ d, where d is the distance of closest approach of a hydrated ion to the sorbent surface. In the case of speciﬁc

ψo

+

ψ

+ +

ψm ψd

+

+

+

+

+ +

+ 0m d

+

+

ψd e + 1/ κ

+

x

Figure 9.8 Gouy–Chapman–Stern electrical double layer with speciﬁc ion adsorption in the Stern layer 0 < x < d.

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154

Chapter 9

adsorption the surface charge s0 is partly compensated by the charge of dehydrated ions that are located at x ¼ m, sm , and partly by the diffusely distributed charge sd , s0 ¼ ðsm þ sd Þ:

ð9:38Þ

As in the molecular condenser cðxÞ drops linearly in the regions 0 < x < m and m < x < d: c0 cm ¼

s0 m em e0

and cm cd ¼

sm ðd mÞ: es e0

ð9:39Þ

A complication in calculating these potential differences is the uncertainty in the values to assign to em and es (the dielectric constants across the regions 0 < x < m and m < x < d, resp.) and, to a lesser extent, the values to assign to m and d. The values for m and d may be approximated as the radii of the bare and the hydrated counterion, respectively. These radii are in the subnanometer range, that is, comparable with the thickness over which water molecules are preferentially oriented in the hydration layer at a surface. It is therefore expected that em and es assume values that are much smaller than that of e in bulk water (e ¼ 80, at 20 C) Reasonable estimates for em and es range from 5 to 20. In the absence of speciﬁc ion adsorption s0 ¼ sd and, in case of speciﬁc adsorption sm ¼ ðs0 þ sd Þ. Analogous to (9.36) it follows that, in the absence, respectively, presence, of speciﬁc adsorption, for low values of cd , s0 ¼ ee0 kcd

and

s0 þ sm ¼ ee0 kcd :

ð9:40Þ

For the diffuse part of the double layer x d, cðxÞ drops exponentially [cf. (9.28)]: cðxÞ ¼ cd exp½kðx dÞ:

ð9:41Þ

Thus, cðxÞ can be derived over the entire double layer if s0 and sd [and, in view of (9.38), sm ] are known. It has been shown in Section 9.1 that s0 can be obtained by titration and, for smooth surfaces, sd may be approximated by sek. Furthermore, the ionic strength should be known to evaluate k. As has been mentioned before and discussed in more detail in Chapter 10, for smooth surfaces cd may be identiﬁed with the electrokinetic potential. It appears that under most conditions cd is much smaller than c0 ; that is, the potential decays for the largest part across the Stern layer. Anticipating the discussions in forthcoming chapters, for the stability of (hydrophobic) colloids as well as in adsorption and adhesion processes at charged interfaces cd plays a more relevant role than c0 . In the models discussed in Sections 9.4.1 through 9.4.3, it was assumed that the charge on the solid is located at its (planar) surface. For various (natural)

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Electrochemistry of Interfaces

155

interfaces this assumption may not hold. For instance, at biological membranes and polymer-like surfaces the charge may extend to a certain depth into the solid phase. Also, the solid may have an open, gel-like, structure which is penetrable for water and counter- and co-ions. The wall of bacterial cells is an example of such surfaces. The electrical double layer at surfaces that are pervious to charges are called ‘‘porous’’ double layers. In Figure 9.9(a) the situation is depicted where the (negative) surface charge is distributed in some fashion in the surface region of the solid. In Figure 9.9(b) not only the surface charge extends over a certain depth in the solid, but also counterions (and co-ions) penetrate this phase. In the second case the charge s0 due to charged groups of the solid needs to be not lower than in the ﬁrst case. On the contrary, it may be much higher (cf., s0 for B. brevis in Figure 9.2), but the surface potential c0 is lower because s0 is to a large extent screened by the countercharge in the solid phase. Furthermore, in the case of Figure 9.9(b) the potential decay in the molecular condenser is less steep than in the case of Figure 9.9(a), simply because the charge density on the plate condenser is reduced to ðs0 sc Þ, sc being the charge density due to the countercharge that has penetrated the solid phase. Taking these modiﬁcations into account, cðxÞ at the solution side of the porous double layer may be derived by applying one or more of the appropriate models discussed in Sections 9.4.1 through 9.4.3. To calculate the potential proﬁle in the solid requires knowledge of the charge distribution in that phase. This depends, among other things, on material properties and it is therefore difﬁcult to make predictions of general validity. In Figure 9.10 the ionic strengths of a variety of natural aqueous systems are summarized, together with indications for the applicability of the different electrical double layer models.

Figure 9.9 Electrical double layers at porous solid surfaces in (a) the absence and (b) the presence of counterion penetration in the solid.

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156

Chapter 9 type of water rain streams and lakes groundwater waste water estuarine sea

type of biofluid saliva milk blood, urine

0

molecular condenser

1

2 3 –log [ionic strength (M)]

Gouy-Chapman-Stern double layer

4

diffuse double layer

Figure 9.10 Applicability of different models for the electrical double layer in various natural and biological environments.

9.5 DONNAN EFFECT; DONNAN EQUILIBRIUM; COLLOIDAL OSMOTIC PRESSURE; MEMBRANE POTENTIAL In Section 9.1, Figure 9.1, we have seen that adjacent to a charged surface there is an excess of counterions and a deﬁcit of co-ions. Both contribute to the neutralization of the surface charge. Let us now focus on the expulsion of the co-ions. The expulsion of co-ions implies a reduced volume available for electrolyte, or, in other words, there is an excluded volume with respect to the presence of electrolyte. This is known as the Donnan effect. For the same reason salt (¼ electrolyte) cannot penetrate in narrow capillaries and pores having charged walls. Based on this phenomenon porous membranes that are permeable for water but not for salt may be used in reversed osmosis (also called ultraﬁltration). Practical applications of reversed osmosis are found in, for example, the production of potable water from seawater, in hemodialysis using artiﬁcial kidneys, and in the concentration of solutions such as fruit juices. In Figure 9.11 the expulsion of co-ions is depicted for a Gouy–Chapman– Stern double layer. Let cþ be the concentration of co-ions at a positively charged surface. In a Gouy–Chapman–Stern double layer the co-ion expulsion per unit surface area, that is, the negative adsorption of co-ions Gþ , is given by ð1 ð1 cþ ðcþ c1 Þdx ¼ c1 1 dx; ð9:42Þ Gþ ¼ c1 d d

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Electrochemistry of Interfaces

157

c+

c+,

0

=c ,

= c( )

d

x

Figure 9.11 Concentration proﬁle of co-ions (in this ﬁgure positively charged ions near a positively charged surface) showing expulsion of electrolyte.

where c1 ð cÞ is the electrolyte concentration in bulk solution. With (9.25) the expression for Gþ becomes ð1 zþ FcðxÞ 1 dx: ð9:43Þ exp Gþ ¼ c 1013 M), this is a reasonable condition. Hence, substituting (9.41) in (9.44) and subsequent integration yields Gþ ¼

zþ Fcd c:
ð9:45Þ

The corresponding amount of expelled charge is sþ ¼ zþ FGþ ¼

z2þ F 2 cd c:
ð9:46Þ

For a symmetrical electrolyte ðzþ ¼ z zÞ, for which (9.29) applies, we can write sþ ¼

ee0 kcd : 2

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ð9:47Þ

158

Chapter 9

Comparing (9.47) with (9.36) or (9.40) reveals that at surfaces of low potential, the expulsion of co-ions accounts for half of the countercharge in the diffuse part of the double layer. Consequently, the other half is due to the excess of counterions. The Donnan effect is at the basis of the Donnan equilibrium, which describes the equilibrium distribution of small ions in a system containing macro-ions such as polyelectrolytes or charged colloidal particles. The Donnan equilibrium may be best illustrated by considering a system where a macro-ion z in a solution containing a low molecular weight electrolyte Mþ X is Mþ z P separated from a solution of only Mþ X. The separation is by a membrane which is permeable to the solvent (water) and Mþ and X, but not to the macro-ion Pz . Figure 9.12 shows a system where a colloidal dispersion (in compartment I) is separated from its medium (in compartment II) by a semipermeable membrane. Let the molar concentration of Mz P in I be c and that of MX in I cI , and the molar concentration of MX in II cII . When Mz P is fully dissociated the total concentration of Mþ in I equals ðcI þ zcÞ. If, initially, the concentrations of MX are the same at both sides of the membrane, that is, cI ¼ cII , then cMþ ;I > cMþ ;II providing a driving force for the diffusion of Mþ from I to II. Electroneutrality requires that X be transported along with Mþ . This again is a manifestation of the Donnan effect: salt is expelled by the charged colloidal particle or polyelectrolyte. As the passage of Mþ X through the membrane proceeds, cX ;II exceeds cX ;I more and more, becoming a driving force for Mþ X transport in the reverse direction. In equilibrium cMþ ;I > cMþ ;II and cX ;I < cX ;II . The number of Mþ X ion pairs being transferred from the one compartment to the other, say, from I to II, scales with cMþ ;I cX ;I . Hence, the Donnan equilibrium is expressed as cMþ ;I cX ;I ¼ cMþ ;II cX ;II ;

ð9:48Þ

which for the system of Figure 9.12 gives ðcI þ zcÞcI ¼ c2II :

ð9:49Þ

+

c Mz Pz– +

+

cI M X–

cII M X–

I

II

Figure 9.12

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Donnan equilibrium.

Electrochemistry of Interfaces

159

Note that the Donnan effect does not require the presence of a semipermeable membrane and therefore the Donnan equilibrium always exists between the electrical double layer around the macro-ion and the bulk of the continuous phase. If no low molecular weight electrolyte is present, which means that Mz P is suspended in pure water, the Mþ ions expelled to the bulk are accompanied by OH ions. As a result, the pH in the bulk solution is higher than near the charged surface of the macro-ion. This phenomenon, the so-called suspension effect, must be taken into account when performing pH measurements in charged colloidal systems of low ionic strength. In Chapter 3, Section 3.6, we derived that a concentration difference across a (virtual) semipermeable membrane gives rise to an osmotic pressure. Similarly, the ion concentration difference, ðcI þ cI þ zc þ cÞ ðcII þ cII Þ, characterizing the Donnan equilibrium leads to an osmotic pressure p, the colloid-osmotic pressure, which follows from (3.35), p ¼
ð9:50Þ

Substituting (9.49) in (9.50) and expressing the Mþ X concentrations in I and II in the dimensionless quantities x cI =zc and y cII =zc [so that (9.49) can be written as the quadratic function x2 þ x y2 ¼ 0 allowing us to solve for x], we obtain p ¼
ð9:51Þ

For y 1 (i.e., low ionic strength), (9.51) reduces to p ¼
1 10– 2 Figure 9.13

10– 1

100

101

y

102

Colloid-osmotic pressure according to the Donnan model.

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160

Chapter 9

The Donnan equilibrium allows the evaluation of the distribution of Mþ and X over both sides of the semipermeable membrane. If electrodes responding to either Mþ or X were inserted at either side of the membrane there would be no potential difference between them. This is a consequence of the system being in equilibrium which implies that no work can be performed. Nevertheless, because of the different ion concentrations the potentials at the respective electrodes c0;I and c0;II, are not equal. Consequently, there must be a compensating potential difference across the membrane cm . If the electrodes respond reversibly to the ion concentrations so that Nernst’s law (9.14) applies, it follows for the membrane potential cm ¼

ð9:52Þ

Needless to say, the membrane potential cannot be directly measured. Determination of cm is only possible by measuring the potential difference between two reference electrodes that are separately connected (by, e.g., a salt bridge) to the other two electrodes. The existence of the Donnan effect, the colloid-osmotic pressure and the membrane potential is not only important for biomedical and biotechnological applications, some of them mentioned in the ﬁrst paragraph of this section, but they are also essential in many biological systems. For instance, the Donnan effect is responsible for maintaining concentration gradients of ions across the membrane surrounding a biological cell in which interior polyelectrolytes are present. Furthermore, biological cells can only survive by a colloid-osmotic pressure that keep the cells under a certain tension so that the cell plasma maintains intimate contact with the enveloping membrane. This is necessary for the passage of nutrients and it should prove clear that the membrane potential plays an essential role in the transfer of ionic components through such membranes. This subject is further discussed in Chapter 19.

EXERCISES 9.1

Comment on the following statements. (a)

The point of zero charge of a polyampholyte containing 10 times as many carboxyl groups ðpKdiss ¼ 5Þ as amino groups ðpKdiss ¼ 9Þ is at pH 6. (b) The molar entropy of ions in the diffuse part of an electrical double layer is higher than in the bulk solution. (c) The potential decay across the electrical double layer is steeper for spherical than for planar surfaces, given the same values for the surface potential and ionic strength.

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Electrochemistry of Interfaces

(d)

(e)

9.2

161

A charged surface has a surface potential of 150 mV and a Stern potential of 70 mV. The potential at a distance k1 from the surface is 55 mV. For a Donnan equilibrium between a polycation in an aqueous solution of NaCl and the corresponding NaCl solution the pH in the polyelectrolyte solution is higher than in the NaCl solution.

The ﬁgure gives a proton titration curve of poly(L-lysin) in aqueous solution.

(a) What is the meaning of a and how is this quantity deﬁned? (b) What is the relation between the dissociation constant of the titratable lysin residues and ðpH þ logða=ð1 aÞÞÞ? (c) Why does ðpH þ logða=ð1 aÞÞÞ increase with increasing value of a? (d) The deviation from linearity of the curve shown in the ﬁgure reﬂects a conformation change in the poly(L-lysin) molecules. Explain how the nonelectric contribution to the Gibbs energy of the conformation change can be calculated from the titration curve. 9.3

An electrical double layer may be conceived as a condenser with a capacitance C, deﬁned as C ¼ s0 =c0 , where s0 and c0 are the surface charge density and the surface potential, respectively. For a condenser with a separation distance d between the plates C ¼ ee0 =d, where ee0 is the dielectric permittivity of the medium across the condenser. (a)

Derive for the diffuse double layer according to Gouy and Chapman that C ¼ ee0 k. (b) Derive an expression for the capacitance of a chargefree Stern layer. The Gouy–Chapman–Stern model of the electrical double layer may

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162

Chapter 9

be understood as two condensers in series, so that for the total capacitance Ct : 1 1 1 ¼ þ : Ct CStern Cdiffuse What are the values of Ct for the limiting cases of extremely high and low ionic strengths? Assume for the Stern layer a thickness of 0:5 nm and a dielectric constant 20 times smaller than in the bulk solution. What would be the values for the concentration of a 1:1 electrolyte to approximate these limiting values of Ct within 10%? 9.4

A volume is divided by two semipermeable membranes in three equal volumes, I, II, and III, as depicted in the ﬁgure. The membrane separating I from II is selectively permeable for anions and the membrane between II and III for cations. Initially, compartment II is ﬁlled with an aqueous solution of NaCl and I and III are ﬁlled with pure water.

(a)

Describe the ion ﬂuxes in the system on its way towards equilibrium. What can be said of the pH in I, II, and III after reaching equilibrium? (b) Derive c2I ¼ Kw1=2 cII , where c is the equilibrium concentration of NaCl in the compartment indicated by the index and Kw the ion product of water. Derive an analogous expression for cIII. The amount of water ﬂowing from one compartment to the other may be neglected. Calculate cI and cIII for cII ¼ 101 M. What is the NaCl concentration in the solution used to ﬁll compartment II? (c) In which direction does the pH in compartments I and III change when a polyanion is added to II?

SUGGESTIONS FOR FURTHER READING M. Blank (ed.). Electrical Double Layers in Biology, New York: Plenum 1986. J. O’M. Bockris, B. E. Conway, F. Yeager (eds.), Comprehensive Treatise of Electrochemistry, I: The Double Layer, New York: Plenum, 1980.

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Electrochemistry of Interfaces

163

A. Kitahare, A. Watanabe (eds.). Electrical Phenomena at Interfaces. Fundamentals Measurements and Applications, New York: Marcel Dekker, 1984. J. M. Kleijn, H. P. van Leeuwen. Electrostatic and electrodynamic properties of biological interfaces, in Physical Chemistry of Biological Interfaces, A. Baszkin and W. Norde (eds.), New York: Marcel Dekker, 2000, Chapter 2.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Lab-on-a-Chip in the Life Sciences

Following the trend in miniaturization of integrated circuits for electronic devices, integrated microﬂuidic circuits are currently being developed and are expected to have a great impact on (bio)chemical synthesis and analysis. Such labs-on-a-chip are glass or plastic plates, typically a few centimeters on a side, with interconnected microreservoirs and pathways on the surface. Minute amounts of liquids and suspended solids are moved around the channels from one reservoir to another. A crucial aspect in the performance of such microreactors is the precise manipulation of ﬂuid ﬂow. Electrokinetic actuation is eminently suited to control the motion of ﬂuids and reagents on the chip. This is effected by strategically placed electrodes creating electrokinetic forces that drive the ﬂuids and reagents through selected pathways. More speciﬁcally, because of its planar velocity proﬁle, electroosmotic ﬂow allows for efﬁcient material transport. Furthermore, electrophoresis provides a powerful separation technique of reactants and products. Thus lab-ona-chip technology, making tools smaller and more integrated and, therefore, less expensive and faster, has great potential use in various ﬁelds of the life sciences, such as genomics, proteomics, clinical diagnostics, and basic biomolecular research. (Figure courtesy of Mesaplus, University of Twente, The Netherlands.)

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10 Electrokinetic Phenomena

Electrokinetic phenomena involve the migration of a charged (colloidal) particle or a charged macroscopic surface relative to the surrounding liquid medium which is an electrolyte solution. They are the result of motion and electric interactions in the electrical double layer. Electrokinetic phenomena may be classiﬁed based on the driving force and the ensuing motion. The driving force is either an externally applied electric potential gradient (electric ﬁeld) or a ‘‘mechanical’’ potential gradient (pressure difference, or a gravitational or centrifugal ﬁeld). With respect to the motion, distinction is made between a mobile phase and a stationary phase. Table 10.1 summarizes the different groups of electrokinetic phenomena. Electro-osmosis is the motion of liquid with respect to a stationary charged solid surface; in electrophoresis charged particles move in a stationary liquid phase. Electro-osmosis and electrophoresis are invoked by an imposed electric ﬁeld. Streaming potential and streaming current may be considered as the opposite of electro-osmosis; they arise from moving an electrolyte solution along a stationary charged surface under the inﬂuence of a pressure difference. The sedimentation potential is due to the electric ﬁeld created by charged particles that sediment in stationary liquid, which is the opposite of electrophoresis.

10.1 THE PLANE OF SHEAR. THE ZETA-POTENTIAL When a (solid) surface moves in a liquid, or vice versa, there is always a layer of liquid adjacent to the surface that moves with the same velocity as the surface. The distance from the surface over which this stagnant liquid layer extends or, in other words, the location of the boundary between the mobile and the stationary

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166

Table 10.1 Electrokinetic phenomena Driving force

Mobile phase

Stationary phase

Measured property

Electrokinetic phenomenon

Electric ﬁeld Electric ﬁeld Pressure gradient

Liquid Dispersed particles Liquid

Porous plug; capillary Liquid Porous plug; capillary

Gravitational or centrifugal ﬁeld

Dispersed particles

Liquid

Motion of the liquid Motion of the particles Potential difference; electric current Potential difference

Electro-osmosis Electrophoresis Streaming potential; streaming current Sedimentation potential

Chapter 10

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Electrokinetic Phenomena

167

phases, the so-called plane of shear or slip plane, is not exactly known. For smooth surfaces the plane of shear is within a few liquid (water) molecules from the surface (see Figure 9.4), that is, well within the electrical double layer. The stagnant layer is probably somewhat thicker than the Stern layer, so that the plane of shear is located in the diffuse part of the electrical double layer. It follows that the potential at the plane of shear, that is, the electrokinetic potential or the zetapotential z, is somewhat lower than the Stern potential cd . Because the largest part of the potential drop in the electrical double layer occurs across the Stern layer, z will not be much lower than cd ; in fact the experimentally accessible value of z is often used to approximate cd . When the surface is coated with a loosely structured (polymeric) layer that is freely penetrable by solvent and small ions the plane of shear is much further outward in the electrical double layer, as is illustrated in Figure 10.1. In that case jzj jcd j. In practice, especially in biological systems such as biological cells, surfaces may be more or less ‘‘hairy’’ rather than smooth. It may be appreciated that electrokinetic phenomena are determined by electric properties at the plane of shear rather than at the real surface. In the following sections of this chapter the relation between the measured property and z is further analyzed. This is done for electro-osmosis, electrophoresis, streaming current, and streaming potential. The sedimentation potential is not discussed any further, because in practice this phenomenon does not play an important role. The electrokinetic charge density sek may then be derived from z using the theory for the diffuse electrical double layer.

10.2 DERIVATION OF THE ZETA-POTENTIAL FROM ELECTROKINETIC PHENOMENA 10.2.1 Electro-Osmosis Consider a capillary, or a narrow slit between two parallel plates, of which the inner walls are charged and which contains an electrolyte solution. See Figure 10.2(a). The surface charge is compensated by a countercharge in the solution. Applying a potential difference between the two ends of the capillary gives rise to an electric ﬁeld E that causes the ions in the solution to move. In the capillary, narrow slit, or pore there are more counterions than co-ions so that a net amount of ions (charge) migrates in one direction. The moving ions drag the liquid along resulting in a ﬂow of liquid in the same direction. Thus, any liquid element, illustrated in Figure 10.2(b), moves under the inﬂuence of an electric force fel . However, the movement is retarded by internal friction in the liquid. It leads to a stationary situation where the friction force ffr just compensates fel , fel þ ffr ¼ 0. Then, the liquid moves at constant velocity. fel scales with the charge in the volume element and is, because of

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168

Chapter 10

(a) ψo

(b) ψo

+ +

ψd ζ

+

+

+

ψd

+

+

+

+

+

+

+

+ +

+

ζ

+

+

+ +

+ +

+ +

+

Figure 10.1 Location of the hydrodynamic plane of shear (– – –) relative to the Stern plane (...) at (a) a smooth surface and (b) a surface coated with highly solvated polymeric molecules. The potentials at the surface, the Stern plane, and the plane of shear are indicated by c0 , cd , and z.

(9.15), related to the potential decay in the solution. In this context it should be noted that when the thickness of the electrical double layer is much smaller than the radius of the capillary the wall can be considered as ﬂat. fel can be expressed as dfel ¼ ErðxÞAdx

ð10:1Þ

in which Adx is the volume of the liquid element shown in Figure 10.2(b). Using (9.15) to substitute rðxÞ, we obtain 2 d cðxÞ dfel ¼ ee0 E Adx: ð10:2Þ dx2 ffr varies proportionally with the viscosity Z of the liquid and it depends on the velocity gradient in radial direction dn=dx. For ffr we write 2 d nðxÞ dx: ð10:3Þ dffr ¼ ZA dx2 From the condition dfel þ dffr ¼ 0 follows that d2 nðxÞ ee0 E d2 cðxÞ ¼ : dx2 Z dx2

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ð10:4Þ

Electrokinetic Phenomena

– – – – – – – – – – +++ + ++ + + + + – + E + + + – – – + + + + + + + + + +

169

– – – + + + – + v

+ + + – – – – – – – – – – – – –

(a)

y x dx

x

fel

ffr (b)

Figure 10.2 Electro-osmosis: (a) distribution of ions in a charged capillary; (b) forces acting on a volume element of liquid.

Integration of (10.4) gives dnðxÞ ee0 E dcðxÞ ¼ þ C1 ; dx Z dx

ð10:5Þ

where C1 is an integration constant. In the center of the (symmetrical) capillary dn=dx ¼ 0 and dcðxÞ=dx ¼ 0, so that C1 ¼ 0. Subsequent integration yields nðxÞ ¼

ee0 E ee E cðxÞ 0 z: Z Z

ð10:6Þ

The second term on the right-hand-side of (10.6) follows from the condition that at the plane of shear c ¼ z and n ¼ 0. Hence, for a given value of z, nðxÞ varies with cðxÞ. In general, the radius of the capillary is at least in the mm range, exceeding by far the thickness of the electrical double layer k1, which over a wide variety of ionic strengths is in the nm range. It means that only in the immediate vicinity of the shear plane cðxÞ 6¼ 0; beyond that region nðxÞ attains its maximum value of ee0 Ez=Z. Now, the electro-osmotic velocity neo is deﬁned as neo ¼ ee0 Ez=Z:

ð10:7Þ

The velocity proﬁle according to (10.6) is depicted in Figure 10.3. Neglecting the deviations near the shear plane the electro-osmotic volume ﬂux Jeo is given by Jeo ¼ neo O;

ð10:8Þ

with O the cross-sectional area of the capillary. Furthermore, according to Ohm’s law I ¼ EOksp ;

ð10:9Þ

where I is the electric current and ksp the speciﬁc conductivity (conductivity per unit length) of the solution. Combining Eqs. (10.7) through (10.9) gives Jeo ¼ zee0 I =Zksp :

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ð10:10Þ

170

Chapter 10

veo

Figure 10.3

Velocity proﬁle of liquid ﬂow induced by electro-osmosis.

Equation (10.10) allows z to be derived by measuring the electrical current and the volume ﬂux through a capillary using an electrolyte solution of known e; Z, and ksp . When the zeta-potential of colloidal particles is sought, the particles may be compressed into a porous plug. The volume ﬂux and the electric current through the plug then provide z, as the pores in the plug may be considered as an ensemble of capillaries.

10.2.2 Electrophoresis As a ﬁrst approximation, the force fel an electric ﬁeld E exerts on a particle containing a charge Q equals QE. The friction force ffr of a spherical particle of radius a that moves with constant velocity n in a medium of viscosity Z is, according to Stokes’ law, 6pZan. Hence, under stationary conditions ð fel þ ffr ¼ 0Þ, n¼

QE : 6pZa

ð10:11Þ

However, the movement of a charged particle is more complicated because of interference by the surrounding counterions. The counterions tend to move in the opposite direction. They drag the liquid along and this slows down the movement of the particle. This effect, which is of hydrodynamic origin, is known as electrophoretic retardation. Some extreme cases present themselves: (a) particles having a large radius a and a thin electrical double layer (small value for k1 ), that is, ka 1, and (b) small particles with a thick double layer, ka 1. For these two extremes the electric ﬁelds around the particles are drawn in Figure 10.4. It is assumed that the particles themselves are nonconducting. In the case of ka 1 the ﬁeld lines are almost parallel to the particle surface tangentially following the contour of the particle. For small ka the ﬁeld lines run practically unperturbed. The counterions move along the ﬁeld lines and it should be clear that the ensuing retardation of the particle is smaller for the situation in Figure 10.4(a) compared to Figure 10.4(b). Moreover, a relatively thin electrical double layer has less inﬂuence on the movement of a large particle compared to a thick double layer on a small particle.

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Electrokinetic Phenomena (a)

κ –1

171 (b)

a a κ –1

Figure 10.4 Electric ﬁeld around particles having (a) large and (b) small values for ka.

The electrophoretic retardation is therefore large for ka 1 whereas it is nearly absent for ka 1. To obtain an expression for the electrophoretic velocity nef in which the retardation effect is taken into account we should consider the force on a volume element of the system, analogous to the procedure followed in deriving neo . The two extreme cases depicted in Figures 10.4(a) and (b) are considered. For ka 1, when the ﬁeld lines run parallel to the particle surface, the situation is similar to that shown in Figure 10.2(b). Hence, Eq. (10.4) applies. Integration, using the proper boundary conditions, that is, dnðxÞ=dx ¼ 0; dcðxÞ=dx ¼ 0; nðxÞ ¼ 0, and cðxÞ ¼ 0 at large distance from the particle, and n ¼ nef and c ¼ z at the shear plane, gives nef ¼ ee0 Ez=Z

ð10:12Þ

and for the electrophoretic mobility uef , deﬁned as the velocity per unit ﬁeld strength uef ¼ ee0 z=Z;

ð10:13Þ

which is known as the Helmholtz–Smoluchowski equation. For ka 1, that is, for an electrical double layer that is (inﬁnitely) thick relative to the particle size, k ! 0. Then, the charged particle may be considered as an isolated charge Q. According to elementary physics the potential cðxÞ at a distance x from the isolated charge is given as cðxÞ ¼ Q=4pee0 x. It follows that at the plane of shear around the particle Qef ¼ 4pee0 za:

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ð10:14Þ

172

Chapter 10

Combining with (10.11) gives 2ee0 z uef ¼ : 3Z

ð10:15Þ

Comparing (10.15) with (10.13) reveals that in the case of ka 1 the electrophoretic velocity is retarded by a factor of 1.5. In practice the value for ka is such that uef ¼ f ðkaÞee0 z=Z; ð10:16Þ with 2=3 f ðkaÞ 1. The functionality f ðkaÞ has been further elaborated by Henry who showed that f ðkaÞ ¼ 2=3 for ka < 0:5 and f ðkaÞ ¼ 1 for ka > 300. The result of Henry’s calculations is represented by the solid curve in Figure 10.5. In addition to hydrodynamic retardation the countercharge has another inﬂuence on the velocity of a charged particle in an electric ﬁeld. This inﬂuence is of an electric nature; it is known as the relaxation effect. The underlying cause is explained in Figure 10.6. When the particle is at rest the centers of the particle surface charge and the countercharge coincide in the center of the particle [Fig. 10.6(a)]. However, because in an electric ﬁeld the particle and the counterions migrate in opposite directions their centers do not coincide anymore [Fig. 10.6(b)]. This charge separation invokes a local electric ﬁeld that counteracts the applied electric ﬁeld and therefore reduces the velocity of the particles. It may be understood that the relaxation effect is very complicated, primarily depending on the values for the zeta-potential, the particle size, and the ionic strength. The relaxation effect has been calculated numerically and the results are presented in the literature in tables and graphs. It turns out that the relaxation effect is most 1 f (κ a)

25 50

2/3

75 ζ = 100 mV 0.1

1

10

100

1000

κa

Figure 10.5 Variation of f ðkaÞ [Eq. (10.16)] with ka for the case where retardation (—) and relaxation effects (- - -) are taken into account.

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Electrokinetic Phenomena

173

+

+

+ +

– – –

+ – – – – – – + + – – + +

–

– – –

+ +

+

+

+ ( a )

+

( b )

Figure 10.6 Electrophoretic relaxation effect. Charge distribution around (a) a particle at rest and (b) a particle in motion: = indicate the centers of the positive and negative charges.

pronounced in the ka range where the retardation effect varies. By way of example, some results are given by the dashed curves in Figure 10.5. This ﬁgure shows that the relaxation effect is negligibly small for both small and large values of ka and, over the whole ka range, if jzj < 25 mV.

10.2.3 Streaming Current and Streaming Potential As in electro-osmosis let us consider a capillary or a plug of dispersed material (i.e., an ensemble of capillaries) ﬁlled with an electrolyte solution. The inner walls of the capillary are charged and, as a consequence, there is an excess of counterions and a deﬁcit of co-ions in the solution. An applied pressure difference p between the two ends of the capillary causes the solution to ﬂow. As a result, the part of the electrical double layer in the mobile phase is displaced tangentially along the plane of shear. An electric current is thus evoked by a mechanical stress. If electrodes placed at both ends of the capillary are connected through a low resistance, a streaming current can be measured. If in the outer circuit between the electrodes a high resistance is placed, the current is blocked and a potential difference, the streaming potential, between the two electrodes is developed. The streaming potential causes a conduction current through the solution which, under steady-state conditions, just compensates the current associated with the migrating ions. The relation between the streaming current Is and z may be obtained as follows. The velocity n of a ﬂuid at radius r in a capillary of radius R and length ‘ is given by Poiseuille’s equation

p 2 R r2 ; nðrÞ ¼ ð10:17Þ 4Z‘

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Chapter 10

where Z is the viscosity of the liquid. Then, for the volume ﬂow V ðrÞ through a cross-sectional area 2pr dr, shown in Figure 10.7, dV ðrÞ=t ¼

p 2 R r2 2pr dr: 4Z‘

ð10:18Þ

The electrical current associated with this volume ﬂow is dI ðrÞ ¼ rðrÞ

dV ðrÞ rðrÞp 2 ¼ R r2 2pr dr; t 4Z‘

ð10:19Þ

where rðrÞ is the space charge density in the volume considered. Now we replace r by x, x being the distance from the plane of shear: x ¼ R r, so that dI ðxÞ ¼

rðxÞp 2Rx x2 2pðR xÞdx: 4Z‘

ð10:20Þ

Assuming that the thickness of the electrical double layer is much smaller than the capillary radius ðkR 1Þ and, therefore, the streaming current takes place near the walls of the capillary, Eq. (10.20) may be approximated taking x R: dI ðxÞ ¼

rðxÞp 2 pR x dx: Z‘

ð10:21Þ

The total current streaming through the capillary is obtained by integration, taking into account the dependency of r on x as given by Poisson’s law, (9.15). Integration by parts yields ( ) R ð R ee0 ppR2 dcðxÞ dcðxÞ Is ¼ x dx : ð10:22Þ Z‘ dx 0 0 dx 2 π r dr

R r dr

Figure 10.7 Cross section through a capillary.

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175

Applying the boundary conditions that c ¼ z at x ¼ 0, and c ¼ 0 and dcðxÞ= dx ¼ 0 at x ¼ R, we obtain Is ee pR2 ¼ 0 z: p Z‘

ð10:23Þ

An expression relating the streaming potential Es to z is easily obtained from the expression for Is, (10.23). In the case where a streaming potential is built up over the capillary Is is compensated by a counterconduction current Ig ; that is, Ig ¼ Is . According to Ohm’s law, Ig ¼

Es pR2 ksp ; ‘

ð10:24Þ

in which ksp is the speciﬁc conductivity of the solution. It follows immediately that Es ee ¼ 0 z: p Zksp

ð10:25Þ

10.3 SOME COMPLICATIONS IN DERIVING THE ZETA-POTENTIAL: SURFACE CONDUCTION; VISCO-ELECTRIC EFFECT The value for z inferred from the various electrokinetic phenomena is not without ambiguity. The reason lies in the assumptions involved in relating z to the measured quantity. Here we pay attention to two major complications: surface conduction and the viscoelectric effect.

10.3.1 Surface Conduction In electro-osmosis, electrophoresis, and streaming potential we deal with conduction and it has tacitly been assumed that the conduction current occurs through the bulk solution. However, ions in the electrical double layer may signiﬁcantly contribute to the overall conduction current. This causes an excess conduction tangential to a charged surface, the so-called surface conduction. Surface conduction plays a more important role when the volume of the electrical double layer approaches the total volume of the system. For a capillary, as used in electro-osmosis and streaming potential measurements, this is when k1 =R is nonnegligible. The surface conduction current in the double layer is proportional to the cross-sectional area of the electrical double layer, that is, to 2pRk1 . In the case of electrophoresis the retardation and relaxation effects are inﬂuenced by surface conduction. In (10.10) and (10.25) ksp , the speciﬁc electric conductivity

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Chapter 10

of the bulk solution, should be replaced by a term accounting for the excess conduction at the surface as well. Taking the streaming potential as an example, the expression for Ig, (10.24), should be modiﬁed to contain a contribution from the surface. Hence, Ig ¼ Igb þ Igs ¼

Es pR2 E 2pRk1 s ksp þ s k ; ‘ ‘

ð10:26Þ

in which ks is the speciﬁc surface conductivity. From Igb þ Igs ¼ Is follows Es ee0 z ¼ : p Zðksp þ 2ks =kRÞ

ð10:27Þ

It goes without saying that surface conduction does not play a role in streaming current measurements because these are performed in the absence of an electric ﬁeld.

10.3.2 Viscoelectric Effect In the discussion of electrokinetic phenomena it has hitherto been assumed that the viscosity Z of the liquid maintains its bulk value right near the shear plane. The notion ‘‘shear plane’’ actually implies a discontinuous jump in the value of Z from nearly inﬁnitely high within the shear plane to the liquid bulk value beyond it. From a physical point of view such an abrupt change is not realistic. Rather, the value of Z changes gradually from the surface to the bulk over a distance of a few molecular layers. Hence for a charged surface the distance over which Z varies lies well within the electrical double layer. The potential decay dcðxÞ=dx (¼ the electric ﬁeld) across the double layer may further inﬂuence the viscosity of the liquid. This is known as the viscoelectric effect. In electrical double layers the electric ﬁeld may be sufﬁciently strong to signiﬁcantly affect the viscosity and, consequently, the motion of the liquid as occurs in electrokinetic phenomena. In order to account for the viscoelectric effect the expressions relating the measured quantity to z should be modiﬁed by writing Z of the liquid as a function of the electric ﬁeld. For a series of (organic) liquids it has been shown that the viscosity is increased by applying an electric ﬁeld according to the empirical relationship ZE Z 0 ¼ f E2 : ð10:28Þ Z0 The subscripts indicate the presence (E) or absence (0) of an electric ﬁeld. The factor f is the viscoelectric constant; for a variety of liquids f has a value of about 2 1016 V2 m2 . It means that the viscosity of the liquid undergoes a 2–300% increase by applying an electric ﬁeld of 107 V m1 to 108 V m1 . In

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177

electrical double layers such ﬁeld strengths are quite realistic. For the diffuse part of the electrical double layer dcðxÞ=dx can be calculated using (9.27) or (9.28). For instance, in a solution of a 1:1 electrolyte of 0:1 M ionic strength the ﬁeld strength at x where c ¼ 50 mV amounts to 6:4 107 V m1 . By way of example, the inﬂuence of the viscoelectric effect on the relation between electrophoretic mobility and z (for ka 1, i.e., when the electrophoretic retardation is negligibly small) is elaborated below. Based on (10.5) with C1 ¼ 0, the relation between nef and z is given by ðz ð0 dcðxÞ ð10:29Þ dnðxÞ ¼ ee0 E Z nef 0 or ðz uef ¼ ee0

dcðxÞ : Z 0

ð10:30Þ

Substituting Z by ZE ; as expressed in (10.28) and with E ¼ dcðxÞ=dx, we obtain ð ee0 z dcðxÞ : ð10:31Þ uef ¼ Z 0 1 þ f ðdcðxÞ=dxÞ2 Because in an aqueous environment, under most conditions, the potential decays for the largest part across the inner region of the electrical double layer (i.e., in the region enclosed by the plane of shear), z usually does not exceed a few tens of millivolts. Equation (9.41) may therefore be used to derive dcðxÞ=dx. When it is further assumed that the plane of shear coincides with the Stern layer and, hence, z ¼ cd , integration of (10.31) gives uef ¼

ee0 z : Z0 ð1 þ f k2 z2 Þ

ð10:32Þ

Under conditions where the second term in the denominator is negligibly small, f k2 z2 1, (10.32) is identical to the Helmholtz–Smoluchowski equation, (10.13). On the other hand, for larger values of k and z the electrophoretic mobility is lower than predicted by (10.13).

10.4 INTERPRETATION OF THE ZETA-POTENTIAL For all electrokinetic phenomena discussed in this chapter the zeta-potential z is the electric potential at the plane of shear. Interpretation of z should therefore start with addressing the question regarding the location of the plane of shear. As mentioned in Section 10.1 for smooth rigid surfaces the plane of shear is situated only a little farther out from the surface than the Stern layer so that cd , which is difﬁcult to establish experimentally, may be approximated by z. However, for

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Chapter 10

irregularly shaped and hairy surfaces the plane of shear is usually farther away from the surface and z is correspondingly lower. It should be realized that z is derived from the measured quantity by invoking a model. Then the applicability of the underlying assumptions may be questioned. It is therefore recommended to compare values of z determined by different electrokinetic methods. The results of electro-osmosis and streaming current or potential may be readily compared because they can be obtained using the same capillary surface. In the case of particles it is most useful to compare electrophoresis of the dispersed particles with streaming potentials (or streaming current) data using a porous plug prepared from the same particles. In Chapter 9, Section 9.1, we deﬁned the point of zero charge as log ccdi (i.e., the pH if Hþ and OH are the charge determining ions) at which the surface charge density s0 is zero. In the absence of speciﬁc ion adsorption and at smooth surfaces s0 ð¼ sd Þ ¼ sek and, hence, the point of zero charge equals the isoelectric point. According to (9.36), at constant surface potential c0 , the surface charge density s0 increases with increasing electrolyte concentration. In view of (9.39) this implies a decrease in cd and, therefore, in z. The dependence of z on the concentrations of the charge determining (cat)ions and the indifferent electrolyte is schematically depicted in the Figures 10.8(a) and (b) for the absence, respectively, presence of speciﬁc adsorption of indifferent electrolyte. If there is no speciﬁc adsorption or, more precisely, if there are no indifferent ions within the plane of shear the isoelectric point is invariant with the concentration of the indifferent electrolyte. In the case of preferential speciﬁc adsorption of

ζ

ζ

– log ccdi+

– log ccdi+

Figure 10.8 Variation of the electrokinetic potential with the concentration of charge determining ions for different concentrations of indifferent electrolyte. (a) Absence and (b) presence of speciﬁc adsorption of counterions. The arrows indicate decreasing concentrations of indifferent electrolyte.

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Electrokinetic Phenomena

179

anions, the concentration of the charge determining cations must increase to reach z ¼ 0, which means that the isoelectric point shifts to lower values of log ccdi .

10.5 APPLICATIONS OF ELECTROKINETIC PHENOMENA Many practical applications using dispersed systems are related to colloidal stability. In some cases one prefers to have the system ﬂocculated as much as possible; in other cases ﬂocculation must be prevented. Whatever the desired situation, the zeta-potential may serve as a useful parameter to indicate colloid stability. All other factors kept constant, the higher z the more stable is the system. For instance, industrial and domestic waste water contains a variety of solid and liquid particles that are colloidally stable, often because they are covered with charged (biological) surface active substances. For most of these particles Hþ and OH are the charge determining ions and, hence, the colloidal stability is sensitive to the pH. By adjusting the pH low z values for most of the materials may be reached so that aggregation readily occurs and the materials can be removed by sedimentation or ﬁltration. In fact, for aggregation to occur, it would be most favorable if some components had a positive value for z and others a negative. The aggregation process is discussed in more detail in Chapter 16. In some fruit juices, such as orange juice or tomato juice, colloid stability contributes to the desired cloudiness. This may be improved by additives that increase z of the dispersed particles. Also in shampoos and hair conditioners addition of macro-ions having a high z may help to increase the hair ‘‘volume’’ by adsorbing to the hairs. Electro-osmosis may be used in environmental remediation, the removal of pollutants from contaminated soil. By applying a potential difference between electrodes placed at both sides of the contaminated soil, injected water is forced to ﬂow through the granular soil taking along the contaminants. Electro-osmosis may also be applied for dewatering of porous materials such as soils and sludges. Furthermore, electrodialysis to desalt aqueous solutions involves electro-osmosis. Under the inﬂuence of an electric ﬁeld an aqueous electrolyte solution is transported across anion- and cation-exchange membranes that are stacked in an alternating array. Ions are accumulated at the membrane that has the same charge sign and in this way the solution is separated in an electrolyte-freed and an electrolyte-enriched fraction. Streaming potentials should be prevented in the transport of (apolar) liquids, for example, oil, through pipelines. In particular in systems with low ksp such a transport may lead to large streaming potentials causing a real danger of explosions. A similar situation exists when grains are transported in elevators.

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180

Chapter 10

Electrophoresis is generally applied in biochemistry and molecular biology laboratories to characterize and=or fractionate mixtures of macro-ions, such as proteins, DNA fragments, and so on. This may be performed in solutions in a capillary, but more often a moist ﬁlter paper or a gel of, for example, polyacryl amide is used as the medium of migration. To improve separation, a pH gradient is applied to the stationary phase, that is, the paper or the gel. Then, each fraction stops migrating at the pH where it is isoelectric. This technique is known as isoelectric focusing. Capillary electrophoresis may further be applied to position molecular species on (bio)analytical microchips. Examples of industrial applications of electrophoresis are electrophoretic coating and electrophoretic imaging. In electrophoretic coating (or electrodeposition) the material to be coated is made into an electrode with a charge opposite to that of the particles to be deposited. This technique may be especially successful in coating surfaces that are otherwise barely accessible, such as image displays, light bulbs, the inner walls of cans for food and drinks, and bodies of cars. Electrophoretic imaging in a liquid environment is a relatively new technique using colloidal particles of pigments dispersed in a liquid to which dyes may be added to improve the contrast. It has found application in ﬂat-panel displays and in photocopying devices. The applications mentioned here are just a few examples taken from a wide variety of processes in which electrokinetic phenomena play a role.

EXERCISES 10.1

Comment on the following statements. (a)

The reduction of the streaming potential observed at increased ionic strength can be ascribed to compression of the electrical double layer. (b) The deviation of the Stern potential cd from the electrokinetic potential z is larger in an aqueous environment than in a nonaqueous environment. (c) Surface conduction affects electrokinetic determination of the isoelectric point. 10.2

Consider the electrophoresis of a negatively charged spherical colloidal particle in a dilute electrolyte solution. The electrokinetic charge Qek of the particle may be calculated using ð1 Qek ¼ 4pr2 rdr; ðE10:1Þ aþd

where r is the space charge density, a the radius of the particle, d the thickness of the hydrodynamic slip plane, and r the distance from the center of the spherical particle.

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(a) Give a sketch for rðrÞ and indicate a and d therein. (b) Explain the factor 4pr2 and the minus sign in Eq. (E10.1). The solution of the integral in (E10.1) is: Qek ¼ 4pee0 að1 þ kaÞz;

ðE10:2Þ

where ee0 is the dielectric permittivity of the medium, k the reciprocal Debye length, and z the electrokinetic potential of the particle. (c)

Calculate the electric force fel imposed on the electrokinetic charge Qek in an electric ﬁeld of strength E. (d) Calculate the friction force ffr on the particle that moves with a velocity vef . (e) Derive vef ¼

2 ee0 E ð1 þ kaÞz 3 Z

ðE10:3Þ

in which Z is the viscosity of the medium. Equation (E10.3) suggests that vef continues to increase with increasing particle size. Experimentally, however, one ﬁnds for large particles that vef is independent of a: vef ¼ ee0 Ez=Z. (f) Which effect is neglected in the derivation of (E10.3)? (g) Why is this neglect allowed for large particles having a compressed electrical double layer? 10.3

A biomedical engineer investigates the transport of apolar ﬂuids through narrow plastic tubings. Such systems may lead to a high streaming potential Es , causing sparks and, therefore, explosion danger. (a)

For what reason(s) are streaming potentials in apolar ﬂuids much higher than in polar ﬂuids? (b) What is the inﬂuence of the viscosity of the apolar ﬂuid on Es , at constant ﬂuid ﬂow? (c) If the plastic tube is replaced by a metal tube of the same dimensions no signiﬁcant streaming potential is built up. Explain.

SUGGESTIONS FOR FURTHER READING A. V. Delgado (ed.). Interfacial electrokinetics and electrophoresis, in Surfactant Science Series 106, New York: Marcel Dekker, 2002. S. S. Dukhin. Non-equilibrium surface phenomena, in Adv. Colloid Interface Sci. 44: 1–134, 1993. R. J. Hunter. Zeta Potential in Colloid Science. Principles and Applications, London: Academic, 1981.

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Liposomes in Gene Therapy

In gene therapy DNA has to be delivered to appropriate cells in order to alleviate symptoms or to prevent the occurrence of a particular disease. To realize these effects (parts of) genes have to be replaced or repaired. The delivery of the high molecular weight DNA across the cell membrane into the cell or cell nucleus is often a severe obstacle. Liposomes, in which lipid molecules are self-assembled in spherical bilayers, have shown to be effective carriers for DNA. The negatively charged DNA fragments bind strongly to cationic lipids. The liposomes may be labeled with ligands that selectively interact with receptors at the membranes of the target cells whereafter the DNA could enter the cell by endocytosis. In the endocytosis process the liposome bilayer merges with the phospholipid bilayer of the cell membrane allowing DNA to be released into the cytoplasm.

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11 Self-Assembled Structures

Self-assembled structures or, as they are otherwise called, association colloids are made up of amphiphilic molecules, molecules that contain a lyophobic and a lyophilic part. In a given solvent and beyond a critical concentration the amphiphilic molecules spontaneously organize themselves to give structures of colloidal dimensions. Self-assembling molecules may form a variety of structures. These structures are widely studied, both for academic and application reasons. In Section 11.9 we address a few applications in the areas of biotechnology and biomedicine. The reason for the self-assembly tendency of amphiphilic molecules is that their lyophobic parts are poorly soluble and tend to separate from the solvent, whereas the lyophilic parts prefer to be solvated. For water as the solvent, hydrophobic interaction is the major cause of aggregation of apolar molecules and molecular fragments. A more detailed discussion of the hydrophobic effect is given in Chapter 4. Amphiphilic molecules supplied to an (aqueous) solution adsorb at available interfaces. For that reason amphiphilic molecules are also called surface active agents or, for short, surfactants. The decrease in interfacial tension g resulting from adsorption is, for a reversible process, given by the Gibbs’ equation, (3.86) or (7.2). Adsorbed layers of surfactants at ﬂuid interfaces are extensively discussed in Chapter 7 and Chapter 14 deals with generic theoretical aspects of the adsorption process. The variation in g, as a function of the surfactant concentration in solution, is schematically depicted in Figure 11.1. The increasing slope of the descending part of the gðln cÞ curve reﬂects an increasing adsorbed amount G. Just before reaching the discontinuity the slope becomes (nearly) constant, which, in view of Eq. (3.86), implies a constant value for G. This corresponds to saturation of the interface with adsorbed surfactant molecules.

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Chapter 11

γ

∂γ

~

∂ lnc

Γ

Γsaturation

ln c Figure 11.1 Interfacial tension as a function of the concentration of a surface active compound in solution.

Beyond adsorption saturation the lyophobic parts of the amphiphilic molecules in solution may associate to avoid unfavorable solute–solvent contacts. Then, upon further addition of the amphiphilic compound to the solution the concentration of free monomeric molecules hardly increases anymore, so that, according to (3.86), g remains essentially constant. Thus the onset of the association process is marked by a discontinuity in the interfacial tension. In addition, a discontinuity is observed in other physical properties, for example, osmotic pressure, light scattering, electric conduction (in the case of ionic amphiphiles), and so on.

11.1 SELF-ASSEMBLY AS PHASE SEPARATION Aggregation of amphiphilic molecules may be regarded as a (micro-)phase separation between a polar (‘‘water’’) and an apolar (‘‘oil’’) phase. In the two-phase system each of the phases contains a fraction of the other component; their compositions are determined by the solubility of oil in water and vice versa. In equilibrium the chemical potential m of the oil in the oil phase equals that of the oil in the aqueous phase, moilðoily phaseÞ ¼ moilðaqueous phaseÞ :

ð11:1Þ

moilðoily phaseÞ can be approximated by moil ; the chemical potential of pure oil, and the solubility of oil in water is so low that the solution can be assumed to behave ideally. Hence (11.1) becomes moil ¼ mooil þ
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ð11:2Þ

Self-Assembled Structures

185

where X is the mole fraction of oil in the aqueous phase and mooil the chemical potential of pure oil ðX ¼ 1Þ as derived by extrapolation from the ideal solution regime. Note that moil 6¼ mooil ðX ¼ 1Þ. Then ln X ¼

moil mooil :
ð11:3Þ

Because moil ð p; T Þ and mooil ( p, T, interaction between oil and water), for a given system at constant p, the functionality X ðT Þ may be given as @ ln X 1 @ðmoil =T Þ @ðmooil =T Þ ¼ @T @T < @T 1 hoil hooil ð11:4Þ ¼þ 2þ 2 T T < ho ¼ sol 2oil ;
stable Tc

Tc stable

T

stable

T

unstable

0

φ

1

Figure 11.2 Phase diagram for an oil=water mixture. The stable areas indicate the conditions at which one phase exists, whereas in the unstable area two phases occur. f refers to the volume fraction of the oil.

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186

Chapter 11

We now discuss the energetic and entropic contributions that are involved in phase separation, that is, the reverse of mixing. Consider a mixture of components A and B in which the interactions A–A, A–B, and B–B are energetically equal. Such a mixture is called ‘‘ideal’’. Ideal mixtures are entropically favorable; mixing increases the conﬁgurational entropies of each component. Per mole mixture the entropy of mixing is m S ¼
ð11:5Þ

and, hence, the Gibbs energy of mixing m Gconfig ¼
ð11:6Þ

where XA and XB are the mole fractions of A and B. As X < 1; ln X < 0 and, hence, m S > 0 and m G < 0. It should be noted that (11.5) and (11.6) are generally valid for ideal mixtures. However, ideal mixing is an exception. As a rule the different types of molecules in a mixture have preferential interactions. In that case, the energy of a mixture is also a function of its composition. To evaluate the Gibbs energy of mixing (without accounting for the change in conﬁgurational entropy), the number of contacts between molecules of different kinds has to be compared with those between molecules of the same kind. For this purpose, the interaction parameter w is deﬁned as the difference in molar Gibbs energy g between ‘‘nonidentical’’ contacts and ‘‘identical’’ contacts, expressed in
fgAB ð1=2ÞðgAA þ gBB Þg :
ð11:7Þ

Assuming a random distribution of the molecules in a mixture (containing one mole) m GintðeractionsÞ ¼
ð11:8Þ

In Figure 11.3 both m Gconfig and m Gint plotted as a function of the composition in a two-component system ðXB X Þ. As already mentioned, m Gconfig is always negative, having an extremum at X ¼ 0:5. The sign of m Gint depends on the sign of w. If AB contacts are preferred over AA and BB, w < 0 and m Gint < 0. This situation is shown in Figure 11.3(a). If, however, AB contacts are relatively unfavorable, w > 0 and, hence m Gint > 0. Then, m Gint and m Gconfig oppose each other. The sum m G ¼
ð11:9Þ

may contain two minima; that is, part of the curve is concave with respect to the X-axis, as is illustrated in Figure 11.3(b). This occurs for w 2. Thus the system separates into two phases if w 2 and at mole fractions between the two minima. The compositions of each of the phases correspond to those of the minima. The

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187

(a) 0

(b)

X (= XB)

1 ∆ m Gint

∆ m Gconfig

∆m G 0

∆ m Gint

X (= XB)

1

∆ m Gconfig ∆m G

Figure 11.3 Schematic representation of the contributions from interaction, m Gint , and conﬁguration m Gconfig , to the Gibbs energy of mixing m G of two components. (a) Interactions between nonidentical molecules are preferred over those between identical molecules; (b) interactions between identical molecules are more favorable.

more positive w is, the wider the separation between the two minima and, hence, the more the compositions of the two phases differ.

11.2 DIFFERENT TYPES OF SELF-ASSEMBLED STRUCTURES In an aqueous medium the apolar parts of the amphiphilic molecules group together, whereas in an apolar environment the polar parts do. The resulting selfassembled structures may be of different types. The most common structures are spheres and rods (usually referred to as ‘‘micelles’’), and bilayer structures in a planar form (‘‘lamellae’’) or in a closed spherical shape (‘‘vesicles’’). Some of these structures are depicted in Figure 11.4. The structures shown in this ﬁgure are cartoon-like representations; in reality they are less ordered, that is, more ﬂuid. At low concentration of amphiphiles micelles are usually formed. Micelles typically contain some 20 to 100 monomeric molecules. In most cases they are spherical, but sometimes they adopt a rod-like shape. In the latter case they contain more monomers. The radius of the sphere or the cylindrical, more or less worm-like, rod is comparable to the length of the apolar parts of the amphiphilic molecule, which is usually in the range of a few nanometers. Spherical micelles are extensively studied and well understood.

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Chapter 11

micelle

inverted micelles

bilayer

bilayer vesicle

Figure 11.4

Some associative structures of amphiphilic molecules.

Figure 11.5 presents a phase diagram of a micelle-forming amphiphilic compound in water. At low temperature the amphiphiles occur as hydrated crystals. Above a certain temperature, the so-called Krafft-temperature, the solubility increases sharply and, at a concentration exceeding the critical micelle concentration (cmc), the amphiphilic molecules form micelles. The temperature at which the solubility equals the critical micelle concentration is called the

T monomers

cmc micelles Krafft-temperature

Krafft-point

crystals c

Figure 11.5 Typical phase diagram for a solution of amphiphilic molecules.

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Self-Assembled Structures

arrays of spherical micelles (volume fraction 0.5 - 0.7)

189

arrays of cylindrical micelles (volume fraction 0.7 - 0.9)

lamellar structures: hydration water and counterions incorporated between the head groups (volume fraction > 0.8)

Figure 11.6 Associative structures formed in systems containing high concentrations of amphiphilic molecules.

Krafft-point. The Krafft-point is characteristic for each amphiphilic compound. Anticipating the discussion in Section 11.5 we mention here that for ionic amphiphiles in an aqueous medium the solubility and the cmc are inﬂuenced by the ionic strength of the medium and, for weak acid or weak alkaline amphiphiles, by the pH as well. At higher concentrations of the amphiphilic substance more complex aggregates are formed. Examples are given in Figure 11.6. The insights obtained from studying micellar systems also appear to be useful for understanding more complex self-assembled structures.

11.3 AGGREGATION AS A ‘‘START–STOP’’ PROCESS. SIZE AND SHAPE OF SELF-ASSEMBLED STRUCTURES Aggregation of, say, the apolar parts of the amphiphilic molecules results in growth of the self-assembled structure. For a spherical micelle the growth occurs in three dimensions. The surface=volume ratio decreases with increasing size of the micelle. As a consequence, for each polar part of the amphiphilic molecule the area at the micellar surface decreases. The polar parts crowd together and this

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190

Chapter 11

hampers their hydration. Effectively, this manifests itself as a repulsion between the polar groups. Moreover, when the polar parts are electrically charged they also repel each other electrostatically when brought together at the micellar surface. The repulsion between the polar parts prevents the self-assembled structures from achieving unlimited growth. The resulting size is a compromise between packing of apolar and polar parts. The molar Gibbs energy of association ass g varies with the number i of monomeric molecules in the micelle, as qualitatively indicated in Figure 11.7. The aggregation process may be compared with a nucleation or a precipitation process, discussed in Chapter 6, Section 6.4. In analogy to (6.26) ass g comprises two terms, one accounting for the aggregation-promoting ‘‘volume contribution’’ which depends on the concentration of monomeric amphiphile molecules in solution and a term accounting for creating the micelle–solution interface. For small i (i.e., small size of the aggregates), the unfavorable ‘‘surface contribution’’ dominates over the favorable ‘‘volume contribution.’’ However, for sufﬁciently large micelles the volume term takes over. Thus the curve for ass gðiÞ would be similar to the one for nucl gðRÞ shown in Figure 6.9, if there were no repulsion between polar groups at the micellar surface. Repulsion, becoming effective at an increased value for i, causes ass gðiÞ to pass through a minimum. The most favorable size of the micelle is determined by this minimum. The ‘‘volume contribution’’ increases with increasing concentration of monomeric amphiphiles and the minimum in ass gðiÞ deepens accordingly. The concentration for which this minimum just equals zero is the critical micelle concentration, cmc. It marks the onset of micelle formation. Thus, when amphiphilic molecules are supplied to a solvent in amounts that exceed the cmc, micelles are spontaneously formed. It may be clear from the proﬁle of the ass gðiÞ curve that beyond the cmc the amphiphilic molecules in the system occur in two states: as monomeric molecules and in micelles of optimum size.

∆ ass g

c < cmc

c > cmc i=1

i

Figure 11.7 Gibbs energy of association as a function of the number (i) of monomers in the aggregate.

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Self-Assembled Structures

191

∆ ass g a m ,o ~ hydrophobic effect a m 1 head group repulsion ~ a

m

am Figure 11.8 molecules.

Attraction and repulsion in self-assembled structures of amphiphilic

effective shape of the surfactant molecule

packing parameter am,0 l / v

am,0

v

aggregate morphology spherical micelles

cone

>3

l

wormlike micelles truncated cone

2-3

bilayers, vesicles cylinder 1-2

inverted micelles inverted (truncated) cone

Figure 11.9

<1

Scheme for various geometries of amphiphilic structures.

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192

Chapter 11

Intermediate states, that is, smaller or larger micelles, are thermodynamically unstable. We have reasoned that self-assembled structures do not grow in three dimensions without compromising hydration of the polar parts of the molecules. However, unlimited growth is, in principle, possible for two dimensions (‘‘lamellae’’) and one dimension (‘‘rods’’). The sizes of such structures are in practice restricted by the fact that contact between their apolar edges and water is unfavorable. For most amphiphilic molecules the polar parts are relatively small ‘‘head groups.’’ As a ﬁrst approximation, these head groups may repel each other as hard spheres; that is, the molar Gibbs energy of the head group repulsion scales with the inverse of the average area am available per head group. If am increases, the contact area between the apolar part of the surfactant and water increases proportionally. In Chapter 4, Section 4.3.2, we have seen that effects of (de)hydration of apolar compounds vary linearly with their area exposed to water. Hence, the Gibbs energy of hydration of the apolar parts scales linearly with am . Figure 11.8 shows that the total molar Gibbs energy, as a function of am , passes through a minimum. This crude model predicts the optimum value for am ð am;0 Þ, but it does not provide information on the shape the self-assembled structure adopts. Prediction of the shape may be based on considering the most favorable packing of the apolar parts at given am;0 . As a rule, the apolar parts are alkyl chains of a certain length l and volume v. The shape of the aggregates then depends on the cross-sectional area v=l of the apolar parts relative to the optimum head group area am;0 . This is illustrated in Figure 11.9. If am;0 > n=l, which is generally the case for single (alkyl)-chain surfactants, micelles will be formed. If am;0 n=l as it is in various phospholipids, lamellar structures (e.g., a bilayer) are most likely, and if am;0 < n=l, as in surfactants containing more than one (branched) apolar chain per polar head group, inverted micelles are the most favorable shapes.

11.4 MASS ACTION MODEL FOR MICELLIZATION Consider a solution of the amphiphilic substance A in which the molecules of A are present as monomers, dimers, trimers, tetramers, and so on. The mole fractions of the amphiphile in the monomeric state and the respective associated states are denoted as XA1 ; XAmon ; XAmon ; . . . ; ðXAi ¼ XAmon =iÞ: 2 3 i Micelles are formed if XA1 > cmc (where the cmc is expressed in mole fraction of A1 , which, for dilute systems, is proportional to the concentration of A1 ). Then mA1 ðaqÞ ¼ mA1 ðimerÞ ;

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ð11:10Þ

Self-Assembled Structures

193

that is, the chemical potential of the free molecule A equals that of A in the associative structures. Equation (11.10) can be written as moA1 ðaqÞ þ

ð11:11Þ

(Note that the conﬁguration entropy of the i molecules of A1 in the i-mer equals =i, that of the i-mer.) Because XAi XAmon i moA1 ðaqÞ þ
XAmon
ð11:12Þ

The standard Gibbs energy for the transfer of a monomer from solution into an i-mer is mon

tr GAo 1 ¼ moA1 ði-merÞ moA1 ðaqÞ ¼

ð11:13Þ

tr GAo 1 is the ith part of the Gibbs energy of the formation of a micelle consisting of i monomers micel Go ¼ itr GAo 1

ð11:14Þ

as iA1 ! Ai ; Kmicel

XAi ðXA1 Þi

¼

XAmon =i i ðXA1 Þi

:

ð11:15Þ

In micelles i is on the order of tens to hundreds. That is the reason why the transition between the monomeric and the micellar solution is so sharp. Upon adding A to the system XAi varies proportionally with ðXA1 Þi . For instance, an increase of XAi by a factor of 2 implies an increase in XA1 by a factor of 21=i . For, say, i ¼ 50 XA1 increases by about 1%. The abruptness of micellization may be illustrated in more detail in the following way. Equation (11.15) can be transformed into XAi =Kmicel ðcmcÞi ¼ ðXA1 =cmcÞi , which is an equation of the type y ¼ xi . Because of the large value for i, y 0 for x < 1, y ¼ 1 for x ¼ 1, and y increases sharply for x > 1. Thus, below the cmc, where ðXA1 =cmcÞ < 1; XAi 0 and above the cmc, where ðXA1 =cmcÞ > 1; XAi increases sharply implying that XA1 is essentially constant and equal to the cmc. The distribution of the amphiphiles over the monomeric and the micellar states, as a function of i, is shown in Figure 11.10. When dealing with ionic amphiphiles the micelles acquire a high charge density at their surface which gives rise to counterion (M) condensation (cf. Section 9.4). A micelle can now be considered as an aggregate of i moles of A1 and j moles of counterions Mð j < iÞ. Then, iA1 þ jM ! Ai Mj

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194

Chapter 11

(a) XA

(b) mon i

i

XA

i XA – XA i

1

i=1

2

5 10

XA

50

cmc

1

cmc

XA

1

mon i

XA

Figure 11.10 Micellization. (a) Mole fraction of micelles XAi as a function of the mole fraction of monomers in solution XA1 for different values i of monomers in the micelles. For each value of i the curves are rescaled allowing a common intersection point. (b) Distribution of the surfactant between micelles and monomers in solution.

with Kmicel ¼

XAi Mj ðXA1 Þi M j

:

ð11:16Þ

The reasoning for ionic amphiphiles is, mutatis mutandis, the same as for nonionics leading to results similar to those shown in Figure 11.10. Because i amounts to at least a few tens, the second term on the right-hand side of (11.13) is almost always negligibly small. Hence, as a good approximation tr GAo 1 ¼
ð11:17Þ

Assuming that the contributions from the apolar and polar parts of the molecule are additive, o o tr GAo 1 ¼ tr Gpolar part þ tr Gapolar part ;

ð11:18Þ

(11.17) may be rewritten as lnðcmcÞ ¼

o tr Gpolar part

þ

o tr Gapolar part

:

ð11:19Þ

For a homologous series of surfactants containing the same polar head group and a varying number n of CH2 -groups in a linear alkyl chain lnðcmcÞ ¼ a þ n b;

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ð11:20Þ

Self-Assembled Structures

195

Table 11.1 Critical micelle concentrations of various amphiphilic compounds in aqueous solutions at 25 C Compound Na-octylsulphate decyl dodecyl tetradecyl hexadecyl K-dodecylsulphate Li-dodecylsulphate Na-hexadecyl-4-sulphate -6-8Na-octylcarboxylate decyl dodecyl Decyltrimethylammoniumbromide Dodecyl Tetradecyl Hexadecyl Octadecyl Dodecyltrimethylammoniumchloride Octy(ethylene oxide)6 ethanol Decyl Dodecyl Tetradecyl

C8 OSO3 Naþ C10 C12 C14 C16 C12 OSO3 Kþ C12 OSO3 Liþ

C8 COO Naþ C10 C12 C10 Nþ ðCH3 Þ3 Br C12 C14 C16 C18 C12 Nþ ðCH3 Þ3 Cl C8(EO)6C2H4OH C10 C12 C14

cmc(M)

ln(cmc)

0.133 0.033 0.0083 0.0021 0.0005 0.0078 0.0105 0.0017 0.0024 0.0042 0.220 0.120 0.028 0.066 0.015 0.0035 0.0009 0.0003 0.020 9.9 103 0.9 103 0.087 103 0.010 103

2.02 3.41 4.79 6.17 7.60 4.85 4.56 6.38 6.03 5.47 1.51 2.12 3.58 2.72 4.20 5.65 7.01 8.11 3.91 4.61 7.01 9.35 11.51

where a¼

o tr Gpolar part

and b ¼

o tr GCH 2

:

The micellization theory given here may be tested by applying Eq. (11.20) to the experimental data collected in Table 11.1.

11.5 FACTORS THAT INFLUENCE THE CRITICAL MICELLE CONCENTRATION The tendency of amphiphilic molecules to aggregate is reﬂected by tr GAo 1, and, in view of Eq. (11.17), in the case of micelle formation by the cmc. We have seen that micellization includes attraction between lyophobic parts of the amphiphiles and repulsion between the lyophilic parts. The relative magnitudes of these

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196

Chapter 11

opposing interactions determine the critical micelle concentration. Along these lines the cmc values for a series of compounds, presented in Table 11.1, are interpreted. In each of the homologous series (alkyl sulphates, alkyl carboxylates, alkylammonium bromides, and the alkyl ðoxyethyleneÞ6 ethanols), lnðcmcÞ varies linearly with the number of CH2 -groups in the alkyl chain, as predicted by o =
ln cmc

0 x

x x

–4

x –0.69

–1.17

–8

T

T

– 12 8

10

12

14

n

16

18

Figure 11.11 Variation of the critical micelle concentration with the number of C-atoms in the hydrocarbon chain of ionic and nonionic amphiphiles: d anionics ðsulphatesÞ; anionics (carbonates); m cationics; j nonionics. (Adapted from Table 11.1.)

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Self-Assembled Structures

197

Table 11.2 Inﬂuence of temperature on the micellization of Na-dodecylsulphate Temperature (K) 283 298 308 318 328 338

cmc (M)

ln (cmc)

0.0086 0.0083 0.0085 0.0090 0.0098 0.0110

4.756 4.791 4.768 4.711 4.625 4.510

head groups of the ionic surfactant molecules is less screened which leads to a larger value for the cmc. Divalent or multivalent ions may bridge between ionic polar parts of the amphiphiles thereby substantially lowering the cmc. Furthermore, the cmc of ionic amphiphiles is depressed at higher ionic strength due to increased screening of electrostatic repulsion. By way of example, Table 11.2 shows the inﬂuence of temperature on the micellization of Na-dodecylsulphate. The data may be analyzed based on Eq. (11.22) which is derived by differentiation of (11.17) with respect to T: d lnðcmcÞ 1 @ðtr GA1 =T Þ ¼ dT < @T o

ð11:21Þ

or d lnðcmcÞ ¼

tr HAo 1 <

dT 1 :

ð11:22Þ

The slope of the curve for lnðcmcÞ versus T 1, as shown in Figure 11.12, equals tr HAo 1 =<. The ﬁgure shows that ð@tr HAo 1 =@T Þp ¼ tr Cp 0, which is characteristic for a process dominated by hydrophobic dehydration (cf. Chapter 4, Section 4.3.2).

11.6 BILAYER STRUCTURES A class of self-assembled structures that deserves special attention is the bilayer. This is a lamellar structure composed of two molecular layers of amphiphilic molecules. Amphiphiles having am;0 l=n close to unity usually assemble into bilayers in which (in aqueous media) the apolar parts of the molecules are directed towards each other. Free-ﬂoating bilayers do not exist; it is too unfavorable to expose the hydrophobic edges to water. The bilayer closes into a

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198

Chapter 11

Figure 11.12 Temperature dependence of the micellization of Na-dodecyl sulphate. (From Table 11.2.)

spherical geometry, the so-called vesicle, or its edges are embedded in a nonaqueous environment. See Figure 11.13. Vesicles of phospholipids are usually called liposomes. Figure 11.14 gives an impression of the molecular distribution in a phospholipid bilayer as obtained by simulation based on estimates of the interactions between the components involved. Of course, the outcome of the simulation is only as realistic as the values of the input parameters are. Nevertheless, the picture demonstrates that the polar head groups of the phospholipid molecules are diffusely distributed in the boundary region between the bilayer and the water, that the water molecules penetrate over quite some distance into the bilayer, and that the apolar inner

Figure 11.13

Amphiphilic molecules in a planar and a spherical bilayer.

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Self-Assembled Structures

199

Figure 11.14 Snapshot of a phospholipid bilayer obtained by molecular dynamics simulation. (From E. Engberts and H. J. C. Berendsen. J. Chem. Phys. 89: 3718, 1988.)

region shows a high degree of disorder. Bilayers of phospholipids form the matrix of biological membranes in animals. In plants and micro-organisms glycolipid bilayers are found. Biological membranes are further discussed in Chapter 19. In technical applications bilayers of dialkyl ammonium molecules are often used. Examples of these bilayer-forming amphiphiles are given in Figure 11.15. The ﬂuidity of amphiphilic bilayers depends on the nature of the apolar chains forming the hydrophobic core of the bilayer and, very strongly, on the temperature. It appears that at lower temperatures the bilayer is in an ordered crystalline gel state where the conﬁguration of the apolar chains is essentially alltrans. The movement in the plane of the bilayer as well as in the normal direction is strongly limited. Upon heating the crystalline gel state is transformed into a liquid crystalline state in which the amphiphiles attain a high lateral mobility and ﬂuidity but their mobility perpendicular to the bilayer surface remains highly restricted. As a result, the diffusitivity in the liquid-like membrane is only slightly smaller than in solution, whereas the rigidity normal to the bilayer causes a barrier to passage. The transition from the gel to the liquid state is a cooperative process, which occurs over a narrow temperature range. It is schematically depicted in Figure 11.16. Note that the transition involves a reduction of the bilayer thickness. The solubility of the monomers of bilayer-forming molecules is usually very low, say, in the range of 105 to 1010 M. Crystals of such amphiphiles immersed in water tend to swell. In this way lamellar liquid crystals

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200

Chapter 11 O

phospholipids

H2 CH3 (CH2)n C O C CH3 (CH2)m C O CH O C O P O X O H2 O +

CH2 CH2 N (CH3)3 X:

+

CH2 CH2 NH3 + CH2 CH NH3

phosphatidyl choline (lecithin) phosphatidyl ethanol amine phosphatidyl serine

COO

O H2 CH3 (CH2)n C O C

glycolipids

CH3 (CH2)m C O CH C O X O H2 X : sugar unit (glucose or galactose)

dialkyl ammoniumsalts CH3 (CH2)n O N

CH3 (CH2)m O

+

CH3 Br

CH3

Figure 11.15 Structure formulae of some bilayer-forming amphiphiles.

Figure 11.16

Schematics of gel–liquid transition in an amphiphilic bilayer.

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Self-Assembled Structures

201

(multilamellar vesicles) made up of bilayers packed in large stacks, separated by water molecules, are usually formed. They reach dimensions of a few thousands of nanometers. These lamellar structures may appear in different forms that readily interchange in response to small variations in temperature or composition. Unilamellar vesicles having a radius of a few tens up to a few hundreds of nanometers are derived from the lamellar liquid crystals by mechanical rupturing such as occurs in ultrasonic treatment, for example. The unilamellar vesicles are thermodynamically unstable and, hence, the properties of a unilamellar vesicle dispersion depends on how it was prepared. The colloidal stability of such a vesicle system is determined by the rate of fusion between two vesicles. This rate, in turn, is governed by the rules of colloidal stability discussed in Chapter 16. Anyway, the colloidal stability of unilamellar vesicles allows their use for in vitro studies of physical and chemical bilayer and membrane properties. Bilayer structures, including biological membranes, may be degraded by micelle-forming amphiphiles. The molecules of the bilayer are then solubilized in the micelles, thus forming mixed micelles.

11.7 REVERSE MICELLES In a nonaqueous medium amphiphilic molecules may associate with their polar parts clustering together and their apolar parts exposed to the continuous phase, as illustrated in the Figures 11.4 and 11.9. Such structures are known as reverse micelles or inverse micelles. Formation of reverse micelles requires that the constituting monomers are characterized by am;0 < n=l but even then their occurrence is far less probable than that of micelles in an aqueous medium. There are two main reasons underlying the difference between amphiphilic aggregation in aqueous and nonaqueous media. The number of amphiphilic compounds that are soluble in an apolar solvent is far less than the number of water-soluble amphiphiles. Well-known examples of reverse micelles forming compounds are Aerosol OT and those of the Triton X-100 series; their structures are given in Figure 11.17. Second, and most important, in contradiction to micellization in an aqueous environment, the solvent does not play an active role in the formation of reverse micelles. More precisely, aqueous micellization is primarily due to hydrophobic interaction which is driven by entropy increase of the water molecules (Chapter 4, Section 4.3.2). In a nonaqueous medium the hydrophobic effect is absent and the amphiphilic molecules themselves are responsible for the association. Hence, reverse micelles are stabilized by favorable interactions between the polar parts of the amphiphiles. Thus, where aqueous micellization is characterized by a large positive entropy change and a minor enthalpy effect (cf. Table 4.6), nonaqueous micellization involves a large negative enthalpy change and an unfavorable entropy effect.

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202

Chapter 11 CH3 CH2 CH3 (CH2)3 CH

CH2

CH3 (CH2)3 CH

CH2

Aerosol OT CH2

O C H C O C O O

OSO3– Na+

CH3

Triton X Figure 11.17 micelles.

CH3 (CH2)7

(O

CH2 CH2)n OH

Structure formulae of surfactants used in the preparation of reversed

Reverse micelles are as a rule much smaller than micelles in water, typically containing a few up to about ten monomers per aggregate. It follows from the theory discussed in Section 11.4 that for a small number of monomers per micelle the onset of micellization is not well deﬁned or, in other words, that the critical micelle concentration is less sharp or less critical. Note that in a nonaqueous environment of low dielectric permittivity the association of ionic surfactants is largely suppressed. Completely anhydrous reverse micelles are difﬁcult to obtain. Uptake of even a relatively small number of water molecules may strongly facilitate reverse micelle formation by favorable hydration of the polar groups. Further solubilization of water in the core of the reverse micelles causes an increase of the size of the aggregate. Such systems approach the domain of microemulsions, which are discussed in Section 11.8.

11.8 MICROEMULSIONS Apolar (‘‘oil’’) and polar (‘‘water’’) liquids do not mix, but the one liquid can be dispersed in the other by the aid of micelles. Oil may be solubilized in the interior of ordinary aqueous micelles and water in the interior of reverse micelles. The micelles swell as the solubilization takes place. Eventually, they become oil particles in water, or vice versa, stabilized by a monolayer of amphiphiles at

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Self-Assembled Structures

203

the oil=water interface. The system thus obtained is called a microemulsion. A microemulsion may therefore be considered as a swollen micelle. Microemulsions differ from macroemulsions (for short, emulsions) by the following. (a)

The size of the particles in a microemulsion is in the range of tens of nanometers whereas in emulsions this is on the order of micrometers. As a consequence, microemulsions are transparant and emulsions are turbid. (b) Microemulsions form spontaneously; that is, they are thermodynamically stable with respect to separation in their components. Emulsions can only be obtained by virtue of energy input (e.g., vigorous stirring) during the emulsiﬁcation process. Thereafter, emulsions are more or less colloidally stable but the oil and the water ultimately separate. Thus, in view of their spontaneous formation microemulsions belong to the family of self-assembled structures and emulsions do not. Emulsions are discussed in Chapter 18. Microemulsions are stabilized by a monolayer of amphiphiles at the oil= water interface. Whether the oil disperses in the water phase or water in the oil phase depends primarily on the shape of the amphiphilic molecules, notably on their value for n=am;0 l. For am;0 > n=l the amphiphile prefers the curvature to be convex towards the water and for am;0 < n=l towards the oil. Accordingly, oil in water and water in oil emulsions are formed in which the dispersed phase occurs as spherical (or sometimes cylindrical) particles as illustrated in Figures 11.18(a), and (b). Only for n=am;0 l 1 is the interface

(a)

(b)

(c) l

l

oil

R

R

water

oil

water water

oil

Figure 11.18 Microemulsions. Dispersions of (a) oil in water, (b) water in oil, and (c) bicontinuous distribution, stabilized by amphiphiles.

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204

Chapter 11

preferably planar resulting in the bicontinuous distribution shown in Figure 11.18(c). In such bicontinuous systems the conduits may easily break and form during which the water and the oil may change places. An unconstrained monolayer will assume a curvature that is primarily determined by the architecture of the amphiphilic molecules. By convention the curvature is deﬁned positive when it is convex towards water. In addition, the curvature is inﬂuenced by the composition of the water and the oil phases. For instance, oil may penetrate between the apolar parts of the amphiphiles thereby decreasing the curvature. The temperature also affects the curvature: increased rotational motions induced by raising the temperature causes more coiling in the apolar tails of the amphiphiles and, hence, a decreased curvature. For ionic amphiphiles electrostatic repulsion between the polar head groups has a large effect on the curvature. Decreased electrostatic repulsion by suppressing the degree of ionization and=or by adding electrolyte (see Chapter 9) makes the interface less convex towards the water, or, for that matter, more convex towards the oil, implying a reduced curvature. It is discussed in Section 16.1.2 that, in an aqueous environment, the Gibbs energy of interaction between charged groups is dominated by the entropy term. The electrostatic interaction therefore increases upon raising the temperature. In the case of ionic amphiphiles the temperature has a dual inﬂuence on the curvature: increased electrostatic repulsion between the head groups causes an increased curvature and coiling of the apolar tails has an opposite effect. For most ionic amphiphiles the inﬂuence of the electrostatic interaction is larger than that of coiling so that the curvature tends to increase with increasing temperature. For microemulsions containing spherical droplets, as depicted in Figures 11.18(a) and (b), the size of the droplets can be easily calculated from the volume fractions f of the different constituents water (w), oil (o), and amphiphile (s). Taking a thickness l for the amphiphilic tails in the monolayer the total interfacial area A in a microemulsion of volume V can be expressed as A ¼ fs V =l:

ð11:23Þ

The area a of a single sphere with radius ðR þ lÞ is given by a ¼ 4pðR þ lÞ2

ð11:24Þ

and the volume v of the dispersed phase in one droplet by 4 v ¼ pR3 : 3

ð11:25Þ

For N spheres in the volume V, N 4pðR þ lÞ2 ¼ fs V =l

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ð11:26Þ

Self-Assembled Structures

205

and 4 N pR3 ¼ fd V ; 3

ð11:27Þ

where fd is the volume fraction of the dispersed phase, that is, fo for oil in water and fw for water in oil. Combining (11.26) and (11.27) yields 2 2 R R fs þ1 ¼ ; ð11:28Þ l l 3fd which allows, for a given value of l, evaluation of the radius ðR þ lÞ. If the inverse of ðR þ lÞ, thus calculated, deviates from the preferred unconstrained curvature the monolayer will relax by adjusting the drop size through expulsion or uptake of liquid to or from the bulk phase. The equilibrium value for ðR þ lÞ can be experimentally assessed, for example, by light scattering techniques. The formation of a (water in oil) microemulsion is depicted in Figure 11.19. From this ﬁgure the contributions to the Gibbs energy change G may be inferred. (a)

Droplet formation increases the oil=water interfacial area. The Gibbs energy of the system increases with gA. For one droplet this amounts to 4pR2 g. (b) Droplet formation increases the conﬁguration entropy of the system by < ln X, where X is the ‘‘mole fraction’’ of the droplets. The corresponding contribution to G (expressed per droplet) equals kB T ln X , which is negative because X < 1. (c) At the concave side of the interface, that is, inside the droplet, there is an excess pressure pL , the Laplace pressure (see Chapter 6) which equals 2g=R. This pressure difference causes a higher value for the chemical potential mH2 O of the water inside the droplet relative to that

oil

water Figure 11.19

Schematic representation of the formation of a microemulsion droplet.

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of bulk water. This would lead to transport of water out of the droplets into the bulk phase. To maintain a thermodynamically stable microemulsion the effect of the pressure difference on mH2 O must be compensated. This is realized by a difference between the mole fractions of water in the droplets and the bulk that causes an osmotic pressure that counteracts the Laplace pressure. The contribution to G is pL n (n being the volume of a droplet), which equals 8pR2 g=3. Summation of the contributions mentioned under (a) through (c) gives G for the formation of one droplet 4 G ¼ kB T ln X þ pgR2 : 3

ð11:29Þ

In practice X ranges between, say, 0.01 and 0.1. This corresponds to gR2 > 2:3 1021 N m and gR2 < 4:5 1021 N m, respectively, to achieve G < 0, which is the condition for spontaneous microemulsion formation. With R-values in the tens of nanometers range it implies that the interfacial tension must be lowered to almost zero. In view of all these conditions with respect to curvature and interfacial tension it may be understood that obtaining a stable microemulsion requires ﬁne-tuning of the system. This may be established by adding one or more cosurfactants (often an alcohol) and=or electrolyte and by adjusting the temperature.

11.9 SELF-ASSEMBLED STRUCTURES IN APPLICATIONS Micelles are widely used in cleansing products. Nonpolar substances can be solubilized in aqueous micelles and, just as well, reverse micelles solubilize polar compounds. Most important for the cleansing activity of amphiphiles is their afﬁnity to adsorb at oil=water interfaces thereby changing the interfacial Gibbs energy and, hence, the wetting behavior of the material to be cleaned. In Chapter 8, Section 8.7.4, the underlying mechanism for the removal of stains and dirt from the surface is explained in more detail. The components released from the surface are then solubilized in the micelles which prevents readhesion. Micelles are not only applied in cosmetic soaps and detergents for laundry and dishwashing but also in solutions to remove oily substances from polluted water and soil. Furthermore, they are used to extract oil from porous rock (‘‘tertiary oil recovery’’) and to collect minerals from ore (‘‘froth ﬂotation’’). Another application of micelles is in catalysis. Solubilization of a component in the micellar interior implies, locally, a tremendous increase in concentra-

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tion. A reagent in the surrounding solution may be attracted, for example, by electrostatic interaction, to the exterior parts of the micelles. Then the increased concentration of both the component (substrate) and the reagent leads to a strongly accelerated reaction. This effect is at a maximum around the critical micelle concentration. Beyond the critical micelle concentration, at higher concentration of micelles, the solubilized substrate is more diluted and the reaction decreases accordingly. In some aspects, in particular, adjoining substrate and reagent, micellar catalysis resembles enzymatic catalysis. Enzymes, however, act much more speciﬁcally. Bilayer structures provide a protective barrier around biological cells and intercellular entities. For that reason they form the matrix of biological membranes. Preparation of vesicles or liposomes in the presence of desired ‘‘guest’’ molecules leads to a fraction of these molecules entrapped inside the vesicles. The vesicles may be separated (e.g., by size-exclusion chromatography) from their surroundings, yielding a preparation of ‘‘functionalized’’ vesicles. Such systems are applied for various purposes, especially in the biomedical ﬁeld. A well-known example is drug targeting and controlled release systems where drugs are encapsulated in vesicles. The vesicles are targeted towards the desired organ by attaching external labels that selectively interact with receptors in that organ. Then the drug should be released by slow continuous transport through the vesicle bilayer. Another example is photodynamic therapy, a method to localize tumors and to cure them using photoactive substances. The selectivity of the therapy is based on photoactive molecules that are preferably accomodated in tumor cells rather than in healthy cells. Thus the vesicles are tagged with a photoactive label that recognizes a receptor in the tumor cell. Then, the photoactive molecules are excited by (laser) light whereafter they react with lipid-like molecules that are solubilized in the vesicle bilayer and with oxygen to form lipid hydroperoxidases. These components destroy cell membranes leading to expiration of the cells. Finally, the use of vesicles in DNA-transfection is mentioned. Complexes of negatively charged DNA fragments with cationic vesicles appear, in one way or another, capable of passing cell membranes. DNA transfection is used in gene therapy to block or to initiate synthesis of desired (poly)peptides. Microemulsions are widely used in cosmetics, cleansing products, foodstuffs, pharmaceuticals, and other products where ultraﬁne dispersion of one phase in another is desired. Other applications are tertiary oil recovery and polymer synthesis in a so-called emulsion polymerization process. In particular, in pharmaceuticals, agrochemicals, foodstuffs, and in biocatalysis functional components are preferably ﬁnely dispersed in an oily or an aqueous environment. Microemulsions are eminently suitable to serve that purpose, provided that the droplets are sufﬁciently large to solvate the functional molecules. By way of example, the relation between the radius of a water in oil microemulsion droplet and the number of water molecules it contains is given in

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208

Chapter 11 Table 11.3 Number of water molecules NH2 O in a microemulsion droplet function of the droplet radius Ra R (nm)

NH2 O

1 4 8 12 16

1:4 103 8:9 104 7:1 105 2:4 106 5:7 106

Volume of a water molecule ¼ 3 103 nm3.

a

Table 11.3. It shows that the droplet should have a size of at least a few nanometers to be able to dissolve (hydrate) polar compounds in its interior.

EXERCISES 11.1

Comment on the following statements. (a)

The critical micelle concentration of ionic surfactants is higher at increased ionic strength. (b) In an aqueous medium the critical micelle concentration of ionic surfactants is lower than that of nonionic surfactants. (c) When the concentration of a surfactant in solution equals its critical micelle concentration half of the surfactant molecules are in the micelles. (d) Amphiphilic molecules in aqueous solution form micelles and in apolar solvents they form reverse micelles. For ionic surfactants the tendency to form micelles is stronger than to form reverse micelles. 11.2 At 20 C a surfactant A in water forms micelles beyond a critical micelle concentration cmc of 5:5 104 M. Calculate the standard Gibbs energy of micellization per mole of surfactant. What is the main driving force for micellization. Is it of enthalpic or entropic nature? (b) What is a vesicle? Describe its structure. Explain why vesicles are not stable upon dilution.

(a)

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(c)

Why does aggregation of di-palmitoylphosphatidyl choline result in (planar) bilayers whereas aggregation of mono-palmitoylphosphatidyl choline does not? (d) What is a microemulsion? Describe the principal differences between a microemulsion and a macroemulsion. What are the main forces driving the formation of a microemulsion? Explain the requirement of an extremely low interfacial tension between the water and oil phases for the spontaneous formation of a microemulsion.

11.3

The table below gives values for the critical micelle concentration cmc of sodium dodecyl sulfate (NaDS) in an aqueous solution of various temperatures T.

T (K) 283 298 308 318 328 338

cmc (M) 0.0086 0.0083 0.0085 0.0090 0.0098 0.0110

Calculate the molar enthalpy of micellization for NaDS at 328 K. Is the change in the heat capacity of the solution due to micellization positive or negative? Why does micellization occur only beyond a certain critical concentration?

11.4

Various alkyl-derived surfactants form spherical micelles. Derive R ¼ 3 v=a, where R is the radius of the micelle, v the volume of the apolar chain, and a the area per polar headgroup. The volume of the head group may be neglected. (b) Calculate the area per headgroup at the surface of a micelle of Na-dodecyl sulfate, CH3 ðCH2 Þ11 OSO3 Naþ . The radius of the micelles is 2 nm and the volume of a CH2 ðor CH3 Þ group is 0:04 nm3 . (a)

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(c) What is the number of monomers N in the micelle? (d) How do R and N change when the alkyl chain length doubles? Does the critical micelle concentration become higher or lower? 11.5

A water-insoluble dye is solubilized in an aqueous solution of an ionic surfactant and in an aqueous solution of a nonionic surfactant. The apolar parts of both surfactants consist of an alkyl chain of the same length. The extinction E (in a 1 cm cuvette) of both solutions at the absorption maximum of the dye is graphically presented as a function of the total surfactant concentration. The molar extinction coefﬁcient em of the dye in a linear alkane (having the same length as the apolar chain of the surfactants) is determined in a separate experiment. The result is em ¼ 2 104 M1 cm1 .

Explain the curves for Eðcsurfactant Þ. What are the physical meanings of the intercept and the slope? Which curve represents solubilization in the ionic surfactant and why? (b) Calculate the aggregation number of both surfactants, assuming that each micelle contains one molecule of the dye. (c) How would the curves shift upon addition of 0.1 M KCl to each of the surfactant solutions?

(a)

SUGGESTIONS FOR FURTHER READING J. H. Clint. Surfactant Aggregation, Glasgow: Blackie & Son, 1992. R. G. Laughlin. The Aqueous Phase Behavior of Surfactants, London: Academic, 1994.

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K. L. Mittal (ed.). Micellization, Solubilization and Micro-Emulsions, New York: Plenum, 1977. Y. Moroi. Micelles, Theoretical and Applied Aspects, New York: Plenum, 1992. M. Rosoff (ed.). Vesicles, in Surfactant Science Series 62, New York: Marcel Dekker, 1996.

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Superabsorbers

What do soft contact lenses, disposable diapers, and sanitary napkins have in common? They all absorb a lot of body ﬂuid. Superabsorbers are made of a crosslinked network of polyacrylate. The network may be considered a highly concentrated salt where sodium ions neutralize the carboxyl groups along the polymer chains. Polyacrylate is well soluble in water and due to osmosis the network sucks up the aqueous body ﬂuid. The sodium ions become hydrated and dissociate from the carboxyl groups. The negatively charged carboxyl ions repel each other which causes the polymer chains to stretch and the network to swell. This creates more space for the ﬂuid to enter. A hydrogel is thus formed in which the amount of absorbed water may exceed 50 times the dry weight. Such superabsorbers could also be applied to stimulate vegetation in (semi)arid regions. They retard evaporation of rain and irrigation water so that a larger fraction of that water is available for the plants. (Figure courtesy of Marijn Besseling.)

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12 Polymers

Polymers belong to the large family of macromolecules. Because of their molecular dimensions macromolecules are part of the colloidal domain. Macromolecules are called polymers when they are constructed of a limited number of different units, the monomers. In polymers the monomers are covalently linked to each other. If each monomer is connected to two others a linear chain is obtained. In some polymers a fraction of the monomeric units share a bond with three (or more) other monomers. In those cases branched chains are developed which usually lead to giant polymer molecules. This may even result in a molecular network in which solvent is entrapped. Such a network is referred to as a polymer gel. Figure 12.1 shows these differently structured polymer molecules. In this chapter we focus mainly on linear polymers. In Section 12.7 attention is paid to polymer gels. Another classiﬁcation of polymers is based on the number of different types of monomers and their distribution along the polymer chain. Thus, polymers are classiﬁed as homopolymers, alternating copolymers, random copolymers, block copolymers, and heteropolymers. They are schematically represented in Figure 12.2. Homopolymers are made up of one type of monomeric unit. Well-known examples of synthetic homopolymers are poly(styrene), poly(vinyl alcohol), poly(vinyl chloride), poly(ethylene), poly(ethylene oxide), and so on. Various natural polysaccharides, such as amylose, cellulose, dextran, chitin, and others, belong to this class as well. Copolymers contain two (or a few more) types of monomers that may be differently distributed along the chain, that is, in an alternating, blocky, or random pattern. Among the synthetic copolymers poly (styrene–butadiene) and poly(ethylene oxide–propylene oxide) may be best known. In nature one ﬁnds agarose, carrageenan, galactomannans, pectin, and

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Chapter 12

Figure 12.1 Differently structured polymers.

nucleic acids. Heteropolymers are built from a variety of monomers. Proteins are well-known representatives of this class. When all the polymer molecules have the same degree of polymerization or, in other words, when all polymer molecules contain equal numbers of monomers, the system is called homodisperse or monodisperse. When the degree of polymerization varies among the polymer population it is called heterodisperse or polydisperse. Synthetic polymers and some biopolymers (e.g., most polysaccharides) are more often than not heterodisperse, whereas other biopolymers such as proteins and nucleic acids are usually homodisperse. Polymers carrying charged groups are referred to as polyelectrolytes. Poly(methacrylic acid) and poly(styrene sulphonate) are examples of industrial polyelectrolytes. Examples from nature are polynucleotides, proteins, and various polysaccharides such as pectin, chitin, hyaluronic acid, and so on. Polymers may possess a wide variety of properties and characteristics. Some polymers are viscous liquids; others are rubbery or glassy. They also vary from very hydrophilic to extremely hydrophobic. These features are not only determined by the

Figure 12.2 Different types of polymers based on their monomer composition and distribution.

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Polymers

215

physical–chemical properties of the monomeric units but also by the distribution of the different units along the chain, the length and the degree of branching of the polymer molecules, and also the type of bond between the monomers. The inﬂuence of monomer–monomer bonding is clearly demonstrated by comparing the polysaccharides amylose and cellulose: both are homopolymers of glucose but the b1;4 bonds in cellulose cause it to be insoluble in water, whereas amylose having a1;4 bonding is readily soluble in water.

12.1 POLYMERS IN SOLUTION Polymer solutions are lyophilic or reversible colloidal systems, which implies that the polymeric material dissolves spontaneously with a decrease in the Gibbs energy of the system: sol G < 0. Dissolution of the polymer involves certain changes: 1. Interactions between segments (monomers) of the polymer and between solvent molecules are disrupted and, on the other hand, interactions between polymer segments and solvent molecules are formed. Suppose that every segment (2) and every solvent molecule (1) is surrounded by z (nearest) neighbors with which it interacts with a molar Gibbs energy g. Then disruption and creation of contacts in the dissolution process involve a Gibbs energy change per mole of segments of zfg12 12 ðg11 þ g22 Þg. Analogous to Eq. (11.7) we now deﬁne an interaction parameter w, the Flory–Huggins parameter, as w

z fg 1 ðg þ g22 Þg:
ð12:1Þ

Thus 0 it is the other way around. For obvious reasons w is a measure of the solvent quality. It is noted that w not only includes the various interaction energies but also entropy effects due to possible changes in orientations of the solvent molecules in the solvation layer around the polymer segments (cf. the discussion in Section 4.3 on the hydration of apolar segments). Dissolution of n2 moles of segments in n1 moles of solvent to yield nð¼ n1 þ n2 Þ moles of solution involves a change in Gibbs energy due to rearrangements of interactions of sol Gint ¼
ð12:2Þ

in which fð f2 Þ is the volume fraction of polymer segments, deﬁned as f n2 n2 =ðn1 n1 þ n2 n2 Þ with ni being the molar volume of component i. Analogously, ð1 fÞ ¼ f1 n1 n1 =ðn1 n1 þ n2 n2 Þ is the volume fraction of the solvent. The product fð1 fÞ is the probability of polymer segment–solvent contact and
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Chapter 12

the Gibbs energy change to create one mole of segment–solvent contacts upon mixing the pure components. 2. Dissolving the polymer in the solvent leads to an increased number of conﬁgurational possibilities and therefore to higher conﬁgurational entropies of both components, that is, to a positive mixing entropy (see Section 3.8). Dissolving n2 moles of polymer segments in n1 moles of solvent resulting in nð¼ n1 þ n2 Þ moles of solution causes a conﬁgurational entropy increase of ( ) n2 sol Sconfig ¼ < ln f þ n1 lnð1 fÞ ; ð12:3Þ Np where Np is the number of segments per polymer molecule and, hence, n2 =Np is the number of moles of polymer. The contribution of the polymer [the ﬁrst term on the right-hand side of (12.3)] to sol Sconfig is, for a given volume fraction, smaller as Np increases. The reason is that in a polymer molecule segments are linked to each other so that these segments cannot independently distribute themselves over the available volume. Because of the large value for Np the mixing entropy is mainly due to the solvent. Consequently, the solubility of polymers is, as a rule, poor as compared to the solubility of its individual monomers. Comparing Eq. (12.3) with (3.52) reveals that in (12.3) the volume fractions of the polymer and the solvent are assumed to be equal to their respective mole fractions. This assumption is valid if the volumes of the segments and the solvent molecules are taken equal; that is, n1 ¼ n2 . Then the change in Gibbs energy due to conﬁgurational possibilities is given by ( ) n2 sol Gconfig ¼ 0 and sol Gconfig < 0. The total Gibbs energy of polymer dissolution sol G is the sum of the interaction and conﬁguration contributions, so that ( ) f sol G ¼ n 0 and not too negative values for sol Gconfig (which is often the case for large values of Np ), the function sol GðfÞ often has two minima, as shown in Figure 12.3. This is a situation similar to that for self-assembling systems, as discussed in Section 11.1 [Fig. 11.3(b)]. Hence, if the total polymer fraction f is between the two minima the solution separates into two phases, one containing a low and the

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Polymers

217

Figure 12.3 Gibbs energy of polymer dissolution as a function of the volume fraction f of the polymer in the solution. The area of phase separation is indicated.

other a high polymer fraction. In practice, the polymer-rich phase is often considered as nondissolved polymer. The quality of the solvent, that is, the preference of polymer–solvent interactions over interactions between the pure components and, consequently, the solubility of the polymer may be altered by changing the composition of the solvent. For instance, for an aqueous solution the solvent quality usually decreases by adding (low molecular weight) electrolytes or alcohols. These additives compete for hydration with the polymer segments causing the polymers to become insoluble. This phenomenon is known as the salting-out effect. Salting-out is more effective when the additives are more strongly hydrated. Referring to the discussion in Section 4.3.1, the salting-out effect is stronger for small and for multivalent ions. The solvent quality may also be sensitive to the temperature. For nonaqueous systems the solubility usually improves on raising the temperature. Aqueous solutions behave in a more complex manner because of the large and rather speciﬁc inﬂuence of the temperature on the water structure (cf. Chapter 4).

12.2 CONFORMATIONS OF DISSOLVED POLYMER MOLECULES We consider dilute solutions where interactions between the individual polymer molecules may be neglected. The monomeric units of one and the same polymer molecule interact with each other and=or with the solvent molecules. For

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Chapter 12

homopolymers and alternating copolymers the interactions between the monomeric units and between the monomers and the solvent are uniform along the chain. The dissolved polymer molecule therefore adopts a regular structure. Depending on the strength of these interactions relative to each other the polymer molecule folds differently. Three extreme shapes may be distinguished. These are illustrated in Figure 12.4. In the compact state monomer–monomer interactions are preferred and the polymer chain folds back on itself to minimize monomer– solvent contacts. The radius of the compact sphere increases with Np1=3 ; Np being the degree of polymerization. In some cases polymers adopt stiff, rod-like shapes. These types of structures may be formed when directive forces are involved which are maximally effective when they are aligned, such as hydrogen bonds in helical and pleated sheet structures (see Section 13.2). The longest linear dimension of a rod varies with Np . When the polymer segment–solvent contacts are favorable the polymer adopts a highly solvated disordered coil-like structure. Such a structure is not ﬁxed. As a result of rotation around its covalent bonds the polymer chain assumes many different spatial arrangements. The structural state can only be characterized by average properties. The size of a coil-structured molecule expressed by its radius of gyration (see Section 12.3) scales with Npa , with 0:5 < a < 0:6. A more detailed discussion on coil-shaped molecules is given in Section 12.3. A block copolymer made from a soluble and a nonsoluble monomer has an amphiphilic character. The nonsoluble blocks aggregate and the soluble parts dangle out in the solution. Random copolymers are expected to have behavior in between that of alternating and block copolymers. Because of their many different types of monomeric units and the almost endless variety of possible sequences along the chain, heteropolymers may adopt all kinds of speciﬁc structures containing elements of the extremes displayed in Figure 12.4. Most soluble synthetic polymers as well as most polysaccharides and

hm

L

2 Rg 2R

Figure 12.4 Three extreme types of polymer conformation: the stiff rod ðL Np Þ, the compact globule ðR Np1=3 Þ, and the coil ðRg Npa , with 0:5 a 0:6).

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219

the nutritional proteins glutelins (from wheat grains) and caseins (from milk) have a predominant coil structure. Collagen, keratin, and double-stranded DNA form rod-like structures, whereas globular proteins are examples of heteropolymers having speciﬁc structures comprising compact spherical, rod-like, and more loosely packed coiled regions.

12.3 COIL-LIKE POLYMER CONFORMATIONS Polymers that have a relatively weak interaction between their segments assume a coiled, ﬂexible, and more or less open structure in solution. The chains of such polymers are in continuous motion. At any instant each individual molecule of a dissolved polymer population has a different shape and, likewise, the shape of each polymer molecule varies in time. It is therefore meaningless to consider the actual shape of the polymer molecule but the conformational state should rather be discussed in terms of the average shape. The average conformation of the polymer is called a statistical coil. The dimensions of such a coil are commonly expressed by the radius of gyration Rg and by the root mean square end-to-end distance hm . See Figure 12.4. The radius of gyration is a kind of average radius of the coil, deﬁned as R2g

Np X ðri rcm Þ2 =Np

ð12:6Þ

i¼1

and the root mean square (indicated by angular brackets) end-to-end distance is deﬁned as h2m ð¼ hh2 iÞ hðrNp r1 Þ2 i

ð12:7Þ

in which ri is the position of the ith segment and rcm the center of mass. For long linear chains a simple relation exists between Rg and hm : R2g ¼

h2m : 6

ð12:8Þ

If the interactions between the polymer segments, between a polymer segment and a solvent molecule, and between solvent molecules are equally favorable all rotational states around the covalent bonds in the polymer chain are equally probable. Assuming that the polymer has no volume the conformation of the polymer molecule can be described by the trajectory of a particle diffusing under the action of a random force. The resulting coil conformation is therefore referred to as a ‘‘random walk’’ or ‘‘random ﬂight’’ conformation. Such a conformation is shown in Figure 12.5. Because the monomeric units are linked to each other by covalent bonds that have more or less ﬁxed valence angles, two consecutive bonds in a polymer

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Chapter 12

lp

Figure 12.5 Random walk conformation of a polymer molecule with chain elements of persistence length lp .

chain cannot assume arbitrary directions. It implies that the motion of the ði þ 1Þth monomer is more or less coupled to that of the ith monomeric unit. In this context the notion of persistence length lp is deﬁned as the length of a chain element over which the orientational correlation disappears. Hence, lp is a measure of the rigidity, or the stiffness, of a polymer chain. Values for lp are in the range of a few nanometers (comprising a few monomeric units) up to tens of nanometers for polymers carrying bulky side groups and for polyelectrolytes in low ionic strength solutions. In the random walk conformation the polymer chain may fold back to itself without any hindrance. In reality this is impossible because the chain has a ﬁnite volume and its elements have to avoid each other. The ‘‘self-avoiding walk’’ gives rise to an exclusion volume which leads to coil expansion. The solvent quality inﬂuences the polymer conformation. In a poor solvent (i.e., for w > 0), the polymer segments effectively attract each other. If the attraction is not too strong the polymer still adopts a coil structure because of the favorable entropy resulting from the high number of rotational possibilities in a coil. It can be deduced that for w ¼ 0:5 attraction between the polymer segments just compensates for the exclusion volume effect. At that condition, the so-called y-condition, the conformation is described by the random walk model. Using the theory for diffusion it can be derived that for the end-to-end distance hm;y , in a random walk conformation of a polymer molecule with segments of length l, h2m;y ¼ Np l 2 :

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ð12:9Þ

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221

Equations (12.8) and (12.9) show that the size of the coil varies proportionally with Np0:5 . It follows that the molar volume np scales with Np1:5 and, hence, the average segmental density rp ð Np =np Þ with Np0:5 . The striking conclusion is that the polymer becomes less dense as the degree of polymerization increases. When the polymer chain grows four times longer the average density reduces by a factor of two. It is stressed again that since Rg is an average quantity, the coil is spherical on average. However, each individual polymer molecule has a nonspherical asymmetrical shape. Furthermore, the monomer distribution in the statistical coil is Gaussian; that is, the density of the polymer segments decreases outward from the center. It can be proven that for a random walk conformation the segmental density distribution rp ðrÞ is given by " # 3=2 Np 3 3r2 rp ðrÞ ¼ exp 2 ; R3g 2Rg 2p

ð12:10Þ

where r is the distance to the center of mass of the coil. The segment distribution is graphically presented in Figure 12.6 for a polymer characterized by l ¼ 0:50 nm, and Np ¼ 1000 and Np ¼ 4000. For these polymers Rg attains values of 6.45 nm and 12.90 nm. Taking a segmental volume of 0.05 nm3, a density of one segment per nm3 corresponds to a volume fraction of 5%. Figure 12.6 reveals that even at the center of mass the coil is very dilute. The difference between the two curves in Figure 12.6 demonstrates the effect of the degree of polymerization on the segmental distribution discussed above.

ρp / nm–3

1.4 1.2 1.0 0.8

(1)

0.6 0.4

(2)

0.2 0

5

10

15

20 r / nm

Figure 12.6 Density distributions in randomly coiled polymer molecules. (1) Np ¼ 1000; l ¼ 0:50 nm and (2) Np ¼ 4000; l ¼ 0:50 nm.

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The distributions shown in Figure 12.6 are for random walk conformations, that is, at y conditions, for which w ¼ 0:5. Note that in a better solvent, w < 0:5, the polymer coil swells which implies even lower polymer volume fractions. The swelling may be accounted for by introducing a linear expansion coefﬁcient a, deﬁned as a ¼ hm =hm;y :

ð12:11Þ

In the random walk conformation the polymer molecule attains maximum rotational freedom along the chain resulting in a root mean square end-to-end distance given by (12.9). The probability O for a polymer chain to deviate from the random coil conformation and, therefore, having an end-to-end distance hm deviating from hm;y is given by a Gauss equation, "

# 3h2m Oðhm Þ exp : 2Np l 2

ð12:12Þ

With Eqs. (3.5) and (12.9) this gives 3 Sðhm Þ ¼ constant kB ðhm =hm;y Þ2 : 2

ð12:13Þ

Thus, swelling the coil by a factor of að hm =hm;y Þ leads to an entropy loss of ð3=2ÞkB a2 per polymer molecule. The corresponding change in Gibbs energy swell Gconfo is ð3=2ÞkB T a2 : On the other hand, ‘‘dilution’’ of the segments in the swollen coil yields a favorable osmotic contribution swell Gosm . Without giving the derivation here, it is stated that swell Gosm scales with ð1 2wÞNp0:5 a3 . At equilibrium both contributions to swell G compensate each other; that is, dswell G=da ¼ 0. From the equilibrium condition follows that for a polymer coil in good solvent a5 ð1 2wÞNp0:5 , showing that a depends on both w and Np . For a given solvent a Np0:1 so that hm ð¼ ahm;y Þ Np0:1 Np0:5 ¼ Np0:6 . As a result of the swelling the volume of the coil increases and, hence, the density decreases by a factor a3 which is proportional to Np0:3 . Although a is only slightly dependent on Np , the large value for Np causes the segment density to decrease substantially. Decreasing the solvent quality below the y-condition (i.e., for w > 0:5) causes collapse of the polymer molecule into a compact structure. For uncharged polymers it results in precipitation but polyelectrolytes may still be colloidally stable by virtue of their electric charge.

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223

12.4 SEMIDILUTE AND CONCENTRATED POLYMER SOLUTIONS Polymer–polymer interactions are neglected in the foregoing sections. This is justiﬁed for very dilute solutions. Because a polymer molecule contains mostly solvent in its coil the molecules start to overlap at a relatively low concentration. At that concentration the separation distances between the coils approach the dimensions of the coils themselves. A solution in which the polymer coils overlap is referred to as a ‘‘nonconnected network.’’ Taking ð4=3ÞpR3g as the coil volume, the concentration c (in mole monomer=dm3) at which the nonconnected network starts to form is given by c ¼

3Np 103 : 4pNAv R3g

ð12:14Þ

In a random walk conformation the average segment density rp;y varies with Np0:5 whereas for a swollen coil rp is proportional to Np0:8 . At c the average segment concentration in solution equals that in an individual coil and, hence, cy Np0:5 and c Np0:8 . The formation of a nonconnected network is accompanied by sudden changes in physical–chemical properties of the system, for instance, an abrupt increase in the viscosity. At polymer concentrations beyond c the chains penetrate into each other making many intermolecular connections. A network having a characteristic mesh size forms. The segments within a volume corresponding to the mesh size are assumed to belong to the same polymer chain. Such chain elements are called ‘‘blobs.’’ Essentially all segment–segment interactions are within the blob. Interactions between blobs rarely occur. As a consequence, properties of the network such as its swelling behavior are exclusively determined by the number of segments in the blob. The blob size ð¼ mesh sizeÞ, in turn, is given by the polymer concentration in the solution. Such a solution of entangled polymer chains is indicated as ‘‘semidilute.’’ Figure 12.7 schematically depicts polymer solutions at various concentration levels: (a) a dilute solution, (b) a nonconnected network, and (c) a semidilute solution. At even higher concentrations, where the volume fractions of polymer and solvent are comparable, the size of the blob reduces to that of the length l of a polymer segment. Then the conformation of the polymer chains can be described again by a random walk. Hence concentrated polymer solutions (and polymer melts) are always at y-conditions.

12.5 POLYELECTROLYTES In aqueous solution polyelectrolytes bear charged groups along their chains. If the charged groups are strong acids or strong bases the charge is essentially invariant

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(a)

(b)

(c)

Figure 12.7 Polymer solutions: (a) dilute; (b) onset of a nonconnecting network; (c) semidilute solution. The dashed circles in (c) indicate the blob size.

with pH and the polymer is called a strong polyelectrolyte. Weak polyelectrolytes contain weak acid or basic groups so that their charge depends on pH. The net charge as well as the types of ionizable groups may be determined by analyzing proton titration data, as discussed in Section 9.1. The presence of charge inﬂuences both inter- and intramolecular interactions. The charged polyelectrolyte molecules are surrounded by a diffuse distribution of counterions (cf. Chapter 9). The molecules repel each other by electrical double layer overlap so that a polyelectrolyte solution may be colloidally stable even when the solvent quality is poor. Intramolecular electrostatic repulsion causes a more stretched conformation of the chain. This can be accounted for by an electrostatic contribution to the persistence length lp : lp ¼ lp0 þ lpel ;

ð12:15Þ

where lp0 is the persistence length of the equivalent uncharged chain and lpel accounts for the electrostatic effect. Because of the electrostatic contribution lp varies with the ionic strength. To estimate the dependency of lpel on ionic strength we have to consider the charge distribution around the chain elements of the polyelectrolyte molecule, the geometry of which usually resembles a cylinder with a very small radius. Analogous to charged interfaces (Chapter 9), in the absence of speciﬁc counterion adsorption the charge distribution and the potential decay around a polyelectrolyte may be described by the Poisson–Boltzmann equation for cylinders. Here, we do not further elaborate on the polyelectrolyte electrical double layer but restrict ourselves to the following discussion. For polyelectrolytes it is customary and convenient to express the charge density in terms of the distance a between two consecutive elementary charges e along the chain. Hence, the linear charge density equals e=a. In analogy with the potential-charge density relationship for interfaces (9.36), for low values of e=a

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225

the potential at the polyelectrolyte surface varies proportionally with e=a, but for high values of e=a that proportionality is lost and, moreover, a considerable fraction of the charge on the polyelectrolyte is compensated by counterions that are in very close proximity to the polyelectrolyte. Eventually, any increase in the polyelectrolyte charge density is essentially compensated by counterions in its immediate vicinity. As a result, the electric potential at a distance farther away becomes rather insensitive to increases in the charge of the polyelectrolyte and can be seen as arising from an almost constant effective charge density, characterized by a parameter aeff . Obviously, at low charge density aeff is simply given by a. At high polyelectrolyte charge densities aeff reaches a critical value acrit below which further increase in charge density is compensated by ion condensation (association of counterions with the ionic groups on the polyectrolyte). For a cylinder acrit ¼

e2 : 4pee0 kB T

ð12:16Þ

In water at room temperature ðe ¼ 80Þ acrit ¼ 0:71 nm. Theory further permits the following expression for lpel, lpel ¼

acrit e2 ðaeff kÞ2 ¼ k2 4 16pee0 kB Ta2eff

ð12:17Þ

with k the reciprocal Debye length [as given by (9.29)]. Equation (12.17) reveals that lpel is proportional to k2 , that is, inversely proportional to the ionic strength. Figure 12.8 shows how lp of DNA varies with ionic strength. At 0.1 M, DNA has

lp / nm 150

100

50

l po

0 10 – 4

10 – 2

1 ionic strength / M

Figure 12.8 The persistence length of DNA as a function of ionic strength. The curve represents the relation lp ¼ lp0 þ lpel (12.17).

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essentially lost its polyelectrolyte character: the polyelectrolyte behaves as an uncharged polymer. In addition to the effect on the rigidity of the chain, electrostatic repulsion also prevents chain sections that are distant along the chain to approach spatially close in the coil. This leads to swelling; the more swelling the stronger the electrostatic repulsion is. As a consequence, for polyelectrolytes the concentration c , where a nonconnected network is formed, is much lower than that for the uncharged equivalent polymer. The situation becomes more complicated, but not principally different, for polyampholytes, that is, polyelectrolytes comprising both anionic and cationic groups. Polyampholytes have a net positive charge at low pH and a net negative charge at high pH. Consequently at a certain and for each polyampholyte characteristic pH, the point of zero charge, they carry as many positive as negative charges. The proton titration behavior and, hence, the charge as a function of pH, is for polyampholytes as fully explained in Section 9.1. Proteins are the most abundantly occuring natural polyampholytes. In addition to their amphoteric character proteins are amphiphilic heteropolymers. The distribution of the monomers (the various amino acid residues) determines the amphiphilicity and charge distribution and is therefore largely responsible for the three-dimensional structure of a protein molecule in aqueous solution. Principles governing the protein structure and structural stability are treated in more detail in Chapter 13.

12.6 PHASE SEPARATIONS IN POLYMER SOLUTIONS: COACERVATION, COMPLEXCOACERVATION, AND POLYMER-INDUCED MICELLIZATION Polymers in solution are, as a rule, thermodynamically stable. However, as has been explained in Section 12.1, when the solvent quality is reduced the polymer solution may separate in two phases, a polymer-rich and a polymer-poor solution, even though sol G is still negative. This phenomenon where the two phases are in equilibrium with each other is called coacervation. It may be clear that coacervation often occurs at the verge of precipitation by adding, for example, alcohol or salt to an aqueous polymer solution. A similar phenomenon occurs when aqueous solutions of a cationic and an anionic polyelectrolyte are mixed. The oppositely charged polyelectrolytes attract each other as a result of which they aggregate and lose a part of their hydration water. However, hydration of the complex may still be sufﬁcient to keep it in solution. This kind of phase separation is referred to as complex-coacervation. Because complex-coacervation is ruled by electrostatic interaction it can be

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227

v / mol surfactant / mol protein

manipulated by varying the concentration of the charge determining ions (usually Hþ ) of the polyelectrolytes and=or the concentration of the indifferent electrolyte. When a (low molecular weight) amphiphile is added to a polymer solution the amphiphiles start to aggregate along the polymer chain at a concentration, the critical aggregation concentration (cac), which is substantially lower than the critical micelle concentration (cmc) for the pure amphiphilic solution. Micellarlike structures develop along the polymer chain thus forming a coiled string of beads. This may even occur when the polymer is a compact globular protein molecule. Various amphiphiles induce protein unfolding, whereafter amphiphilic aggregation takes place along the unfolded polypeptide chain. As a typical example, Figure 12.9 shows graphs for the binding of a cationic surfactant at a positively charged and a negatively charged globular protein. In the case of favorable electrostatic interaction surfactant molecules bind to oppositely charged groups at the protein surface. At higher surfactant concentrations cooperative binding takes place with concomitant surfactant aggregation along the unfolding polypeptide chain. Under electrostatically adverse conditions the initial binding step is clearly absent. In contrast to ionic surfactants, binding of some nonionic surfactants (e.g., Tween, CHAPS) does not destroy the integrity of the globular protein structure. With ionic amphiphiles electrostatic repulsion is responsible for some separation distance between the beads. After saturating the polymer with micelles further addition of amphiphiles increases the amphiphile monomer concentration

100 80 60 pH 10

pH 7

40 20 0

5

10 cac

15 cmc

20

25

csurfactant × 103 / M

Figure 12.9 Binding of the cationic surfactant dodecyl pyridinium chloride at the protein lysozyme. The net charge on lysozyme is positive at pH 7 and negative at pH 10. Ionic strength 0.025 M; T ¼ 25 C.

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Chapter 12

in solution until the cmc is reached and ordinary micelles in solution are formed. The tendency of the amphiphilic molecules to assemble at the polymer chain depends on various factors. The water-soluble polymer may wrap around the micellar surface and screen unfavorable carbohydrate–water contacts. Also, electrostatic interaction between the amphiphiles and the polymer affects aggregation. In special cases, in particular when the polymer main chain carries not too short apolar side chains, these side chains may penetrate into the apolar micellar core. Then, it is not unlikely that micelles link two or more polymer chains thereby forming a network. This manifests itself by a dramatic increase in the viscosity of the system. In practice, a variety of systems contains both amphiphilic molecules and polymers. These are, for instance, found where detergent is used to remove polymeric (proteinaceous) deposits from a fouled surface, for example, in food-processing equipment, teeth, contact lenses, and other biomedical appliances. By aggregation of the amphiphiles at the polymers the latter ones are ‘‘solubilized’’ and subsequently released from the surface. This process is schematically depicted in Figure 12.10.

Figure 12.10 Removal of polymeric material from a surface via binding of surfactants and subsequent solubilization.

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229

12.7 POLYMER GELS Entangled chains in semidilute polymer solutions, discussed in Section 12.4, are still able to move relative to each other. As a consequence, the system behaves as a liquid, albeit a viscous liquid. When the polymer chains are connected to each other by stable crosslinks, either chemical or physical ones, the whole system is stationary although the chain elements between the crosslinks are in rapid motion. On a macroscopic scale the system is solid-like whereas on a microscopic scale it behaves as a liquid. Such a system is called a gel. Thus, a gel is a system in which colloidal particles are interconnected and form a coherent macroscopic structure that is permeable for the solvent. Polymer gels can be formed by physical connections or by chemical crosslinks. A physical polymer gel may be formed when solvent is added to a lyophilic polymer but in insufﬁcient amount to completely dissolve the individual polymer molecules. Then crystalline domains still remain and connect the ﬂexible chains. Also, changing environmental conditions such as pH, ionic strength, solvent polarity, and the like may trigger the transition from a molecular solution into a gel, or vice versa. In those cases the gelation process is often caused by a coil-to-helix transition in some parts of the polymer molecules. The helical parts tend to associate because they have a low solubility due to their relatively low conformational entropy in solution. Some proteins may form physical gels. The most well-known example is gelatin. Various polysaccharides are able to form physical gels as well, for instance, pectin, carrageenan, and agarose. The structure of a physical gel is schematically represented in Figure 12.11. Such gels are usually reversible; that is, they can be formed and disrupted by changing conditions in opposite directions. Gel formation is not restricted to lyophilic colloids; under appropriate coagulation conditions lyophobic particles may form a gel as well, a so-called particle gel. See Figure 12.11(b). Polymer chains that are covalently connected may tend to

Figure 12.11 Polymer gels stabilized by (a) crosslinks (left) and association of helixes (right); (b) gels formed by aggregated lyophobic particles.

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dissolve but the crosslinks prevent complete molecular dissolution. Unvulcanized rubber dissolves completely in benzene, whereas vulcanized rubber cannot dissolve, but only swell, because of the S–S crosslinks. Various globular proteins that are soluble in their native state may form crosslinked gels when they are heated. Intermolecular disulﬁde bonds are usually responsible for the crosslinking. For instance, ovalbumin causes gelation of the egg white upon boiling an egg. The reasons for polymer gels to become solvated are the same as those for dissolution discussed in Section 12.1. They are changes in the mutual interactions between segments and solvent molecules, expressed in Eq. (12.1), and an increase in the conﬁguration entropy due to mixing, expressed in (12.3). The swelling of a gel is ruled by conformational and osmotic contributions, explained in Section 12.3. The swelling pressure of a gel is the osmotic pressure difference between gel and equilibrium ﬂuid outside the gel. This pressure may attain values as high as 1000 bar ð108 N m2 Þ. Because of the high swelling pressure gels (usually made of crosslinked polyelectrolytes) are used in superabsorbers such as disposable diapers, sanitary napkins, and the like. In nature, the high swelling pressure allows desert plants to take up moisture from arid soils. A characteristic feature of a gel is its elastic behavior: the gel deforms by an imposed force and it relaxes to its original state after releasing the force. Elastin, a crosslinked polypeptide network takes care of the elasticity of human and animal connective tissues, such as skin, ligaments, and arterial walls. For a more extensive discussion on the rheological properties of gels, the reader is referred to Chapter 17. Gels may take up and expel solvent in a more or less reversible manner. When the gel has a rather rigid structure the uptake of solvent in the pores does not cause much swelling. This phenomenon is called imbibition. Uptake of water in a sponge, in a germ cell, and of gravy in boiled rice are examples of imbibition. Sometimes a swollen gel shrinks spontaneously and irreversibly under the expulsion of water. This is known as syneresis. Examples are the excretion of serum by clotted blood, the ‘‘sweating’’ of cheese, and the ‘‘bleeding’’ of lubricants. In addition to the applications already mentioned, gels are widely used in food, pharmaceuticals, and biomedical products, mostly to provide the material with the desired consistency and texture (e.g., mouth feel, deformability, etc.) and as matrix material in drug delivery systems. Furthermore, gels are used as a carrier in electrophoresis and as a matrix in size exclusion chromatography and in ﬁlter membranes.

EXERCISES 12.1

Comment on the following statements. (a)

The solubility of a polymer increases with an increasing degree of polymerization.

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231

(b)

The dimensions of a coil conformation of a dissolved polymer molecule is solely determined by the chemical structure of the polymer. (c) Segments of a polymer molecule attract each other in a y-solvent. (d) The segment density in the center of mass of a polymer coil is inversely proportional to the root mean square of the degree of polymerization. (e) Polyelectrolyte gels shrink with increasing ionic strength. 12.2

Consider a heterodisperse polymer in a y-solvent. It is known from the literature that the molecules in a homodisperse fraction of molar mass ðM Þ 106 g mol1 have a radius of gyration ðRg Þ of 100 nm. Derive a simple expression for Rg ðM Þ for this polymer.

12.3

Calculate the segment density distribution in a random coil of polyethylene oxide (PEO) at y-conditions. Assume for the persistence length 1 nm ( three ethylene oxide units). Do the calculations for ½CH2 CH2 O500 and ½CH2 CH2 O5000 .

12.4

The swelling of a gelatin gel depends on the pH of the medium. Give a sketch reﬂecting the variation in the swelling with pH in case the pH is adjusted by (a) HCl or (b) H2SO4. The point of zero charge of gelatin is at pH 5.

12.5 (a)

Explain the phenomenon of complex-coacervation. Why does it occur only over a limited pH range? (b) Aggregates formed by complex-coacervation may dissociate by adding NaOH. Upon subsequent lowering of the pH by adding HCl the aggregates are formed again. Is it possible to continuously repeat this cycle of aggregate dissociation and formation? (c) Gelatin and gum arabic form a complex-coacervate at 2 < pH < 4; aggregation is at a maximum at pH 3. Indicate and explain the order in which the electrolytes NaCl, BaCl2, and Na2SO4 destabilize the aggregates at pH 2.5, 3.0, and 3.5, respectively. (d) Consider a protein having its point of zero charge at pH 4. Do you expect complex-coacervation when solutions of this protein of pH 3 and pH 5 are mixed?

SUGGESTIONS FOR FURTHER READING P. G. de Gennes. Scaling Concepts in Polymer Physics, New York: Oxford University Press, 1979. J. des Cloizeaux, G. Jannink. Polymers in Solution, New York: Clarendon, 1990. P. J. Flory. Principles of Polymer Chemistry, Ithaca, NY: Cornell University Press, 1953. P. Molyneux. Water Soluble Synthetic Polymers: Properties and Behaviour, Boca Raton, FL: CRC Press, 1983. C. Tanford. Physical Chemistry of Macromolecules, New York: Wiley, 1961.

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Conformational Diseases

The functional behavior of a protein molecule is intimately and speciﬁcally related to its spatial structure—its conformation. Enzymes, transport-, and immunoproteins become inactive when the conformation is perturbed. The structure–function relationship may be most dramatically demonstrated for proteins that are held responsible for so-called conformational diseases. The classical example is sickle cell anemia. Due to a slight, genetically determined, modiﬁcation of its composition the oxygen transporting protein hemoglobin forms oblong aggregates causing morphological and physiological disruption of the erythrocytes. Patients suffer from severe anemia. Subtle conformational disorders in certain protease inhibitors lead to a variety of diseases. For instance, familial thromboembolic disease is caused by a conformational change in the protein antithrombin. Another class of conformational diseases shares the feature that the protein involved undergoes a structural rearrangement leading to intermolecular b-sheet linkages. As a result, the protein aggregates to form ﬁbrillar structures and deposits in the tissue. Neurodegenerative diseases such as Huntington’s, Alzheimer’s, and CreutzfeldJakob’s diseases are thus explained. The pathological conformation could follow from a genetic mutation. However, the conformation transition may also occur in normal protein molecules where it is triggered by exposure to the abnormal form. The pathological conformation then acts as a template for the transformation. This mechanism, schematically presented in the picture, is in particular proposed for the propagation of pathological prion proteins between individuals or even species.

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13 Proteins

Proteins rule life. Essentially all chemical reactions that sustain life are catalyzed by enzymes, which are proteins. Other proteins play a vital role in transport processes, ranging from electron transfer to carrying oxygen, hormones, and other substances. Membrane proteins control the passage of ions and metabolites across membranes that separate biological cells and cell organelles from their surroundings. Immunoproteins, the antibodies, defend the organism against foreign invaders. In muscles proteins take care of converting chemical energy into mechanical work and the elasticity and structure of connective tissues and bones is also provided by proteins. The overwhelming variety of biological functions requires a corresponding diversity within the three-dimensional structure protein molecules adopt. Proteins are heteropolymers of some 20 different amino acids linked together in one or more polypeptide chains. A number of amino acids along the polypeptide chain contain an anionic or cationic group. This makes the protein a polyampholyte. The various amino acid side groups differ greatly in polarity so that the protein is amphiphilic. The sequence of the amino acids in the polypeptide chain ultimately determines the folded spatial architecture, that is, the three-dimensional structure of the protein molecule. The three-dimensional structure is the net result of interactions between segments within the protein molecule but also between segments of the protein and the environment, which, except for membrane proteins, is usually an aqueous medium. Based on the three-dimensional structure the following division may be made. 1. Protein molecules that are highly solvated and ﬂexible, resulting in a disordered coil-like structure. This group comprises some proteins of which

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234

Chapter 13

the natural function is nutritional, such as glutelins in wheat grains and caseins in milk. 2. Protein molecules that have a very regular structure (e.g. helices, pleated sheets), the so-called ﬁbrous proteins. Among others, collagen, keratin, and myosin belong to this category. These proteins are usually insoluble in water; they are mainly found in muscle and connective tissues. 3. By far the greatest proportion of the protein species (but only a small fraction of the protein mass on earth) contains different structural elements, that is, a-helices, b-pleated sheets, and parts that are unordered, which are folded together into a compact dense globule: the globular proteins. In an aqueous environment the apolar amino acid residues are mainly located in the interior whereas the polar residues are primarily found at the periphery of the molecule. The globular proteins have evolved to fulﬁll speciﬁc functions, such as biocatalysis (enzymes) and immunoreactions (antibodies). An almost countless number of different kinds of globular proteins exist, each kind having its own speciﬁc biological function related to its own characteristic three-dimensional structure. In this chapter we focus on the structure of compact globular proteins and their unfolding into highly hydrated, expanded coil structures.

13.1 THE AMINO ACIDS IN PROTEINS The general structure of the amino acids occurring in proteins is: H2N*CHRCOOH (except for the amino acids proline and 4-hydroxy proline; see Table 13.1). They are all a-amino acids. The asterisk at the a-C atom indicates the asymmetry around this atom. The amino acids in proteins are L-stereo isomers. Table 13.1 summarizes the various amino acids that occur in proteins. They are usually indicated by their three letter abbreviation or by their one-letter symbol. Amino acids are assembled in polypeptide chains by polycondensation.

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Proteins

235

Table 13.1

Amino Acids Occuring in Proteins

Name

Abbreviation Symbol

Structure

Remarks

Glycine

Gly

G

a–C atom is not asymmetric

Alanine

Ala

A

Hydrophobic side groups promote compact structure in aqueous environment

Valine

Val

V

Leucine

Leu

L

Isoleucine

Ile

I

Proline

Pro

P

4-Hydroxy proline

Hyp

Serine

Ser

S

Threonine

Thr

T

Aspartic acid

Asp

D

Polar, anionic side-groups

Glutamic acid

Glu

E

pK 4 5

Asparagine

Asn

N

Polar side groups converted into Asp and Glu at extreme pH values

Branching promotes dense packing of protein interior

Cyclic structures: the side-group folds back to the N-atom Somewhat polar sidegroups

Continued

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236

Chapter 13

Table 13.1

Continued

Name

Abbreviation Symbol

Structure

Remarks

Glutamine

Gln

Q

Lysine

Lys

K

Arginine

Arg

R

Histidine

His

H

Nucleophilic side group pK 6 7 often in active sites of enzymes

Phenylaline

Phe

F

Large aromatic side groups

Tyrosine

Tyr

Y

pKTyr 10

Tryptophan

Trp

W

Cysteine

Cys

C

S-containing side groups pKCys 8:5

Methionine

Met

M

Oxidation of SH groups leads to SS bonds: CysSSCysCystine

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side group contains polar and apolar parts, and is cationic pKLys 10 pKArg 12

Proteins

237

The polypeptide backbone is composed of repeating units that consist of three atoms and varying side-groups, R.

This unit is called the peptide unit. Figure 13.1 shows the geometry of the peptide backbone. Due to mesomery the peptide unit is stabilized by a resonance energy of 80–90 kJ mol1 :

O 0.12 nm

R

H 0.13 nm

121.1o

123.2o

C m 5n 0.1

Cα

R

Cα

121.9o

m

4n

115.6o 119.5o

N

0.1 118.2o

0.1 nm

H H

Figure 13.1 The geometry of the peptide backbone, with a trans-peptide bond showing all atoms between two C a -atoms of adjacent amino acid residues. The peptide bond is shaded.

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The mesomeric structure has a number of consequences: 1. 2.

3.

The peptide unit has a large dipole moment. There is no rotational freedom around the peptide (o) bond. Therefore, the atoms depicted below have a strong tendency to be located in one plane. Two conﬁgurations can be distinguished:

The trans-conﬁguration is the most stable one because of the least steric hindrance between the R-groups that are linked to the Ca -atoms. In other words, the trans-conﬁguration has a larger rotational freedom around the c and F bonds. The C0 –N bond (the peptide bond) is 10% shorter than normal; the C0 –O bond is longer than those in aldehydes and ketones, for example. The C0 –Ca bonds have normal lengths.

In the polypeptide chain, each of the amino acid residues has a number of distinguishable stable conformations in the backbone (due to rotations around the c- and the F-bonds) as well as in the side group R that, in general, contains a number of single covalent bonds. It has been inferred that within the polypeptide backbone each peptide unit may attain about four conformations and for the side group R an average of six conformations is estimated. Thus, the number of conformations for a polypeptide consisting of 100 amino acids adds up to 10100. It leads to a large conformational entropy Sconfo. According to Eq. (3.5): Sconfo ¼ kB ln O;

ð13:1Þ

where O stands for the number of distinguishable conformations. For the example given here Sconfo would amount to 1914 J K1 mol1 . However, in a globular protein molecule the polypeptide adopts a speciﬁc, more or less ﬁxed, structure. Such a compact inﬂexible architecture is stable only if interactions within the protein molecule and between the protein and its environment compensate for the loss of conformational entropy.

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Proteins

239

13.2 THE THREE-DIMENSIONAL STRUCTURE OF PROTEIN MOLECULES IN AQUEOUS SOLUTION The 21 amino acids mentioned in Table 13.1 allow an almost inﬁnite number of different polypeptide molecules, just as an almost inﬁnite number of words can be written using the the 26 letters of the alphabet. The sequence of the amino acid residues in the polypeptide chain is referred to as the primary structure of the protein. The primary structure ultimately determines the three-dimensional (3-D) structure of the protein in a given environment. As the number of conformations that the polypeptide chain can adopt is extremely large and external factors are involved in selecting the preferred conformation, it is very difﬁcult and, as yet, impossible to reliably predict the 3-D structure solely on the basis of the amino acid sequence. The folding of the polypeptide chain into the 3-D structure of the native protein is believed to proceed along narrow paths avoiding Gibbs energy barriers and kinetic traps. This is illustrated in Figure 13.2. As guides along the folding path, nature uses molecular chaperones, molecules that temporarily bind to certain sites of the polypeptide to prevent or promote undesired interactions in the folding chain. Hence, the native protein structure corresponds with one of the local Gibbs energy minima, not necessarily the deepest one. Other nonexisting (because unreachable) states might be thermodynamically more stable. The suggestion that protein folding is a directive, kinetically determined process may be further supported as follows. For a polypeptide consisting of 150 amino acids it would take 1069 years to sample all, say, 4150, possible conformations of

Figure 13.2 Gibbs energy landscape of protein structure stability. (From K. A. Dill and H. S. Chan. Nature Struct. Biol. 4: 10, 1997.)

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Chapter 13

the backbone, assuming that the transition from the one conformation to the other takes 1013 s. Within the 3-D structure different levels may be distinguished, referred to as the secondary, the tertiary, and the quaternary structures. The secondary structure is the spatial arrangement of the polypeptide backbone ignoring the side groups. There are a few generally occurring secondary structures in proteins, notably the a-helix and the b-pleated sheet. The a-helix is a spiral-like structure, having 3.6 amino acid residues and a pitch of 0.54 nm per turn; see Figure 13.3. The a-helix allows a dense packing of atoms. Each peptide unit is involved in two hydrogen bonds (C¼O . . . HN). These hydrogen bonds are well-aligned, almost parallel to the axis of the helix. The dipole moments of the peptide units along the helix have a favorable interaction with each other. The side groups are directed outwards from the helix so that they hardly interfere sterically with the helix formation. Proline and hydroxyprotine do not ﬁt in the helical structures, because the R-group folds back

(a)

(b) N

O

C

N

C R

O

C

C N

R C

O

C N

R

C

O

C N

O

R

C

C

N

R C

O

n×d

C R

N C

r

O C N

R

O

C

2π r

C

N

O

C N R R

d

R C

O

C

C O

N C O

C R

Figure 13.3 The a-helix: (a) spatial and (b) projected representations. Helix parameters: r ¼ 0:23 nm, n ¼ 3:6 amino acids per turn; d ¼ 0:15 nm per peptide unit.

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Proteins

241

to the N-atom in the backbone (see Table 13.1), leaving no N–H available for hydrogen bonding, and the F-bond is blocked for rotation. In the b-pleated sheet the polypeptide backbone is almost fully stretched. The chains aggregate side by side through hydrogen bonding between peptide units (C¼O . . . HN). The hydrogen bonds are well aligned and there is a favorable interaction between the dipolar peptide units. This results in a dense packing. A sheet usually contains two to six chains, as illustrated in Figure 13.4. The orientation of the amino acids in adjacent chains are in the same (parallel) or in opposite (antiparallel) directions. Most globular proteins contain 40–80% a-helix and=or b-sheet structures. These conformations are mainly found in the interior of the molecule, where they are stabilized by hydrogen bonds and also by interactions between hydrophobic side groups (cf. Sections 4.3 and 13.3). Hydrogen bonding between the peptide units in the a-helix and the b-sheet structures hampers the rotational mobility of the peptide chain around c and the F-bonds and, hence, reduces the conformational entropy. Assuming four possible conformations per peptide unit (excluding the side groups) in the expanded coil structure and only one in the a-helix and bsheet, the loss in conformational entropy per peptide unit equals R ln 14 ¼ 11:53 J K1 mol1 . Thus a large entropy decrease, 577 J K1 per mole of protein, corresponding to a Gibbs energy increase of 173 kJ mol1 at 300 K, would result from the folding of a protein consisting of 100 amino acids, of which 50% of the amino acids are involved in a-helices and b-sheets. Additional losses in conformational entropy will result from the ‘‘freezing’’ of other parts of the

Figure 13.4 The b-pleated sheet in an (a) antiparallel and (b) a parallel orientation.

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polypeptide backbone (and probably some side chains) into densely packed random structures within the interior of the protein molecule. The tertiary structure refers to the overall topology of the polypeptide chain, that is, the way in which folded segments of the polypeptide chain are mutually arranged in space with each other and the mutual ordering of side groups. The notion quaternary structure is used for the noncovalent association of independent tertiary structures. Proteins show a wide variety of three-dimensional structures. Globular proteins in an aqueous environment have a number of structural characteristics in common:

They are more or less spherical, with molecular dimensions in the range of a few to a few tens of nanometers. Hydrophobic side groups tend to be buried in the interior of the molecule where they are shielded from contact with water. As a result, part of the hydrophilic hydrogen bond forming polypeptide backbone must also be located in the interior. Therefore, one important property of secondary structures such as a-helices and b-sheets is the efﬁcient matching of hydrogen bond donors and acceptors between internal polar groups of the polypeptide backbone. Charged groups are almost always located in the aqueous periphery of the protein. Any charged groups in the interior occur as ion pairs since dissociation is strongly opposed by the low local dielectric permittivity. The atoms are densely packed, with most adjacent atoms in Van der Waals contact (Section 13.3). Internal atomic packing densities average around 75% (v=v), which is similar to the maximum packing density of equally sized hard spheres. For comparison, water and cyclohexane have packing densities of 58% (v=v) and 44% (v=v) respectively, at 298 K and 1 bar. For not too large globular protein molecules, that is, those with a molar mass M 30;000 D, the volume V and the water-accessible surface As (see Figure 13.5) may be approximated using the following empirical equations, V ¼ 1:27 103 M As ¼ 11:12 10

2

M

ðnm3 Þ 2=3

ð13:2Þ

ðnm Þ: 2

ð13:3Þ

The water-accessible surface area of a completely unfolded polypeptide chain is given by As ¼ 1:45 102 M

ðnm2 Þ:

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ð13:4Þ

Proteins

243

water molecule accessible surface protein

protein

inside

Figure 13.5 The water-accessible surface As of a (protein) molecule is deﬁned as the surface described by the center of a water molecule that rolls over the exterior of the (protein) molecule. A water molecule is considered to be a sphere of 0.14 nm radius. The surface in narrow clefts may not be accessible for water.

By way of example Figure 13.6 presents computer graphic images of native bovine pancreas ribonuclease (RNase). The left image shows the folding pattern of the polypeptide backbone; the right image is a space-ﬁlling model showing the compact packing of the protein molecule. For a few small protein molecules, the composition of the water-accessible surface is given in Figure 13.7. For larger (or more spherical) proteins, with lower As =V ratios, the percentage of apolar atoms on the surface is usually less. The apolar content of the interior of small globular proteins is about 60%.

Figure 13.6 Computer graphics images of bovine pancreas ribonuclease, showing (left) the polypeptide backbone made up of a-helices (spirals, seen, for example, in the lower right of the molecule), b-sheets (upper left) and ‘‘unordered’’ parts, and (right) a spaceﬁlling model showing the compact packing of a globular protein molecule.

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244

Chapter 13 Ribonuclease-S side chain: C,S 41%

side chain: N,O 38%

non-polar 46% polar 54%

main chain: C,S 5%

main chain: N,O 16%

Lysozyme

Myoglobin side chain: C,S 34%

side chain: N,O 42%

main chain: N,O 17%

side chain: N,O 40%

main chain: C,S 7%

side chain: C,S 42%

main chain: N,O main chain: C,S 6% 12%

non-polar 41%

non-polar 48%

polar 59%

polar 52%

Figure 13.7 Polar and nonpolar contributions made to the water-accessible surface of different proteins.

13.3 NONCOVALENT INTERACTIONS THAT DETERMINE THE STRUCTURE OF A PROTEIN MOLECULE IN WATER Protein folding requires that the large conformational entropy opposing the folded state be outweighed by the sum of the enthalpic and other entropic factors that affect the stability of the folded state. Below, the most important interactions determining the protein structure in water are brieﬂy reviewed.

13.3.1 Hydrophobic Interaction The principles and characteristics of hydrophobic interaction were discussed in Section 4.3.2. In order to estimate the contribution from hydrophobic interactions to the stabilization of the compact native structure (N), taking the fully expanded and hydrated structure (D) as the reference state, the degree of hydrophobicity of the various residues R of the amino acids along the polypeptide chain should be

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245

known. These hydrophobicities may be deﬁned as the change in the standard molar Gibbs energy tr gRo on transferring R from a nonpolar medium to water. o This, in turn, can be obtained by subtracting tr ggly from tr g o amino acid [glycine (gly) contains no R-group]:

o o tr gRo ¼ tr gamino acid tr ggly :

ð13:5Þ

o Values for tr gamino acid are calculated from partitioning the amino acid in a water=nonpolar two-phase system, according to:

o tr gamino acid ¼ RT ln

cwater amino acid cnonpolar amino acid

:

ð13:6Þ

In this way various hydrophobicity scales for amino acids (or, for that matter, R-groups) have been proposed, based on using different apolar solvents. An R-group is called hydrophobic if tr gRo > 0 and hydrophilic if tr gRo < 0. Figure 13.8 shows the relation between the hydrophobicity and the wateraccessible surface area for a number of amino acids. The Gibbs energy of transfer from a nonaqueous to an aqueous medium is 9–11 kJ mole1 nm2 . Note that the value for tr gRo is less positive for the amino acids containing a polar group.

∆tr go / kJ mol– 1

20 Trp Phe 10

Tyr

Leu Val Thr

Ala 0

Met His

Ser 0

0.5

1.0 1.5 2.0 accessible surface area / nm2

Figure 13.8 Gibbs energies of transfer of amino acids from a nonaqueous medium to water. (From F. M. Richards. Ann Rev. Biophys. Bioeng. 6: 151, 1977.)

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246

Chapter 13

The following example may serve as an indication of the contribution from hydrophobic interaction to the stabilization of a compact protein structure.

N

D

For relatively small proteins the change in As resulting from the N ! D transition may be approximated by applying the equations (13.3) and (13.4). Hence, for a protein of M ¼ 10;000 D, N!D As ¼ 93 nm2. As mentioned before, the interior of such small proteins usually consists of approximately 60% of hydrophobic side chains. Combining these data a value of approximately 560 kJ per mole of protein is calculated for N!D Ghydrophobic int.

13.3.2 Electrostatic Interactions a. Ion Pair Formation The Gibbs energy G of interaction between two point charges Qi and Qj , separated over a distance rij , is given by G¼

ð rij

Qi Qj Qi Qj dr ¼ ; 2 4pee0 rij 1 4pee0 r

ð13:7Þ

where er e0 is the dielectric permittivity of the medium (e is the dielectric constant of the medium and e0 that of vacuum). For water at room temperature, e ¼ 80 and for the interior of the protein e ¼ 5 are taken. It is not possible to give a reliable indication for the value of e at the protein–aqueous solution interface. Charged groups are almost always located at the protein’s surface. If charges are present in the interior of the molecule, they occur as ion pairs. Upon unfolding, such ion pairs are disrupted and the ionic groups become hydrated. This is schematically depicted below.

+ +

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247

The change in the Gibbs energy of the Coulomb interaction due to the disruption of one ion pair in such an unfolding process equals: N!D Gion pair ¼ NAv

ð rij ðDÞ

Qi Qj Qi Qj dr ¼ NAv 2 4pe e r 4pe0 rij ðNÞ r 0

1 1 d e r rij ðNÞ r

ð rij ðDÞ

ð13:8Þ per mole of protein. During an unfolding process in which rij changes from 0.5 nm to 10 nm, eðrÞ varies from 5 to almost 80. Taking, for the sake of simplicity, a constant value of, say, 10 for er (the largest contribution to AN!D GCoul is produced at relatively close proximity between Qi and Qj ) and with Qi ¼ Qj ¼ 1:6 1019 C, it follows that N!D Gion pair ¼ 26 kJ per mole of protein. The Gibbs energy of ion hydration (upon ion transfer from an apolar medium to water) depends on the type of ion; for most ions it is in the range of a few to a few tens of kJ per mole. Hence, the unfavorable change in Coulomb interaction due to ion pair disruption is more or less compensated by the favorable hydration effect upon exposing the isolated ionic groups to the aqueous environment. Moreover, protein molecules usually contain only a few internal ion pairs. For these reasons, ion pair formation in the protein’s interior does not substantially affect the Gibbs energy of stabilization of a compact protein structure. More often ion pairs are found at the aqueous periphery of the protein molecule, where the ions are still (partly) hydrated. Then, ion pair formation will signiﬁcantly contribute to the stabilization of a folded structure. b. Charge Distribution In the folded native structure (N) essentially all the charge is located at the exterior of the protein molecule, whereas in the unfolded denatured structure (D) the charge is more or less homogeneously distributed in the fully hydrated expanded coil. Rg

a R

The electrical part of the Gibbs energy of a given charge distribution Gel equals the reversible isothermal, isobaric work required to place all charges on the originally uncharged molecule (cf. Section 9.3). Reversibility implies that at any

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248

Chapter 13

time of the charging process the charge distribution (including counterions and co-ions) is fully relaxed. ðQ Gel ¼ c dQ; ð13:9Þ Q¼0

where c is the electrostatic potential and Q is the charge of the protein molecule. To solve (13.9) the relation between c and Q must be known. Expressions for cðQÞ are derived for a few models. For example, assuming that the charge is smeared out over the spherical surface, (13.9) gives NAv Q2 kR ð13:10Þ Gel ¼ 1 8pee0 R 1 þ ka and for the charge homogeneously distributed in a spherical volume, NAv Q2 3 9 2 2 2 Gel ¼ ½k R 1 þ ð1 þ kRÞ expð2kRÞ ; 8pee0 R k2 R2 2k5 R5 ð13:11Þ where in (13.10) and (13.11) R is the radius of the sphere, a the distance of closest approach of a counterion to the center of the protein molecule, and k the reciprocal Debye length. By way of example, values of Gel , thus calculated, are given in Table 13.2 for a protein having a molar mass of 40,000 D. According to the values presented in Table 13.2 Gel ¼ 0 for Q ¼ 0 (the isoelectric point). However, under isoelectric conditions the protein, as a rule, still carries charged groups, that is, equal numbers of positive and negative charges. If these charges were homogeneously distributed, this would result in a negative value for Gel (the more so for the NTable 13.2 Gibbs Energies, Gel (kJ mole1 ), of Charge Distributions of a Protein in the Folded (N) and in the Unfolded (D) Statea

Number of charged groups

Q (1018 C)

0 10 20 30 40 a

0 1.60 3.20 4.80 6.40

N-state: R ¼ 2:5 nm, a ¼ 2:7 nm; r ¼ 1:37 g cm3

D-state: R ¼ 5 nm; r ¼ 1:04 g cm3

N!D Gel

0.01 M

0.15 M

0.01 M

0.05 M

0.01 M

0.15 M

0 20 81 182 323

0 10 41 92 163

0 7 27 62 110

0 1 4 10 17

0 13 54 120 213

0 9 37 82 147

T ¼ 298 K; e ¼ 78:5.

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Proteins

249

state than for the D-state). Including discrete charge effects (which are very difﬁcult to account for) would give more realistic results, especially around the isoelectric point of the protein. However, in the case of larger net charges on the protein molecule the difference between the discrete and the smeared-out charge models becomes very small. Hence, an excess of positive or negative charge readily leads to negative values for N!D Gel, implying promotion of the D-state. As expected, the dependence of N!D Gel on pH is reduced at higher ionic strength, because ions shield electrostatic interactions.

13.3.3 Dipolar Interactions In dipolar molecules centers of partial charges dþ and d can be distinguished. If d is the distance between dþ and d, the dipolar moment is given by: ~ ¼ d d~ : m

ð13:12Þ

~ and d~ are from dþ to d . The dipolar moment of a water The directions of m molecule is 1.8 Debye units (1 Debye unit ¼ 3:336 1030 C m) and that of an amide group is 3.7 Debye. Below, equations for the Gibbs energy of interaction for some special cases are given. 1.

Charge–dipole interaction:

Gchdipole ¼ Q

m~ ~r ; 4pee0 r3

ð13:13Þ

where r is the distance between the point charge Q and the dipole moment m~ . 2.

Dipole–dipole interaction (Keesom):

Gdipoledipole ¼ 3.

~ 2 3ð~m1 ~rÞð~m2 ~rÞ=r2 m~ 1 m : 4pee0 r3

ð13:14Þ

Charge-induced dipole interaction:

Gchind:dipole ¼

aQ2 ; ð4p0 Þ2 r4

ð13:15Þ

where a is the polarizability of the molecule. 4.

Dipole-induced dipole interaction (Debye):

Gdipoleind:dipole ¼

am2 : ð4pee0 Þ2 r6

ð13:16Þ

Upon folding the D-structure into the N-structure, dipolar interactions between groups of the protein and water molecules are disrupted, and dipolar interactions between groups of the protein and between water molecules are formed. As a result, dipolar interactions only marginally affect the protein structure. Because of

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250

Chapter 13

the relatively dense packing in the N-state (and, consequently, a somewhat shorter distance between the dipoles) the N-state is probably slightly favored.

13.3.4 Dispersion Interactions Close proximity between atoms leads to synchronization of their orbiting electrons. This causes the induction of dipoles that attract each other (London– Van der Waals interaction). At closer approach, the electron ‘‘clouds’’ overlap, giving rise to repulsion (Born repulsion). The variation of the Gibbs energy of dispersion interactions Gdisp between two atoms with their separation distance r is given by Gdisp ¼

A B þ 12 ; 6 r r

ð13:17Þ

where A and B are constants. Figure 13.9 gives a graphical representation of Gdisp ðrÞ. The most favorable separation is r0 (the so-called Van der Waals distance), which equals the sum of the Van der Waals radii of the interacting atoms. Because of the compactness of the N-state (relative to that of water) dispersion interactions are likely to make a signiﬁcant contribution to the stability of the folded state; however, the overall effect on the protein structure stability is difﬁcult to assess quantitatively.

13.3.5 Hydrogen Bonding Interaction through hydrogen bonds may be classiﬁed between dipole–dipole and covalent interactions (cf. Section 4.2). In proteins by far most of the hydrogen bonding is between an amide and a carbonyl group: Nd Hdþ Od ¼ Cdþ :

Gdisp

ro

r

Figure 13.9 Gibbs energy of interaction between two atoms as a function of their separation distance r; r0 refers to the most favorable separation ( ¼ Van der Waals distance).

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251

The distance between the H-atom and the O-atom is 0.19 nm, whereas on the basis of Van der Waals radii it would be 0.27 nm. Hence, the hydrogen bond involves an ‘‘extra’’ dipole and an ‘‘extra’’ dispersion interaction. Hydrogen bonds are most effective when the dipoles are aligned. For the amide–carbonyl groups GH ¼ 12 kJ mol1 and for hydrogen bonding between water molecules GH ¼ 16:7 kJ mol1 . In the fully hydrated D-state the hydrogen bonds are between peptide and water, and a lesser number between amino acid side groups (R) and water. When the protein folds into the N-structure most of these bonds are disrupted in favor of the formation of hydrogen bonds between peptide units, peptide–R, R–R, and water–water. Peptide units dominate intramolecular hydrogen bonding. The unique number of 3.6 residues, 0.54 nm translation per turn of the a-helix (see Figure 13.3) is due in large part to the strong hydrogen bonds formed between the carbonyl oxygen atom of the ith peptide unit and the amide group of the ði þ 4Þth peptide unit in the polypeptide backbone. The parallel and antiparallel b-sheet structures are also largely determined by favorable H-bond formation (Figure 13.4). Although hydrogen bonding plays an important role in the formation of secondary and tertiary structures of proteins, its inﬂuence on the stabilization of the N-state (relative to the D-state) is not clear at all. In most textbooks it is stated that hydrogen bonds make a large contribution to the stabilization of a compact protein structure. However, experiments with model compounds do not support this view. For instance, dimerization of N-methylacetamide (H3C–CO–N–CH3) in water and in CCl4 to mimic peptide– water and peptide–peptide hydrogen bonding, respectively, reveal that amide– water hydrogen bonds are preferred over amide–amide hydrogen bonds in CCl4. This suggests that peptide–water hydrogen bonding is more favorable than intramolecular peptide–peptide bonding, so that hydrogen bonding by itself does not stabilize the N-state. If, however, for other reasons (such as hydrophobic interactions between R-groups) peptide units are forced into the nonaqueous interior of the protein, formation of hydrogen bonds is strongly promoted, thereby stabilizing secondary (and, possibly, tertiary) structures of the molecule.

13.3.6 Bond Lengths and Angles It is probable that in a tightly packed compact conformation not all the covalent bonds attain the most favorable lengths and angles. Indeed, energy-minimalization calculations point to distortion of covalent bonds in (crystallographic) globular proteins that signiﬁcantly opposes the folded conformation, possibly up to several kJ per mole.

13.4 STABILITY OF PROTEIN STRUCTURE IN AQUEOUS SOLUTION The three-dimensional structure of a protein molecule is the net result of the covalent structure (primary structure), noncovalent interactions, and conforma-

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252

Chapter 13

Table 13.3 Interactions that Determine the Structure of a Protein Molecule in an Aqueous Environment N!D G

Remarks

Electrostatic

> <0

Hydrogen bond Dipole Dispersion

0

0 0

Hydrophobic dehydration

0

Distortion of bond lengths and angles Rotational freedom along the polypeptide chain

<0

Depending on the pH relative to the isoelectric point of the protein Formation of protein–protein and water–water bonds compensated by loss of protein–water bonds Atomic packing densities in compact protein molecules higher than in water Entropy increase of water molecules released from contact with hydrophobic side groups of polypeptide chain Some bonds are under stress in the folded structure

Type of interaction

0

Folding reduces the conformational entropy of the polypeptide chain and, possibly, the side groups

tional entropy. Table 13.3, summarizing the various interactions, compares the compact native (N) state with the completely unfolded denatured (D) state. Which state is promoted depends on the sign of N!D G, which contains an enthalpic and an entropic term: N!D G ¼ N!D H T N!D S:

ð13:18Þ

A negative value for N!D G implies that the N-state is less stable than the Dstate and vice versa. Various contributors to N!D G counteract each other. Hydrophobic interaction and changes in the rotational freedom of the polypeptide backbone, both dominated by entropy effects, are the main competing factors. Hence, the existence of a compact protein molecule in an aqueous environment is a demonstration of the constructive power of chaos: the highly ordered protein structure is preferred because of increased disorder in the surrounding water. As a result of the compensating contributions the native structure is only marginally stable from a thermodynamic point of view. For single-domain proteins maximum values for N!D G are usually some tens of kJ mole1 ; see Tables 13.4 and 13.5. Because of this marginal stability, none of the other factors affecting protein folding are unimportant. As a consequence, (small) changes in environmental conditions such as pH, ionic strength, temperature, or presence of other substances may induce structural rearrangements in the protein molecule. The uncertainties in estimating the contributions of the various interactions are rather large, that is, on the same order of magnitude as the net Gibbs energy of the stabilization of the one structure (N or D) relative to the other. Hence, on the basis of such estimates it is, as a rule, not possible to predict the structure of the protein.

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253

Table 13.4 Gibbs Energy for Stabilization of the N-State, Relative to the D-State, as Estimated in Gu HCl Equilibrium Studiesa Protein 3-Phosphoglycerate kinase Ribonuclease Lysozyme a-Chymotrypsin Penicillase a

N!D G (kJ mol1 ) 20 62 53 43 25

T ¼ 298 K.

It is, however, possible to predict in which ‘‘direction’’ the protein structure changes as a result of changing conditions, such as the following. —

—

pH: Changing the pH away from the isoelectric point the net charge density on the protein molecule increases, and, as a result, intramolecular repulsion increases. This promotes the D-structure. Temperature: Because of the large positive contribution from hydrophobic hydration to the heat capacity (see Section 4.3.2) both N!D H and N!D S become more positive with increasing temperature. This so-called ‘‘enthalpy–entropy compensation’’ results in a relatively small value for N!D G. Furthermore, since N!D Sconfo has a large positive value, T N!D Sconfo becomes proportionally more negative with increasing temperature. As a result, the N-structure becomes less stable at high temperature. In Section 13.5 it is demonstrated that at low temperature the N-state becomes unstable as well. Hence, the Nstate structure is only stable at intermediate temperatures. However, because of the much lower reaction rate the N ! D transition is much slower at low temperature. Table 13.5 Gibbs Energy for Stabilization of the N-State, Relative to the D-State, as Calculated from Differential Scanning Calorimetry Dataa Protein Ribonuclease Lysozyme a-Chymotrypsin Cytochrome-c Myoglobin a-Lactalbumin a

T ¼ 298 K.

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N!D G (kJ mol1 ) 50 60 54 38 54 20

254

Chapter 13

Figure 13.10 Patterns for cooperative unfolding of proteins induced by changing environmental conditions, that is, pH, temperature, and concentration of guanidinium chloride.

—

Denaturants: Addition of denaturants such as guanidinium chloride, (H2N)2C¼N2 þ Cl , or urea, H2NCONH2, increases the solubility of apolar components in water, and consequently, promotes the formation of the D-structure.

Because of the thermodynamically marginal stability of the one structure relative to the other, the N !D transition occurs in a relatively narrow trajectory of changing conditions. It is a highly cooperative process: the disruption of any signiﬁcant portion of the folded structure leads to the unfolding of all the rest. This cooperativity is illustrated in Figure 13.10.

13.5 THERMODYNAMIC ANALYSIS OF PROTEIN STRUCTURE STABILITY When applying reversible thermodynamics, the reversibility of the N !D transition should be checked. This transition generally is reversible, unless the

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number of molecules

Proteins

255

T

∆T

N

T

N

Figure 13.11

T

D

∆T

D

Conformational populations.

denaturing conditions are pushed too far or when the system is kept in the denatured state for too long a period of time. Under such circumstances new (intermolecular) bonds may be formed, preventing the protein from returning to its native state. For various proteins, especially the smaller ones, denaturation follows a two-state transition; that is, it implies that only two states (or, more precisely, two populations of states) N and D exist. Any intermediate state between N and D is not thermodynamically stable. This is illustrated in Figure 13.11. Experimental data such as shown in Figure 13.10 allow the determination of the concentration of N and D, so that the equilibrium constant K of the transition can be determined as K¼

½D ½N

ð13:19Þ

and N!D Go can be calculated according to (3.45) and the (differential) enthalpy of the transition from the Van’t Hoff equation, (3.46). Comparing the differential enthalpy with the integral enthalpy (as determined by calorimetry) provides a test for the two-state assumption: for a two-state transition these enthalpies are equal. When studying protein stability by adding a denaturant the data must be extrapolated to zero concentration of the denaturant. This introduces an uncertainty, the more so because N!D G usually does not depend linearly on the denaturant concentration. As an example Figure 13.12 shows data for 3phosphoglycerate kinase corresponding to the curve shown in Figure z13.10(c). For some proteins, values for N!D G thus estimated, are given in Table 13.4.

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Chapter 13

–10

0

0.5

∆N

D

Go /kJ mol– 1

256

1.0 1.5 cGu HCl / M

10

20

Figure 13.12 Gibbs energy of the unfolding of 3-phosphoglycerate kinase in guanidinium chloride solutions. (From C. Tanford. Adv. Protein Chem. 24:1, 1970.)

Cp / J k– 1

The most detailed thermodynamic analysis of protein structure stability is based on differential scanning calorimetry (DSC). In a DSC experiment the heat capacity Cp of a sample is monitored while heating (or cooling) the sample. Figure 13.13 shows a typical DSC thermogram for heat-induced denaturation of a protein in solution. The thermodynamic observables are the temperature of denaturation Td (the temperature at half peak area), the enthalpy change N!D HðTd Þ involved in the denaturation process (the area under the peak), and the change in the heat capacity N!D Cp of the solution (the shift of the

∆N

D

H

∆N

D

Cp

Td T/ K Figure 13.13 Typical thermogram for thermally induced protein denaturation obtained by differential scanning calorimetry, in which the denaturation temperature Td , the enthalpy of denaturation N!D H, and the change in the heat capacity due to denaturation N!D Cp can be observed.

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Proteins

257

baseline). The thermodynamic difference functions describing the N !D transition are related to these observables by ðT N!D HðT Þ ¼ N!D HðTd Þ þ N!D Cp dT ; ð13:20Þ Td

N!D SðT Þ ¼

N!D HðTd Þ þ Td

ðT N!D Cp d ln T ;

ð13:21Þ

Td

Td T þ Td ðT ðT N!D Cp dT T N!D Cp d ln T :

N!D GðT Þ ¼ N!D HðTd Þ

Td

ð13:22Þ

Td

In principle one could calculate all of these thermodynamic difference functions from a single DSC thermogram of the protein and the well-established assumption that, at least for not too high temperatures, N!D Cp is independent of temperature. However, it is extremely difﬁcult to obtain accurate values for N!D Cp this way. Instead, N!D Cp can be more reliably determined from the slope of the curve relating N!D HðTd Þ to Td . Figure 13.14 shows such a N!D HðTd Þ functionality for lysozyme and a-lactalbumin. The shifts in Td and N!D H were generated by varying the pH of the solution. Both proteins show a linear relation between N!D H and Td, indicating that N!D Cp is independent of temperature so that N!D HðT Þ ¼ N!D HðTd Þ þ N!D Cp ðT Td Þ:

ð13:23Þ

Such a linear relationship is generally found for single domain proteins.

(a)

(b)

350

300

500

∆N

D

H /kJ mol– 1

600

250 400 200 330

340

350 Td / K

330

340

350 Td / K

Figure 13.14 Dependence of the denaturation enthalpy on the denaturation temperature for (a) lysozyme and (b) a- lactalbumin, both dissolved in 0.05 M phosphate buffer pH 7. (From C. A. Haynes and W. Norde. J. Colloid Interface Sci. 169: 313, 1995.)

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258

Chapter 13

For the proteins lysozyme and a-lactalbumin N!D HðT Þ, T N!D SðT Þ, and N!D GðT Þ, as obtained according to the procedure described above, are given in Figure 13.15. N!D H and T N!D S are strongly temperature-dependent, resulting in a pronounced nonlinearity in N!D GðT Þ. These features reﬂect the relative importance of hydrophobic hydration (cf. the curves shown in Figure 4.6). It can be observed in Figure 13.15 that the curve for N!D GðT Þ adopts the shape of a pseudoparabola. It implies the occurrence of denaturation at some low subzero temperature. In recent years, cold denaturation of proteins has been established as a real phenomenon. In view of the two-state approach and because N!D G is a continuous function of T , the cold- and heat-induced D-states are expected to be identical. However, more experimental results, in particular for the low-temperature transition, are required to test this hypothesis. It is furthermore observed in Figure 13.15 that, even under optimum conditions, the native globular protein structure is only marginally stable, that is, about 60 kJ mol1 for lysozyme and 20 kJ mol1 for a-lactalbumin. For a number of proteins N!D G values, at 298 K and at a pH where maximum stability is reached, are summarized in Table 13.5. Note that for the various proteins the values for

80

800 pH 5.3 ∆N

D

40

T∆N

D

S

(a)

–20 –40

0 pH 5.4

400 ∆N

D

20

G / kJ mol– 1

0

200

D

DH

∆N

H or T∆N

60

20 ∆N

D

∆N

G

400

D

S / kJ mol– 1

600

G 0

200 ∆N

D

H –20

T∆N

0 273

293

D

(b)

S

313

333

353 T/K

–40 373

Figure 13.15 Denaturation enthalpy (N!D H), entropy (N!D S) and Gibbs energy (N!D G) for (a) lysozyme and (b) a-lactalbumin, both in 0.05 M KCl solution. (From C. A. Haynes and W. Norde. Colloids Surfaces B: Biointerfaces 2: 517, 1994.)

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N!D G are in the range of a few tens of kJ mol1 , which is not more than the Gibbs energy change associated with the formation of only a few H-bonds in the protein’s interior. So far, the discussion on protein denaturation may be an oversimpliﬁcation. It is based on the assumption of a reversible two-state transition, N !D. Protein molecules that are built of one folded domain (most proteins having a molar mass of, say, < 30,000 D) usually satisfy the two-state assumption. Larger protein molecules often comprise two or more domains that may unfold more or less independently. This may show up by two (or more) peaks in the DSC thermogram and a corresponding number of transitions in various physical properties associated with the protein structure. Although the denaturation process may be reversible, experimental veriﬁcation of the reversibility is often obstructed by irreversible interactions following unfolding of the protein molecules. This phenomenon is further explained in Section 13.6.

13.6 REVERSIBILITY OF PROTEIN DENATURATION. AGGREGATION OF UNFOLDED PROTEIN MOLECULES In many cases protein denaturation seems to occur irreversibly, that is, after release of the denaturing condition (extreme temperature and=or pH, denaturing agent, etc.) the protein does not regain its original 3-D-structure. Prevention of refolding can have different causes. The most probable ones are: —

Upon unfolding the protein molecule exposes many hydrophobic residues. This leads to a decreased solubility in water and a tendency to aggregate through hydrophobic interaction. When the unfolded protein contains free SH (cysteine residue) of which at least a part is present in the ionized form (at pH > 6), reshufﬂing of disulﬁde bridges may occur. Interchange of disulﬁde bridges among different protein molecules produces aggregates.

—

A simple model that is consistent with the observed irreversibility is a reversible unfolding step followed by an irreversible step that locks the unfolded protein in a state from which it does not refold: k1

k2

N ! D ! A; k1

where A is the ﬁnal (aggregated) state of the protein arrived at from the reversibly denatured state D and the ks are the ﬁrst-order kinetic constants. If k2 k1 essentially all molecules in state D are converted into state A instead of returning to the native state N.

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For various proteins the formation of aggregates or even a macroscopic protein gel has been observed. Moreover, the structure of the aggregates or the gel may depend on the denaturing conditions, notably on the magnitude of the kinetic constants relative to each other. When k1 k2 , which is the case at a temperature substantially below the denaturation temperature Td , at a pH sufﬁciently away from the denaturing pH, and so on, the concentration of molecules in the D-state is so low that the aggregation will proceed slowly. At a higher rate of denaturation aggregation also will be faster. It could well be that at high aggregation rates denatured molecules are incorporated in the aggregates before they have sufﬁcient time for complete unfolding. Knowledge of the kinetics of these processes is highly relevant for optimizing the structure of a protein gel, which, in turn, determines its rheological properties, for example, its possible penetration by micro-organisms and proteolytic enzymes.

EXERCISES 13.1

Comment on the following statements. The a-helix structure of a polypeptide dissolved in water is stabilized by adding tetra chlorocarbon (CCl4) to the solution. (b) Cold-denaturation of proteins in aqueous solution is accompanied by an entropy increase of the system. (c) The secondary structures a-helix and b-sheet in globular proteins dissolved in water become more stable with increasing pressure.

(a)

13.2

Globular protein molecules in an aqueous environment are sometimes called ‘‘nature’s own oil droplets.’’ (a)

What is the main similarity between an oil droplet in water and a globular protein molecule in water? In fact, globular protein molecules in water resemble (spherical) surfactant micelles in water rather than dispersed oil droplets. (b) Indicate two relevant similarities between a micelle and a globular protein molecule in an aqueous medium. Also indicate two principal differences. (c) Consider the processes of micellization and protein folding. Discuss for both processes the main driving force and the main opposing force. (d) At room temperature, protein folding generally is an exothermic process, whereas micellization of various anionic surfactants proceeds athermally. Discuss the principal reasons for this difference. 13.3

Globular proteins dissolved in water may be denatured (N ! D) by exposing the solution to high but also to cold temperature. For a solution

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261

of myoglobin the functionality ND GðT Þ is graphically presented in the ﬁgure.

(a) What are the principal differences between the N- and D-states? (b) What is the physical meaning of ND G? (c) What are the most important factors determining the stability of the native structure? How do the contributions of each of these factors change in the rising part of the right-hand side of the curve? (d) At which temperatures are the concentrations of the N- and D-states the same? (e) Under most conditions the structure of the protein molecule is the result of enthalpy–entropy compensation. Suppose the following empirical relations ND H ¼ 1:56 102 T 2 þ 1427 J mol1 ; ND S ¼ 3:12 102 T þ 9:49 J K1 mol1 : Derive the temperature range over which the native state is promoted both by enthalpy and entropy effects. (f)

What are the signs of the entropy changes in heat denaturation and in cold denaturation?

SUGGESTIONS FOR FURTHER READING T. E. Creighton. Proteins, Structures and Molecular Properties, 2nd edition, New York: W.H. Freeman, 1993. G. D. Fasman (ed.). Prediction of Protein Structure and the Principles of Protein Conformation, New York: Plenum, 1990. F. Franks (ed.). Protein Biotechnology, Totowa, NJ: Humana, 1993. R. B. Gregory (ed.). Protein-Solvent Interactions, New York: Marcel Dekker, 1995.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Charcoal as a Life Saver

Activated charcoal contains an internal network of very ﬁne pores that constitute an extensive surface area. One gram of this material typically exposes a few thousand square meters of surface area. The surface has a high afﬁnity for adsorbing all kinds of substances. For this reason activated charcoal has a longstanding reputation for use in air and water ﬁlters, in gas masks, and most eminently, as a universal antidote to drugs, chemicals, and toxins. Charcoal is inert to the body. In other words, it slides through the stomach and intestines without being absorbed or metabolized. On its way it binds (toxic) substances, carrying them throughout the digestive system so that they can be eliminated from the body. By doing so, activated charcoal has saved the lives of people that have consumed lethal doses of poisons such as microbial toxins in contaminated food, household chemicals, sedatives, party drugs, and pain killers. (Figure courtesy of Norit N.V., Amersfoort, The Netherlands.)

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14 Adsorption

Adsorption is the spontaneous accumulation of matter at an interface. We came across this phenomenon in different contexts in foregoing chapters, for instance, in Section 3.11 where the Gibbs adsorption equation is derived and in Section 7.3 where adsorbed monolayers (Gibbs monolayers) at liquid interfaces are distinguished from spread monolayers (Langmuir monolayers). The term ‘‘adsorption’’ is not only used to indicate the process of interfacial accumulation but it often refers to the amount of accumulated material as well. Alternatively, the amount of material accumulated at an interface, expressed per unit area, may be denoted as ‘‘interfacial concentration’’ or ‘‘interfacial load.’’ The adsorbed substance is called adsorbate and the material at which adsorption occurs is the (ad)sorbent. Adsorption may take place from a gas phase onto a solid or a liquid (i.e., at a S=G or a L=G interface) or from solution onto a solid or a liquid phase (at a S=L or a L=L interface). The major difference between adsorption from a gas and from a solution is that in the latter case it is an exchange process: molecules of the adsorbing solute displace solvent molecules from sites at the sorbent surface. The reasons for the tendency of a compound to accumulate at an interface may be various. Whatever interactions are involved, for spontaneous adsorption to occur, the Gibbs energy ads G (at p and T constant) or the Helmholtz energy ads F (at V and T constant) should be negative. According to the Gibbs’ adsorption equation (3.84) or variations thereof, this manifests itself by a decrease of the interfacial tension. For instance, any gas adsorbs at any surface by virtue of Van der Waals interactions between adsorbate and adsorbent. Clearly, the translational freedom of a molecule in the adsorbed state is more restricted than in the gaseous state and so is the translational entropy. Adsorption from the

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Chapter 14

gas phase must therefore be driven by lowering the energy of the system: gas adsorption is always an exothermic process. Applying Le Chatelier’s principle it follows that the afﬁnity for adsorption decreases with increasing temperature. At high temperature the entropy term T ads S may be so large that it outweighs the reduction in energy ads U, resulting in a positive value for ads F. ads Fð¼ ads U T ads SÞ > 0 implies that, at constant T and V, adsorption does not take place spontaneously. In passing it is noted that for gas adsorption the condition of constant volume is much more practical than constant pressure. For adsorption from solution both pressure and volume are usually (essentially) constant and, hence, ads U ¼ ads H and therefore ads F ¼ ads G. Adsorption from solution is a much more complex process because it is the result of exchanging solvent (1) for solute (2): u1 ð1Þs þ u2 ð2Þ ! u1 ð1Þ þ u2 ð2Þs ; where the us are the stoichiometric coefﬁcients and the superscript s indicates the interface. The change in Helmholtz energy for exchanging 1 against 2 at the surface is composed of four contributions: ads F21 ¼ u2 f2s þ u1 f1 u2 f2 u1 f1s ; or for the molar Helmholtz energy of adsorption of component 2, u u ads f21 ¼ f2s þ 1 f1 f2 1 f1s : u2 u2

ð14:1Þ

ð14:2Þ

Equations (14.1) and (14.2) reveal that the driving force for the adsorption of component 2 may originate from preferential interaction of that component with the sorbent material but it could as well be due to unfavorable interactions between solute and solvent. The adsorbate may desorb (i.e., leave the interface) on changing conditions such as temperature, pressure, concentration of solute, and so on. If adsorption and desorption occur along the same path but in opposite directions the (ad)sorption process is reversible. However, with various systems, for instance, with polymers and proteins, adsorption and desorption are usually not each other’s reverse, at least not during the time of observation. The adsorption process is then considered to be irreversible. Adsorption may be subdivided into physisorption and chemisorption. In physisorption the interactions are of physical nature, whereas chemisorption involves the formation of chemical bonds between adsorbate and adsorbent. Hence, the interactions in chemisorption are much stronger than in physisorption. In this chapter we deal with physisorption only. Phenomena that are closely related and, in fact, analogous to adsorption are adhesion and (physical) binding. Where ‘‘adsorption’’ refers to molecules at an interface, the term ‘‘adhesion’’ is used for the accumulation of (micro)particles at an interface and ‘‘binding’’ is used for small molecules (ligands) at a larger one.

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265

The adsorption theories discussed in this chapter may therefore, mutatis mutandis, also apply to adhesion and binding processes. Gas adsorption is often applied to determine the speciﬁc surface area ( ¼ surface area per unit mass) of a porous material. From the known size of the gas molecules, the adsorbed mass in a complete monolayer and the mass of sorbent material, the speciﬁc surface area can be derived. Adsorption of compounds from solution, having well-deﬁned dimensions, may also be used to determine the speciﬁc surface area. In many biological and technological processes adsorption from solution is of crucial importance. Most of these processes take place in aqueous media. Examples are the adsorption of toxic or other adverse components on active carbon in the production of potable water, adsorption of contaminants from the gastrointestinal tract on medical carbon, adsorption and desorption of herbicides and pesticides on plant and soil materials, and adsorption of surfactants to inﬂuence the wettability of surfaces. Applications of adsorption are given further attention in Section 14.5. Because of its characteristic features and also because of its relevance in a wide variety of biotechnological and biomedical applications adsorption of polymers, in particular, proteins, is dealt with in a separate chapter.

14.1 ADSORBENT–ADSORBATE INTERACTIONS Gas adsorption is universal. At sufﬁciently low temperature, where the loss in translational entropy of the gas molecules does not outrule the favorable Van der Waals interactions between adsorbent and adsorbate, any gas adsorbs at any surface. In addition to Van der Waals interactions, other interactions such as hydrogen bonding may contribute as well. Clearly, the energy and entropy effects involved vary from system to system. The adsorption entropy and, in many cases, the adsorption energy depend on the degree of coverage of the sorbent surface by the adsorbate. The positioning of the adsorbate molecules affects the adsorption entropy. Random packing is entropically least unfavorable, but sorbent surface heterogeneity and=or lateral interaction between the adsorbed molecules may interfere with such packing. Many surfaces are heterogeneous with respect to energy effects; that is, the energy decrease associated with the ﬁrst adsorbing molecules is larger than that with the later ones. Adsorption from solution is always a competitive process: as indicated by (14.1), for adsorption to occur the interaction between the solute molecules and the sorbent material is preferred over the interaction between solvent and sorbent and=or the solvent is of poor quality. Various types of interactions may be involved in the exchange process. 1.

Van der Waals interactions (including Debye, Keesom, and London interactions). Dispersion ( ¼ London–Van der Waals) interactions are

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266

Chapter 14

2.

3.

Γ / mmol g– 1

4.

always operating. Dipolar interactions (Debye– and Keesom–Van der Waals interactions) occur with polar and polarizable components (cf. Section 13.3). Hydrogen bonds may be effective at sorbent surfaces where hydroxyl, amino, or other electron-donating or electron-accepting groups are present. In aqueous media the contribution of hydrogen bonding to the adsorption afﬁnity is often marginal because of the compensation of adsorbent–adsorbate and water–water contributions, and those of adsorbent–water and adsorbate–water. Coulomb ( ¼ charge–charge) interactions are effective when both the sorbent surface (including the adsorbate) and the adsorbing solute molecules are electrically charged. Coulomb interactions play a central role in the formation of the electrical double layer at a charged interface (Chapter 9). Hydrophobic interaction due to dehydration of apolar parts of the sorbent and solute surfaces occurs exclusively in an aqueous environment. For a more detailed description of hydrophobic interaction please refer to Section 4.3.2. The contribution from hydrophobic interaction often dominates over those from other types of physical interaction. The involvement of hydrophobic interaction is, for instance, reﬂected in the adsorption isotherms of a series of hydrocarbon acids from aqueous solution on active carbon. They are displayed in Figure 14.1. The adsorption afﬁnity gradually increases

4

4

5

3

3

2 n =1

2 1

0

20

40

60

80 c / mmol dm– 3

100

Figure 14.1 Adsorption isotherms of hydrocarbon acids, H(CH2)n COOH from water on active carbon at 25 C.

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Adsorption

267

with increasing size of the hydrocarbon moiety. Anticipating the discussion in Section 14.3 it is mentioned that the adsorption isotherms can be described by a Langmuir equation. It then allows calculation of the increment per CH2 group in ads f ð ads gÞ which amounts to 2.8 kJ mol1 . This is less than the molar Gibbs energies for transfer of these compounds from water to their own medium (g ¼ 3:8 kJ per mol CH2; see Section 4.3.2), but more than for micellization of an ionic surfactant (g ¼ 1:7 kJ per mol CH2; see Section 11.5).

14.2 ADSORPTION KINETICS The rate of adsorption comprises two steps: transport of the gas or solute molecules towards the interface and interaction with the sorbent surface. These steps are considered separately in the following. For the sake of simplicity a solution containing one adsorbing component is considered. The same reasoning applies to adsorption from the gas phase. In that case, the composition should be expressed in (partial) pressure instead of concentration.

14.2.1 Transport Towards the Interface The basic mechanisms of transport towards the interface are diffusion and convection by laminar or turbulent ﬂow. As adsorption proceeds the solution near the interface, the subsurface region, becomes depleted. In the absence of convection the ﬂux J from the bulk solution towards the subsurface region is given by J ¼ ðcb cs Þ

1=2 D ; pt

ð14:3Þ

where cb and cs are the solute concentrations in the bulk solution and in the subsurface region, respectively, D is the diffusion coefﬁcient, and t the contact time between the solution and the sorbent surface. However, most transport processes take place under steady-state convective diffusion, driven by a (linear) concentration gradient. This results in J ¼ ktr ðcb cs Þ;

ð14:4Þ

where ktr is a transport rate constant that depends on the hydrodynamic conditions and the diffusion coefﬁcient.

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268

Chapter 14

ka

kd

Figure 14.2 Molecule interacting with an interface through adsorption and desorption.

14.2.2 Interaction with the Interface: Attachment and Detachment Consider the simple case depicted in Figure 14.2. The adsorbate molecules may attach at, and detach from, the sorbent surface. This gives rise to two ﬂuxes, one forward, dG=dtjþ , and one backward, dG=dtj . The net ﬂux ( ¼ adsorption rate) equals dG dG dG ¼ : dt dt þ dt

ð14:5Þ

The forward ﬂux responds to cs and the fraction of unoccupied sorbent surface area, (1 y), as dG ¼ ka ð1 yÞcs ; dt þ

ð14:6Þ

where ka is the attachment rate constant and y G=Gmax, Gmax being the adsorbed amount when the sorbent surface is saturated with adsorbate. Hence, dG=dtjþ varies linearly with dy=dtjþ . This deﬁnition of y holds for molecules that do not change their conformation when they adsorb. This is the case for small rigid molecules. Polymers, including proteins, may show adsorption-induced conformational changes. The relation between G and y then usually is more intricate; this is discussed in Section 15.1. The value of ka is lowered by any repulsive barrier to attachment. The origin of such a barrier might be electrostatic repulsion, solvation effects, or it could be that the residence time at the sorbent surface for a fraction of the molecules is too short for attachment. Because the repulsive barrier may be affected by the degree of coverage of the surface by the adsorbate, ka may vary with y and, hence, with time.

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269

The backward ﬂux is approximated as dG ¼ kd y; dt

ð14:7Þ

where kd is the detachment rate constant. As y increases, the backward ﬂux increases. Eventually, a steady-state (dG=dtjþ dG=dtj ¼ 0) is reached which implies that ka ð1 yÞceq ¼ kd y;

ð14:8Þ

where ceq is the equilibrium concentration of solute corresponding to the equilibrium value of G. Then cb ¼ cs ¼ ceq . The ratio between the rate constants for the forward and backward ﬂuxes is deﬁned as the equilibrium constant for adsorption Kð ka =kd Þ. Off-equilibrium dG=dtj is still given by ka ð1 yÞceq, provided that dG=dtj is uniquely determined by kd and y. Hence, we may write for the net adsorption rate dG dG dG ¼ ¼ ka ð1 yÞðcs ceq Þ: dt dt þ dt

ð14:9Þ

Combining (14.4) and (14.9), with ðdG=dtÞ ¼ J , we derive cb ceq ðyÞ dG ¼ : dt 1 1 þ ktr ka ð1 yÞ

ð14:10Þ

To obtain an explicit expression for dG=dt the adsorption isotherm ceq ðyÞ, or for that matter ceq ðGÞ, must be known. Equations describing adsorption isotherms are derived in Section 14.3. Anticipating the discussion in Section 14.3 we mention here that for an ideal gas or an ideally diluted solution ceq ðyÞ may be expressed as ceq ¼ ð1=KÞðy=ð1 yÞÞ. After substituting this dependency in (14.10) it can be inferred that at low yðy ! 0Þ dG=dt is determined by ð1=ktr þ 1=ka Þ, which means that if ktr ka the transport is the rate-limiting step and if ktr ka the attachment determines the adsorption rate. At high y the probability for an arriving molecule to ﬁnd an empty adsorption site is very low so that for y ! 1 the adsorption rate is governed by the attachment step.

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Chapter 14

14.3 ADSORPTION EQUILIBRIUM: A THERMODYNAMIC APPROACH Then molecules are distributed between the ‘‘free’’ state and the adsorbed state, the equilibrium situation generally obeys Boltzmann’s law h u u i Ni;a Oi;a i;b ¼ exp i;a ; ð14:11Þ Ni;b Oi;b RT where Ni is the number of molecules of component i, ui is its partial molar energy, and Oi the degeneracy of the adsorbed or free state ( ¼ bulk phase) indicated by the subscripts a and b, respectively. The degeneracy Oi , is deﬁned as the number of states accessible to the molecules of i in the system, at given ui . Equation (14.11) can be written as ui;a þ RT ln

Ni;a N ¼ ui;b þ RT ln i;b : Oi;a Oi;b

ð14:12Þ

Deﬁning the partial molar entropy si as si R lnðNi =Oi Þ, we obtain ui;a Tsi;a ¼ ui;b Tsi;b

ð14:13Þ

and with ui Tsi fi , where fi is the partial molar Helmholtz energy of i, fi;a ¼ fi;b :

ð14:14Þ

At constant temperature and volume fi ¼ mi , where mi is the chemical potential of i, so that mi;a ¼ mi;b :

ð14:15Þ

Equation (14.15) expresses the criterion for adsorption equilibrium. In ideal solutions (and gases) ui depends on the interaction of i with its environment but is independent of the composition of the system. However, si depends on the interaction with the environment but also on the composition which is expressed in the mole fraction or the concentration of i. The latter dependency is the result of the conﬁgurational possibilities of the molecules of i in the system.

14.3.1 Conﬁguration Entropy Assume an interfacial area which comprises N0 independent and identical adsorption sites. If Ni;a molecules of i are adsorbed the number of distinguishable is given by conﬁgurations of i at the interface Oconfig i;a Oconfig ¼ i;a

N0 ! : Ni;a !ðN0 Ni;a Þ!

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271

The integral conﬁguration entropy of the adsorbate can be calculated using Eq. (3.5), which after using the Stirling approximation (ln x! ¼ x ln x x, for large values of x) results in config Si;a ¼ kB N0 fyi ln yi þ ð1 yi Þ lnð1 yi Þg

ð14:16Þ

with yi Ni;a =N0 . Because ni;a ¼ ðN0 =Ni;a Þ y, with ni;a the number of moles of adsorbed i, the molar conﬁguration entropy si;a , deﬁned according to (3.19), can be expressed as ¼ R ln sconfig i;a

yi : 1 yi

ð14:17Þ

A similar reasoning with respect to the bulk leads to sconfig ¼ R ln i;b

Xi ; 1 Xi

ð14:18Þ

where Xi is the mole fraction of i. Now, (14.11) can be rewritten, and yi Xi ¼ exp½ads ui =RT 1 yi 1 Xi

ð14:19Þ

may be derived, where ads ui ui;a ui; f , the change in molar energy due to adsorption. Thus Eq. (14.19) describes equilibrium adsorption under conditions where entropy effects resulting from changes in interactions are negligible. Below, we include the inﬂuence of interaction entropy as well.

14.3.2 Interaction Entropy When molecules adsorb, they may do so in a preferred orientation. Also, changes in the interaction with solvent molecules may occur, for instance, when adsorption is driven by hydrophobic dehydration. Therefore, degeneracies not only contain a conﬁgurational contribution but also a contribution from interactions: Oint Oi ¼ Oconfig i . It follows that i ! Ni;a 1 sa ¼ R ln config þ ln int ¼ sconfig þ sint ð14:20Þ i;a : i;a Oi;a Oi;a int Because for ideal systems sint i;a is independent of Ni;a , si;a is combined with ui;a int (which is also independent of Ni;a ): ui;a Tsi;a fi;a , being the partial molar Helmholtz energy of i (without taking into account the contribution from the conﬁgurational entropy). Following the same reasoning, ui;b Tsint i;b fi;b . With ð fi;a fi;b Þ ads fi , (14.19) becomes yi Xi f ¼ exp ads i : ð14:21Þ 1 yi 1 Xi RT

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Chapter 14

Thus, unlike (14.19), Eq. (14.21) accounts for the change in the interaction entropy due to adsorption. The foregoing discussion is based on applying the Boltzmann equation to only one component. However, in practice, we usually deal with mixtures, for example, solutions, in which at least two and often more components are present. Starting from (14.11), for each component 1 and 2 in a two-component system: N1;a ¼ N1;b exp½ads f1 =RT and

(14.22) N2;a ¼ N2;b exp½ads f2 =RT :

As discussed above, ads f does not include the conﬁguration entropy. The excess at the interface of component 2 relative to that of component 1 may be expressed as N2;a =N1;a , for which N2;a N2;b ads f ; ð14:23Þ ¼ exp RT N1;a N1;b in which ads f ads f2 ads f1 . Equation (14.23) applies to any pair of components, that is, also to 1 and 3, 1 and 4, and so on. The interfacial concentration of any component may be expressed relative to that of Pa reference = component (for which the solvent is usually taken). Using N 1;b i Ni;b Xi P P P and X ¼ 1, and N = N y and y ¼ 1, it follows for the i i;a i;a i i i i i adsorption of a solute (component 2) from solution in a solvent (component 1), y X ads f ¼ exp ; ð14:24Þ 1y 1X RT where y and X refer to the degree of interfacial coverage and mole fraction in solution, respectively, of component 2. Equations (14.19), (14.21), and (14.24) are theoretical descriptions of adsorption isotherms; they are referred to as Langmuir equations. For dilute solutions, X 1, the Langmuir equation approaches y ¼ KX ; 1y

ð14:25Þ

where K is the equilibrium constant for adsorption and K is related to the Helmholtz energy of adsorption: RT ln K ¼ ads f :

ð14:26Þ

According to (14.25) the Langmuir adsorption isotherm assumes a curved shape as shown in Figure 14.3(a). Equation (14.25) may be linearized into y1 ¼ 1 þ K 1 X 1

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ð14:27Þ

Adsorption

273

(a)

(b) θ –1

1

θ=1

Γ = Γmax

θ

slope: K –1= exp +

X

∆ ∆ ads f RT

X–1

Figure 14.3 (a) Langmuir adsorption isotherm and (b) linearized Langmuir adsorption isotherm.

so that a plot of y1 against X 1 yields a straight line with slope K 1 ; see Figure 14.3(b). Substituting G=Gmax for y (where G is the adsorbed mass per unit interfacial area and Gmax its maximum value), (14.27) becomes 1 1 ; G1 ¼ G1 max þ ðGmax KÞ X

ð14:28Þ 1

indicating that Gmax can be derived from the intercept of a plot of G versus X 1 . It should be realized that ﬁtting the Langmuir model to the experimental data makes sense only if the conditions underlying the Langmuir theory are fulﬁlled. These are adsorption equilibrium and identical and independent adsorption sites. It implies localized adsorption and no interaction between the adsorbed molecules. It may be obvious that in particular the last-mentioned condition is often not met in practice. Below, a modiﬁcation of the theory is discussed that takes interaction between adsorbed molecules into account.

14.3.3 Nearest-Neighbor Interactions When the interfacial coverage increases adsorbate molecules may occupy neighboring adsorption sites. As a rule, interaction between adsorbate and solvent molecules is different from those between adsorbate molecules and between solvent molecules. As a ﬁrst approximation only interactions between nearest neighbors are considered. Nearest-neighbor (nn) interactions may be visualized in a lattice, as is two-dimensionally shown in Figure 14.4. The number of contacts a central molecule (d) has with other molecules of each type (s and 3 in a bicomponent system) has to be counted. If the system is divided in layers, numbered 1; 2; . . . ð‘ 1Þ; ‘; ð‘ þ 1Þ; . . . , the central molecule in layer ‘ has z neighbors of which a fraction l0 is in the same layer ‘ and fractions l1 in each of the neighboring layers (‘ 1) and (‘ þ 1). For instance, in a cubic lattice l0 ¼ 4=6 and l1 ¼ 1=6.

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Chapter 14

layer number l+1 l l–1

Figure 14.4 Nearest-neighbor interactions on a cubic lattice.

Assuming that in each layer the molecules are randomly distributed, the average surrounding hyið‘Þ i, of the central molecule in layer ‘ by molecules of component i is given by hyið‘Þ i ¼ l1 yið‘1Þ þ l0 yið‘Þ þ l1 yið‘þ1Þ ;

ð14:29Þ

where yi is the degree of occupation of the layer indicated by the index, by molecules of type i. In bulk solution the components are homogeneously distributed over the lattice, that is, yið‘1Þ ¼ yið‘Þ ¼ yið‘þ1Þ Xi , so that hyið‘Þ i ¼ l1 Xi þ l0 Xi þ l1 Xi ¼ Xi . At a surface we deal with an asymmetric situation. Deﬁning the ﬁrst layer at the surface as layer ‘ and assuming that the surface is not penetrable for i, which implies that layer ð‘ 1Þ does not exist, it follows that hyið‘Þ i ¼ l0 yið‘Þ þ l1 yið‘þ1Þ . For a two-component system (e.g., a solution of a single component), it is common to refer to the solvent as component 1 and to the solute as component 2. Assuming that component 2 adsorbs only in the ﬁrst layer ‘, and that the solution is diluted such that l1 Xi is negligibly small, it follows that at the surface hyð‘Þ i ¼ l0 yð‘Þ l0 y. Next, we deﬁne the w-parameter that indicates whether contacts between molecules of the same kind or contacts between molecules of different kinds are preferred [cf. (11.7) and (12.1)], z 1 f ð f þ f22 Þ ; ð14:30Þ w RT 12 2 11 where f is the molar Helmholtz energy of interaction between the components indicated by the subscripts. w > 0 implies that contacts between molecules of the same kind are preferred. The change in the Helmholtz energy due to altered nearest-neighbor interactions, resulting from adsorption, may now be calculated for the solvent as nn ¼ RT wðhyð‘Þ i X Þ fsolvent

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ð14:31Þ

Adsorption

275

and for the adsorbate (solute) as nn ¼ RT wfhð1 yÞð‘Þ i ð1 X Þg: fadsorbate

ð14:32Þ

During adsorption solvent molecules at the interface are replaced by molecules of the adsorbate; therefore ads f nn ¼ RT w½fhð1 yÞð‘Þ i ð1 X Þg fhyð‘Þ i X g:

ð14:33Þ

Because hyð‘Þ i þ hð1 yÞð‘Þ i ¼ 1 l1 , hy‘ i hð1 yÞ‘ i ¼ 2l0 y l1 1. Substituting these equalities in (14.33) gives ads f nn ¼ 2wðl0 y X Þ þ l1 w; RT

ð14:34Þ

which for dilute solutions (X 1) approaches ads f nn ¼ 2wl0 y þ l1 w: RT

ð14:35Þ

The contribution ads f nn =RT is taken up in the Boltzmann factor in (14.21) and, hence in (14.24), y X ads f ¼ exp þ 2l0 wy l1 w : ð14:36Þ 1y 1X RT Equation (14.36), known as the Frumkin–Fowler–Guggenheim equation, reveals the inﬂuence of nearest-neighbor interactions on the adsorption isotherm. Preferred interactions between molecules of the same kind (w > 0), results, at low surface coverage (l1 w > 2l0 wy), in a less positive value for the exponential factor and therefore in a lower value for yðX Þ. At high surface coverage (l1 w < 2l0 wy), yðX Þ increases due to preferred interactions between molecules of the same kind. Following similar reasoning, preferred interactions between molecules of different kinds causes enhanced adsorption in the initial part of the isotherm and suppressed adsorption when the surface becomes crowded with adsorbate molecules. The inﬂuence of nearest neighbor interactions is zero for 2l0 wy ¼ l1 w; that is, for y ¼ l1 =2l0. These effects are shown in Figure 14.5.

14.3.4 Cooperativity It is not exceptional that adsorption of the ﬁrst molecules promote adsorption of the subsequent ones, and so on. This phenomenon is called co-operative adsorption. It may be the result of lateral attraction between adsorbed molecules or by adsorption-induced changes in the interface leading to exposure of sites having a higher afﬁnity for the adsorbate. In such a co-operative process adsorption of the next molecule is coupled to that of the previous one. Let us discuss this phenomenon on the basis of lateral attraction between adsorbed molecules.

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276

Chapter 14

1 χ>0

θ

χ=0

Langmuir χ<0

0 X

Figure 14.5 Frumkin–Fowler–Guggenheim adsorption isotherms for different interactions between solvent and adsorbate molecules.

Attraction between adsorbed molecules leads to aggregate formation at the interface. The number p of molecules per aggregate is a compromise of the favorable lateral interaction and the unfavorable reduction of the conﬁguration entropy of the aggregated molecules in the adsorbed state. Because, with respect to conﬁgurational possibilities, the adsorbed aggregates should be considered as single entities, the molar conﬁguration entropy of the adsorbate reduces with a factor p. Consequently, (14.17) should be modiﬁed to become ¼ ðR=pÞ ln y=ð1 yÞ, and this leads to sconfig a p y X p ads f ¼ ; ð14:37Þ exp 1y 1X RT the so-called Hill equation. The value of p, which is a measure of the degree of cooperativity, can be derived by plotting ln y=ð1 yÞ versus ln X =ð1 X Þ, or, for dilute solutions, versus ln X . By way of example, Figure 14.6 shows Hill plots for the binding of oxygen at myoglobin and at hemoglobin. Hemoglobin binds oxygen in a cooperative manner, whereas that is not the case for myoglobin.

14.3.5 Adsorption of Ions For the adsorption of charged species (i.e., ions), an electrical term must be included in the criterion for adsorption equilibrium (14.15), mi;a þ zi Fca ¼ mi;b þ zi Fcb ;

ð14:38Þ

where ca and cb are the electrostatic potentials at the interface and in bulk solution, respectively. Deﬁning cb 0 and ca c, y X ads f þ zFc ¼ exp : ð14:39Þ 1y 1X RT

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Adsorption

277

log

2 θ 1– θ 1 0

myoglobin p=1

–1

hemoglobin pmax = 3

–2 –1

0

1

2 log pO

2

Figure 14.6 Hill plots for the binding of oxygen to hemoglobin and myoglobin. pO2 is taken in mm Hg. (From J. D. Rawn. Biochemistry, Burlington, NC: Neil Patterson, 1989, Chapter 6.)

Equation (14.39) is an implicit equation, because c is a function of s, which, in turn, is directly related to the degree of surface coverage y. Furthermore, c is inﬂuenced by the ionic strength of the medium. Evaluation of cðsÞ requires a model for the charge distribution in the electrical double layer. The models most currently applied are presented in Section 9.4. The diffuse double layer according to Gouy and Chapman gives 2RT zF s arc sinh c¼ ; ð14:40Þ zF 2RT ee0 k which, for low values of s (and, hence for low values of c, say, c 50 mV), reduces to s : ð14:41Þ c¼ ee0 k For the deﬁnition of k please refer to (9.29). The interfacial charge density s is determined by ion adsorption s ¼ y y*; ð14:42Þ smax where smax is the maximum charge density, that is, the difference between the charge densities corresponding to y ¼ 0 and y ¼ 1, respectively; y* is the surface coverage for which s ¼ 0. It is obvious that for surfaces containing only one type of ion-adsorbing group y* ¼ 0 or y* ¼ 1. For amphoteric surfaces 0 < y* < 1. Combining equations (14.39), (14.41), and (14.42) gives y X ads f þ zFðy y*Þsmax =ee0 k ¼ exp : ð14:43Þ 1y 1X RT

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278

Chapter 14

For 0 < y* < 1 it can be derived that zFc y y* X* X ¼ ln ln þ ln ln ; ð14:44Þ RT 1y 1 y* 1 X* 1X where X * is the mole fraction for which y ¼ y*. In the case of a surface that attains a high value for smax (e.g., most metal oxide surfaces, silica, glass, quartz, silver halides, etc.) at y relatively close to y* zFc already reaches a high value. Further charging will then be severely hampered by the increasing potential. Under such conditions and for dilute solutions (both X and X * 1):

zFc

ln X * ln X RT

ð14:45Þ

or RT RT ln X * þ ln X ; ð14:46Þ zF zF which is known as the Nernst equation for the surface potential [cf. (9.14)]. c¼

14.4 BINDING OF LIGANDS Buiding of small molecules to large ones (macromolecules) could, in principle, be treated using the adsorption theories discussed in the foregoing sections. However, it is more common to present ligand binding in terms of the multiple equilibrium theory. Let P be a polymer molecule containing j independent identical sites to bind ligand A. Then, for the ﬁrst molecule of A that binds to P: P þ A ! PA

with

K1 ¼

½PA ½P½A

(the square brackets indicate concentration) and for the subsequent molecules of A that bind: PA þ A ! PA2 PA2 þ A ! PA3 PAj1 þ A ! PAj

½PA2 ½PA1 ½A ½PA3 with K 3 ¼ ½PA2 ½A ½PAj : with K j ¼ ½PAj1 ½A with

K2 ¼

Because the binding sites are independent and identical, K1 ; K2 ; . . . ; Kj assume one single value, that is, independent of the site where A binds. In general, it is impossible to determine the concentrations of each individual species P, PA, PA2 ; . . . ; PAj , but the average number v of moles of A bound per mole P is usually experimentally accessible, v ¼

½PA þ 2½PA2 þ 3½PA3 þ þ j½PAj : ½P þ ½PA þ ½PA2 þ þ ½PAj

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ð14:47Þ

Adsorption

279

[PAi ] can be expressed in the equilibrium constants K1 ; K2 ; . . . ; Kj , [A] and [P], as follows: ½PA ¼ K 1 ½P½A ½PA2 ¼ K 2 ½PA½A ¼ K 2 K 1 ½P½A2 ½PAj ¼ K j K j1 . . . K 1 ½P½Aj so that v ¼

ðK1 ½A þ 2K1 K2 ½A2 þ þ jK1 K2 . . . Kj ½A j Þ½P ð1 þ K1 ½A þ K1 K2 ½A2 þ þ K1 K2 . . . Kj ½A j Þ½P

and for the degree of occupancy y, of P by A, ( ) 2 v 1 K1 ½A þ 2K1 K2 ½A þ þ jK1 K2 . . . Kj ½A j y ¼ ; j 1 þ K1 ½A þ K1 K2 ½A2 þ þ K1 K2 . . . Kj ½A j j

ð14:48Þ

ð14:49Þ

which is known as the Aldair equation. If the binding sites were not identical, but all different, j different forms of PA would exist. For PA2 there would be jð j 1Þ=2! distinguishable forms, and for PAj that number would be jð j 1Þð j 2Þ . . . ð j ð j 1ÞÞ=j! However, because of the assumption of identical sites, binding of A to any one of the j sites is equally probable. It implies that ½PA ¼ jbPA*c where * indicates a speciﬁc site out of the j sites on P. Let Ka be the binding constant for each individual site, then ½PA ¼ K1 ½P½A ¼ jbPA*c ¼ jKa ½P½A, and, hence K1 ¼ jKa :

ð14:50Þ

Similarly, the second molecule of A binds in jð j 1Þ=2! equally probable ways to P, so that ½PA2 ¼ K1 K2 ½P½A2 ¼ f jð j 1Þ=2!g½PA** ¼ f jð j 1Þ=2!gKa2 ½P½A2 : 2 It is thus derived that K1 K2 ¼

jð j 1Þ 2 Ka 2!

ð14:51Þ

and so on. Introducing these expressions for K1 ; K2 ; . . . in Eq. (14.48) yields v ¼

Ka ½A jð1 þ Ka ½AÞ j1 jKa ½A ¼ j 1 þ Ka ½A ð1 þ Ka ½AÞ

ð14:52Þ

or, written differently v ¼ Ka ½A; j v

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ð14:53Þ

280

Chapter 14

which can be modiﬁed into y ¼ Ka ½A: ð14:54Þ 1y For ideal solutions of A, [A] is proportional to the mole fraction of A, so that y ¼ Ka0 X : ð14:55Þ 1y Equation (14.55) is identical to the Langmuir equation for dilute solutions, expressed in (14.25). This similarity was to be expected because the conditions underlying the Langmuir adsorption theory and the multiple equilibrium theory are the same, that is, binding (adsorption) equilibrium, and independent and identical binding (adsorption) sites. Analogous to the adsorption theory, the multiple equilibrium theory may be modiﬁed to include co-operativity, nearest-neighbor interactions, electrostatic effects, and so on. In particular for ﬂexible macromolecules being responsive to ligand binding co-operativity is often observed, as for example, for the binding of oxygen to hemoglobin. See Figure 14.6.

14.5 APPLICATIONS OF ADSORPTION Widespread adsorption occurs when a gas or a solution meets an interface. Here we mention only a few cases where adsorption is of practical interest. Adsorption from the gas phase is commonly applied in determining the speciﬁc surface area of ﬁnely dispersed materials. For that purpose assumptions have to be made concerning the dimensions of the gas molecules and the structure of the adsorbed layer under saturation conditions (fully packed monolayer, multilayer, etc.). Small gas molecules may enter pores and capillaries in porous materials. Hence, by comparing the surface area determined by gas adsorption with the ‘‘outer’’ surface area obtained from, for example, electron microscopy, the porosity of the material can be estimated. Moreover, by using different types of gas having different molecular dimensions an impression of the pore size distribution may be obtained. Gas adsorption is also applied to aid mechanical pumps in achieving ultrahigh vacuum. Furthermore, surfaces may act as catalysts for reactions that occur relatively slowly in the gas phase. For example, hydrogen (H2) and oxygen (O2) coexist for a long time in the gas phase because the activation energy to form water (H2O) is very high. In the presence of platinum the formation of water occurs explosively. Hydrogen and oxygen are (chemi)sorbed to platinum in their monoatomic form so that the gases do not have to be dissociated before reacting. This causes an enormous reduction of the activation energy and a corresponding acceleration of the reaction. In a similar way the formation of ammonia (NH3) from nitrogen (N2) and hydrogen (H2) is catalyzed by a proper mixture of metals and metal oxides. Activated charcoal is often used as an adsorbent in air and water ﬁlters as well as in medical preparations to eliminate poisonous substances.

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Adsorption

281

Various applications of adsorption are based on the concomitant reduction of the interfacial tension. The inﬂation of the lungs of newborns requires a large ﬂuid–air interface. Formation of that interface is facilitated by the adsorption of the lung surfactant that is released to the alveoli. Other examples of interfacial tension reduction are found in wetting phenomena such as impregnation, ﬂotation, cleansing, and so on. These phenomena are explained and discussed in Section 8.7. Finally, adsorption of surfactants at gas–liquid and liquid–liquid interfaces to prepare foams and emulsions should be mentioned (see Chapter 18). Different types of chromatography such as ion exchange, hydrophobic, and gas chromatography are based on adsorption. In these applications the adsorbate– adsorbent interaction should not be too strong because the adsorbed component should be readily released from the adsorbent material by changing external conditions. Biological ﬂuids, blood, urine, saliva, milk, fruit juices, and the like, contain a complex mixture of solutes usually including polymeric ones. The adsorption afﬁnity of polymers is generally higher than that of low molecular weight components. Even if the adsorbent surface may initially be covered with the low molecular weight components (because they may be present in higher molar concentrations and because they diffuse faster), the (bio)polymeric components often take over and ﬁnally populate the interface. Therefore, in biotechnological and biomedical applications (bio)polymer adsorption is often more relevant than adsorption of small molecules. Adsorption of polymers and biopolymers, in particular, proteins, is the topic of Chapter 15.

EXERCISES 14.1

Comment on the following statements. (a)

Spontaneous adsorption from solution may occur under the evolution of heat. (b) Unfavorable nearest-neighbor interactions between adsorbate molecules lead to higher sorbent surface occupancy at low degrees of surface coverage. (c) The rate of adsorption of a component at an interface is determined by its transport towards the interface rather than by its interaction with the interface. (d) Ion adsorption at an interface which is uncharged in its pristine state is enhanced by increasing the ionic strength. 14.2

The table below gives data for the binding of a ligand A at a macromolecule that has many binding sites for A; cA is the concentration of A in solution and y is the degree of occupancy of the sites by ligand. cA (mM) y

0.5 0.08

1.0 0.15

2.0 0.26

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5.0 0.47

10.0 0.64

20.0 0.78

282

Chapter 14

(a)

Show (graphically) that the binding sites are identical and independent. (b) Determine the binding constant. 14.3

For a solution perpendicularly ﬂowing towards a surface the ﬂux J of the dissolved component i at the stagnation point is given by J ¼ 0:776ðavÞ1=3 ðD=RÞ2=3 ci

ðE:1Þ

where a is a dimensionless parameter, v the velocity of the ﬂuid, R the radius of the cylindrical nozzle of the inlet, and D and ci the diffusion coefﬁcient and the concentration of i. (a) What is the dimension of J ? (b) At which conditions does J equal the rate of adsorption of i at the surface? (c) Equation (E.1) may be simpliﬁed to become J ¼ kD2=3 ci

ðE:2Þ

Suppose that the adsorption of i is described by (E.2). What value for ci must be taken to achieve an adsorbed amount of i of 0.5 mg m2 , given a value of 104 m s1 for kD2=3 ? 14.4

(a)

Draw in one ﬁgure adsorption isotherms GðcÞ for small uncharged molecules and for small negatively charged ions at an uncharged interface. The Gibbs energies of adsorption are the same for the uncharged and the charged molecules.

(b)

Under what conditions does a substance adsorb according to the Langmuir model? What is meant by cooperative adsorption? For what reason(s) does cooperative adsorption deviate from Langmuirian adsorption? Draw in one ﬁgure isotherms for the adsorption of a component for the cases where it adsorbs co-operatively and according to the Langmuir model, respectively.

(c)

The table presents data for the binding of oxygen gas to hemocyanin (an oxygen binding protein) in mollusks. Determine the cooperativity coefﬁcient p in the Hill equation. pO2 (mmHg)

y

pO2 (mmHg)

y

5.6 7.7 31.7 71.9 100.5 123.3

0.01 0.02 0.08 0.19 0.33 0.48

136.7 166.8 262.2 452.8 736.7

0.56 0.67 0.79 0.87 0.91

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Adsorption

283

pO2 is the partial oxygen pressure in a gas mixture. The total pressure in the gas is 760 mm Hg. The molefraction XO2 of oxygen in the gas may be approximated by pO2 =p. 14.5

The Langmuir equation for adsorption of a component i from a solution is often described by KXi yi ¼ ; ðE:3Þ 1 þ KXi where yi is the degree of coverage of the sorbent surface by i and Xi the mole fraction of i in solution. (a)

Show that (E.3) only applies for very dilute solutions. A solution consists of three components, A, B, and the solvent O. The mole fractions of A and B are very small: XA 1 and XB 1. A, B, and O adsorb at an interface. The degrees of coverage are yA , yB , and yO, and the molar Gibbs energies of adsorption are ads gA , ads gB , and ads gO . (b) Apply Boltzmann’s equation to express yA in terms of XA , yO , and ads gA . (c) Show that KA XA yA ¼ 1 þ KA XA þ KB XB with KA ¼ exp½ðads gA ads gO Þ=RT ½ðads gB ads gO Þ=RT . (d)

and

KB ¼ exp

Calculate yA , yB and yO for XA ¼ XB ¼ 0:01, KA ¼ 900, and KB ¼ 4000.

SUGGESTIONS FOR FURTHER READING D. H. Everett. Thermodynamics of adsorption from solution, Part 1. Perfect systems, Trans. Faraday Soc. 60:1803–1813, 1964. Part 2. Imperfect systems, Trans. Faraday Soc. 61:2478–2495, 1965. A. B. Mersmann, S. E. Scholl (eds.). Fundamentals of Adsorption, New York: United Engineering Trustees, Inc., 1991. R. H. Ottewill, C. H. Rochester, A. L. Smith (eds.). Adsorption from Solution, New York: Academic, 1983.

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Improving Diagnostics

Immunoglobulins are special kinds of proteins that function as antibodies. They bind speciﬁcally to ‘‘foreign’’ substances, the antigens. Because of their high selectivity and speciﬁcity immunoglobulins are excellent probes to detect viruses (e.g., hepatitis or HIV), hormones (e.g., the pregnancy hormone) and so on. In many of such diagnostic test systems the immunoproteins are immobilized to solid surfaces to increase their local concentration and to facilitate the visualization of the antibody–antigen complexes. The most common immobilization method is physical adsorption. However, adsorption may induce structural changes in the protein and a decreased accessibility of the antigen binding sites, both resulting in a loss of immunological activity. To minimize these adverse effects tricks may be played to force the protein molecules to adsorb in the desired orientation and to conﬁne their native shape. With immunoglobulins, which are Y-shaped, this may be realized by grafting polymer chains on the surface that leave interstitial spaces allowing the leg of the Y to enter but that are too small to accommodate the arms of the Y. With the arms reaching outward from the surface, the antigen binding sites (white spots in the picture) are accessible for the antigens to bind.

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15 Adsorption of (Bio)Polymers, with Special Emphasis on Globular Proteins

Polymer adsorption occurs ubiquitously. Among the biopolymers proteins are the most surface active. The interaction between proteins and interfaces attracts attention from various disciplines, ranging from soil and food science to medical sciences. In nature, protein adsorption manifests itself in various areas. For instance, even in the prebiotic stage of the genesis of terrestrial life, accumulation of proteins and proteinaceous components at soil–water interfaces may have played an essential role. In soils the activity of extracellular enzymes may be affected as a result of their adsorption to clay particles, which, in turn, inﬂuences the microbial life in the soil. Pancreatic lipases and phospholipases control the digestion of alimentary fats in the duodenum. These fats are insoluble in water; they are present as emulsiﬁed globules. Before the dissolved enzymes can exert their action they have to adsorb at the surface of the globules. Furthermore, intravascular thrombosis is an interfacial process in which adsorption of proteins at the blood vessel wall enhances the adhesion of blood platelets. In biotechnological and biomedical systems protein adsorption often is involved in the early development of biofouling, that is, the formation of a layer of organic material (including bacteria or other biological cells) on a surface. Protein adsorption is the initial event in the fouling of cardiovascular implants, teeth and dental restoratives, artiﬁcal kidney membranes, contact lenses, catheters, blood bags, and so on. Similar processes occur at pipelines, heat exchangers, and separation membranes in (food) processing equipment and, in a marine environment, at desalination units and ship hulls. In other applications proteins are adsorbed on purpose, for example, as immobilized enzymes in biosensors and

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286

Chapter 15

1

5

1*

5*

1**

5**

2**

4**

6**

2

4

6* 2*

4*

3

3*

Figure 15.1 Schematic presentation of the protein adsorption process.

bioreactors, immunoglobulins in immunoassays, drugs in drug targeting and controlled release systems, and as stabilizers of dispersions, emulsions, and foams in foodstuffs, pharmaceuticals, and cosmetics. Protein puriﬁcation systems such as displacement, hydrophobic, and ion exchange chromatographies require well-controlled adsorption and desorption. When a polymer molecule adsorbs from solution at an interface, it changes its environment which, more often than not, is accompanied by structural rearrangements. For proteins this implies a change in biological functioning as well. Hence, in the various applications one tries to optimize the adsorbent–protein interaction aiming at preservation of biological activity after adsorption or prevention of subsequent adverse events, for instance, in biofouling. Figure 15.1 schematically depicts the consecutive steps through which an adsorbing and desorbing polymer molecule passes: (a) transport towards the interface, (b) deposition and attachment at the interface, (c) structural rearrangements (relaxation) of the adsorbed molecule, (d) detachment from the interface, (e) transport away from the interface, and (f) possible restructuring of the desorbed molecule. The asterisks indicate the degree of relaxation of the adsorbed molecule. Each of these steps is considered in more or less detail in the following sections.

15.1 ADSORPTION KINETICS The theory presented in Section 14.2 may, as a ﬁrst approximation, apply to polymer adsorption as well. Interpretation of dG=dt, as expressed in (14.10), cb ceq ðyÞ dG ¼ 1 1 dt þ ktr ka ð1 yÞ

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ð15:1Þ

Adsorption of (Bio)Polymers

287

Γ

time Figure 15.2 Typical example of kinetics of polymer adsorption.

is for polymers less straightforward than for low molecular weight components. The reason is that, as a rule, when polymers adsorb they do so with a high afﬁnity, characterized by an extremely high value of the adsorption equilibrium constant Kð¼ ka =kd Þ. This feature is further discussed in Section 15.4. From a high afﬁnity adsorption isotherm, as displayed in Figure 15.12, it can easily be derived that for adsorbed amounts below adsorption saturation the equilibrium concentration ceq of the polymer in solution is extremely low (usually below the detectable limit). Upon reaching adsorption saturation, reﬂected by the (semi)plateau in the adsorption isotherm, ceq ðyÞ suddenly increases steeply and approaches cb . It implies that near saturation dG=dt drops strongly. This corresponds with the sharp kink in the curve for GðtÞ shown in Figure 15.2. Away from saturation, where ceq 0 Eq. (15.1) can be rewritten in the form cb =Gmax 1 1 ¼ þ : ktr ka ð1 yÞ dy=dt

ð15:2Þ

If ktr ka ð1 yÞ the second term in the denominator of (15.1) may be neglected. Then, the net rate of adsorption dG=dt, or, for that matter dy=dt, is fully determined by the transport towards the interface, independent of y. In that case ktr can be directly obtained from the initial linear part of the GðtÞ curve. See Figure 15.2. If ktr ka ð1 yÞ, a condition that may apply when there is a barrier for attachment at the interface and=or at high surface coverage, dG=dt is determined by the rate of attachment and, hence, depends on y. The value of ka , as a function of y, can be derived from GðtÞ using (15.2). Because of the extremely high value of K and the fact that, for physical reasons, ka can not attain very high values, the value of kd must be small. In other words, the rate of desorption must be very low. It may be explained by the low probability of simultaneous detachment of all the attached segments of the

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288

Chapter 15

1.5 Γ / mg m– 2

Γ / mg m– 2

1.5 Jf 1.0

LSZ+

0.5

Jf 1.0

0.5

α LA

LSZ+

α LA

0

100

200 300 400

0

time / s

100

200 300 400 time / s

Figure 15.3 Adsorption of positively charged lysozyme (LSZþ) and negatively charged a-lactalbumin (aLA ) on surfaces of negatively charged polar silica (left) and negatively charged apolar polystyrene (right). The dashed line indicates the limiting maximum ﬂux of the proteins towards the sorbent surface. (Adapted from W. Norde et al., Biofouling 4: 37, 1991.)

adsorbed polymer molecule. We return to this in Section 15.4. By way of example, Figure 15.3 shows experimental results for the rate of adsorption of the globular proteins lysozyme and a-lactalbumin. For the polar silica surface dG=dt reﬂects electrostatic interaction between the protein and the surface: dG=dt strongly reduces under electrostatically repulsive conditions. However, even when the electrostatic interaction is attractive dG=dt is signiﬁcantly lower than the limiting maximum ﬂux Jf [reached for cs ¼ 0; see (14.4)], towards the interface, suggesting some nonelectrostatic adsorption barrier. With the apolar polystyrene surface the rates of adsorption are higher and, remarkably, the inﬂuence of electrostatic interaction is masked. Apparently, factors other than electrostatic ones, such as hydrophobic dehydration and=or, anticipating the discussion in Section 15.4, structural rearrangements in the protein molecules, dominate the adsorption process. So far, the discussion is based on the simple model shown in Figure 14.2. However, in most cases the interaction with the interface induces a conformational adaptation (i.e., a relaxation) of the adsorbed (bio)polymers. This situation is depicted in Figure 15.4, where the native conformation is denoted ‘‘n-state’’ and the perturbed, adapted conformation ‘‘p-state.’’ The rate constant for relaxation is kr and the rate constants for desorption from the n-state and p-state are kd;n and kd;p, respectively. It then follows that dyn ¼ ka;n cs ð1 yn yp Þ kd;n yn kr yn dt

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ð15:3Þ

Adsorption of (Bio)Polymers

ka,n

289

kd,n

kd,p kr

n-state

p-state

Figure 15.4 Molecule interacting with an interface through adsorption and desorption, where, after adsorption, the molecule may be transformed into another state from which it may desorb as well. After desorption the molecule may or may not return to its original state.

and dyp ¼ kr yn kd;p yp : dt

ð15:4Þ

Now, the condition for steady-state, that is, dG=dt ¼ 0, is ka;n ð1 yn yp Þ ceq ¼ kd;n yn þ kd;p yp :

ð15:5Þ

Contrary to the situation for the simple model (Figure 14.2) the rate of desorption dG=dtj is not a unique function of the desorption rate constants but depends on the ratio yp =yn as well. This ratio changes during the course of the adsorption process. Because relaxation of the adsorbed polymer molecule implies optimization of interaction with the sorbent surface, the p-state molecules desorb much more slowly than the n-state molecules. The smaller the kd;p =kd;n ratio is, the larger is the fraction of perturbed molecules in the steady-state and if the value for kd;p approaches zero, steady-state is not reached and the adsorbed layer eventually consists of molecules that are all in the p-state. It is evident that when adsorptioninduced conformational changes are taken into account the expression for dG=dt is much more complicated than the one given in (15.1). It is a rule rather than an exception that adsorption-induced adaptation leads to an increased area per adsorbed molecule: the molecule more or less spreads at the interface. See Figure 15.5. The area per adsorbed molecule in the n-state is An , and in the p-state Ap . The ratio Ap =An a 1. The maximum number of molecules in the n-state per unit surface area is N0 and that in the p-state N . Hence, N0 ¼ aN and N0 An ¼ 1. Furthermore, yn nn =N0 and yp np =N0 . Note that both yn and yp are expressed in the number of adsorbate molecules per ﬁxed

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290

Chapter 15

ka,n

kd,n

kd,p kr

n-state

p-state

Figure 15.5 Molecule interacting with an interface in a similar way as depicted in Figure 15.4, but where the transition at the interface leads to an increased interfacial area occupied by the adsorbed molecule.

number of sites per unit surface area. This implies that (yn þ yp ) scales with the adsorbed mass per unit surface area. It follows that dyn ¼ ka;n cs ð1 yn ayp Þ kd;n yn kr yn f ða; yn ; yp Þ dt

ð15:6Þ

in which f ða; yn ; yp Þ accounts for the fact that the enlargement of the area per adsorbed molecule, resulting from the n ! p transition, puts restraints on the possibility for other molecules of the adsorbate to subsequently change from the n- to the p-state. Obviously, f ða; yn ; yp Þ equals unity for a ¼ 1 and it decreases for larger values of a, the more so the higher the fraction of the interface that is covered by the adsorbate (yn þ ayp ) is. Similarly, dyp ¼ kr yn f ða; yn ; yp Þ kd;p yp : dt

ð15:7Þ

This model explains the remarkable phenomenon of transient adsorption sometimes observed with polymers: if kd;p < kd;n and a > 1, ðyn þ yp Þ and, hence, the adsorbed mass per unit surface area may pass through a maximum during the course of the adsorption process. The steady-state for this model is given by ka;n ð1 yn ayp Þceq ¼ kd;n yn þ kd;p yp ;

ð15:8Þ

which is, in fact, indentical to (15.5), because (1 yn ayp ) and (1 yn yp ) in the respective equations both stand for the fraction of sorbent surface area covered by adsorbate. In Section 15.3 some phenomenological aspects of adsorption-induced structural relaxation are addressed. Most theories for polymer adsorption kinetics start from (combinations of) the models discussed above. Other theories, often proposed for (bio)polymer adsorption, are based on the random sequential adsorption (RSA) model. According to this model the adsorbate molecules arrive randomly at the interface

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291

Figure 15.6 Excluded interfacial area for deposition of molecules that adsorb according to the random sequential adsorption model.

and they stick where they hit. It implies that both desorption and tangential motion of the adsorbate at the interface are absent. Because the center of a newly arriving spherical molecule can not be accommodated within the shaded areas enclosed by the dashed circles shown in Figure 15.6, only the unshaded fraction f of the surface is available for adsorption. It is obvious that f is a function of y, the fraction of the surface that is covered by the adsorbate. For sphere-like molecules y ¼ npR2 (R being the radius of the molecule). The following expressions for the available surface function fðyÞ can be derived from the RSA theory. 2 7 y 7 ð15:9Þ fðyÞ ¼ exp 2 þ þ lnð1 yÞ ð1 yÞ2 8 ð1 yÞ2 8 which can be developed into the series pﬃﬃﬃ 6 3 2 fðyÞ ¼ 1 4y þ y þ 2:4243y3 þ : p

ð15:10Þ

The third term of the right-hand side of (15.10) accounts for the overlap of shaded areas in Figure 15.6, the fourth term for double overlap, and so on. In Figure 15.7 curves for fðyÞ, thus calculated, are presented. Next, (1 y) in Eq. (14.6) is replaced by f and kd in (14.7) is taken as zero, which results in dy ¼ ka cs fðyÞ: dt

ð15:11Þ

Figure 15.8 shows curves for yðtÞ for spheres and ellipses having different aspect ratios. Clearly, according to the RSA model the fraction of the interface covered by adsorbed molecules increases with decreasing aspect ratio but even for spheres it does not exceed 50%.

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Chapter 15

1.0 φ

0.8 0.6 0.4 0.2 0

0.1

0.2

0.3

0.4

θ

0.5

Figure 15.7 Available interfacial area for adsorption of spheres as a function of surface coverage, according to the random sequential adsorption model. The full-line curve represents fðyÞ calculated using (15.9), whereas the other curves are calculated using (15.10) taking two, three, and four terms of the series into account.

15.2 MORPHOLOGY OF THE INTERFACE In the foregoing it is tacitly assumed that the surface at which the polymer adsorbs is more or less smooth. However, in more than a few cases surfaces are ‘‘hairy.’’ For instance, at biological surfaces (e.g., those of biological membranes and bacterial cells) natural polymers such as polysaccharides are often present. When these surface polymers reach out in the surrounding medium with some

0.6 a / b = 1 (spheres)

θ

2 4 6 10 15

0.4

b a

0.2

0

5

10

15

t

20

Figure 15.8 Adsorption kinetics according to the random sequential adsorption mechanism for spherical and ellipsoidal particles having varying aspect ratios (a=b).

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relative rate of adsorption

Adsorption of (Bio)Polymers

293

0.08 0.06 0.04 0.02

0

20

40 60 80 100 pre-adsorbed PEO (% of saturation)

Figure 15.9 Relative rate of adsorption of a lipase on a surface as a function of the preadsorbed amount of polyethylene oxide, PEO. (From C. G. P. H. Schroe¨n et al., Langmuir 11: 3068, 1995.)

ﬂexibility, the surface will dynamically respond to polymer adsorption. It offers the possibility of optimizing contact by conforming to the shape of the adsorbing polymer molecule, but squeezing the surface polymers (the ‘‘hairs’’) between the surface and the adsorbed layer will cause repulsion, so-called steric repulsion, due to increased osmotic pressure and decreased conformational entropy. Grafting or preadsorbing water soluble polymers at the sorbent surface is often used to tune the adsorption of polymers, especially that of proteins. In particular, applying polyethylene oxide (PEO) to surfaces has proven to be succesful in making them protein-repelling. The protein resistance is determined by the length of the PEO-chains and their density at the surface. Thus, relatively short chains of PEO, consisting of, say, less than 10 monomer units do not severely hamper protein adsorption, but they do prevent intimate contact between the protein and the underlying surface. As a consequence, the adsorbed protein molecules may be less perturbed so that they retain more biological activity. Longer PEO chains more effectively repel proteins. As an example, Figure 15.9 gives the relative rate of adsorption of a lipase as a function of the coverage of the surface by PEO chains that contain an average number of 127 monomers. The ﬁgure shows a strong retardation of the adsorption process and almost complete suppression at surface coverages beyond 50%.

15.3 RELAXATION OF THE ADSORBED MOLECULE After the polymer molecule has attached at the surface it tends to relax towards its equilibrium structure, that is, towards the structure corresponding to the lowest Gibbs energy in the adsorbed state. For a simple ﬂexible polymer that structure will be relatively ﬂat implying that there is a driving force towards spreading.

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Such spreading becomes more difﬁcult as the polymer–surface interaction is stronger because nonequilibrium states tend to become quenched. With polymers having a (strong) internal coherence, as in globular proteins, the rate and the degree of spreading may be relatively low. Obviously, the degree of spreading depends on the balance of creating favorable polymer–surface contacts versus disrupting favorable internal contacts in the polymer molecule. Spreading leads to a larger number of polymer–surface contacts and, consequently, it becomes more difﬁcult for the whole polymer molecule to detach. Spreading may cause a structural heterogeneity in the adsorbed layer: molecules arriving at an early stage ﬁnd sufﬁcient area available for spreading, whereas this is not the case for the molecules that arrive when the surface becomes crowded with polymer molecules. For proteins this implies that the structure of the later-arriving molecules deviates less from their native, biologically active, structure. Another consequence is that the outcome of the adsorption process depends on the rate of attachment and the rate of spreading, relative to each other. When spreading occurs quickly, the adsorbed molecules will be more ﬂattened. If, however, the polymer ﬂux Jf to the surface increases, the adsorbed molecules retain a more coil-like or globular conformation and, therefore, the adsorbed mass per unit surface area is higher. Hence, when the rates of attachment and spreading are of the same order, saturation adsorption, Gmax , increases with increasing ﬂux. Figure 15.10 shows adsorption kinetics for a ﬂexible polymer, polystyrene, and for a globular protein, subtilisin, illustrating this phenomenon. In the ﬁgure G is plotted versus Jf t. Transport-controlled adsorption rates result in curves with unit slopes. For polystyrene it is observed that adsorption saturation is invariant with Jf , implying that the rate of spreading is much faster than the transport to the

2.20

1.5

(b) 1.10

11.7 x 6.9 4.2

x

1.0

0.44

x

0.22

1.5 1.0

Γ / mg m– 2

Γ / mg m– 2

(a)

1.4 Jf × 102 (mg m2 s– 1 )

0.5

0

1

2 3 Jf × t / mg m– 2

Jf × 102 (mg m2 s– 1 ) 0

2

4

6 8 Jf × t / mg m– 2

0.5

10

Figure 15.10 Effect of polymer concentration on the kinetics of adsorption onto a silica surface for (a) polystyrene from decalin and (b) subtilisin from water. The polymer ﬂuxes Jf towards the surface are indicated in the ﬁgures. (From W. Norde and C. E. Giacomelli. Macromol. Symp. 145: 125, 1999.)

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295

interface. For subtilisin, however, adsorption saturation increases with increasing ﬂux. From the dependency of Gmax on Jf , a relaxation time of about 100 s is estimated for the spreading of subtilisin.

15.4 ADSORPTION AFFINITY; ADSORPTION ISOTHERM 15.4.1 Polymers As discussed in Section 12.3, ﬂexible polymers in solution possess a high conformational entropy resulting from the various states each of the many segments in the polymer chain can adopt. The expansion of a polymer coil is determined by the quality of the solvent, expressed by the w-parameter deﬁned in Eq. (12.1). The better the solvent the more expanded the coil is and the higher its conformational entropy. Adsorption leads to a reduction of this conformational entropy. Hence, adsorption takes place only if the loss in conformational entropy is compensated by sufﬁcient favorable interaction between polymer segments and the interface. Analogous to polymer–solvent interaction, polymer–interface interaction may be characterized by a ws -parameter, deﬁned as ws

ads gsolvent ads gpolymer segment : RT

ð15:12Þ

According to this deﬁnition, ws > 0 implies a net segment–interface attraction. To achieve spontaneous adsorption a critical positive value of ws is required to compensate for the loss of conformational entropy of the polymer. That critical value typically is a few tenths of RT per mole of segments. Even if ws is only slightly larger than the critical value, the whole polymer molecule adsorbs tenaciously, with an extremely high afﬁnity. This is because the contribution from each adsorbing segment adds to the Gibbs energy of adsorption of the whole polymer molecule. Figure 15.11 illustrates how the segments of an adsorbed ﬂexible polymer molecule may be distributed over trains, loops, and tails. Trains account for the attached segments; trains are rarely very long and they do not completely cover the entire surface, leaving about 20–30% of the surface uncovered. Generally, loops account for most of the adsorbed mass. Their occurrence limits the reduction of conformational entropy. Their extension is primarily determined by the solvent quality. A high loop density is tolerated only if the solvent quality is relatively poor (w > 0). Then the maximum amount of polymer that can be accommodated in an adsorbed layer typically is in the range of 2 to 5 mg m2 . For a good solvent (w 0) that amount is in the range of 0.5 to 2 mg m2 . For entropic reasons tails usually extend relatively far in solution. High-afﬁnity adsorption is reﬂected in the shape of the adsorption isotherm Gðceq Þ: the initial part of the isotherm practically merges with the G-axis, because at low polymer

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Chapter 15 tails

loops

loops

trains

trains

Figure 15.11 Conformations of a ﬂexible polymer molecule on a surface adsorbed from (left) poor solvent and (right) good solvent.

supply all of the polymer is adsorbed until the interface is saturated. See Figure 15.12. With ﬂexible polymers the plateau value of the adsorption isotherm Gplateau is usually not well deﬁned; a pseudoplateau is observed in which G slightly increases with increasing polymer concentration in solution. Apparently, when adsorbed from a higher concentration in solution a larger fraction of the adsorbed polymer is in loops. Furthermore, higher (pseudo)plateaus are generally observed for polymers having a higher degree of polymerization. The effect of the degree of polymerization (or, for that matter, the molecular mass) on Gplateau depends on the solubility of the polymer. The effect is most pronounced in poor solvents where longer loops are allowed. As a consequence, adsorption isotherms of polydisperse polymer samples usually have a more or less rounded shape, as illustrated in Figure 15.13, because at higher supply the larger molecules gradually displace the shorter ones from the interface. Along the isotherm the surface becomes enriched with the larger polymer molecules. In other words, upon adsorption fractionation of the polymer takes place.

Γ χ>0

~0 χ< ceq

Figure 15.12

High-afﬁnity adsorption isotherms, typical for polymer adsorption.

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297

Γ

ceq

Figure 15.13 Schematic representation of adsorption (!) and desorption ( isotherms for a polydisperse polymer solution.

––)

15.4.2 Polyelectrolytes According to the principles discussed in Section 15.4.1 the adsorption behavior of ﬂexible polyelectrolytes may be predicted as well. Because of the charge they carry, polyelectrolytes are strongly expanded in aqueous solution; in other words, water is an excellent solvent for ﬂexible polyelectrolytes. As a consequence the formation of loops is strongly suppressed. Hence, polyelectrolytes tend to adsorb in thin layers up to only a few tenths of mg m2 . As with uncharged polymers, polyelectrolytes also require some critical attractive interaction with the surface to become adsorbed. In addition to an electrostatic contribution the Gibbs energy of adsorption may also comprise a nonelectrostatic component. Depending on the charge signs of the polyelectrolyte and the surface the electrostatic contribution is attractive or repulsive; it may or it may not outweigh the nonelectrostatic contribution. If too strongly repulsive, it prevents the polyelectrolyte from adsorbing. For the sake of comparison, the electric part of the molar Gibbs energy of a monovalent ionic group in an electric ﬁeld is approximately 1 RT for every 25 mV and the contribution to the Gibbs energy of adsorption from dehydration of a CH2 group is approximately 1:1 RT (cf. Section 14.1), both at room temperature. Hence, polyelectrolytes with some hydrocarbon groups in their chain may readily adsorb on a hydrophobic surface against an unfavorable electric potential. As opposed to uncharged polymers, the adsorption of polyelectrolytes is very sensitive to variations in the ionic strength. Electrolytes exert a dual effect: (1) they screen the intramolecular repulsion between charged segments, which manifests itself in water becoming a poorer solvent. Therefore, the addition of salt promotes the formation of loops and, hence, results in more adsorbed mass per unit surface area; and (2) they also screen electrostatic interactions between a polymer segment and the surface. Attachment of the segments to the surface is

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Chapter 15 ( a ) low ionic strength

( b ) high ionic strength

Γ

Γ

same charge signs opposite

opposite charge signs

same charge signs ceq

Figure 15.14 electrolytes.

ceq

Inﬂuence of electrostatic interactions on adsorption isotherms of poly-

promoted=opposed by electrolytes if this interaction is repulsive=attractive. These effects are reﬂected in adsorption isotherms as indicated in Figures 15.14(a) and (b). Along similar lines the inﬂuence of the pH on the adsorption of polyelectrolytes containing weak ionic groups (e.g., carboxyl and=or amino groups), shown in Figure 15.15, may be explained. At a pH where such a polyelectrolyte is uncharged it adsorbs in a relatively thick layer; the adsorbed amount is then high and independent of ionic strength. However, at a pH where the ionic groups are fully charged the adsorbed amount is low, but it increases with increasing salt concentration.

Γ

opposite charge signs

same charge signs

pH (charge density on the polyelectrolyte)

Figure 15.15 electrolytes.

Inﬂuence of electrostatic interactions on the adsorption of weak poly-

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299

For similar reasons amphiphilic polyelectrolytes show maximum adsorption around their isoelectric point. This maximum is less pronounced at higher salt concentrations.

15.5 DRIVING FORCES FOR ADSORPTION OF GLOBULAR PROTEINS The adsorption behavior of globular proteins shares some features with that of ﬂexible polymers, but in various ways it is different. The adsorption kinetics, as discussed in Sections 15.1 and 15.3, apply to both ﬂexible polymers and globular proteins. Like the polymers, proteins tend to spread over the surface. However, due to their relatively strong internal coherence the spreading rate is far lower than that of the ﬂexible polymers [compare Figures 15.10(a) and (b)]. Anticipating the forthcoming discussion in this section on rearrangements in the protein structure, it is mentioned here that adsorbed globular proteins, as a rule, do not unfold into a loosely structured polypeptide chain that is freely penetrable for water and ions. In other words, although structural rearrangements take place, the adsorbed protein molecules remain compact. Apparently, exposure of apolar amino acid residues to water, as would occur upon unfolding, is too unfavorable. Although a compact tertiary structure is retained, adsorption may involve a loss in ordered secondary structure. This, as explained in Section 13.2, is accompanied by a signiﬁcant increase in the conformational entropy of the protein molecule. Hence, unlike ﬂexible polymers for which adsorption leads to a decrease in conformational entropy, protein adsorption may be promoted by an increase in conformational entropy! Because of the relatively rigid structure of adsorbed globular protein molecules, the adsorption isotherms display well-deﬁned plateau values. The adsorption pattern, the effects of pH (i.e., charge of the protein), and ionic strength are in agreement with those of a polyampholyte: the adsorbed mass generally is at a maximum around the isoelectric point of the protein=surface complex, that is, at conditions where the charges on the protein and the surface just compensate each other. Below, the primary contributions determining protein adsorption afﬁnity, reﬂected by the Gibbs energy of adsorption ads G are considered in more detail. The major contributions originate from (1) redistribution of charged groups (ions) when the electrical double layers around the protein molecule and the surface overlap, (2) dispersion interaction between the protein and the surface, (3) changes in the hydration of the surface and of the protein molecule, and (4) structural rearrangements in the protein. Although these factors are discussed more or less separately, it may be clear that their actions are interdependent, being either synergistic or antagonistic. For instance, the structural ﬂexibility of an

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Chapter 15

adsorbed protein molecule strongly affects the electrostatic and hydrophobic interaction between the protein and the surface.

15.5.1 Interaction Between Electrical Double Layers Generally, both the protein molecule and the surface are electrically charged and surrounded by counterions and co-ions. A fraction of the countercharge may be bound to the surface and=or the protein molecule and the other part is diffusely distributed in the solution (see Section 9.4). The Gibbs energy Gel to invoke a charge distribution can be calculated as the isothermal isobaric reversible work Gel ¼

ð s0

c00 ds00 ;

ð15:13Þ

0

where and c00 and s00 are the variable surface potential and surface charge density, respectively, during the charging process. Solving this equation requires knowledge of c00 ðs00 Þ and this functionality can be derived from models for the electrical double layer (Section 9.4). Charge distributions for the system before and after adsorption are schematically depicted in Figure 15.16. Under most (practical) conditions the Debye length, the separation distance over which charges interact, is considerably smaller than the thickness of the adsorbed protein layer. For instance, in a solution of 0.1 M ionic strength the Debye length is about 1 nm whereas the thickness of the adsorbed protein layer in which the molecules retain a compact conformation is at least a few nm. Hence,

+

+ +

+ + +

+

+ +

+

+ + +

+ +

+ +

+ +

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+

+ + +

+ +

Figure 15.16 Schematic representation of charge distributions before (left) and after (right) protein adsorption. The charge on the surface and the protein molecule are indicated by þ=. The low molecular weight electrolyte ions are indicated by =.

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301

proton charge

16 8 0

–8 – 16 2

4

6

8

10

12

pH

Figure 15.17 Proton charge (net number of charged groups per protein molecule) as a function of pH for a-lactalbumin in solution (——) and adsorbed on a negatively (---) and positively ( ) charged surface. (From W. Norde et al. Polym. Adv. Technol. 6: 518,

such a compact protein layer shields the protein–surface contact region from electrostatic interaction with the solution. To prevent an excessively high electrostatic potential, the charge density in the protein–surface contact region should be (nearly) zero. Charge neutralization has been conﬁrmed experimentally by the shift in proton charge of the protein upon adsorption at a charged surface. Examples are shown in Figure 15.17. Adjustment of the surface charge may occur as well. Apart from adjustments in the protein and the surface charge, neutralization may be further regulated by the incorporation of indifferent ions from solution into the protein–surface contact region. This has been demonstrated by electrokinetic data and, more directly, by tracing radiolabeled ions. Trends, derived from electrokinetics, are shown in Figure 15.18; they clearly follow the charge antagonism between the protein molecules in solution and the bare surface. As a consequence of the charge regulation the contribution from changes in the charge distribution to the Gibbs energy of adsorption ads Gel is only moderately sensitive to the charge densities of the protein and the surface and it usually does not exceed a few tens of RT per mole of adsorbing protein. Its sign and value depend on the charge distributions and the dielectric constants of the electrical double layers before and after adsorption, respectively. It should be realized that the transfer of ions from the aqueous solution into the nonaqueous protein–surface environment is chemically unfavorable. In other words, the chemical effect of ion incorporation opposes the overall adsorption process. This explains why maximum protein adsorption afﬁnity is reached when the charge density on the protein just matches that on the surface so that no additional ions have to be incorporated.

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Chapter 15

∆ ads σ ek / µC cm–2

8

4

0

–4 2

4

6 α LA

8

10 pH

12 LSZ

Figure 15.18 Variation in the electrokinetic charge density resulting from plateau adsorption of lysozyme (s) and a-lactalbumin (d) on negatively charged polystyrene particles. The arrows indicate the isoelectric points of the two proteins. (From W. Norde et al. Polym. Adv. Technol. 6: 518, 1995.)

15.5.2 Dispersion Interaction In the case of loosely structured, highly solvated train-loop-tail-like conformation of the adsorbed layer, as is the case with ﬂexible polymers, the density of the adsorbed layer approaches that of the bulk solution and, hence, the contribution from dispersion (London–Van der Waals) interactions may be negligibly small. However, for the formation of a compact adsorbed protein layer dispersion interactions have to be taken into account. For a sphere interacting with a planar surface the contribution from dispersion interaction to the Gibbs energy of adsorption ads Gdisp can be approximated by ads Gdisp

A132 a a h þ þ ln ; ¼ 6 h h þ 2a h þ 2a

ð15:14Þ

where A132 is the Hamaker constant for the interaction between (1) the ﬂat surface and (2) the spherical protein molecule, (3) across the (aqueous) medium, a is the radius of the sphere, and h the separation distance between the sphere and the surface. Under most conditions h a, so that (15.14) simpliﬁes to ads Gdisp ¼

A132 a : 6h

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ð15:15Þ

Adsorption of (Bio)Polymers

303

Values for Hamaker constants (of the individual components) are given in several references and a compilation is also made in Table 16.1. The Hamaker constant for the system can be derived from the individual ones according to the rules: A132 ¼ A12 A13 A23 þ A33

ð15:16Þ

which, based on the approximation Aij ðAii Ajj Þ1=2 gives 1=2 1=2 1=2 A132 ¼ ðA1=2 11 A33 ÞðA22 A33 Þ:

ð15:17Þ

In aqueous media usually A11 > A33 and A22 > A33, so that, according to (15.17), A132 > 0 and, hence, ads Gdisp < 0, which implies attraction. The Hamaker constant for interaction across water is about 6:6 1021 J for globular proteins, (1–3) 1019 J for metals and (4–12) 1021 J for synthetic polymers such as polystyrene, teﬂon, and the like. According to (15.15), ads Gdisp varies proportionally with the dimensions of the protein molecule and it drops off hyperbolically with increasing distance between the protein and the surface. Thus, for a globular protein molecule of 3 nm radius at 0.15 nm distance from the surface ads Gdisp amounts to ð1 3Þ RT per mole at a synthetic polymer surface and to ð4 7Þ RT at a metal surface. Due to the various approximations involved these values are only indicative. Deriving more accurate values is practically impossible because of the irregular shapes (adsorbed) protein molecules may adopt in real systems. Moreover, rearrangements in the protein structure induced by adsorption may affect the Hamaker constant in an unknown way.

15.5.3 Changes in the State of Hydration Polar groups interact favorably with water molecules, mainly through electrostatic interaction, including hydrogen bonding. These interactions overcompensate the strong cohesion between the water molecules rendering polar components readily soluble in water. Apolar groups, which do not offer the possibility for such favorable interactions with water, are expelled from an aqueous environment. This mechanism is at the basis of hydrophobic interaction, as discussed in Section 4.3.2. If the protein molecule and the surface are polar it is probable that some hydration water is retained between the surface and the adsorbed protein layer. However, if (one of) the surfaces (is) are apolar, dehydration would be a driving force for adsorption. Although the apolar residues of a globular protein in water tend to be buried in the interior of the molecule, the water accessible surface of the protein may still comprise a signiﬁcant apolar fraction, even up to 40–50% (cf. Section 13.2). In this context it should be realized that apart from the polarity of the outer shell the overall polarity of the protein could be relevant for its adsorption behavior. The overall polarity inﬂuences the protein structural stability

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Chapter 15

(cf. Section 13.3.1) and, hence, the extent of structural perturbation upon adsorption. This, in turn, affects the adsorption afﬁnity, as discussed below. The effect of the polarity of the surface is also difﬁcult to assess, because changing the polarity usually involves a change in the surface electrostatic potential as well. A fair estimate of the contribution from hydration changes to the Gibbs energy of adsorption ads Ghydr may be inferred from partitioning (model) components in water=nonaqueous two-phase systems. It has thus been estimated that, at room temperature, dehydration of apolar surfaces involves a lowering of the Gibbs energy of about 10 to 20 mJ m2 (cf. Figure 13.8). For a protein of 15,000 Da molar mass that adsorbs to about 1 mg m2 it results in ads Ghydr ranging between 60 RT and 120 RT per mole of adsorbed protein. It is obvious that apolar dehydration dominates over the effects from overlapping electrical double layers and dispersion interaction.

15.5.4 Rearrangements in the Protein Structure As discussed in Section 13.4, the three-dimensional structure of a globular protein molecule in an aqueous environment is only marginally stable so that interaction with an interface may induce changes in that structure. However, as compared to ﬂexible polymers the conformational changes in adsorbing protein molecules are usually small. It is commonly experimentally observed that the thickness of a monolayer of adsorbed protein is comparable to the dimensions of the native protein molecule indicating that structural rearrangements do not lead to unfolding into a loose, highly hydrated loop-and-tail structure. After adsorption, at one side of the protein molecule the aqueous environment is replaced by the surface. As a consequence, intramolecular hydrophobic interaction becomes less important as a structure-stabilizing factor; that is, apolar parts of the protein that are buried in the interior of the dissolved molecule may become exposed to the surface without making contact with water. Hydrophobic interactions between amino acid residues support the formation of a-helices and b-sheets; hence a reduction of these interactions tends to destabilize such secondary structures. A reduction of the a-helix and=or b-sheet content is indeed expected to occur if the peptide units released from the helices and sheets can form hydrogen bonds with the surface, as is the case at polar surfaces. Then, the decrease in ordered (secondary) structure would result in an increased conformational entropy of the protein and, hence, an increased adsorption afﬁnity. The contribution from increased conformational entropy to a negative value for ads G may amount to some tens of RT per mole of protein (cf. Section 13.2). However, adsorption at an apolar, nonhydrogen bonding surface may stimulate intramolecular peptide– peptide hydrogen bonding resulting in an increased order in the protein’s structure. Whether extra hydrogen bonding within the protein molecule occurs depends on the outcome of the opposing effects of the energetically favorable hydrogen bonds and the unfavorable change in the conformation entropy.

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Needless to say, the degree of perturbation of the structural integrity has its impact on the biological functioning of the adsorbed proteins. As a rule, except for certain classes of lipases, the adsorption-induced structural changes lead to a decreased biological activity, generally the more so the more hydrophobic the surface is. On the other hand, the structure of the protein in the adsorbed state is often more stable against thermal unfolding, so that under conditions of elevated temperatures it may be advantageous to use adsorbed enzymes. The synergistic and antagonistic effects of the interactions discussed above are indicated in Figure 15.19. Based on their mutual inﬂuences it is to be expected that proteins adsorb from aqueous solution at apolar surfaces, even under conditions of electrostatic repulsion. With polar surfaces distinction must be made between structurally stable (‘‘hard’’) and structurally labile (‘‘soft’’) proteins. The hard proteins adsorb at polar surfaces only if they are electrostatically attracted. The soft proteins undergo considerable structural rePROTEIN-SORBENT INTERACTIONS dispersion interaction

+ redistribution of charged groups (electrochemical effect) opposite charges on protein and sorbent

similar charges on protein and sorbent

+ dehydration of the sorbent surface and part of the protein surface hydrophobic surface

hydrophilic surface

+ structural rearrangements in the protein affecting intra-molecular H-bonding increased

decreased

+

decreased but compensated by protein-sorbent H-bonding

structural rearrangements in the protein affecting conformational entropy decreased

increased

increased

+

+

Figure 15.19 Interdependency of the major subprocesses that are involved in the overall protein adsorption process. Adsorption-promotion is denoted by a þ sign and adsorption-opposition by a sign.

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Chapter 15

arrangements (i.e., a decrease in ordered secondary structure) resulting in a conformational entropy gain that is large enough to make them adsorb at a polar, electrostatically repelling surface.

15.6 REVERSIBILITY OF THE PROTEIN ADSORPTION PROCESS: DESORPTION AND EXCHANGE Phenomenologically, a system is in equilibrium if no changes take place at constant surroundings. At constant pressure p and temperature T the equilibrium state of a system is characterized by a minimum value of the total Gibbs energy G. Any other state, away from this minimum, is in nonequilibrium and there will be a spontaneous transition towards the equilibrium state provided that the Gibbs energy barriers along this transition are not prohibitively large. By deﬁnition, a process is reversible if during the whole trajectory of the process the departure from equilibrium is inﬁnitesimally small, so that in the reverse process the variables characterizing the state of the system return through the same values but in the reverse order. Because a ﬁnite amount of time is required for the system to relax to its equilibrium state, investigating the reversibility of a process requires that the time of observation exceed the relaxation time. If both the adsorption and desorption isotherms are of the high-afﬁnity type it is difﬁcult, if not impossible, to verify reversibility because, apart from the slowness of the desorption process (as discussed in Section 15.1), it would require a method to determine the protein concentration in an almost inﬁnitely diluted solution. However, protein adsorption isotherms sometimes display a nonhigh afﬁnity character. Also in those cases dilution usually does not lead to detectable desorption. In other words, Gðceq Þ rarely, if ever, follows the same path backwards. See Figure 15.20. The deviation between adsorption and desorption remains even when the observation time is extended to several days and is therefore much longer than the relaxation time of the proteins at the surface (cf.

Γ

ceq Figure 15.20

Protein adsorption (——) and desorption (---) isotherms.

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Adsorption of (Bio)Polymers

307

Section 15.3). The occurrence of such hysteresis indicates that at a given protein concentration in solution two metastable states exist: one on the adsorption isotherm and one on the desorption isotherm, each being characterized by local minima of G which are separated by a Gibbs energy barrier that prevents the transition from the one state into the other. The fact that the adsorption and desorption isotherms represent different metastable states implies that between adsorption and desorption a physical change has occurred in the system that may or may not be restored after desorption. Because of the poor desorbability of proteins upon dilution it is virtually impossible to investigate molecules that are desorbed by dilution. However, adsorbed proteins may be (partially) desorbed by varying the pH and=or ionic strength of the solution. Furthermore, they may be readily exchanged against other surface-active components being of the same or of another kind. Permanent structural changes have indeed been observed in such exchanged protein molecules.

15.7 COMPETITIVE PROTEIN ADSORPTION Most practical systems, for example, essentially all biological ﬂuids, are multiprotein systems. The various proteins compete with each other (and with other surface-active components) for adsorption at any interface present. As a rule, the interface will at ﬁrst become covered by the molecules that have the highest rate of arrival (i.e., the smaller ones that have the highest diffusion coefﬁcient and the ones that occur most abundantly in the solution). At later stages the initially adsorbed molecules may be displaced in favor of other molecules that have a higher afﬁnity for the surface. Competitive adsorption between monomers, dimers, higher aggregates of the same type of protein is easily understood in terms of more anchoring segments per adsorbing entity as the number of segments in that entity increases. Thus, preferential adsorption of aggregates, relative to monomers, has been reported for serum albumin, ﬁbrinogen, and insulin, for example. Competition between proteins of different types is more complicated. The molecular size may still play an important role but, in view of the discussion in Section 15.5, other variables such as polarity, electrical charge density, and structural stability should be taken into account as well. Most of the experimental data on competitive protein adsorption refer to blood proteins. In mixtures containing albumin, g-globulins, and ﬁbrinogen it has been observed that albumin adsorbs ﬁrst, followed by g-globulin, and that ﬁbrinogen ﬁnally takes over. This sequence corresponds to that of the molar masses (67,000, 150,000, and 340,000 D, resp.) and, hence, to that of the diffusion coefﬁcients. The same sequence has been found with blood plasma and with whole blood, where high molecular weight kininogen, that occurs in relatively low concentration in blood, covers the surface at last. The transient adsorption of ﬁbrinogen from diluted blood plasma on glass surfaces is shown in

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308

Chapter 15 (a) normal plasma x x

0.8

x

0.25 % x

x

0.6 x

0.5 % 0.12 %

x

x 0.4

minutes

1.0 Γ / mg m– 2

1.0 Γ / mg m– 2

(b)

1 x x

0.8

5

0.6 x

0.4 x

2.5 %

0.2 x

x

0.2

0

20

0

40 60 t / minutes

20

120 10 20 30 plasma concentration (% normal)

Figure 15.21 Adsorption of ﬁbrinogen from diluted blood plasma on glass. The ‘‘Vroman effect.’’ (From L. Vroman and A. L. Adams. J. Colloid Interface Sci. 111: 391, 1986.)

Figures 15.21(a) and (b). The (higher molar mass) compounds that displace ﬁbrinogen from the surface occur in relatively low concentrations in plasma, which explains the effects of time and plasma concentration. The occurrence of a maximum in GðtÞ of a protein resulting from displacement by (an)other protein(s) is called the ‘‘Vroman effect.’’ The inﬂuence of physicochemical characteristics such as electrical charge, hydrophobicity, and structural stability of protein molecules on their competitive adsorption may be revealed by investigating systems containing equal concentrations of various kinds of proteins that differ with respect to the above-mentioned characteristics but that have comparable molar masses and dimensions. Because

Γ / mg m–2

PS + 2

1

2

α LA

α -Fe2O3 +

PS α LA MGB + RNase +

1 LSZ ++

MGB + RNase + LSZ ++

α -Fe2O3

2

2

1

1 α LA MGB + RNase ++ LSZ +++

α LA MGB RNase +

LSZ +

Figure 15.22 Competitive adsorption between lysozyme (LSZ), ribonuclease (Rnase), myoglobin (MGB), and a-lactalbumin (aLA) on surfaces of apolar polystyrene (PS) and polar ion oxide (a–Fe2O3). The þ and signs indicate the electrical charge of the components. The solutions supplied to the sorbent surfaces contain equal concentrations (0.3 g dm3 ) of each of the proteins. (From T. Arai and W. Norde. Colloids Surfaces 51: 17, 1990.)

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Adsorption of (Bio)Polymers

309

of the similar concentrations and sizes, any inﬂuence of the rate of transport is practically eliminated. Figure 15.22 shows results, obtained under such conditions, for competitive adsorption from solution containing four different proteins. At both the polar and the apolar surface the adsorption preference tends to reﬂect electrostatic interaction, except for the structurally relatively labile a-lactalbumin which adsorbs in disproportionately high amounts at surfaces of the same charge sign. It is therefore concluded that protein structure stability is an important and often even a dominating factor in competitive protein adsorption, resulting in preferential adsorption of the soft proteins over the hard ones.

EXERCISES 15.1

Comment on the following statements. (a) Proteins usually adsorb reversibly with respect to variations in the protein concentration in solution. (b) The rate of polymer adsorption from an aqueous solution onto a solid surface increases with increasing hydrophobicity of that surface. (c) The conformational entropy of a protein molecule decreases as a result of adsorption. (d) The ‘‘loops’’ in the conformation of an adsorbed ﬂexible polymer molecule extend farther in the solvent when the quality of the solvent is better. (e) Fractionation of a polydisperse polymer sample can be achieved by adsorption.

15.2

Upon attachment at the sorbent surface adsorbed polymer molecules try to relax whereby they undergo structural rearrangements. Sketch in one ﬁgure the courses of the adsorbed amount G with time t for polymer solutions of three different concentrations c1 , c2 , c3 , for which c1 ¼ 2c2 and c2 ¼ 2c3 : (a) in the case where the structural relaxation is much faster than the polymer ﬂux towards the sorbent surface; (b) in the case where the rate of structural relaxation is comparable with the rate of arrival at the surface.

15.3

The rate of adsorption dG=dt of a negatively charged protein from solution of concentration cp on the wall of a ﬂow cell is given by dG fk ¼ c : dt f þk p f is the transport coefﬁcient in solution; it is determined by the diffusion coefﬁcient of the protein and the ﬂow rate of the solution through the cell. k is the adsorption rate constant and is determined by the activation Gibbs energy for adsorption.

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310

Chapter 15

(a)

Under what circumstances does the protein ﬂux J towards the cell wall equal the adsorption rate? (b) How is J related to f ? (c) Deﬁne a retardation factor for adsorption expressed in f and k. (d) For f ¼ 104 m s1 , what would be the minimum value of cp to reach an adsorption of 0.3 mg m2 within 1 minute? (e) The bare sorbent surface is positively charged. How does the electric surface potential change during the course of the adsorption process? Give a sketch for dG=dt as a function of G. 15.4

(a)

For the adsorption from solution of a ﬂexible polymer the adsorbed amount per unit interfacial area G is determined by ws ¼

ads gsolvent ads gpolymer segment RT

and gpolymer segmentsolvent 12 ðg polymer segmentpolymer segment þ g solventsolvent Þ w¼ : RT Make drawings for Gðws Þ for a good solvent and for a poor solvent. (b) In the case of a polyampholyte G varies with pH and ionic strength. Sketch a curve for adsorption saturation Gmax as a function of pH at a negatively charged surface for low ionic strength and for high ionic strength. The pH trajectory must include the point of zero charge of the polyampholyte. (c) A bioengineer immobilizes an enzyme to be able to carry out a biocatalytic reaction in a continuous process. The immobilization is performed by physical adsorption of the enzyme at a solid matrix. Which properties of the matrix surface and the solution from which the adsorption takes place should be chosen to minimize adsorptioninduced reduction of the enzyme activity? 15.5

Two globular proteins A and B have their isoelectric points at pH6. The plateau values Gmax of the adsorption isotherms are determined as a function of pH. The results are given in the ﬁgure below.

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Adsorption of (Bio)Polymers

311

(a) What is the charge sign of the surface? (b) Is the surface hydrophilic or hydrophobic? (c) Which one of the two proteins has the highest stability of the globular structure? The adsorption G of a protein from a unknown solution is monitored as a function of time. See the ﬁgure below.

(d) (e)

Give two possibilities to explain the course of GðtÞ. The solution is diluted ﬁve times. What do you expect for GðtÞ for the diluted solution for each of the two possibilities given under (d)?

SUGGESTIONS FOR FURTHER READING J. L. Brash, P. W. Wojciechowski. Interfacial Phenomena and Bioproducts. New York: Marcel Dekker, 1996. G. J. Fleer, M. A. Cohen Stuart, J. M. H. M. Scheutjens, T. Cosgrove, B. Vincent. Polymers at Interfaces, Boca Raton, FL: Chapman and Hall, 1993. C. A. Haynes, W. Norde. Globular Proteins at Solid–Liquid Interfaces. Colloids Surfaces B: Biointerfaces 2:517–566, 1994. Y. Lvov, H. Mo¨hwald (eds.). Protein Architecture, New York: Marcel Dekker, 2000. M. Malmsten (ed.). Biopolymers at Interfaces, 2nd Edition, in Surfactant Science Series, vol. 110, New York: Marcel Dekker, 2003.

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Colloidal Stability in Natural Waters

Colloids are ubiquitous. They are present, for instance, in natural waters such as oceans, surface waters, ground waters, and aquifers. The colloids in these systems are diverse: various minerals, clays, oxides, and biological cells. The colloidal particles may be (partly) covered by organic molecules. Colloidal stability, or, in other words, the rate at which particles settle, in those waters has great environmental impact. Lakes, and to a lesser extent rivers, are settling basins for particles thus forming a sediment over the years. Pollutants as well as nutrients may bind to the sedimenting particles which implies that colloidal stability plays an important role in regulating the biological quality of those waters. In estuaria, where rivers ﬂow into oceans, delta formation takes place through aggregation and subsequent sedimentation of riverborne colloids in an environment of increased salinity. When colloidally stable particles exist in seawater they are often stabilized by adsorption of organic compounds such as biopolymers.

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16 Stability of Lyophobic Colloids Against Aggregation

In the introductory chapter, Section 1.1, we deﬁned colloidal systems as systems in which particles dispersed in a medium are subjected to both thermal motion and motion due to external forces (e.g., gravity). This deﬁnition leads directly to the concept of ‘‘stability’’ of a colloidal dispersion. A colloidal dispersion is considered to be stable if no rapid phase separation occurs through sedimentation (if the density of the particles is higher than that of the medium) or creaming (if the density of the particles is lower than that of the medium). Thus colloidal stability refers to the ability of a dispersion to resist aggregation into larger entities that then would segregate from the medium. Colloidal stability is the key issue in applications of colloidal systems in biology and technology. In some cases stable dispersions are desired, for instance, in biological ﬂuids such as blood, milk, and cloudy fruit juices and in processed foodstuffs such as butter and mayonnaise, and also in paints, lubricants, cosmetic creams, pharmaceutical ointments, drug-delivery systems, and immunolatexes. As a matter of fact, the requirements with respect to colloidal stability of a product depend on its shelf-life; for cosmetic and medical products this is usually longer than for milk or fruit juices, for example. On the other hand, in applications such as wine clariﬁcation, water puriﬁcation, and soil amelioration aggregation is preferred. Lyphobic colloids are, by deﬁnition, insoluble in the surrounding medium and they are therefore thermodynamically unstable: the lyophobic particles tend to minimize their contact with the medium in which they are dispersed and their ﬁnal destination is the aggregated state. Lyophobic colloids owe their stability to a

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314

Chapter 16

repulsive interparticle (Gibbs) energy barrier. The height of the interaction barrier relative to the kinetic energy due to thermal motion and due to other forces applied to the particles determines the stability of the colloidal system. Discussion of the stability of lyophobic colloids therefore focuses on the origin of the interaction energy barrier. Irrespective of the nature of the stabilizing forces, for the net force f between two surfaces at a separation distance h, we can write @G @H @S ¼ þT ; f ¼ @h p;T @h p;T @h p;T

ð16:1Þ

where G, H, and S are the Gibbs energy, the enthalpy, and the entropy of the system. By convention, a repulsive force has a positive sign. For two surfaces approaching each other across a liquid medium, the liquid must leave the region between the two surfaces. Assuming noncompressibility of the liquid, (16.1) can be rewritten as f 1 @G @G 1 @G ¼ ¼ ¼ ; A A @h p;T @V p;T n1 @n1 p;T

ð16:2Þ

where A is the area of the interacting surfaces, n1 the number of moles of the liquid leaving the interparticle region, and n1 the molar volume of the liquid. Based on the theory given in Sections 3.5 and 3.6 the right-hand term of (16.2) equals the osmotic pressure p for the liquid between the surfaces. Thus, for two approaching surfaces, the force per unit area can be regarded as an osmotic pressure. This pressure is also referred to as the disjoining pressure. So far we tacitly assumed that the colloidal particles are rigid so that they do not deform upon approach. This assumption more or less holds for solid particles. However, for ﬂuid particles, as present in emulsions and foams, deformations may easily occur and they invoke additional mechanisms affecting their stability. In this chapter we restrict ourselves to nondeformable particles. Emulsions and foams are discussed in Chapter 18.

16.1 FORCES OPERATING BETWEEN COLLOIDAL PARTICLES The stability of lyophobic colloids is governed by long-range forces, that is, forces that operate over a distance of at least a few nanometers. These forces can, in principle, accurately be formulated provided there is a well-deﬁned geometry and known composition of the particles and the medium.

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Stability of Lyophobic Colloids

315

16.1.1 London–Van der Waals Forces or Dispersion Forces In Sections 13.3.4 and 15.5.2 we already encountered the role of dispersion interaction in, respectively, globular protein structure and protein adsorption. Here we discuss dispersion forces in somewhat more detail. In each atom the nucleus carries a positive charge. Negatively charged electrons orbit around the nucleus. When two atoms come in close proximity the electron orbits inﬂuence each other so that the one atom induces a small dipole moment in the other, and vice versa. This gives rise to electromagnetic attraction between the atoms, known as London–Van der Waals interaction or, as it is otherwise called, dispersion interaction. The corresponding Gibbs energy for two spherical atoms separated in vacuum over a distance r between their centers is given by Gdisp ¼ b=r6 :

ð16:3Þ

The minus sign stems from the convention that attractive interaction is counted negative. The constant b is proportional to a2 , where a is the polarizability of the atom. The induced dipole is larger for larger values of a and, consequently, the interaction energy is stronger. Note that (16.3) no longer holds for distances at which the electron orbits of the respective atoms interpenetrate. In that case a strong repulsive force (so-called Born repulsion) results. The interaction between two (colloidal) particles may be considered as between two ensembles of atoms, each containing q3 atoms. As a ﬁrst approximation it is assumed that the dispersion forces between a pair of atoms are additive. Thus, the attraction between two particles separated over a distance equal to the particle diameter originates from q3 q3 ¼ q6 atom–atom interactions which are at a distance of q times the atom diameter. Therefore, as a rule, the dispersion interaction between two particles at a separation of the particle diameter is the same as that between two individual atoms separated over a distance of the atomic diameter. However, at shorter separation between the particles the dispersion interaction is much stronger. Even when the particles are spherical, at separations much shorter than the particle diameter, the particles ‘‘see’’ each other as thick plates and their surfaces may be regarded as planar. Such a geometry is now considered in more detail; see Figure 16.1. The total dispersion interaction between the particles is obtained by summation of all atom-pair interactions. The interaction between the atoms in the disk-like volume dV1 in plate 1 and the ring-like volume dV2 in plate 2 is found by applying (16.3) to each atom-pair interaction n2 b dV1 dV2 dGdisp ¼ ; ð16:4Þ ðr2 þ R2 Þ3 where n is the number of atoms per unit volume. To calculate the interaction between a column of cross-sectional area O in plate 1 and the whole plate 2 we

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316

Chapter 16

dR h

dV1

dr

z r

R

x y

1 Figure 16.1

2

dV2

Dispersion interaction between two thick plates.

have to integrate (16.4) four times, that is, in the x-direction in plate 1 and in the x-, y-, and z-directions in plate 2. It results in A Gdisp ðhÞ ¼ ð16:5Þ 12ph2 in which A, known as the Hamaker constant, is deﬁned as A p2 bn2 :

ð16:6Þ

A is expressed in energy units. Similarly, for two spheres of radius a at a separation h a, Aa : ð16:7Þ Gdisp ðhÞ ¼ 12h Note that in (16.7) Gdisp ðhÞ is expressed in energy units, whereas from (16.5) it is obtained as energy per unit surface area. It is thus derived that for a given geometry Gdisp ðhÞ is determined by the value of A, which, in turn, is proportional to the square of the atom density in the particles and the square of the polarizability of the atoms. So far, the discussion has been restricted to particles in vacuum. When the particles are immersed in a ﬂuid, which is generally the case for colloidal systems, the same reasoning applies, but instead of the polarizability of the atoms in the particles we must now consider the difference between the polarizibility a1 of the atoms in the particles and the polarizibility a2 of the atoms in the medium. The Hamaker constant for the particles (1) interacting across the medium (2), A121 , varies linearly with ða1 a2 Þ2 , so that 1=2 2 A121 ¼ ðA1=2 11 A22 Þ ;

ð16:8Þ

where A11 and A22 are the Hamaker constants for 1–1, respectively, 2–2, interactions across vacuum. It follows that A121 is always positive and, consequently, that Gdisp is always attractive. Moreover, the attraction between two particles of material 1 dispersed in material 2 equals the attraction between two particles of material 2 dispersed in material 1, provided identical geometries.

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Stability of Lyophobic Colloids

317

Table 16.1 gives values of Hamaker constants for various materials in vacuum and in water. Their assessment is rather complicated and difﬁcult; they are usually inferred from theoretical interpretation of experimental data. A more advanced and, in principle, more accurate computation of dispersion interactions between mesoscopic bodies is based on quantum-electrodynamics of continuous media, as presented in the Lifshitz theory. However, the simpler treatment, given above, in which the interaction energy is obtained by pairwise summation of the energies between all atoms of the interacting particles does not deviate more than, say, 30% from the absolute value. Because the materials in the system are often not sufﬁciently well-deﬁned to allow for accurate quantum-dynamic computations, it is reasonable to utilize the more simple theory. Figure 16.2 displays Gdisp ðhÞ between polystyrene particles and between bacterial cells (both having a radius of 500 nm), immersed in water, calculated using (16.7) and taking values of 1:0 1020 J and 0:1 1020 J, respectively, for the Hamaker constant. It shows that Gdisp attains large values at short separation, but it fades away when the particles are farther apart. For the polystyrene spheres the dispersion interaction equals the energy of thermal motion (at 25 C) at 100 nm and for the bacteria this is at 10 nm. These examples demonstrate that, even if gravitational effects can be neglected, the dispersed system is not stable. The combined effects of thermal motion and dispersion forces would cause attractive collisions between the particles leading to the formation of aggregates and, subsequently, segregation. Colloid stability therefore requires the action of repulsive interparticle forces. Table 16.1 Approximate Values (in 1020 J) of Hamaker Constants for Individual Materials Interacting Across Vacuum (A11 and A22 ) and Across Water (A121 ) Material Metals Quartz (fused) Silica (fused) Carbon (graphite) Alkanes Teﬂon Poly(styrene) Poly(vinyl chloride) Globular proteins Biological cells (bacteria, blood platelets, etc.) Water

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A11

A121

15–50 6–7 5–8 40–50 4–5 4–6 6–9 8 — A22

4–25 0.5 0.5 20–25 0.1 0.1 0.3–1.0 0.8 1–2 0.02–0.3

4

—

318

Chapter 16

0

20

40

60

80

100

h / nm 120 140

160

2 –5 Gdisp / kBT

1

– 10

– 15 – 20

Figure 16.2 Gibbs energy of dispersion interaction between two identical spheres (a ¼ 500 nm) as a function of their separation distance. (1) A121 ¼ 1:0 1020 J; (2) A121 ¼ 0:1 1020 J. T ¼ 298 K.

16.1.2 Electrical Double Layer Forces In Chapter 9 we discussed the distribution of charge in an electrical double layer at a charged interface. Binding of all counterions to the surface and expelling all co-ions into the bulk solution would minimize the energy. On the other hand, entropy maximalization tends to a uniform ion distribution throughout the system. However, the counterions cannot leave the charged surface because that would imply complete charge separation which is energetically too costly. Therefore, in a relaxed electrical double layer the counterions and co-ions are nonuniformly diffusely distributed, representing the optimal compromise between the competing energy and entropy effects. Such a distribution is illustrated in Figure 9.1 and, together with the ensuing electric potential proﬁle, in Figures 9.7 and 9.8. It has also been shown in Section 9.4.2, that the distance over which the electric potential c decays across the double layer depends on the ionic strength. This dependency is quantitatively given by (9.27) or (9.28). Thus, c drops off to c=e over a distance of (approximately) the Debye length k1 , deﬁned by (9.29) or (9.30). For intermediate ionic strengths, say, between 101 M and 103 M, the Debye length is in the nm range. When two charged surfaces approach each other the diffuse parts of their double layers start to interpenetrate, which implies overlap of the respective electric ﬁelds. Such overlap is schematically illustrated in Figure 16.3 for two identical surfaces. If the surfaces have different charge densities and, consequently, different cðxÞ-proﬁles, the overlap pattern would of course not be symmetrical. In view of the distance over which electrical double layers extend into the solution, interactions between double layers become effective at a separation

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Stability of Lyophobic Colloids

(a)

ψ

319

(b)

ψ

ψx = h / 2

0 h

Figure 16.3 surfaces.

h

h/2

0

Potential decay at two identical (a) noninteracting and (b) interacting

between the surfaces of at least a few nm. Hence, electrical double layer forces can be considered as long-range forces; they operate over distances comparable to those over which dispersion forces between particles of colloidal dimensions reach signiﬁcant magnitudes. When the two double layers overlap ions are conﬁned in a smaller volume and this leads to an entropy reduction. At the same time, additional charge is brought in an electric ﬁeld of opposite sign, which is energetically favorable. The resulting net force the double layers exert on each other can, in principle, be calculated from these two contributions. It is most conveniently done at the plane where the electric ﬁeld dcðxÞ=dx in the overlapping double layers is zero that is, for identical surfaces, at half separation (see Figure 16.3), and ion redistribution does not include an electric effect. In that case the disjoining pressure due to electrical double layer overlap equals the osmotic pressure resulting from increased ion concentration at half separation or, for nonidentical surfaces, at the separation where dcðxÞ=dx ¼ 0. For the same reason, a charged surface is repelled by a neutral one. P The ion concentration i ci in the diffuse part of the double layer at x ¼ h=2 follows from Boltzmann’s equation (9.25) relating ci ðxÞ to cðxÞ: X zi Fcx¼h=2 h ¼ c x ¼ c exp ; i i i i;1 RT 2

X

ð16:9Þ

where ci;1 is the ion concentration in the bulk solution. In forthcoming equations in this chapter the bulk concentration of i is denoted ci . Using Eq. (3.35) we obtain for the disjoining pressure f =Að¼ pÞ, X zi Fcx¼h=2 f ¼ RT c exp c i : i i RT A

ð16:10Þ

For not too high potentials, as in the outer parts of the double layers, for both the counterions and the co-ions the exponent may be expanded in a power series.

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320

Chapter 16

Then, for weakly overlapping double layers, where only the outer parts of the double layers interpenetrate, it follows that 2

ee0 k2 cx¼h=2 f

A 2

ð16:11Þ

in which k is the reciprocal Debye length deﬁned by (9.29) and hence (16.11) is only valid for symmetrical electrolytes. In the case of weak double layer overlap cx¼h=2 may be approximated by superposition of the Gouy–Chapman potentials, expressed in (9.27) of the two double layers in a ﬂat geometry: 4RT kh gd exp cx¼h=2 ¼ 2 ð16:12Þ zF 2 with gd tanhðzFcd =4RT Þ. Note that gd is a function of cd , the Stern potential, and not of the surface potential c0 . Indeed, cd rather than c0 is relevant for the potential decay across the diffuse part of the double layer. Substituting (16.12) in (16.11) yields f ee ¼ 32ðRT Þ2 2 02 k2 g2d exp½kh z F A

ð16:13Þ

which, combined with (9.29), can be written as f ¼ 64RTcg2d exp½kh; A

ð16:14Þ

where c refers to the bulk concentration of the symmetrical electrolyte. The Gibbs energy of interacting electrical double layers Gedl is obtained by integrating f =A from x ¼ 1 to x ¼ h, ð 1 h Gedl ðhÞ ¼ f ðhÞ dx ð16:15Þ A 1 which yields Gedl ðhÞ ¼ 64RTck1 g2d exp½kh:

ð16:16Þ

For two nonidentical surfaces (1) and (2) having different potentials, g2d should be replaced by gdð1Þ gdð2Þ. For two spheres it can be derived that Gedl ðhÞ ¼ 64pRTcak2 g2d exp½kh:

ð16:17Þ

Here again, Gedl in (16.16) is expressed in energy per unit surface area for the ﬂat surfaces, and in (16.17) for the spheres. Calculation of Gedl according to (16.15) assumes reversible double layer interaction. This may not be completely true. In fact, we can distinguish between two limiting cases, namely, constant surface potential and constant surface charge density. If the surface is charged because of adsorption or desorption of charge

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Stability of Lyophobic Colloids

321

determining ions, double layer relaxation re-establishes ionic equilibrium at the surface, so that the potentials are kept constant as the surfaces approach each other. However, if the approach is very rapid the double layers may not be fully relaxed when they interpenetrate. If the surface charge is due to ionization, charge regulation by the incorporation or expulsion of ions usually occurs in order to prevent the surface potential from reaching a too high value. Although for most cases it is not known to what extent such charge regulation occurs, the constant potential model rather than the constant charge model is usually chosen to calculate the interaction Gibbs energy. In this context it should be mentioned that the constant potential model and the constant charge model give different results only for very small values of h. The equations derived for Gedl ðhÞ, (16.16) and (16.17), show that Gedl drops off exponentially with increasing separation distance and that the decay steepens with increasing value of k, that is, with increasing ionic strength. Figure 16.4 presents curves for Gedl ðhÞ for two identical spheres (a ¼ 500 nm) interacting across an aqueous solution containing a 1 : 1 electrolyte. Curve (1) is for cd ¼ 25 mV and c ¼ 0:025 M. For a higher value of cd , but otherwise unaltered conditions, Gedl is larger at any separation distance, as indicated by curve (2). At higher ionic strength but the same value of cd Gedl is higher at small separation but drops off much more steeply, as we observe by comparing curves (1) and (3). At larger separation distances, where the outer parts of the double layers interpenetrate, Gedl is considerably suppressed by increasing the ionic strength. The functionality tanhðxÞ can be approximated by x for small values of x and it approaches unity for large x. Hence, for zFcd =4RT 2, gd may be

140 120 Gedl / kBT 100 80 2

60 1

40 3

20 0

2

4

6

8

10

12 h / nm

14

Figure 16.4 Gibbs energy of electrical double layer overlap of two identical spheres (a ¼ 500 nm) as a function of their separation distance. (1) cd ¼ 25 mV, 25 mM ionic strength; (2) cd ¼ 50 mV, 25 mM ionic strength; (3) cd ¼ 50 mV, 100 mM ionic strength. T ¼ 298 K.

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replaced by zFcd =4RT and for zFcd =RT 8, gd reaches unity. In a 1 : 1 electrolyte solution at room temperature, the ﬁrst condition applies if cd 50 mV and the second if cd 200 mV. In the latter case the interaction between the electrical double layers becomes insensitive for the Stern potentials. When the charges at the interacting surfaces have opposite signs double layer interaction is attractive. The one surface may be considered as a giant counterion for the other so that upon approach the original counterions are released into the bulk solution with a concomitant entropy increase. When the opposite charge densities are equal in magnitude, that is, s1 ¼ s2 , the interaction is attractive down to immediate contact between the surfaces, h ¼ 0. If, however, the opposite charges do not exactly cancel each other, s1 þ s2 6¼ 0, some counterions must remain between the approaching surfaces. Then, at close proximity the counterions concentrate in the narrow gap and this manifests itself as a repulsive force. It explains why oppositely charged particles may aggregate without making direct contact with each other.

16.1.3 Short-Range Forces At very short separations, say, 0.5 nm, other interactions become effective as well. These could be hydrophobic interaction, hydrogen bonding (sometimes referred to as Lewis acid-base interactions), ion pairing, and interactions between dipoles. Because water molecules are involved in most of them (cf. Chapter 4), these short-range interactions are particularly signiﬁcant in aqueous systems. They may be attractive or repulsive, depending on the types of interacting surfaces. The short-range forces are relevant as to the strength by which the particles stick together in aggregates. However, they are irrelevant for the stability of lyophobic colloids because the repulsive barrier between interacting surfaces is determined by long-range forces.

16.2 DLVO THEORY OF COLLOID STABILITY The DLVO theory, named after its founders Derjaguin, Landau, Verwey, and Overbeek, provides a quantitative description of the stability of lyophobic colloids. According to the DLVO theory, colloid stability is determined by long-range particle interactions. These are the London–Van der Waals or dispersion interaction and the interaction resulting from overlapping electrical double layers. Hence, the total interaction Gibbs energy between two charged particles as a function of their separation distance int GðhÞ, comprises two contributants Gdisp ðhÞ and Gedl ðhÞ that are assumed to be additive: int GðhÞ ¼ Gdisp ðhÞ þ Gedl ðhÞ:

ð16:18Þ

As explained in the preceding Section, Gdisp ðhÞ and Gedl ðhÞ, and therefore int GðhÞ, depend on the Hamaker constant A, the dielectric permittivity ee0 , the

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Stability of Lyophobic Colloids

323

reciprocal Debye length k (i.e., the ionic strength of the medium), the Stern potentials cd of the interacting surfaces, and, in the case of spheres, the radius a. All these quantities are more or less (experimentally) accessible and the value of cd is usually approximated by that of the electrokinetic potential z (see Chapter 10). Then, for a given system, curves representing Gdisp ðhÞ, Gedl ðhÞ, and int GðhÞ may be constructed for different ionic strengths. Results are schematically shown in Figures 16.5(a) to (c) for surfaces having the same charge sign. At large and, especially, at very short separations Gdisp dominates overGedl, but at intermediate values of h the electric double layer interactions may give rise to a repulsive barrier. In the previous section we saw that Gedl is suppressed by increasing the ionic strength whereas Gdisp is essentially unaffected. As a result, at low electrolyte concentration (small value of k) a high Gibbs energy barrier ðint GÞmax has to be overcome for the surfaces to come into close contact in the so-called primary minimum where int GðhÞ reaches its lowest value. [See Figure 16.5(a).] In other words, the system is colloidally stable. At high electrolyte concentration the Gibbs energy barrier is eliminated and the particles are attracted to each other [Figure 16.5(c)]. At intermediate ionic strengths the barrier still exists but is much lower. It can be crossed by a signiﬁcant fraction of the colliding particles. More speciﬁcally, the fraction of collisions that result in primary minimum contact is proportional to expbðint GÞmax =kB T c. In biological ﬂuids the ionic strength is sometimes high (about 0.7 M in sea water), often intermediate (0.15 M in blood serum and lachrymal ﬂuid, 0.10 to 0.20 M in urine, and about 0.07 M in milk), and seldomly low (0.03 to 0.07 M in saliva and about 103 to 102 M in ground and surface water). It is emphasized that the thermodynamically stable state of a lyophobic colloidal system is the state where it is aggregated in the primary minimum, corresponding with the minimum value for int GðhÞ. Thus colloidal stability is of G

(a)

G

(b)

G

(c)

Gedl (∆intG)max h h0

∆intG

h

h

Gdisp

Figure 16.5 Gibbs energy of interaction between two identical particles according to the DLVO theory. The suspending medium has (a) low, (b) intermediate, and (c) high ionic strength.

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Chapter 16

kinetic nature. It depends on the height of the barrier. However, irrespective of that height every lyophobic colloid ends up in the aggregated state, given sufﬁcient time. From that state it cannot be redispersed anymore: aggregation in the primary minimum, also referred to as coagulation, is an irreversible process. The separation distance between the colloidal particles h0 at which the top of the barrier occurs can be established for the case where ðint GÞmax ¼ 0. At the maximum of int GðhÞ, dint GðhÞ dGdisp ðhÞ dGedl ðhÞ þ ¼ ¼ 0: dh dh dh

ð16:19Þ

Taking (16.5) for Gdisp and (16.16) for Gedl, for the case of planar surfaces, the differentiation yields 2 Gdisp ðhÞ kGedl ðhÞ ¼ 0; h

ð16:20Þ

which, combined with int G ¼ Gdisp þ Gedl ¼ 0 at h ¼ h0 gives h0 ¼ 2k1 :

ð16:21Þ

Similarly, for two interacting spheres it can be derived that h0 ¼ k1 :

ð16:22Þ

For a colloidally stable system, for which ðint GÞmax > 0, the top of the barrier is at a separation larger than those given in (16.21) and (16.22). Equation (16.21) indicates that, for two approaching charged planar surfaces the energy barrier is encountered at, at least, two times the thickness of the diffuse part of the electrical double layer. The lower value of h0 in the case of a spherical geometry is due to the divergence of the electrical double layer with increasing distance from the surface. Under certain conditions, especially at intermediate ionic strengths, int GðhÞ shows a shallow secondary minimum at a separation beyond the repulsive barrier, as is shown in Figure 16.5(b). For most systems the depth of the secondary minimum is only a few kB T. Particles settled in that minimum can therefore easily be redispersed, for instance, by merely shaking. However, for systems having a high value of the Hamaker constant (see Table 16.1) and=or for relative large particles, say, in the mm range, the secondary minimum may reach 10 to 20 kB T , which is sufﬁciently deep to cause aggregation. Emulsion droplets usually aggregate in the secondary minimum and this may also happen with bacteria and other biological cells when they aggregate or adhere to surfaces. Particles in the secondary minimum are not in direct contact with each other but remain at some distance. They still have the possibility of moving and sliding along each other. Consequently, the structures of an aggregate in the primary and secondary minima are different, as we discuss in Section 16.6. Unlike in the primary minimum, aggregation in the secondary minimum is reversible with

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Stability of Lyophobic Colloids

325

respect to variation of the ionic strength: lowering the ionic strength leads to spontaneous redispersion of the colloidal system.

16.2.1 Critical Coagulation Concentration The height of the repulsive barrier between two surfaces of the same charge sign is suppressed by increasing the electrolyte concentration. At higher ionic strength a larger fraction of the colliding particles aggregates in the primary minimum. The transition from the stable to the unstable state of a lyophobic colloidal system is not sharp and we need an (arbitrary) criterion to deﬁne the stability. The rate of coagulation continues to increase with increasing ionic strength until the situation is reached where ðint GÞmax ¼ 0. Then, and beyond that particular ionic strength, each collision between the particles leads to coagulation. It is therefore plausible, and convenient for mathematical reasons, to deﬁne stability in terms of the socalled critical coagulation concentration (ccc), being the electrolyte concentration at which the maximum coagulation rate is just reached and, hence, for which ðint GÞmax ¼ 0. From Gdisp þ Gedl ¼ 0 together with (16.19) and (16.21) or (16.22), and (9.29) we can derive an expression for the ccc, which for planar surfaces reads 6 ccc ¼ 2:13 105 ðee0 Þ3 ðRT Þ5 g4d A2 121 ðzFÞ :

ð16:23Þ

In the case of spheres the numerical constant of 2:13 105 has to be replaced by 0:88 105 . Equation (16.23) reveals how the stability of a colloidal system depends on the various experimental variables. —

—

—

—

The critical coagulation concentration is inversely proportional to the square of the Hamaker constant, or, in other words, to the square of the dispersion interaction between the particles at a given separation. At given cd (and hence gd ) and A, charged colloids are far less stable in a medium of low dielectric permittivity ee0 because electrostatic interactions are much stronger in a low dielectric medium. Equation (16.23) suggests that the ccc varies with T 5 . However, the temperature-dependency of the colloidal stability is far less because the dielectric permittivity reduces with increasing temperature so that ðee0 Þ3 T 5 is usually only slightly dependent on the temperature. The inﬂuence of the valency of the counterion on colloid stability is ruled by g4d z6. At high value for cd , where gd approaches unity, the ccc scales with z6 . On the other hand, at low values for cd ; gd is proportional to zcd resulting in a ccc that varies with c4d z2 . Thus, at constant cd , the ccc decreases with the square of the valency. However, even at low values of cd (which is often the case at ionic strengths corresponding to the ccc), a stronger inﬂuence of the valency is often experimentally observed. The reason is that cd is not constant

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326

Chapter 16

but reduces with increasing valency (i.e., increasing value of k), as discussed in Section 9.4.3, Eq. (9.40). For most cases the ccc varies with zx with 2 x 6. If there is an increased speciﬁc adsorption of counterions, that is, adsorption within the Stern layer (see Figure 9.8) with increasing valency, the valence-dependency of the ccc may go even beyond the sixth power. In Table 16.2 values of critical coagulation concentrations are collected for different aqueous colloidal dispersions. Within a series of counterions having the same valency a small but systematic inﬂuence of the type of ion is observed. This nonelectrostatic and, therefore, speciﬁc inﬂuence seems to be related to the size of the hydrated ion. The larger the ions, the less they are hydrated and the smaller their hydrated size. These ions can approach the particle surface more closely. In view of the Stern model, presented in Section 9.4.3, a larger value of cd and, hence, a higher ccc, is expected. The observation of the opposite trend may be explained by a stronger speciﬁc adsorption of the weaker hydrated ions. This systematic variation in ccc, related to the ion size and hydration, is known as the lyotropic series. In practice, the critical coagulation concentration is usually determined after allowing the formed aggregates some time to settle. Clearly, a lower value for the coagulation concentration is obtained when the elapsed time before the occurrence of coagulation is judged or chosen to be longer. Experimental assessment of the ccc, deﬁned as the electrolyte concentration at which ðint GÞmax ¼ 0 is reached, must be based on establishing the onset of the maximum coagulation rate. The kinetics of coagulation is further discussed in Section 16.4. Finally, it is emphasized again that the DLVO theory predicts the stability of lyophobic colloids. It addresses the question of whether aggregation occurs in the Table 16.2 Critical Coagulation Concentration (in 103 moles per liter) for Mono-, Di-, and Trivalent Counterions at Positively and Negatively Charged Colloidal Particles Positively charged Fe2O3 particles Electrolyte KCl

ccc 100

CaSO4

6.6

KFe(CN)6

0.65

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Negatively charged Agl particles Electrolyte

ccc

LiNO3 NaNO3 KNO3 RbNO3 Mg(NO3)2 Ca(NO3)2 Ba(NO3)2 Al(NO3)3 La(NO3)3

165 140 136 126 2.60 2.40 2.26 0.067 0.069

Stability of Lyophobic Colloids

327

primary minimum. However, as it does not include short-range interactions, the DLVO theory cannot be used to describe the behavior of the particles in the aggregates.

16.3 THE INFLUENCE OF POLYMERS ON COLLOID STABILITY In many lyophobic colloidal systems, especially the biological ones, polymers are present in solution and=or are attached to the particle surface. These polymers could be uncharged or charged (polyelectrolytes), highly solvated or more compact (polymers below y-conditions; see Section 12.3). For instance, most micro-organisms have a rather thick layer of polymeric material associated with their surfaces. Also, micro-organisms may excrete polymers into the surrounding solution. The inﬂuence of polymers, either attached to the particle surface or not, on colloidal stability is subtle and intricate. Under certain conditions they stabilize the dispersion whereas under other circumstances they induce aggregation. Polymer-mediated aggregation is usually referred to as ﬂocculation as distinct from coagulation for salt-induced aggregation. Polymers are often applied to promote either stabilization or aggregation. Examples of polymer-induced aggregation are found in water puriﬁcation, clariﬁcation of wine and fruit juices, and in papermaking. Stabilization of colloidal particles by polymers is applied in foodstuffs, paints, and dyes, in the dispersion of particles on magnetic tapes and photographic paper, as well as in many cosmetic and medical salves and lotions. A rather recent development is the use of polymers to reduce or prevent the deposition of protein molecules and biological cells on surfaces in order to avoid biofouling and, hence, improve the biocompatibility of these surfaces. Because of the complex inﬂuence of polymers on colloidal stability, we treat this subject on a qualitative or, at best, semiquantitative level.

16.3.1 Nonadsorbing Polymers: Depletion Flocculation Polymer molecules dissolved in the medium surrounding the particles may inﬂuence colloidal stability. A nonadsorbing polymer molecule with radius R cannot come within a distance R of the particle surface. For a highly ﬂexible, coily polymer with a radius of gyration Rg this might be possible, but the coil must then be deformed and that can only occur at the expense of conformation entropy (cf. Section 12.3). Therefore, the concentration of polymer close to the surface is lower than that farther away in the solution. As depicted in Figure 16.6 the particles are enveloped by a depletion region with a thickness d (¼ RÞ that is forbidden for the polymer. Because there are many particles in the system the total depletion volume Vdepl is far from negligible. For instance, for a 1% (w=v)

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328

Chapter 16

d R

Figure 16.6 Polymer depleted layers around particles.

dispersion of particles with a speciﬁc surface area of 50 m2 g1 , the volume from which polymer molecules having a radius of 50 nm are excluded amounts to 25 cm3 per dm3 of the dispersion. Due to the polymer concentration difference between the depletion zone and the bulk solution the dissolved polymer molecules exert an osmotic force on the particles with a corresponding increase in Gibbs energy Gdepl equal to Vdepl ppol , where ppol is the osmotic pressure of the polymer solution. As the osmotic force acts on all sides of the particles it does not result in a net force. When particles approach each other at a separation distance h 2d the depletion zones of the individual particles start to overlap. The total depletion volume and, consequently, Gdepl ðhÞ decreases. Based on this mechanism, particles are spontaneously driven together until the overlap of depletion zones is at a maximum. Maximum overlap is reached when the particles contact each other. Figure 16.7 illustrates this phenomenon, which is called depletion ﬂocculation. The range over which depletion attraction operates equals 2R. In particular for highly swollen polymers, Rg may reach values of some tens of nm and, hence, the depletion forces may be effective over separation distances between particles that exceed the range of dispersion and double layer forces (cf. Section 16.1). On the other hand, the osmotic forces are relatively weak. Depletion ﬂocculation occurs when the molar polymer concentration is sufﬁciently high, which is more readily achieved by using polymers of a relatively low degree of polymerization. h

d

d

Gdepl

Figure 16.7 Attraction between particles due to overlapping depletion regions. Gibbs energy of depletion attraction as a function of particle separation.

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Stability of Lyophobic Colloids

329

When the mole fraction of polymer in the medium approaches unity, that is, in a polymer melt, osmotic effects are absent. Colloidal particles dispersed in a polymer melt may be stable because aggregation causes deformation with a decreased conformational entropy of the polymer molecules.

16.3.2 Adsorbing Polymers: Bridging Flocculation and Steric Stabilization Adsorbed polymers have strong effects on particle interactions. First of all, adsorbed polymers may alter Gdisp ðhÞ and Gedl ðhÞ. The general situation is depicted in Figure 16.8. To calculate Gdisp ðhÞ we have to include the Hamaker constant A3 of the adsorbed layer. In the case of a compact adsorbed layer such as the ones that are usually formed by globular protein molecules (cf. Chapter 15), A3 may signiﬁcantly differ from the Hamaker constant of the supporting medium A2 . With respect to Gedl ðhÞ, the electrokinetic potentials corresponding to the potentials at the peripheries of the polymer-coated particles have to be considered. Now, for compact adsorbed layers of thickness d, int GðhÞ can still be calculated using (16.18), provided that the above-mentioned modiﬁcations are taken into account and that h is replaced by (h 2d). Thus polymers that adsorb in a compact or ﬂat orientation may either improve or worsen colloidal stability, depending on their inﬂuences on the potentials and, usually to a lesser extent, on the Hamaker constant of the whole system. When the adsorbed polymer forms a loosely structured train-loop-tail-like structure, as shown in Figure 15.11, the polymer segment density in the adsorbed layer is usually low so that A3 A2 . Furthermore, as the adsorbed layer is highly solvated and (nearly) freely penetrable for counterions the electrostatic potential at x ¼ d is relatively low. Such ﬂexible, loopy polymer layers may protrude into the surrounding medium over a distance exceeding the electrical double layer thickness; that is, d > k1 . For example, in blood of ionic strength 0.15 M double layer overlap starts at 1 to 2 nm whereas the extension of adsorbed polymer layers may readily reach a few tens of nm.

A3 A2 A1

A1

a

d

d h

Figure 16.8 Polymer-covered particles.

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330

Chapter 16

If only one of the two interacting surfaces is covered with a polymer layer of thickness d > 2k1 and the other one is bare, or if both surfaces are not more than partly covered, one and the same polymer molecule may attach to both surfaces and bridge between them with a concomittant Gibbs energy Gbr . For sufﬁcient afﬁnity between the polymer and the particle surface and sufﬁcient anchoring at both surfaces it causes the particles to aggregate due to a so-called bridging ﬂocculation process. See Figure 16.9. Needless to say, bridging ﬂocculation of charged particles is enhanced by increasing the ionic strength. In the aggregates thus formed the particles are not in direct contact with each other. Their separation compares with the polymer layer thickness because closer separation would be detrimental to the polymer conformational entropy and, hence, cause Gbr ðhÞ to become less negative. When there is an excess of polymer each particle is saturated with a layer of adsorbed polymer. Then, upon approach between particles, the outermost fringes of the loops and tails ﬁxed onto each particle begin to interpenetrate resulting in a steric repulsion with a Gibbs energy effect Gst . The onset of this effect occurs at a separation h ¼ 2d which is for most ﬂexible polymers far more than the separation at which electrical double layers overlap. Also, the adsorbed polymer layers keep the particles apart which weakens the contribution of Gdisp . Altogether, as can be seen in Figure 16.10, the minimum in int GðhÞ equals Gdisp ðhÞ at h ¼ 2d. In the case of thick adsorbed layers the particles are kept apart at a separation where Gdisp is negligibly small. Consequently, the colloidal suspension is stabilized by steric repulsion between the adsorbed polymer layers. The steric repulsion comprises two effects. In the region where the (outer parts of the) adsorbed layers interpenetrate the polymer segment density is higher. It causes a locally increased osmotic pressure which tends to push the particles apart. Furthermore, upon overlap the polymer molecules cannot adopt all conformations that are otherwise available to them. This limited freedom reduces the conforma-

G

Gedl h ∆intG Gbr Gdisp

Figure 16.9 Polymer bridging between two particles. Gibbs energy of interaction between particles that are linked by polymer bridges as a function of their separation distance.

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Stability of Lyophobic Colloids

331

G Gst Gedl

h

∆intG Gdisp

Figure 16.10 Steric stabilization between polymer-coated surfaces. Gibbs energy of interaction between particles that are sterically stabilized.

tional entropy of the polymers with an equivalent increase in the Gibbs energy of the system. In practice it is not always easy to achieve steric stabilization by polymers. It takes time for polymer adsorption to reach saturation. Therefore, for a certain period of time the surfaces are only partly covered and bridging ﬂocculation may take place. The ﬂocs may be redispersed if the dynamics of attachment and detachment of polymer segments are such that all bridges are disrupted, which is not likely to happen. An efﬁcacious way of stabilizing colloidal particles is to use polymer of relatively low molecular mass. The risk of bridging is negligible since at any stage of the adsorption process the thickness of the adsorbed layer is less than the double layer thickness. Then, after polymer adsorption saturation is reached the system is stable because at high ionic strength steric stabilization takes over. The distribution of polymer segments between trains, loops, and tails may be adjusted when the polymer-covered surfaces encounter each other and this, in turn, may interfere with the steric stabilization mechanism. The polymers best performing as stabilizers are block copolymers made of two kinds of monomers that are clustered in long sequences of one kind. The sequence of the one kind of monomer should have a high afﬁnity for adsorption and, hence, ﬁrm attachment at the surface whereas the sequence of the other type should prefer to stay in solution giving extended loops and tails. With homopolymers afﬁnity for the sorbent surface and extension into the solution are compromising tendencies. As, under most conditions, polymer interpenetration is dominated by adverse entropy effects, the corresponding Gibbs energy Gst becomes less positive when lowering the temperature. If the enthalpy of interpenetration is negative, Gst becomes negative at some low temperature resulting in attraction between the polymer layers. Indeed, various sterically stabilized dispersions are observed to ﬂocculate on lowering the temperature. Interaction between the polymer-coated surfaces is affected by changing the quality of the solvent (i.e., the solvent–polymer interaction). Improving the

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332

Chapter 16

polymer solubility by adding a co-solvent and=or changing the temperature, for example, could result in release of the polymer from the surfaces. Bridging could thus be eliminated and steric stabilization could turn into depletion ﬂocculation. Of course, the reverse changes might be induced by lowering the solvent quality. Furthermore, decreasing the solvency below y-conditions induces intra- and intermolecular attraction between polymer segments (cf. Section 12.3). As a result, polymer bridges contract leading to more compact aggregates and steric repulsion turns into steric attraction and, hence, ﬂocculation. When the polymers are polyelectrolytes the situation is even more complicated. The inﬂuence of polyelectrolytes arises from steric and electrostatic interactions, together referred to as the electrosteric effect. Most ﬂocculants used in practice to aggregate colloidal dispersions are polyelectrolytes. By selecting a polyelectrolyte with a charge sign opposite to that of the colloidal particles the main effect of the polyelectrolyte is charge neutralization. In that case the polyelectrolyte adsorbs in a relatively ﬂat orientation but large polyelectrolytes may also form bridges between the particles. Small oppositely charged divalent ions may bridge between charges on the polyelectrolyte and the surface, so that charged particles may be ﬂocculated by a polyelectrolyte of the same charge sign. A superequivalent amount of polyelectrolyte at the surface may again stabilize the dispersion both by electrical double layer repulsion and steric repulsion. A highly solvated adsorbed layer of uncharged polymer in which the polymer segment density is low would barely inﬂuence the electrostatic potential proﬁle and the dispersion forces between the interacting particles. Therefore int GðhÞ may be approximated as Gdisp ðhÞ þ Gedl ðhÞ þ Gst ðhÞ. However, with polyelectrolytes the additivity is less obvious because the charge distribution in the electrical double layer affects the segment density distribution of the polyelectrolyte in the adsorbed layer. In conclusion, the way polymers inﬂuence the stability of lyophobic colloids is far more complicated than the way low molecular weight electrolytes do. Whether polymers stabilize or destabilize the dispersion is delicately determined by the properties and composition of the system (adsorption afﬁnity, solvent quality, particle size, degree of polymerization, charge densities on the particle and the polymer, particle–polymer ratio, ionic strength, presence of divalent ions, etc.) and external conditions such as the temperature.

16.3.3 Polymer Brushes A particularly interesting and effective way steric stabilization can be achieved is by grafting polymers at one end onto the particle surface leaving the other part dangling in the solution. For high grafting densities this results in stretching of the polymer chains out from the surface to produce a brush. The best results are obtained by using diblock copolymers with one block having a strong afﬁnity for the surface and the other for the solvent. Such a polymer brush is shown in Figure 16.11.

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Stability of Lyophobic Colloids

Figure 16.11

333

Polymer brush at a surface.

If the grafting density, that is, the number of polymer chains per unit surface area, can be controlled (which usually is not an easy task) and using polymers of known degree of polymerization, brushes on particles and, more often, on macroscopic surfaces may be tailormade to serve special applications. For instance, brushes having ‘‘hairs’’ of hydrophilic polymers may be used to obtain antifouling surfaces to which protein molecules, bacteria, and other biological cells do not adhere. Furthermore, brushes act as lubricants since they considerably reduce friction between two slipping surfaces. This phenomenon may be utilized when designing artiﬁcial joints. In various ways the behavior of polymer molecules in brushes is different from that in dilute solutions. As we discussed in Section 12.3 the radius of gyration of a polymer molecule in a random coil conformation increases with the square root of the degree of polymerization Rg Np0:5 . However, in a dense brush the polymer molecule can adjust its size only in one dimension, namely, normal to the supporting surface. The thickness d of the brush therefore scales linearly with Np . Another striking difference between a polymer in a brush and in solution is observed for the case where the polymer is a polyelectrolyte. Because of screening of intramolecular electrostatic interactions the coil-like conformation of a polyelectrolyte in solution shrinks when the ionic strength is increased. This may be different in a polyelectrolyte brush. The high polymer segment density in the brush suppresses the charge density along the polymer chains. Adding low molecular weight electrolyte now primarily causes an increase of the polyelectrolyte charge density which makes the brush swell.

16.4 AGGREGATION KINETICS Imagine an unstable colloidal dispersion in which two single primary particles collide and stick together, thereby forming a doublet. Subsequently, doublets collide with primary particles or with other doublets forming triplets and

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334

Chapter 16

quadruplets which, in turn, encounter other kinetic particles, and so on. Finally, macroscopic aggregates are formed that segregate from the medium. In the course of the aggregation process the number N ð¼ N1 þ N2 þ N3 þ N4 þ Þ of kinetic particles per unit volume (equivalent to a concentration) steadily decreases. The subscripts refer to the number of primary particles in the aggregate. In sufﬁciently diluted dispersions essentially all those collisions, a few of which are represented in Figure 16.12, are of binary nature. Therefore, for dilute dispersions the rate of aggregation may be approximated by second order kinetics, described by dN =dt ¼ kN 2 ;

ð16:24Þ

in which k is the rate constant for the aggregation process. Integration of (16.24) yields N ðtÞ ¼

N ðt ¼ 0Þ : 1 þ N ðt ¼ 0Þkt

ð16:25Þ

The half-time for aggregation t1=2 , deﬁned as the time it takes to reduce N by a factor of two, is then given by t1=2 ¼

1 ; kN ðt ¼ 0Þ

ð16:26Þ

which implies that the stability of a colloidal dispersion is inversely proportional to the particle concentration. The rate constant k is determined by the interaction between the particles. Let us ﬁrst consider the simple case of particles that do not interact except at zero separation where they form a permanent bond. The aggregation rate of such k11

+

+

k12

+ k22 k13

+

Figure 16.12

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Particle aggregation.

Stability of Lyophobic Colloids

335

indifferent sticky particles is governed by the rate of transport toward each other. The transport may be dominated by an imposed external force ﬁeld or it may be the result of only diffusion. In the former case the aggregation process is called orthokinetic and in the latter perikinetic. Here, we focus on perikinetic aggregation. During aggregation the number of kinetic particles adjacent to the aggregate is decreased and the resulting concentration difference is responsible for the diffusion of kinetic particles towards the growing aggregate. For two identical spherical particles it has been derived that k ¼ 8pDa;

ð16:27Þ

where D is the diffusion constant of a sphere with radius a. For two different spheres i and j k ¼ 4pðDi þ Dj Þðai þ aj Þ:

ð16:28Þ

Combining (16.27) with D¼

kB T ; 6pZa

ð16:29Þ

where kB is Boltzmann’s constant and Z the viscosity of the medium, we obtain k¼

4kB T ; 3Z

ð16:30Þ

showing that the rate of aggregation slows down as the viscosity of the medium increases. Another remarkable consequence of (16.30) is that the rate constant does not depend on the particle size. This independence results from the cancellation of the increased probability of larger particles colliding by the decreased collision probability due to a smaller value of the diffusion coefﬁcient. When the particles attract each other, for example, by a dominating dispersion force, polymer bridging, or depletion ﬂocculation, the k-value will be only slightly larger than the ones given by (16.27) or (16.30) which are based on diffusion down to zero separation. The reason is that, at not too high particle concentration, by far the largest part of the distance to be traveled by the encountering particles is beyond the range over which interparticle forces act. The diffusion-controlled aggregation rate is referred to as rapid aggregation and the rate constant is usually denoted kr . For a 1% (v=v) dispersion of particles having radii of 100 nm in an aqueous medium of 20 C (Z ¼ 0:001 N m2 s) it can thus be calculated that the half-time of the aggregation is less than a tenth of a second. When the particles encounter a repulsive energy barrier ðint GÞmax, only a fraction of the collisions are effective. This is reﬂected in a lower value for the aggregation rate constant. Clearly, the lower the value of the rate constant the

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336

Chapter 16

more stable the dispersion is. The stability of the dispersion may be characterized by W

kr ; k

ð16:31Þ

which is called the stability ratio. The reciprocal value of W may be considered the sticking probability for two colliding particles; that probability is practically zero for a stable dispersion and unity for rapid aggregation where upon each collision the particles stick together. The rate constant k may therefore be interpreted as the product of the collision rate and the sticking probability: k ¼ kr ð1=W Þ. The sticking probability is related to the height of the barrier for deposition: W expbðint GÞmax =kB T c. For dispersions stabilized by electrical double layer repulsion W may be derived using the DLVO theory. We do not go into detail here but just give an approximate expression that provides a simple way to estimate W : 1 ðint GÞmax exp W ¼ : ð16:32Þ 2ka kB T For the same system as mentioned above in a 103 M solution of a 1 : 1 electrolyte (so that k ¼ 101 nm1 ) but now with a Gibbs energy barrier of 20 kB T between the particles, we ﬁnd that the half-time of aggregation goes up to more than three weeks. Depending on acceptible storage time or shelf life, the system may now be regarded as stable or not. The rate of aggregation and, consequently, the value of W can be experimentally determined. In view of (16.32) and (9.29) and applying the DLVO theory to relate ðint GÞmax to the electrolyte concentration c, it follows that for an electrical double layer stabilized dispersion log W varies linearly with log c until the suppression of Gedl ðhÞ results in ðint GÞmax ¼ 0 and rapid aggregation (W ¼ 1) is reached. Hence, a plot of log W against log c consists of two linear parts, as shown in Figure 16.13. The value of c where the linear parts intersect is the critical coagulation concentration, as deﬁned in Section 16.2.1. When the dispersion is stirred or when it ﬂows, diffusion is not the only mechanism determining the collision probability. Under such conditions the shear rate, that is, the velocity gradient, normal to the particle surface, dnðxÞ=dx (see Section 17.1.3) enhances the collision frequency so that the orthokinetic aggregation rate is faster than the perikinetic aggregation rate. Under conditions of rapid aggregation it can be derived that krortho krperi

¼

4Za3 dnðxÞ=dx kB T

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ð16:33Þ

Stability of Lyophobic Colloids

337

log W

rapid aggregation 0

log (ccc)

Figure 16.13

log c

Rate of particle aggregation as a function of electrolyte concentration.

with (16.30), krortho ¼

16 3 a dnðxÞ=dx: 3

ð16:34Þ

As opposed to krperi [see (16.30)], krortho is strongly dependent on the particle size a, so that for relatively large particles in a ﬂowing system the aggregation rate may be dominated by velocity gradient-induced collisions.

16.5 MORPHOLOGY OF COLLOIDAL AGGREGATES Colloidal aggregates may be formed by different mechanisms. For instance, particles surrounded by an electrical double layer coagulate either in the primary or secondary minimum by the addition of salt. Polymer-coated particles may ﬂocculate by polymer bridging or by changing the composition of the medium such that the polymer coats become insoluble. The ﬁnal structure of the aggregate depends on the way the particles associate and also on the possible rearrangements that occur after the initial association. Salt-induced coagulation in the secondary minimum leads to a rather compactly packed aggregate which, as explained in Section 16.2, may be readily redispersed on lowering the ionic strength. The reason is that in the coagulum the particles are not physically attached to one another and move around to position themselves at the most optimal mutual separations. Aggregation in the primary minimum gives irreversible particle–particle contact. In case of rapid aggregation, that is, in the absence of any barrier, the colliding particle is locked into the aggregate at the position of ﬁrst contact. This leads to much less dense fractal aggregates.

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Chapter 16

In contrast to a compact aggregate in which the number N of primary particles varies with the third power of the radius R of the aggregate, in a fractal aggregate N scales with R f . The exponent f is the so-called fractal dimension. Figure 16.14 illustrates the nature of a fractal aggregate. Close inspection of such objects reveals self-similarity. By way of example, in Figure 16.14 primary particles stick together in clusters of 5 and in a similar array 5 of these clusters form a cluster of 25 particles. Subsequently, 5 of the 25 particle clusters associate in a similar pattern to give a 125 particle cluster, and so on. For this twodimensional example increasing the number of particles in the aggregate by a factor of 5 corresponds to growth by a factor of 3. The fractal dimension is obtained from 5 ¼ 3 f , yielding f ¼ 1:465. For an aggregate that grows linearly, f ¼ 1 and if the particles arrange themselves in a space-ﬁlling square array, f ¼ 2. The structure of a fractal aggregate is intermediate between those of a linear and a compact object. Hence, the fractal dimension is a (noninteger) number between 1 and 2 for a two-dimensional aggregate and between 1 and 3 for a three-dimensional aggregate. Computer simulations have shown that a hit-and-stick aggregation based on diffusion-limited kinetics leads to aggregates with a fractal dimension of 1.8. In the case of slow aggregation, when a barrier must be overcome, an incoming particle collides with the aggregate several times before it sticks. It is then allowed to ﬁnd a more than average favorable contact with the aggregate. This leads to a somewhat more compact structure with a fractal dimension of about 2. It may be understood that the density of polymer-mediated aggregates is further affected by the ﬂexibility and the thickness of the adsorbed polymer layers. Fractal aggregates become less dense the larger they grow. Finally, one highly tenuous aggregate may ﬁll the entire volume: a particle gel has formed. This occurs when the volume fraction of the particles in the aggregate is reduced until it reaches the volume fraction of the particles in the system before aggregation [cf. the formation of an unconnected polymer network described in Section 12.4, Eq. (12.14)]. The model thus predicts a certain maximum fractal aggregate size.

Figure 16.14

Fractal aggregates illustrating self-similarity.

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Figure 16.15 Particle gel showing (from left to right) progressive aging stages.

After it has been formed the gel may still undergo further changes. First, because of the low density of contacts the network in a particle gel is not strong. It can easily be disrupted by stirring or even the inﬂuence of gravity. The gel fragments then settle as a voluminous sediment. When the particles in the gel still have some freedom to move, the particle strings are ﬂexible, allowing the formation of extra bonds which strengthens the gel. Such an aging process renders a coarser gel with a densiﬁed solid matrix and increased pore sizes, as illustrated in Figure 16.15. Gels may expel or take up solvent. These phenomena, known as syneresis and imbibition, are brieﬂy described in Section 12.7.

EXERCISES 16.1

16.2

Comment on the following statements: (a) Lyophilic colloids in aqueous dispersion do not aggregate at electrolyte concentrations below the critical coagulation concentration. (b) The electrical double layer interaction between two identical charged particles that are separated over a distance equal to the electrical double layer thickness is independent of the ionic strength. (c) The stability factor (W ) equals unity for stable colloidal dispersions. (d) Steric stabilization of spherical colloidal particles by an adsorbed polymer layer is independent of the particle size. (e) The electrical double layer interaction between two charged particles in water is stronger than between the same particles in the air. Using advanced equipment the interaction force f between two molecularly smooth surfaces immersed in an aqueous electrolyte solution can be measured as a function of the separation distance h down to a few nanometers. (a) What is the relation between f and the Gibbs energy of the interaction between the two surfaces (at constant temperature and pressure)?

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Chapter 16

(b)

In the case of identical charged surfaces having a relatively low Stern potential cd , the electrical double layer contribution to f is given by fedl ¼ 2ee0 k2 c2d exp½kh;

ðE16:1Þ

where k is the reciprocal Debye length and ee0 the dielectric permittivity of the medium. Derive (E16.1) and check the dimension of fedl . (c) Under a certain condition (E16.1) can be written as fedl ¼

2s20 exp½kh; ee0

ðE16:2Þ

where s0 is the surface charge density. Which condition allows transformation of (E16.1) into (E16.2)? (d) Calculate fedl in a 103 M NaCl solution and charge densities of 3 mC m2 on each surface. (e) Show in one ﬁgure curves for fedl in 103 M and 101 M NaCl. Assume a constant value for the surface charge density. 16.3

16.4

(a)

A colloidal dispersion of negatively charged particles ﬂocculates upon the addition of a polymer solution. Mention two possible mechanisms for the ﬂocculation. How can you experimentally infer which of the mechanisms applies without determining whether the polymer adsorbs on the particles? (b) Describe the procedure to sterically stabilize charged colloidal particles against salt-induced aggregation. (c) How far should highly solvated ﬂexible polymers adsorbed on colloidal particles extend into a surrounding 0.1 M NaCl solution to prevent aggregation? (d) Why are the contributions from steric and electrical double layer interactions to the total Gibbs energy of interaction between two particles not additive? The ﬁgure shows the electrokinetic potential z as a function of pH, for colloidal particles in 102 M KNO3 and 102 M LiNO3.

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Stability of Lyophobic Colloids

341

(a)

16.5

Give a qualitative interpretation for the curves, assuming that in KNO3 speciﬁc adsorption is absent. The critical coagulation concentration ccc of the dispersion for KNO3 is 0.15 M at pH 4 and 0.12 m at pH 11. (b) What do you expect for the ccc for LiNO3 at those two pH values as compared to KNO3? The ccc for Ca(NO3)2 is 0.075 M at pH 4 and 0.002 M at pH 11. (c) Explain the differences in the ccc values for KNO3. (d) Calculate the ratio of the Stern potentials in solutions of Ca(NO3)2, respectively, KNO3 at their cccs at pH 4 and pH 11. Assume that (i) at conditions of rapid coagulation the Stern potentials are relatively low and (ii) the ccc for an electrolyte is solely determined by the valency of the counterion. The aggregation rate of a polymer latex is determined by measuring the particle concentration as a function of time. 10 cm3 of the latex is instantaneously mixed with 10 cm3 of a polyoxyethylene (PEO) solution having a mass concentration Cp and the aggregation rate np is immediately measured. PEO is an uncharged highly soluble ﬂexible polymer and, in the Cp range considered, it does not signiﬁcantly inﬂuence the viscosity of the solution. In another experiment 10 cm3 of the polymer latex is mixed with 10 cm3 1 M NaCl solution and the aggregation rate ns determined. The ﬁgure shows a curve for logðns =np Þ as a function of Cp . (a) Does ns =np approximate the stability ratio W as deﬁned in (16.32)? Indicate over which Cp range rapid aggregation occurs. (b) Give a physical interpretation of the descending branch of logðns =np Þ versus Cp . Why does logðns =np Þ reach negative values? What is the reason for the increase of ns =np at higher values of Cp ?

SUGGESTIONS FOR FURTHER READING M. Elimelech, J. Gregory, X. Jia, R. A. Williams. Particle Deposition and Aggregation, Oxford: Butterworth-Heinemann, 1995. F. Family, D. P. Landau (eds.). Kinetics of Aggregation and Gelation, Amsterdam: North Holland, 1984. D. H. Napper. Polymeric Stabilization of Colloidal Dispersions, London: Academic, 1983.

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Mouthfeel

Texture, as much as ﬂavor, determines the oral experience of foods. The consumer has a conditioned expectation of the mouthfeel of a product and if he or she does not sense what is expected, the product is often considered unacceptable: ice cream should be soft and creamy, pasta stretchy, potato chips crispy, and chocolate should melt slowly and smoothly in the mouth. Moreover, textural contrasts on a plate often add to the pleasure of eating. Most textured foods are built of a network of colloidal particles and molecules held together by a variety of forces. The structure, that is, consistency and texture, of such heterogeneous systems is strongly inﬂuenced by the strengths of these forces. The mechanical response of the food product during mastication, in particular its resilience against deformation, is a key factor for the in-mouth sensation. Therefore, rheological properties play a crucial role in the sensorial acceptance of food.

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17 Rheology, with Special Attention to Dispersions and Interfaces

Rheology deals with ﬂow and deformation of materials. More precisely, rheology is the study of the deformation of matter in response to an applied force, as a function of time. The rheological properties of a material reﬂect the relation between the deformation and the force. The following extremes may be distinguished. 1.

2.

The material deforms (ﬂows) at constant rate as long as the force is imposed and the deformation remains when the force is released. Materials showing such rheological behavior are called viscous. Gases and most liquids are viscous ﬂuids. The physical explanation is that the molecules in the viscous material move to different positions under the inﬂuence of the applied force. They slide along each other whereby intermolecular interactions (physical bonds) are disrupted and reformed. The energy required for the disruption is supplied by the exerted force and the energy released by the reformation of bonds is dissipated as heat. The material deforms as long as the force is applied but it returns to its original state when the force is released. As a rule, the extent of deformation is proportional to the imposed force. This is known as elastic behavior. Examples are (metal) springs, rubber bands, arterial walls, and (human) skin. The force brings the molecules of the elastic material into a state of higher Gibbs energy (distortion of bond lengths and angles between the atoms in the metal spring, stretched polymer chains in the rubber band, etc.). This state is maintained while the force

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is applied but on release the deformation is undone; that is, the original atomic and molecular arrangement is restored, during which the material exerts a corresponding force on its surroundings. Thus the energy added by the applied force is stored in elastic materials. Whether a material behaves as viscous or elastic depends on the properties of its intermolecular bonds, notably their strengths and lifetimes relative to the strength and duration of the applied force. It is not surprising that many materials behave partly viscous and partly elastic. Such materials are referred to as viscoelastic. Interactions between structural elements in soft condensed matter such as concentrated dispersions, coagulated and ﬂocculated colloidal particles, and particle and polymer gels often give rise to viscoelastic behavior. By way of example, the chains of the polymer molecules in semidilute and concentrated solutions (see Sections 12.4 and 12.7) are able to slide along each other but the chains may become entangled to form temporary crosslinks. A certain amount of time is needed for the disentanglement allowing the material to ﬂow. Therefore, the rheological behavior of the system depends on the timescale of the measurement, that is, on the duration of the applied force. In general, if no bonds are broken during the time the force is applied, a material behaves fully elastic; if all bonds are disrupted the material is viscous. Viscoelasticity is related to the ease with which bonds are broken or, more precisely, with which a material is able to relax the force that is needed to maintain a constant deformation. In this context a characteristic time, the relaxation time, is deﬁned as the time in which the required force f has decayed to 1=e of its original value. In a purely elastic system no bonds are broken and the force is constant in time. For a purely viscous substance in which the bonds are instantaneously broken the force is reduced to zero directly after application. In the case of a viscoelastic material the bonds are gradually disrupted so that the required force decreases gradually and, usually, exponentially during the time of observation. These different characteristics are graphically represented in Figure 17.1. Elastic materials are characterized by a very long relaxation time tr , whereas for viscous materials tr is extremely short. More precisely, whether a material presents itself as elastic or viscous depends on the relaxation time relative to the time of observation tabs (¼ timescale of the experiment). This ratio tr =tabs is known as the Deborah number De. Now the rheological behavior may be classiﬁed on the basis of De. If De 1 the material is viscous, if De 1 it is elastic, and for viscoelastic materials De is on the order of unity. For instance, under essentially all practical conditions liquid water, in which the average lifetime of intermolecular bonds is on the order of 1013 s, is viscous, whereas solids, in which intermolecular bonds may live longer than 1010 s, are elastic. However, given sufﬁcient time, even solids are viscous: glass windows in medieval cathedrals often are thicker at the bottom than at the top and over

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345

elastic

force

f

f e

visco-elastic viscous

τr

time

Figure 17.1 Force required to maintain a constant deformation in time.

geological timespans even rocks and mountains ﬂow. On the other hand, liquids, like water, may behave elastically in (molecular) processes that take place on very short timescales. Because of the interparticle interactions concentrated lyophobic and lyophilic colloidal systems are often viscoelastic. This shows up in such products as paint, medical and cosmetic ointments and lotions, and a wide variety of foodstuffs. The practical relevance of the rheological properties of such products lies primarily in quality control. A food product should have the right texture. Creams and pastes should ﬂow by rubbing or by pressing the tube in which they are contained and paints should ﬂow by the force exerted by the brush, but thereafter these products should quickly become solid. Furthermore, during processing such as boiling, cooking, drying, freezing, and so on, a material may change its rheological properties drastically. Monitoring these properties may therefore help to control the progress of the process. Also, when designing and constructing processing equipment the rheological properties of the materials to be processed should be taken into account.

17.1 RHEOLOGICAL PROPERTIES Consider a cubical volume-element of material with dimensions that are large relative to its structural units and small relative to the size of the whole sample. Any deformation of that volume-element may be caused by forces acting on each of the six sides of the cube. Each of these forces has its own magnitude and direction, and description of the relation between the overall deformation and the forces is very complex. For the sake of clarity we consider three simple deformations: equal compression at all sides of the cube, dilation in one direction, and shear.

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Chapter 17

17.1.1 Compression at All Sides When all-sided compressed, as indicated in Figure 17.2, the volume of the element changes but the shape remains the same. If the material behaves fully elastically, the relation between the variation in the exerted force f per unit area A, s, and the relative change in the volume, dV =V ð¼ d ln V Þ, is given by ds ¼ Kd ln V ;

ð17:1Þ

in which K is the compression modulus. It is characteristic for each material and it has a constant value for small volume changes. The value of K for condensed phases, solids and liquids, is on the order of 109 N m2 but for gases it is much smaller, say, about 105 N m2 . Compression of condensed phases is energetically highly unfavourable as the short-range intermolecular and interatomic repulsive forces, especially the Born repulsion (see Section 13.3.4), steeply increase with decreasing separations. Conversely, when the volume-element is isotropically expanded (dilation), Eq. (17.1) applies as well, with K now being called the dilation modulus.

17.1.2 Elongation in One Direction The length l of a volume-element changes due to an elongational stress sð f =AÞ that acts normal to two opposite sides of the element; see Figure 17.3. In the case of a totally elastic material, ds ¼ Ed ln l:

ð17:2Þ

The proportionality factor E, which is constant for small changes of l, is the elongation modulus.

f A

Figure 17.2 All-sided compression of a volume.

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Rheology, with Special Attention to Dispersions and Interfaces

347

A f

f

l Figure 17.3 Elongation of a volume in one direction.

For a viscous material the deformation continuously changes in time, according to s ¼ Ze

d ln l dt

ð17:3Þ

which deﬁnes the elongation viscosity Ze .

17.1.3 Shear This type of deformation is caused by a force that is directed parallel to two opposite sides of the volume-element, as depicted in Figure 17.4. The shear stress sð f =AÞ causes a deformation gð¼ tgaÞ of the element but the volume is essentially unaltered. For a purely elastic material ds ¼ G dg;

ð17:4Þ

deﬁning the shear modulus G. The value of G is constant for small deformations. In the case of viscous behavior the deformation continues to change in time: layers of the material slide along each other in a parallel mode giving rise to laminar ﬂow. The deformation rate dg=dt equals the shear rate dvðxÞ=dx, being the

A

α

f

Figure 17.4

Volume subjected to shear.

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f

348

Chapter 17

ﬂow velocity gradient normal to the direction of the ﬂow. If the material behaves fully viscous s¼Z

dg dvðxÞ ¼Z ; dt dx

ð17:5Þ

where Z is the (dynamic) viscosity. For a less ﬂuid liquid, that is, a liquid having a higher viscosity, a larger shear stress is required to obtain a given shear rate. The shear stress is needed to disconnect interactions between structural units, or, otherwise stated, to compensate for the friction force between the layers that slide along each other. Hence, at a given shear rate, more energy is dissipated in a liquid of higher viscosity. Rheological properties may be described using analogons consisting of one or more springs and=or damping vessels. The spring [Figure 17.5(a)] represents an elastic component of length x, given by x x0 ¼ af , where x0 is the length at rest (¼ zero force) and a a spring constant related to the elasticity modulus. The damping vessel, a liquid in a cylinder with a piston [Figure 17.5(b)], mimics a viscous component. The displacement velocity of the piston is given by dx=dt ¼ bf , where the constant b is inversely proportional to the viscosity Z. Viscoelastic materials are modelled by a spring and a damping vessel in series [Figure 17.5(c)] to form a so-called Maxwell element. It is assumed that upon applying a force the spring stretches instantaneously to its equilibrium length but that the movement of the piston lags behind. Thus, at a short timescale only the deformation of the spring is observed, whereas at a long timescale it is the displacement of the piston that is presented. In mathematical terms: dx=dt ¼ adf =dt þ bf , so that the force to maintain constant deformation ðdx=dt ¼ 0Þ follows from adf =dt þ bf ¼ 0. The solution is f ¼ f0 exp½t=tr ;

ð17:6Þ

where f0 is the force at t ¼ 0 and the relaxation time tr ¼ a=b. Equation (17.6) indicates the exponential decay of f ðtÞ for viscoelastic material, as shown in Figure 17.1 and, furthermore, that tr attains a large value when elasticity

x

(a) Figure 17.5

(b)

(c)

Simple models for different rheological behavior.

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349

dominates ða bÞ and a small value ða bÞ when the material is mainly viscous. More complex viscoelastic behavior may be modelled by other combinations of springs and damping vessels.

17.2 CLASSIFICATION OF MATERIALS BASED ON THEIR RHEOLOGICAL BEHAVIOR The classiﬁcation of materials given here is based on the rheological properties discussed in the foregoing section. Elastic behavior is independent of the duration of the deformation but with viscous systems time-independent and time-dependent behavior may be distinguished.

17.2.1 Time-Independent Behavior Different types of time-independent rheological behavior are presented in Figures 17.6 (a) to (c) for liquids (predominantly viscous) and in Figures 17.7(a) to (c) for solids (predominantly elastic). Figure 17.6(a) shows linearity between the shear stress s and the shear rate dg=dt, which implies a value of Z that is independent of the shear rate (and hence of the shear stress). Liquids obeying such behavior are called Newtonian liquids. Essentially all simple liquids and dilute solutions as well as some concentrated solutions are Newtonian. They are characterized by a unique value of Z (at given composition, temperature, and pressure). For instance, water at 20 C and 1 bar has a viscosity of about 0.001 N m2 s (cf. Section 4.1). Liquid colloidal systems in which interactions between the particles inﬂuence the ﬂow are often non-Newtonian: the relation between s and dg=dt is nonlinear or, in other words, the viscosity depends on the shear rate and is referred to as apparent viscosity: Zapp ¼ f ðdg=dtÞ. If Zapp decreases with increasing shear rate the liquid is called ‘‘shear thinning’’ or ‘‘pseudoplastic’’ the reverse dependency is denoted ‘‘shear-thickening’’ or ‘‘dilatant.’’ Both types of behavior are displayed in Figure 17.6(b). Many concentrated polymer solutions as well as dispersions of lyophobic rod-shaped colloids are shear thinning. With increasing shear rate the particles orient themselves parallel to the ﬂow ﬁeld thus facilitating ﬂow. Shear thickening may be observed for concentrated dispersions in which the particles are pressed together by increased stress thereby obstructing further displacement. Another rheological phenomenon is that the shear stress has to exceed a certain minimum value before ﬂow sets in. Beyond this minimum value, the socalled yield stress ss , the liquid shows more or less plastic ﬂow, that is, a reduced viscosity with increasing shear rate. See Figure 17.6(c). The yield stress reﬂects the minimum force that is required to disrupt bonds between particles. Lubricants are well-known examples; they should be ﬂuid under stress but not drip at rest in the gravitational ﬁeld. Figure 17.7(a) shows the behavior of a fully elastic material of which the deformation g varies linearly with the shear stress s, which

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350

Chapter 17

σ

(a)

Newtonian fluids: η

1

2

σ is constant d γ / dt

1. high viscosity 2. low viscosity

d γ / dt σ

(b)

1 non-Newtonian fluids: ηapp 2

dσ d (d γ / dt)

1. pseudoplastic (shear thinning) 2. dilatant (shear thickening)

d γ / dt σ

(c) 3 3. plastic (shear thinning) σ s : yield stress

σs

d γ / dt

Figure 17.6 Rheological classiﬁcation of ﬂuids.

according to (17.4) implies a constant value for the shear modulus G. Nonlinear elastic behavior is illustrated in Figure 17.7(b). If after release of the stress the material does not return to its original state, as is shown in Figure 17.7(c), the material is viscoelastic. The rheological behavior displayed in Figure 17.7(c) is compatible with that in Figure 17.6(c). It shows plastic ﬂow: the stress increases with increasing deformation until a yield stress is reached beyond which the material continues to deform without further increase of the stress (at constant dg=dt).

17.2.2 Time-Dependent Behavior For some ﬂuids the relation between the shear rate and the shear stress depends on the duration of the ﬂow or, more precisely, on the kinematic history of the

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Rheology, with Special Attention to Dispersions and Interfaces

σ

(a) 1

linear elastic: G

2

351

σ is constant γ

1. high shear modulus 2. low shear modulus

γ σ

(b)

σ non-linear elastic: G app d dγ

γ σ σs

(c) plastic flow

γ

Figure 17.7 Rheological classiﬁcation of solids.

ﬂuid. We can distinguish two categories: thixotropy and rheopexy. Thixotropy is the most relevant time-dependent rheological behavior. In thixotropic ﬂuids interparticle interactions break down under the inﬂuence of a stress and the viscosity decreases. Hence, it is shear thinning. Upon reducing the shear stress interactions are restored with a concomitant increase in viscosity. Figure 17.8 illustrates this behavior. If, from an arbitrary equilibrium state 1, s1 is instantaneously increased to s2 , situation 20 is reached without a change in viscosity. Then at constant s2 it takes some time for interactions to be disrupted (i.e., structure to be broken down) and therefore for dg=dt to reach its equilibrium value given by state 2. If, conversely, starting from state 2, s2 is abruptly changed to s1 , the response of dg=dt passes through state 3 on its way to reach the equilibrium state 1. Like plastic systems, a thixotropic system is transferred from a gel to a ﬂowing system by exerting a stress s > ss , the yield stress. This may be achieved by simply shaking or stirring. In contrast to plastic behavior, after releasing the

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Chapter 17

Figure 17.8 Thixotropic behavior.

stress thixotropic systems need a ﬁnite time to recover the gel state. Many food, cosmetic, and pharmaceutical products are thixotropic. An example where thixotropy is required is paint. The paint should be ﬂuid when applied under stress and thereafter it must become solid but a certain recovery time is desired to let the paint ﬂow to form a homogeneous even layer. When shear-thickening systems need some time to re-establish sðdg=dtÞ equilibrium, they are called rheopectic. Thus, rheopexy is the relatively slow increase in Zapp due to gradual interaction and structure formation. Rheopexy is a rather rare and not very relevant phenomenon.

17.3 VISCOSITY OF DILUTED LIQUID DISPERSIONS The presence of (colloidal) particles in a liquid medium always increases the viscosity of the system. The reason is that the particles interrupt the stream lines (i.e., inside the particles there is no or less shear) so that a higher stress is required to achieve the same shear rate.

17.3.1 Compact Particles The most simple situation is that of incompressible, impermeable, noninteracting particles dispersed in a ﬂuid continuum. For such (hypothetical) systems Einstein derived that the viscosity Zs increases proportionally with the volume fraction f of the dispersed particles, Zs ¼ Z0 ð1 þ kfÞ;

ð17:7Þ

where Z0 is the viscosity of the medium and k a constant depending on the shape of the particles. According to (17.7), for a given value of f, Zs does not depend on

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353

25 k 20 prolate

15 10

oblate 5 0 1

5

10

15 aspect ratio

Figure 17.9 shape.

The value of k [Eq. (17.7)] for ellipsoidal particles as a function of their

the particle sizes, as long as they are large in comparison to the molecules of the medium. Particles rotate in a (laminarly) ﬂowing medium. For spheres this rotation does not contribute to the dissipation of energy but for anisotropic particles it does. The value of k increases with increasing anisotropy, as is shown in Figure 17.9. Anisotropic particles tend to align themselves in the direction of the ﬂow, the more so the higher the ﬂow rate. This orientational effect lowers the viscosity relative to that obtained for random particle orientation. Dispersions of oblong particles are therefore shear thinning. Because of the underlying assumption that the particles are noninteracting, which implies here that the ﬂow pattern around any particle is not affected by the presence of the others, (17.7) applies to very dilute dispersions only. Roughly speaking, (17.7) is valid for Zs =Z0 < 1:03. For more concentrated systems Zs can be approximated in an expansion of powers of f: Zs ¼ Z0 ð1 þ kf þ k 0 f2 þ k 00 f3 þ Þ:

ð17:8Þ

The constants k 0 ; k 00 ; . . . are related to the interaction between two particles, three particles, and so on. For hard spheres, for which k ¼ 2:5, a value of 6.2 has been calculated for k 0. It is common to compare the viscosity of the dispersion Zs with that of the medium Z0 . The relations between Zs ; Z0 , and the mass concentration C of the dispersed phase are expressed in different functional forms. The most usual ones are Zr

Zs ; Z0

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ð17:9Þ

354

Chapter 17

which is called the relative viscosity or viscosity ratio of the dispersion, and Z Z0 ¼ Zr 1; ð17:10Þ Zsp s Z0 the so-called speciﬁc viscosity. It represents the contribution from the dispersed particles to the relative viscosity. Furthermore, Zred

Zsp ; C

ð17:11Þ

which is known as the reduced viscosity or viscosity number and ½Z lim Zred ¼ C!0

Z sp

C

C!0

;

ð17:12Þ

the intrinsic viscosity, which, because of the extrapolation to zero concentration, is a characteristic of the dispersed particles. For nonsolvated particles the mass concentration C is related to the volume fraction f by C ¼ frd , where rd is the density of the dispersed material. It follows that Zred ¼

k k0 þ 2 C þ rd rd

ð17:13Þ

and ½Z ¼

k : rd

ð17:14Þ

The inﬂuence of C or, for that matter, f, on the viscosity may be analyzed on the basis of a plot of Zred versus C. See Figure 17.10. Curve 1 represents nonsolvated particles that do not interact. For interacting particles curve 2 is obtained and the departure from curve 1 is determined by the third and higher terms on the righthand side of (17.8). If the effective volume fraction feff ð¼ C=reff d Þ of the particles exceeds C=rd , for instance, due to solvation, ½Z ¼ k=reff > k=rd . For such d systems curves 10 and 20 are found, depending on whether the particles interact.

17.3.2 Uncharged Polymers Compactly shaped polymer molecules (e.g., globular proteins) may be treated as compact particles, as described above. However, in solution many polymers adopt a more or less swollen, coil structure (cf. Chapter 12). Such structures are at least partially permeable for the solvent and the question is whether the solvent in the

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355

Figure 17.10 Reduced viscosity of a dispersion as a function of the mass concentration of the dispersed phase. Curves 1 and 10 refer to noninteracting particles and 2 and 20 to interacting particles. Solid curves: nonsolvated particles; dashed curves: solvated particles.

polymer coil moves along with the coil or with the bulk of the solvent. Even though the monomer segment density in the coil may be rather tenuous, in particular for polymers with a high degree of polymerization (cf. Section 12.3), every solvent molecule trapped in the coil will ﬁnd several monomer segments within a nanometer distance. It results in a strong immobilizing effect on the solvent molecules and the coil can therefore be considered to be largely impermeable for the solvent. In a ﬁrst approximation, the coil may be treated as an impermeable solvated sphere having an effective radius Re and, hence, ½Z ¼ 2:5=reff ¼ 2:5Vm1 , with V ¼ 4=3pR3e and m ¼ M =NAv, M being the molar mass of the polymer. It was discussed in Section 12.3 that the root mean square end-to-end distance hm of a coil is a measure of its degree of swelling and this is true for Re as well, so that hm =Re is essentially constant. It then follows that ½Z ¼ Kh3m =M ;

ð17:15Þ 1

in which K is a universal constant. Its value is about 2:5 10 mol , if ½Z is expressed in m3 kg1 , hm in m and M in kg mol1 . For a y solvent for which according to (12.9) hm is linearly dependent on M 1=2 , we derive 23

½Zy ¼ ky M 1=2 :

ð17:16Þ

For a better solvent hm ¼ ahm;y and, as discussed in Section 12.3, the expansion factor a increases with increasing M: a M q with 0 < q < 0:1. We then obtain ½Z ¼ kM a

ð17:17Þ

with 0:5 < a < 0:8. Equation (17.17) is the well-known Mark–Houwink relation. The constants k and a depend on the solubility of the polymer but they are independent of the polymer molecular mass. Thus, if for a given system k and a

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356

Chapter 17

are known, the molar mass of a polymer may be obtained by applying (17.17) to the viscometry data. In the case of a heterodisperse polymer the molar mass thus derived is the viscosity-average molar mass, deﬁned as P a 1=a i C i Mi P Mv : i Ci

ð17:18Þ

For a heterodisperse sample both Zsp and C are the sum of the contributions from molecules P in various molecular massP classes. We can therefore write P Zsp ¼ kMva i Ci and Zsp ¼ i Zsp;i ¼ k i Ci Mia . Equating these two expressions for Zsp results in (17.18). The value of Mv is between those of the number average molar mass Mn and the mass average molar mass Mw , but closer to Mw . For a ¼ 1, that is, in the case of a fully permeable coil, Mv ¼ Mw . For Mv to become equal to Mn a should attain the value of 1 and this is unrealistic since the lower limit for the permeability of a coil is reached in a y solvent, for which a ¼ 0:5.

17.3.3 Polyelectrolytes The presence of electric charge along the polymer chain complicates the rheological behavior. In particular, the viscosity of a polyelectrolyte solution is very sensitive to the ionic strength. At high ionic strength electrostatic interactions between the charged groups are effectively screened so that the polyelectrolyte behaves as an uncharged polymer. On decreasing the ionic strength the polyelectrolyte molecules swell due to intramolecular electrostatic repulsion. Moreover, the charge on the molecules promotes intermolecular repulsion through overlap of the electrical double layers (see Section 16.1). The inﬂuence of the ionic strength on the intra- and intermolecular interactions is reﬂected in the viscosity of the polyelectrolyte solution. This is known as the electroviscous effect. In view of the foregoing we may distinguish between the primary and the secondary electroviscous effects. The primary electroviscous effect is due to the swelling of the molecules which causes a larger effective volume fraction and, hence, a higher viscosity. As this effect is of intramolecular origin it is independent of the polyelectrolyte concentration. The secondary electroviscous effect is associated with the overlap of electrical double layers giving rise to intermolecular repulsion. This implies deviation from ideal behavior with a corresponding increase in viscosity. The primary and secondary electroviscous effects can be evaluated by plotting Zred versus C for solutions of different concentrations of low molecular weight electrolyte c. A typical result is qualitatively shown in Figure 17.11. The primary electroviscous effect is reﬂected by the change in ½Zð¼ ðZred =CÞC!0 Þ and the secondary effect by the variation in

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Rheology, with Special Attention to Dispersions and Interfaces

η red

357

low c

high c

C Figure 17.11

Electroviscous effects for an aqueous polyelectrolyte solution.

the slope of the curve, both with varying ionic strength. A remarkable feature is observed at low polyelectrolyte concentration in a low ionic strength medium. At low concentration of low molecular weight electrolyte the polyelectrolyte itself contributes signiﬁcantly to the overall ionic strength. Consequently, dilution of the polyelectrolyte results in a lower ionic strength which, in turn, causes the polyelectrolyte molecules to swell and the viscosity to increase. Obviously, in the case of isoionic dilution Zred ðCÞ follows the dashed curve. Electrostatic interactions are more effectively screened and the viscosity of a polyelectrolyte solution suppressed accordingly by counterions of higher valency. The inﬂuence of the valency of the counterions on the viscosity of a polyampholyte solution is indicated in Figure 17.12. For a solution of a given concentration of polyampholyte curves for Z (pH) are shown at equal concentrations (in equivalents per unit volume) of counterions. At the point of zero charge (pzc), where the net charge on the polyampholyte is zero, there is only a slight inﬂuence, if any, of the valency of the low molecular weight ions. Away from the pzc the inﬂuence of the valency of the counterions is clearly observable as for pH < pzc the cations and for pH > pzc the anions are the counterions.

17.4 INTERFACIAL RHEOLOGY Interfaces exist by virtue of the adjoining bulk phases; in other words, interfaces are not autonomous. The interface and the bulk phases are mechanically coupled, which implies that any deformation or ﬂow in either adjacent bulk phase induces a motion in the interface and vice versa. Interfacial rheology deals with the relationships among a stress applied on an interface, the corresponding interfacial deformation, and the ﬂow in the adjacent bulk ﬂuid. Interfacial rheology is of great relevance for various biological and technological processes. Examples are the formation and stabiliza-

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358

Chapter 17

Figure 17.12 Electroviscous effects for an aqueous solution of a polyampholyte (e.g., a protein). Electrolyte: —— NaCl; – – – BaCl2 ; . . . . . . Na2 SO4 .

tion of emulsions and foams in food, cosmetic, pharmaceutical, and medical products, emulsiﬁcation of alimentary fats in the duodenum, phagocytosis, expansion and contraction of alveoli, motion of amoebae, aeration of waste water, and so on. As in bulk phases, the most relevant rheological properties of an interface are the viscosity and the elasticity. Interfacial viscosities and interfacial elasticities are excess quantities; that is, they are in excess of the corresponding bulk properties of the adjoining phases that are assumed to be constant up to the dividing plane (cf. the ﬁnition of the Gibbs dividing plane in Section 3.9). Hence ﬁnite values for the interfacial rheological properties implies the presence of a monolayer of amphiphiles in the interface which is either of the Langmuir (insoluble) or of the Gibbs (soluble) type. Two types of interfacial deformation are considered, dilation or compression, and shear. See Figure 17.13.

A+∆A

constant A

A

(a)

Figure 17.13

(b)

Interfacial deformations: (a) dilation or compression; (b) shear.

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359

17.4.1 Dilation or Compression Consider the relative change in interfacial area A=A brought about by an external stress. Dilation leads to dilution of the monolayer which, in turn, results in a rise of the interfacial tension g. Conversely, reducing the interfacial area causes g to decrease. The response of the monolayer to the imposed deformation may be more or less elastic or viscous. Elastic behavior is expected for monolayers in which the amphiphilic molecules are interconnected forming a two-dimensional gel. Also, when the rate of deformation is too high to allow for relaxation back to equilibrium by, for example, adsorption or desorption of amphiphiles to or from the interface, or by reorientation and=or reconformation of the molecules in the monolayer (especially in the case of polymers and proteins), the monolayer responds partly elastically. Analogous to the three-dimensional situation (cf. Section 17.1) the interfacial dilation (or compression) modulus K s is deﬁned through dg ¼ K s d ln A:

ð17:19Þ

K s is a measure of the resilience of the monolayer against changing the interfacial tension upon expansion or compression. Note that K s is the inverse of the compressibility of the monolayer as deﬁned by Eq. (7.6). If the monolayer is purely viscous the change in the interfacial tension would be related to the rate at which the interfacial area is changed, as g ¼ Zsd

d ln A ; dt

ð17:20Þ

where Zsd is the interfacial dilational viscosity. It reﬂects the response of the interfacial tension to the rate of changing the relative interfacial area. The value of Zsd strongly depends on the time of observation relative to the relaxation time. Therefore, there may be a great difference in magnitude between Zsd in expansion and compression, as the respective relaxation processes occur at very different timescales. Monolayers seldom, if ever, show an entirely viscous behavior. They always present an elastic contribution. Interfacial dilation or compression causes a change in the interfacial tension which, after releasing the stress, relaxes with a characteristic time towards equilibrium. Thus the interfacial tension change induced by changing the interfacial area is determined by an elastic and a viscous contribution that are likely to be additive: g ¼ K s

A d ln A þ Zsd : A dt

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ð17:21Þ

360

Chapter 17

17.4.2 Shear When the interface is deformed by shear parts of the monolayer shift positions relative to other parts, at constant interfacial area. Consider a Langmuir monolayer at rest, as illustrated in Figure 17.14(a). In such a situation the interfacial tension g has the same value all over the interface. When a tangential ﬂow is imposed in the adjacent ﬂuid, as indicated in Figure 17.14(b), the monolayer is dragged along in the same direction thereby developing an interfacial tension gradient. As a consequence the monolayer tends to move from B to A to level the interfacial tension gradient. In this way the interface responds to the shear stress exerted by the ﬂowing ﬂuid. For the interface to be in mechanical equilibrium, dg dvx ðzÞ ¼ Z ; ð17:22Þ dx dz z¼0 where Z is the viscosity of the bulk phase and ðdvx ðzÞ=dzÞz¼0 is the velocity gradient normal to the interface at z ¼ 0. It implies that (17.22) applies exclusively to the interface. In passing it is noted that the development of an interfacial tension gradient requires the presence of a monolayer. In the absence of a monolayer dg=dx ¼ 0 and hence ðdvx ðzÞ=dzÞz¼0 ¼ 0. Under that condition the interface moves at the same speed as the adjacent ﬂuid. In contrast to the case discussed above, an imposed gradient in the interfacial tension causes a viscous ﬂow in the adjoining bulk phase(s). See (a)

(b) A

B x

z

γ

γ

Figure 17.14 (a) Monolayer at rest corresponding to a uniform interfacial tension; (b) gradient in a monolayer, causing an interfacial tension gradient, imposed by a tangential ﬂow in the adjoining liquid.

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361

surfactant x

z

γ

Figure 17.15 Liquid ﬂow generated by an interfacial tension gradient.

Figure 17.15. For instance, if the interfacial tension is locally lowered by applying an amphiphilic compound the monolayer moves away from that place to annihilate the interfacial tension gradient. This phenomenon is called the Marangoni effect. The ensuing shear rate in the bulk phase is such that (17.22) is satisﬁed. When the monolayer is of the Gibbs type, that is, if the amphiphilic molecules are partitioning between the monolayer and the bulk solution, the situation is more complicated. Then any gradient in the interfacial tension will be counteracted by amphiphilic molecules taken up in, or released from, the monolayer. The transport rate of the amphiphiles to or from the interface relative to the tangential movement of the molecules in the monolayer determines whether the monolayer behaves Gibbs- or Langmuir-like. The coupling of the motions of the interface and the adjacent bulk phase is of great importance for the stability of emulsions and foams. This is further discussed in Chapter 18.

EXERCISES 17.1

Comment on the following statements. (a) Thixotropic systems remain ﬂuid for a longer period of time compared to pseudoplastic systems, after release of a shear stress. (b) The yield stress in concentrated colloidal systems results from interparticle repulsion.

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362

Chapter 17

The viscosity Z of a dilute polymer solution in a y solvent is given by Z ¼ a þ bC, where C is the mass concentration of the polymer in the solution and a and b constants. (N.B. The interaction between polymer molecules in a y solvent is negligibly small.) (d) The interfacial dilational viscosity of a monolayer is larger for low molecular weight surfactants than for polymers. (e) Marangoni effects are more pronounced for Langmuir monolayers than for Gibbs monolayers. (f) The Deborah number (De) of a viscous material is much larger than one (De 1). Consider a globular protein with a density (in the dry state) of 1.3 g cm3 . The point of zero charge of this protein is at pH 5. The relative viscosities Zr of aqueous solutions of this protein are determined at pH 5 and pH 7. The results are as follows. (c)

17.2

17.3

C (g dm3 )

5

10

15

Zr at pH 5 Zr at pH 7

1.010 1.013

1.022 1.032

1.036 1.057

(a) What is the reason for the different results between pH 5 and pH 7? (b) Give a plot from which the interaction between the dissolved protein molecules can be evaluated. (c) Calculate the intrinsic viscosity ½Z for both pH values and derive the water volume fraction in the hydrated protein. (N.B. The density of the hydrated protein is a linear combination of those of dry protein and water.) The rheological behavior of two types of applesauce (indicated A and B) can be described by s ¼ aD1=2 þ b; where s is the shear stress, Dð¼ ðdvðxÞ=dxÞ ¼ ðdg=dtÞÞ the shear rate, and a and b are empirical constants. ða 0 and b 0Þ. (a) Do the applesauces behave Newtonian? (b) What are the physical meanings of the constants a and b? The value of a is larger for type A than for type B, and for type B, b ¼ 0. (c) Give qualitative graphical representations of sðDÞ for both types of applesauce. By what names would you typify the different rheological behavior? (d) Express the apparent viscosity Zapp as a function of s and give a graphical presentation of Zapp ðsÞ for type B.

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17.4

363

Consider a liquid ﬁlm vertically suspended in air. The ﬁlm is stabilized by a monolayer of surfactant. The height and the width of the ﬁlm are 3 cm. It has a thickness of 50 mm and it contains an aqueous solution of surfactant. The density of that solution is 1000 g dm3 and the viscosity 103 N m2 s. What is the difference in surface tension between the upper and lower sides of the ﬁlm required to immobilize the surfaces?

SUGGESTIONS FOR FURTHER READING H. A. Barnes, J. F. Hutton, K. Walters. An Introduction to Rheology, Amsterdam: Elsevier, 1989. P. Joos. Dynamic Surface Phenomena, Zeist, the Netherlands: VSP, 1999. R. Miller, R. Wu¨stneck, J. Kra¨gel, G. Kretzschmar. Dilational and shear rheology of adsorption layers at liquid interfaces, Colloid Surfaces A 111:75–118, 1996. T. G. M. van de Ven. Colloidal Hydrodynamics, London: Academic, 1989.

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Milk and Its Derivatives

Milk comes from different types of mammals, cow’s milk being most common. Over the centuries it has been used in various ways. This has resulted in the development of a number of milk products such as cream, butter, cheese, and yogurt. Milk is an emulsion, a suspension of tiny fat droplets in water that further contains proteins (predominantly occurring as casein micelles), lactose, and some minerals. The fat droplets are coated with an adsorbed layer of proteins that helps to keep the droplets suspended. Production of yogurt and cheese involves destabilization of the milk emulsion with the assist of microbial activity. In yogurt the milk is coagulated and soured by lactic acid produced by bacteria. Cheesemaking starts with an enzymatic modiﬁcation of the casein micelles allowing them to coprecipitate with fat droplets thus forming the cheese curd. Cream is obtained by collecting the fatty fraction of the milk in a fat-rich phase leaving skimmed milk behind. Like milk, cream is still a fat-in-water emulsion. When the cream is whipped air bubbles are incorporated into the cream. The emulsion is transformed into a foam. The bubbles are stabilized by a layer of denatured milk proteins and surrounded by fat globules. Further whipping results in collapse of the foam structure and destruction of the fat globules. Fat becomes the continuous phase in which water droplets are dispersed. The emulsion, as it was in milk and cream, has inverted: butter is formed.

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18 Emulsions and Foams

Oil, or any other apolar liquid, and water mix poorly if they mix at all. Also, gases usually have a limited solubility in liquids. However, in various systems in nature as well as in technological products, we ﬁnd these compounds intimately blended in emulsions and foams. Emulsions are ﬁne dispersions of droplets of one type of liquid in another that forms the continuous phase. In foams gas bubbles are ﬁnely dispersed in a liquid that, thereafter, may be solidiﬁed. Emulsions and foams belong to the lyophobic colloids and are therefore thermodynamically unstable. Colloidal stability is achieved by adding one or more compounds that adsorb or otherwise accumulate at the interface between the dispersed and the continuous phases, the so-called emulsiﬁers, foaming agents, or, more generally, stabilizers. Emulsions and foams occur ubiquitously in food products. Milk, butter, margarines, sauces, and soups are examples of emulsions. Foamy foods are whipped cream, mousse, milkshakes, bread, and so on. In pharmaceutical and cosmetic ointments, lotions, and creams, water-insoluble active components are often supplied as emulsions. An advanced application is that of emulsiﬁed ﬂuorocarbons as a blood substitute. It goes without saying that alimentary and biomedical emulsions and foams should be qualiﬁed as safe. For these products proteins rather than synthetic substances are often used as stabilizers. Plant protection agents such as pesticides and insecticides are more often than not insoluble in water. They are usually dissolved in emulsiﬁed oil droplets in water. In view of environmental sustainability, in the newer generation of paints and lacquers the apolar pigments are dissolved in the oily droplets of emulsions so that these paints can be diluted and manipulated using water instead of organic solvents.

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366

Chapter 18

Emulsions and foams are also used in construction materials such as in bitumen for coating roads and roofs and in foam concrete in lightweight constructions. Foams are applied in isolation material (e.g., styrofoam) as well, and, because of its elastic properties, in the soles of sports shoes. Finally, we should not forget to mention nature’s own emulsions and foams. Examples of natural emulsions are rubber latex produced by Hevea brasiliensis, fat droplets in milk stabilized by proteins, fats from the diet emulsiﬁed in the duodenum and stabilized by bile acids and salts, and blood too may be regarded as an emulsion. Where (micro)organisms living in an aqueous environment produce gaseous metabolites foams are often formed. This may be desirable, as in beer, or undesirable, as in installations for the treatment of waste water. All these examples illustrate that emulsions and foams are encountered in everybody’s everyday life.

18.1 PHENOMENOLOGICAL ASPECTS As indicated above, emulsions and foams are thermodynamically unstable dispersions. They are distinguished from the dispersions discussed in Chapter 16 in that their dispersed phase is ﬂuid and not solid. This has a number of consequences of which the most important ones are (1) the particles are deformable, (2) the interface between the dispersed and the continuous phases is deformable which may give rise to interfacial rheological phenomena, and (3) the particles may coalesce. These features play important roles in the formation and subsequent stability of emulsions and foams. Notwithstanding the characteristics they have in common, emulsions and foams differ in various ways. In foams the density of the dispersed phase differs from the continuous phase more than it does in emulsions. As a consequence, the diameter of emulsion droplets is in the 0.1 to 50 mm range, whereas the size of foam bubbles is rarely below 100 mm (cf. Section 6.3). For these reasons foams cream very rapidly, say, within a few seconds, and form a top layer in which the bubbles are closely packed leaving only a thin ﬁlm of the continuous phase between the bubbles. Emulsions as well may cream or sediment (depending on the density of the dispersed phase relative to that of the continuous phase), but at a much slower rate. Because of the relatively large particle size and, hence, low Laplace pressure (see Section 6.1) and the strong buoyancy force, gas bubbles in foam may change shape to form a hexagonal structure as this structure represents minimum interfacial area between the dispersed and continuous phases. See Figures 18.1, 5.2, and 5.3. The foam thus formed may be piled up; it has a certain solidity. Perturbation of the hexagonal structure of the foam involves an increase in interfacial area and, consequently, in Gibbs energy of the system. Upon release of the stress the hexagonal form is restored. Thus, for not too large deformations,

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Emulsions and Foams

Figure 18.1

367

Schematic representation of (left) a spherical and (right) a hexagonal foam.

foams are elastic. In general, the behavior of emulsions is quite different from that of foams. The densities of the dispersed and continuous phases are more similar and the droplets are smaller. As a consequence, the deforming buoyancy force is considerably smaller. Moreover, the higher Laplace pressure in the much smaller droplets strongly resists deformation.

18.2 EMULSIFICATION AND FOAMING To make an emulsion or a foam droplets or bubbles have to be generated from the interface between the two immiscible phases. This is usually done in a homogenizer where droplets or bubbles continuously break up and recoalesce. The droplet or bubble size is ultimately determined by the timescales of the breakup and recoalescence processes. Foams may also be produced by supersaturating a liquid with a gas. This can be realized by gas-producing micro-organisms or by dissolving gas in a liquid under pressure, whereafter the pressure is released as happens when opening a bottle of beer. A crucial step in preparing emulsions and foams is to break up larger particles into smaller ones. The interfacial area increases with a concomitant increase in Gibbs energy. The adsorption of a surface active compound (a low molecular weight amphiphile, a polymer, or protein) at the interface between the two phases lowers the interfacial tension and hence facilitates the formation of interfacial area. For instance, the formation of a 10% (v=v) oil-in-water emulsion with droplets of 106 m in diameter and in which the interfacial tension is 10 mN m1 requires 6 kJ per m3 emulsion formed. However, the energy input to make the emulsion is much higher, say, about three orders of magnitude. The reason is that not only more interfacial area has to be created but also the Laplace pressure in the droplets that opposes the deformation and subsequent breakup must be overcome. This process is illustrated in Figure 18.2. According to Eq. (6.4), the Laplace pressure pL for the example given above equals 4 104 Pa ð¼ 0:4 bar). The stress exerted on the droplet has to exceed pL . Such stresses may be achieved by (a combination of) very high shear rates and

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Chapter 18

Figure 18.2 Scheme for particle deformation and breakup during emulsiﬁcation or foaming.

strong turbulence and these can only be generated by a high energy input. Because pL varies linearly with the interfacial tension g, the energy needed to break up the droplets or bubbles decreases with decreasing value of g. Thus the reduction of the interfacial tension is essential for the formation of emulsions and foams, but it is by no means sufﬁcient to obtain dispersions with better than transient stability. If this were the case stable emulsions and foams could be prepared from pure low surface tension liquids, in the absence of surface active agents, and this is found to be impossible in practice. Breakup involves rapid and substantial stretching of droplets and bubbles (see Figure 18.2) and, consequently, in the presence of a monolayer of low or high molecular weight surfactant the interfacial tension may be far from equilibrium. The interfacial tension gradients invoked by the emulsiﬁcation or foaming process prevents, or at least retards, droplets or bubbles to recoalesce. Figure 18.3 shows how an interfacial tension gradient induces a ﬂow of liquid (cf. Section 17.4.2) thereby developing a resistance against local thinning of the liquid ﬁlm between emulsion droplets or foam bubbles. The interfacial tension gradient will be neutralized by lateral diffusion (spreading) of the surfactants in the monolayer and by transport of such molecules from the bulk phase towards the interface. The spreading process is usually relatively slow, typically in the range of 0.1 m s1 . Under most conditions transport from the adjoining phase is faster. For the stabilizing mechanism to be effective the gradient must exist long enough to drag a signiﬁcant amount of liquid into the ﬁlm between the droplets or bubbles. This is realized only if the surface active agent is dissolved in the continuous phase. The thin ﬁlm between two nearby particles is easily depleted with surfactant so that the adsorbing molecules have to be transported over a longer distance which takes time. If the surfactant were dissolved in the dispersed phase the interfacial gradient would disappear almost instantaneously. This is the basis of Bancroft’s rule, saying that in an emulsiﬁcation process the continuous phase is the phase in which the surfactant dissolves. If the surfactant is soluble in

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Emulsions and Foams

369

γ

Figure 18.3 Interfacial tension gradients causing a liquid ﬂow that opposes coalescence of emulsion droplets or foam bubbles.

the aqueous phase an oil-in-water (o=w) emulsion will be obtained and oil-soluble surfactants yield water-in-oil (w=o) emulsions. In this context surfactants may be characterized by a HLB number. The HLB number represents the hydrophile–lipophile balance of the surfactant. It is deﬁned in such a way that HLB ¼ 7 for a compound with equal solubility in the aqueous and the oily phase. Hence, w=o emulsions are formed using surfactants with HLB < 7 and o=w emulsions when HLB > 7. The HLB number of a surface active agent may be calculated by linear addition of the contributions of the constituting chemical groups. As a rule, the HLB number is sensitive to temperature variation. For most surfactants HLB increases by raising the temperature so that at a certain temperature a phase transition may take place. Thus, a w=o emulsion may invert into an o=w emulsion upon increasing the temperature. Around the phase inversion temperature HLB 7 which results in a rather unstable emulsion.

18.3 EMULSION AND FOAM STABILITY Following the formation stage emulsions and foams are subject to various processes that inﬂuence their long-term stability, notably their colloid stability, their particle size, and their particle size distribution. The major processes, which are often correlated, are brieﬂy reviewed.

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370

Chapter 18

18.3.1 Sedimentation and Creaming For spherical noninteracting particles the driving force fs for sedimentation or creaming is given by 4 fs ¼ pa3 ðr1 r2 Þg; 3

ð18:1Þ

in which a is the particle radius, g the acceleration due to the gravitational (or centrifugal) force, and ðr1 r2 Þ the density difference between (1) the dispersed and (2) the continuous phases. For r1 > r2 the direction of fs is the same as that of the external force ﬁeld and the particles sediment; for r1 < r2 the directions are opposite and the particles move opposite to the direction of the external ﬁeld; that is, the particles cream. The moving particles are retarded by friction with the continuous phase. According to Stokes the friction force ffr is expressed as ffr ¼ 6pZ2 an;

ð18:2Þ

where Z2 is the viscosity of the continuous phase and n the velocity of the particle. In equilibrium fs ¼ ffr , so that the rate of sedimentation or creaming can be derived as n¼

2a2 ðr1 r2 Þg : 9Z2

ð18:3Þ

Because of their relatively large size emulsion droplets and, even more so, foam bubbles, sediment or cream with a noticeable velocity, provided that the densities between the dispersed and the continuous phase differ signiﬁcantly. This last-mentioned condition is certainly the case for foams but for emulsions the densities could be rather similar. Results of calculations using (18.3) are presented in Table 18.1, taking jr1 r2 j ¼ 100 kg m3 . It shows that in the earth’s gravitational ﬁeld the particles should be at least about 1 mm for the sedimentation or creaming not to be disturbed by Brownian motion. Of course, the minimum size of the particles required for noticeable settlement varies linearly with the density contrast. For an aqueous foam Z1 Z2 1000 kg m3 and because the radius of the gas (air) bubbles in the foam usually ranges between 104 and 103 m, the creaming rate under gravity varies from 5 103 to 0:5 m s1 , indicating that creaming occurs very rapidly after foam formation. In practice the rates observed in emulsions and foams are usually somewhat lower than predicted by (18.3), mainly because of particle–particle interaction and convectional perturbation. The sedimentation or creaming rate can be kept low when the particles are small, the density difference between the two phases is small, and the viscosity of the continuous phase is high. The last two conditions may be realized by supplying additives to one or both phases. Another way to prevent dispersed

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Table 18.1 Sedimentation or Creaming Velocities of Particles Having a Density that Differs 100 kg m3 from the Density of the Continuous Phase (Water at 25 C). Values are in 106 m s1 . For Comparison the Brownian Displacement (in 106 m) per Second is Added in the Bottom Row Radius (106 m) Motion under the inﬂuence of Gravity (g ¼ 9:81 m s2 ) Centrifuge (g ¼ 9:81 103 m s2 ) Centrifuge (g ¼ 9:81 105 m s2 ) Brownian displacement

103

5

5 10

102

101

100

10

5 106 5 103

5 104 0.5

5 102

5

0.03

0.01

3 103

5 103

0.5

0.33

0.11

particles from traveling through the surrounding liquid is to select a continuous phase that has a yield stress. The driving force for sedimentation or creaming is exerted on an area of pa2 of the continuous medium, which, using (18.1), corresponds to a stress of ð4=3Þaðr1 r2 Þg. For a particle of 1 mm and jr1 r2 j ¼ 100 kg m3 the stress amounts to about 0.001 N m2 when it is subjected to the earth’s gravitational ﬁeld. Hence, a yield stress of the continuous phase as small as 0.001 N m2 sufﬁces to prevent sedimentation or creaming.

18.3.2 Drainage It has already been mentioned in Section 18.1 that the buoyancy force may deform the droplets or bubbles in densely packed sedimented or creamed emulsion and foam layers, and that larger particles (having a lower Laplace pressure) are more susceptible to such deformations. As a consequence, particularly in foams, hexagonal structures are observed where the liquid ﬁlms between the bubbles are lamellar. See Figure 18.1. Zooming in on Figure 18.1 reveals a pattern as shown in Figure 6.7 and which is repeated in Figure 18.4 but here with the surfactant molecules indicated. Due to the Laplace pressure difference between the planar and the curved parts of the ﬁlm, liquid ﬂows out of the lamellar into the surrounding Plateau borders and from there under the inﬂuence of gravity down into the bulk phase. As discussed in Section 17.4.2 the movement of the liquid along the interface containing the monolayer invokes an interfacial tension gradient that counteracts the tangential ﬂow of liquid. Under conditions of mechanical equilibrium with respect to the interface the induced interfacial tension gradient just compensates the shear stress due to the ﬂowing liquid, as formulated in (17.22). Hence, the liquid in the lamellae ﬂows between two immobile interfaces and this results in a parabolic ﬂow velocity proﬁle, as shown in Figure 18.5. According to such a proﬁle the rate of drainage scales with the third power of the ﬁlm thickness and it therefore decreases strongly as the ﬁlm thins.

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Plateau borders

L G

G

G

Figure 18.4 Liquid lamellae in drained foam showing the stabilizing surfactant molecules.

The ﬁnal thickness of the liquid ﬁlm is primarily determined by the disjoining pressure between the dispersed particles, which determines whether aggregation takes place (cf. Sections 16.2 and 16.3). Obviously, if the interfaces do not contain surface active compounds or if such compounds could be nearly instantaneously supplied (from the dispersed phase) it would be impossible to create an interfacial tension gradient. The interface would then move along with the liquid and the ﬁlm would be completely drained in one split second. Drainage is retarded by increasing the viscosity of the continuous phase and it can even be prevented if aggregated solid particles or polymeric molecules in the continuous phase form a network. For instance, in whipped cream such a network is formed by aggregated fat particles.

Figure 18.5 Velocity proﬁle of a liquid ﬂowing between immobile interfaces.

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18.3.3 Aggregation When, after sedimentation or creaming, possibly followed by drainage, the droplets or bubbles are in close proximity the disjoining pressure across the ﬁlm between the particles determines whether aggregation takes place. In Chapter 16 we have seen that the disjoining pressure may comprise electrostatic, dispersion, and steric contributions. As foams and emulsions belong to the lyophobic colloids the principles of colloidal stability discussed in Chapter 16 are equally applicable here. When bubbles or droplets aggregate in the deep primary minimum the thin liquid ﬁlm is not able to withstand the strong attractive disjoining pressure and the particles fuse. Because of the relatively large size of the droplets and bubbles the disjoining pressure in emulsions and foams is likely to yield a secondary minimum at a certain interparticle separation [see Section 16.2, Figure 16.5(b)]. The depth of the secondary minimum and the distance between the particles where it occurs can be calculated according to the principles presented in Chapter 16. At intermediate ionic strengths the secondary minimum for emulsions and foams may readily reach some tens of kB T and this is more than sufﬁcient to cause aggregation. Note that in the case of aggregation in the secondary minimum the droplets or bubbles do not coalesce but a thin liquid ﬁlm remains between them. Aggregation in the secondary minimum can be prevented by long-range steric stabilization (cf. Section 16.3) using polymers that extend into the continuous phase over a distance that keeps the particles at a separation at which the attractive dispersion interaction is reduced to less than a few kB T -units.

18.3.4 Disproportionation Disproportionation, also called ‘‘Ostwald ripening’’ or ‘‘isothermal distillation’’ is a type of instability that occurs when the disperse phase is slightly soluble in the continuous phase. The Laplace pressure in bubbles and droplets causes an increased solubility of the disperse phase molecules in the continuous phase. The reason for this effect is explained in Section 6.3. The larger the Laplace pressure, that is, the smaller the dispersed particles, the larger the solubility is. This phenomenon is known as the Kelvin effect (for gas in liquid) or the Ostwald effect (for liquid in liquid). Both effects are quantitatively treated in Section 6.3. Because of these effects smaller particles are better soluble than larger ones and consequently disperse phase material diffuses from the small to the large particles. The large particles grow at the expense of the small ones. The size distribution in the foam or emulsion becomes wider: the system disproportionates and the dispersion coarsens. However, towards the end of the process when the small particles have disappeared the size distribution is more narrow. The rate of disproportionation is ruled by the rate of diffusion of the disperse phase molecules through the continuous phase. This, in turn, is determined by the concentration gradient and the diffusion coefﬁcient.

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The concentration gradient is proportional to the solubility difference between the small and large particles, respectively. Therefore disproportionation proceeds faster in a more heterodisperse system. For a given particle size distribution the difference in solubilities can be modulated by selecting the surface active agent. The more the surfactant reduces the interfacial tension, the smaller the Laplace pressure and, hence, the lower the solubility of the dispersed phase material is. Furthermore, the absolute value of the concentration difference near the small and the large particles depends, of course, on the solubility in the bulk of the continuous phase. Most oils are hardly soluble in water, if at all, and the corresponding emulsions disproportionate at an extremely low rate. Many foams consist of air in water and because of the relatively high solubility of the air they show fast disproportionation yielding a rather uniform coarse foam. Similar behavior is observed for soda and beer, for example, where carbon dioxide is dispersed in an aqueous phase. Hence, if a stable foam is desired as is the case for shaving cream, for example, a poorly soluble gas should be used. Apolar gases such as butane and pentane may serve this purpose. The concentration gradient of the disperse phase molecules in the continuous phase is furthermore inversely proportional to the distance between the particles in the dispersion. Disproportionation therefore proceeds quickly when the emulsions or foams are sedimented or creamed, and even more so when they are drained or aggregated. There are a few possibilities to retard or even stop disproportionation. One is to add a component that is well-soluble in the dispersed phase but essentially insoluble in the continuous phase. Shrinkage of the droplet or bubble then leads to a higher concentration of the additive therein and in the growing particles the additive is diluted. As a result, the osmotic pressure difference between the small and large particles increases. This may ﬁnally lead to termination of the disproportionation. For the situation (depicted in Figure 18.6) of a small dispersed particle and a very large particle (which is approximated by a planar interface having zero Laplace pressure), disproportionation stops when the Laplace pressure pL in the small particle is just compensated by the osmotic pressure p in the small particle relative to the large one ( bulk solution). Using (3.35) for the osmotic pressure of ideal solutions and (6.4) for the Laplace pressure, we obtain c ¼ 2g=RTa;

ð18:4Þ

where c is the difference in concentration of the additive in the small and the large particles, g the interfacial tension, and a the radius of the small particle. This method of stopping disproportionation may be successfully applied for emulsions in particular because of the possibility of selecting additives that dissolve in the dispersed phase only. With foams it is often more problematic. Nevertheless, it is applied in foams as well. For instance, adding nitrogen to beer reduces disproportionation and hence delays coarsening of the foam layer.

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dispersed phase containing low molecular weight solute

∆pL + π = 0

continuous phase

Figure 18.6 Disproportionation of solute-containing particles is ruled by differences in the Laplace pressure and osmotic pressure.

A second mechanism to suppress disproportionation is based on the dynamics of the surfactant monolayer in the interface between the dispersed and the continuous phases. If in the timescale of the disproportionation process the surfactant molecules do not desorb from the shrinking particles into the continuous phase to let the monolayer relax towards equilibrium with the surrounding solution the interfacial tension of these particles decreases. This situation is readily realized using polymers or proteins as stabilizers (cf. Chapter 15). The increase in the Laplace pressure due to the decreasing radius is then counteracted by the reduction due to the decreased interfacial pressure. More quantitatively, variation of the particle radius a with an amount da, giving rise to a variation dg in the interfacial pressure, causes a net change in the Laplace pressure dpL given by @pL @pL 2g 2 ð18:5Þ dpL ¼ da þ dg ¼ 2 da þ dg: a a @a g @g a Disproportionation stops when dpL ¼ 0, from which it is derived that dg : ð18:6Þ d ln a As for spheres 2d ln a ¼ d ln A (A is the interfacial area of the particle), (18.6) can be written as g¼

g ¼ 2K s ; s

ð18:7Þ

where K is the interfacial elasticity modulus with respect to compression or dilation, deﬁned by (17.19). Thus the value of g at which disproportionation stops is determined by the interfacial elasticity modulus. Polymeric surfactants, including proteins, are known for their high values of K s as explained in Section 17.4.1.

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18.3.5 Coalescence Coalescence occurs when the ﬁlm between the droplets or bubbles ruptures. Subsequently, the Laplace pressure is responsible for fusing of the particles, forming a larger single particle, and so on. This process eventually results in the disappearance of the dispersion, that is, in a complete segregation into two bulk phases. Coalescence requires that the ﬁlm separating the particles be thin and therefore it is much more likely to happen when the emulsion or foam is creamed (or sedimented) or drained and, even more so, when it is aggregated. The rate of coalescence could, in principle, be determined by (1) the rate at which a thin ﬁlm is formed and (2) the rate at which the ﬁlm is ruptured. Generally, ﬁlm rupture is the rate-determining step so that coalescence is a ﬁrstorder process. Film rupture can be caused by different mechanisms. Thermal ﬂuctuation may induce a hole in the ﬁlm; see Figure 18.7. Hole formation across a ﬁlm leads to a temporarily increased interfacial area with a corresponding increase in the Gibbs energy of the system. This increase is larger for thicker ﬁlms. The probability that thermal ﬂuctuations ( kB T ) overcome the interfacial Gibbs energy increase and hence cause rupture is therefore much higher for thin ﬁlms. Moreover, a lower interfacial tension enhances coalescence because of the smaller increase in interfacial Gibbs energy when making a hole in the ﬁlm. This reveals the dual role of surfactants in emulsions and foams: to facilitate emulsion and foam formation the surfactant monolayer should reduce the interfacial tension as much as possible, but a lower interfacial tension makes the dispersion less stable against coalescence. Thus the most effective emulsifying or foaming agent lowers the interfacial tension substantially and at the same time sees to it that the ﬁlm remains sufﬁciently thick. This last requirement may be achieved by electrostatic and=or steric repulsion (cf. Sections 16.2 and 16.3). Disturbances that lead to a symmetrical transversal wave in the ﬁlm, as shown in Figure 18.8, could be another reason for ﬁlm rupture. In most cases the wave will be quenched because of the adverse effect of the increased interfacial area on the Gibbs energy of the system. The tendency to quench is stronger when the interfacial tension is higher and also when the particles or, more precisely, the ﬁlm radius is smaller. This is because larger ﬁlm radii can accommodate longer waves which, for a given amplitude, results in a lower interfacial area increment.

h

Figure 18.7 Liquid ﬁlm between gas bubbles or emulsion droplets in which a hole is created, for example, by thermal ﬂuctuations.

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Emulsions and Foams

A

377

B

Figure 18.8 Symmetrical transversal wave in a liquid ﬁlm between gas bubbles or emulsion droplets.

Other effects, however, may overrule the tendency to quench the waves. The sinusoidal shape of the wave implies different Laplace pressures at A and B in Figure 18.8. Under certain conditions this may lead to further constriction of the ﬁlm, in analogy to the mechanism causing capillary instability of thin liquid jets (see Section 6.2, Figure 6.8). When a monolayer of stabilizer is present in the interface the situation is much more complicated. First, when a wave develops, the liquid in the ﬁlm moves but the monolayer prevents the interface from moving along [cf. Section 17.4.2, Eq. (17.22), and Section 18.3.2]. Second, in thin ﬁlms colloidal interaction forces are effective, together resulting in a total Gibbs energy of interaction int G between the dispersed particles that varies with particle separation h. This subject has been discussed in some detail in Sections 16.2 and 16.3. When the perturbation is so severe that across the thin region of the ﬁlm the interparticle separation is such that dint GðhÞ=dh > 0 the ﬁlm thins until rupture. The use of polymeric and=or proteinaceous stabilizers that form intermolecular bonds provides the interface with a high dilation modulus, that is, a high resistance against ﬁlm thickness ﬂuctuation. The presence of small solid particles in the ﬁlm could as well affect the stability against coalescence. Particles that are poorly wetted by the ﬁlm liquid enhance coalescence. The curvature at the contact angle by which the solid is wetted causes a Laplace pressure that is higher than in the more planar ﬁlm regions and, hence, causes the liquid to ﬂow away from the solid particles until the ﬁlm breaks. This is depicted in Figure 18.9. According to the same principle solid particles in the liquid ﬁlm protect the emulsion or foam against coalescence. This is the case when the particles are not preferentially wetted by either one of the two ﬂuid phases. It is known as ‘‘Pickering stabilization,’’ which was discussed in more detail in Section 8.7.3. Finally, if for one reason or another the molecules of the surface active agent are not evenly distributed over the interface an interfacial tension gradient exists. In the process of leveling the gradient molecules of the ﬁlm liquid are dragged along in the direction of the higher interfacial tension (Marangoni effect; see Section 17.4.2). For a strong interfacial tension gradient and a high rate of the ensuing liquid ﬂow a hole in the ﬁlm could be created. This situation may occur when surface active material spreads from a single particle. See Figure 18.10. As a rule, emulsions are far more stable against coalescence than foams. Emulsions may be kept over periods of weeks or months but most foams

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378

Chapter 18 poorly wetted particle

Figure 18.9

Film rupture caused by a poorly wetted particle.

disappear in minutes or, at most, a few hours. The reasons are related to the various aspects discussed above. In particular, the very thin ﬁlms in (the top layers of) creamed, drained foams and the relatively large ﬁlm dimensions around bubbles that allow long-wave ripples are responsible for the low coalescence stability of foams. Moreover, the rupture of a given number of ﬁlms has a much more dramatic effect on the foam. For example, 1 cm3 of drained foam with bubbles of 1 mm radius contains about 2500 bubbles, whereas the same volume of a 50% (v=v) emulsion contains 1010 droplets having a radius of 5 mm. It is obvious that rupture of, say, 103 ﬁlms results in almost complete loss of the foam whereas the existence of the emulsion is not severely affected. Thus stability against coalescence is inﬂuenced by various factors, some of them playing simultaneous roles. Interpretation of experimental data may easily

Figure 18.10 Film rupture caused by a high interfacial tension gradient, for example, invoked by spreading a solid surfactant.

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be obscured by the different underlying mechanisms. Nevertheless, a few general leading principles can be formulated. Coalescence is strongly retarded by reducing the size of the emulsion droplets or foam bubbles. Furthermore, stabilizers forming monolayers that electrostatically and=or sterically repel each other and, hence, keep the dispersed particles apart may prevent coalescence. For this reason, polyelectrolytes, including proteins, are good long-term stabilizers. Addition of an excess of low molecular weight surfactant to a protein-stabilized emulsion increases the coalescence rate. The low molecular weight surfactant displaces the protein from the interface with a concomitant loss of steric stabilization. By using a nonionic surfactant electrostatic repulsion is lost as well. Moreover, as the reduction in interfacial tension usually is substantially larger for the low molecular weight surfactant than for the protein or polymer this further enhances coalescence.

18.4 MODULATION OF THE COARSENESS AND STABILITY OF EMULSIONS AND FOAMS In most practical systems, such as those mentioned in the introductory part of this chapter, stable foams and emulsions are preferred. However, sometimes the development of a foam or emulsion is inconvenient, such as in oil drilling, in fermentation tanks, and in infusion preparations. It could also be that we prefer a foam or emulsion of intermediate stability, for instance, when compounds such as ﬂavors in foodstuffs or drugs in medical products have to be released from the dispersed phase at a desired rate. In the foregoing sections we have seen that adsorption dynamics, that is, the rate by which the interfacial tension decreases upon adsorption of the emulsifying or foaming agent, as well as the interfacial rheological properties play key roles in the formation and subsequent long-term stability of emulsions and foams. The coarseness and the stability of these dispersions may therefore be modulated on the basis of adsorption dynamics and interfacial rheological properties. The choice of the type of emulsifying or foaming agent has dramatic consequences. The basic functions of the surface active materials are (1) to help make drops or bubbles small, that is, to facilitate breakup of larger particles into smaller ones and to retard recoalescence during the emulsiﬁcation or foaming stage; and (2) to keep the bubbles or droplets small, that is, to retard or prevent disproportionation and=or coalescence. As a rule, low molecular weight surfactants reduce the interfacial tension more than polymeric or protein molecules do. Furthermore, because of the difference in molecular size the rate of adsorption is much faster for the low molecular weight surfactants. The low molecular weight surfactants may also be more effective in retarding recoalescence during the emulsiﬁcation or foaming process, because the faster adsorption allows establishment of interfacial tension gradients on the timescale of particle breakup. Polymers, in particular, polyelectrolytes and proteins, perform better in main-

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Chapter 18

taining long-term stability. The bulkiness of these macromolecules causes steric (and, if the polymers are charged, electrostatic) stabilization, which keeps the particles farther apart. As a consequence the concentration gradient of dispersed phase molecules in the continuous phase between two particles of dissimilar size is smaller. This, together with the longer traveling distance for these molecules through the continuous phase slows down disproportionation. In addition, the higher interfacial dilation modulus obtained with polymers and proteins further retards or even stops the disproportionation process. Because they give thicker ﬁlms, polymeric surfactants also reduce the probability of coalescence. In addition, the fact that a polymeric monolayer usually has a higher interfacial tension than a monolayer of low molecular weight surfactants implies a stronger resilience against bubble or droplet deformation and, hence, at a given ﬁlm thickness, a lower probability of coalescence. From the foregoing it follows that an ideal emulsifying or foaming agent contains two types of surfactant: low molecular weight and polymeric (preferably a polyelectrolyte or protein). The low molecular weight surfactant molecules adsorb at a higher rate and promote a transiently stable dispersion during the formation stage. The later-arriving polymer molecules displace, at least partially, the smaller surfactants from the interface thereby promoting the dispersion’s long-term stability. It goes without saying that the combination of these two types of surfactant is only effective if the ratio is chosen properly. A too-high level of low molecular weight surfactant will obstruct polyelectrolyte or protein adsorption and consequently reduce the long-term stability of the emulsion or foam.

EXERCISES 18.1

Comment on the following statements. (a) The creaming rate of oil droplets in an oil-in-water emulsion increases when the droplets coalesce. (b) Drainage of a foam occurs through the Plateau borders. (c) Disproportionation is retarded by increasing the viscosity of the continuous phase. (d) The interfacial tension of bubbles in a foam is generally higher than that of droplets in an emulsion. (e) The role of a surface active agent as emulsiﬁer is to lower the interfacial tension of the liquid=liquid interface.

18.2

(a)

Make a qualitative sketch indicating the time-dependence of the size of a shrinking gas bubble as a function of time during disproportionation. Give an explanation using the theory presented in Chapter 6.

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What is the concentration of NaCl in a water droplet of 10 mm in a water-in-oil emulsion in equilibrium with pure bulk water? The interfacial tension of the water=oil interface is 35 mN m1 and the temperature 20 C. (c) If the droplets in the emulsion mentioned in Part (b) consisted of pure water, what would be the minimum value of the interfacial compression (or dilation) modulus of the stabilizing surfactant monolayer to prevent ongoing disproportionation? Calculate the repulsion between the two surfaces of a drained water ﬁlm containing 102 M (1 : 1) electrolyte and having an equilibrium thickness of 2.5 nm. The ﬁlm is stabilized by a low molecular mass surfactant. The contribution from dispersion forces may be neglected. The electric potential at the surfaces is 40 mV, the temperature 25 C, and the dielectric constant in the ﬁlm is 78.5. Consult the theory of Chapter 16. Consider an oil-in-water emulsion. The radius of the oil droplets is 10 mm. The density of the oil is 0.9 g cm3 and of water 1.0 g cm3 . The viscosity in the continuous phase is 103 N m2 s. (a) Calculate the creaming rate of the droplets. (b) How high should the yield stress in the aqueous phase be to prevent the oil droplets from creaming? (c) Explain why the calculations under Parts (a) and (b) are irrelevant for oil droplets as small as 0.1 mm in radius. A mixture of 30 cm3 oil and 70 cm3 water is emulsiﬁed using a polymeric surfactant as emulsiﬁer=stabilizer. The result is an emulsion representing 30 m2 interfacial area. The interfacial tension of the monolayer at the oil=water interface is 20 mN m1 . The emulsiﬁcation occurs in an insulated container in which the temperature rises from 20 C to 65 C. Calculate the total interfacial Gibbs energy of the created interface and compare this with the heat dissipation during emulsiﬁcation. The density of water is 1.00 g cm3 and that of the oil 0.85 g cm3 , and the heat capacities are 4.18 J K1 g1 and 2.5 J K1 g1 for the water and the oil, respectively. (b)

18.3

18.4

18.5

SUGGESTIONS FOR FURTHER READING P. Becher. Emulsions, Theory and Practice, New York: Reinholdt, 1965. E. Dickinson, J. M. Rodrı´guez Patino. Food Emulsions and Foams, Cambridge: The Royal Society of Chemistry, 1999. D. Exerowa, P. M. Kruglyakov. Foam and Foam Films, Amsterdam: Elsevier, 1998. K. Larsson, S. Friberg (eds.). Food Emulsions, New York: Marcel Dekker, 1990. R. K. Prud’homme, S. A. Khan (eds.). Foams: Theory, Measurements, and Applications. Surfactant Science Series 57, New York: Marcel Dekker, 1996. D. Weaire, S. Hutzler. The Physics of Foam, New York: Clarendon, 1999.

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Structural Rigidity of Plants

Nonwoody parts of plants contain a lot of water. Most of the free water is in the vacuoles, membrane-bound cavities inside the cells. The sap in the vacuoles contains sugars, proteins, and salts. Because of the semipermeability of the tonoplast, the membrane around the vacuole, water is allowed to enter the vacuole by osmosis. The cell becomes swollen and is said to be turgid. In this turgid state the protoplasm exerts a force on the (slightly) elastic cell wall and this is called turgor pressure. Turgor supports the rigidity of the plant. Plants lacking in turgor, for instance, due to drought or to loss of membrane permeability (e.g., by boiling), visibly wilt. (Figure courtesy of Department of Plant Sciences, Wageningen University, The Netherlands.)

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19 Physicochemical Properties of Biological Membranes

Biological membranes, called biomembranes for short, are nature’s most abundant interfaces. Or, more precisely, as their matrix is made of a self-assembled bilayer of phospholipids (see Chapter 11), biomembranes rather form interphases with interfaces at both sides of the bilayer. This is clearly pictured in the classical generalized presentation of the membrane structure, proposed by Singer and Nicolson in the 1970s and shown in Figure 19.1. Biomembranes enclose biological cells creating, maintaining, and protecting a proper intracellular environment required for the life processes to take place. Besides enveloping whole cells, subcellular particulates such as the nucleus, mitochondria, chloroplasts, and other organelles are enshrined by biomembranes as well. See Figure 19.2. Thus biomembranes mediate between natural ‘‘microbioreactors’’ and their ‘‘outside world.’’ They act as a physical barrier preventing molecules from moving freely in and out of the cell and its organelles but at the same time they should be selectively permeable allowing controlled transport of nutrients and metabolites. Selective ion passage through the membrane is at the basis of bioenergetics, transduction of nerve impulses, and other signaling. The biomembrane further accommodates various biomolecules having speciﬁc functions. Several enzymes, for example, those involved in oxidative phosphorylation and in photosynthesis, are attached to the (mitochondrial) membranes, and receptor molecules for biological recognition, often glycoproteins and glycolipids, are located at the outer membrane surface. See Figure 19.1.

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Chapter 19

Figure 19.1 Cartoon of a biological membrane, showing the lipid bilayer matrix, intrinsic and extrinsic proteins, glycocalyx, and cytoskeleton. (From G. L. Zubay, W. W. Parson, and D. E. Vance. Principles of Biochemistry, Dubuque, IA: Wm. C. Brown, 1995, Ch. 17.)

Various scientiﬁc disciplines are concerned with research on biomembranes. Altogether this has generated a vast amount of literature. In this chapter we only brieﬂy consider a few generic physical–chemical aspects of biomembranes related to their structure and dynamics, their electrical properties, and their permeability for compounds, particularly ions. Understanding these characteristics is invaluable in many areas of biotechnology, biomedical, and pharmaceutical sciences.

19.1 STRUCTURE AND DYNAMICS OF BIOMEMBRANES Biological membranes are heterogeneous containing a variety of compounds. The main ones are phospholipids and proteins. The protein=lipid ratio varies strongly between say, 0.25 and 2.5 depending on the species and, for a given species, the type of cell or organelle and, therewith, the functions. The current view of the structure and dynamics of a biomembrane starts from the ﬂuid mosaic model depicted in Figure 19.1. According to this model the membrane consists of a dynamic mosaic of different types of protein molecules embedded in, or attached to, a bilayer of phospholipids (in animals) or glycolipids (in plants). The basic structures of these lipids are presented in Section 11.6, Figure 11.15. These lipids are highly amphiphilic with aliphatic fatty acid chains mostly in the C16 –C24 range. In an aqueous medium the glyco- and phospholipids

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385

Figure 19.2 Eucaryotic cell. (From G. L. Zubay, W. W. Parson, and D. E. Vance. Principles of Biochemistry, Dubuque, IA: Wm. C. Brown, 1995, Ch. 1.)

spontaneously self-assemble. Depending on the geometry of the molecules, notably the packing factor n=am;0 l (see Section 11.3) planar or curved bilayers are formed. The thickness of these bilayers typically is between 5 and 10 nm. If the value for n=am;0 l is around unity planar structures are favored, and for n=am;0 l < 1, corresponding to cone-shaped molecules, spherical membranes (i.e., liposomes) result (see Section 11.6). Biomembranes of eukaryotic organisms contain another class of lipids as well. These are steroids; cholesterol (see Figure 7.1) may be the most famous example. The steroids are incorporated in different membranes to different extents. Because of their rigid structure they increase the stiffness of the core of the membrane and they reduce the permeability for water.

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Chapter 19

The lipid membrane matrix may be considered as a ‘‘two-dimensional solvent’’ for the membrane proteins. In view of the way the protein molecules are organized in the membrane we can distinguish three different situations. First, there are the integral or intrinsic membrane proteins. Some of these have the function of inﬂuencing (either increasing or decreasing) the order and strength of the membrane structure. They expose mainly apolar residues to the lipid bilayer and many of them proceed as glycoproteins to the outer surface of the membrane. Their carbohydrate parts extend into the extracellular environment and together with the carbohydrate moieties of the glycolipids they form a loose external layer, the glycocalyx, around the membrane. The glycocalyx may reach out into the outer solution over distances of a few tens of nm, much farther than the thickness of the lipid bilayer. Other intrinsic proteins take care of transporting polar compounds across the membrane. These transmembrane proteins usually adopt helical structures with an apolar exterior facing the lipid bilayer and a polar interior forming a channel enabling polar compounds, especially ions, to traverse the membrane. Another class of proteins that occur in many, but not all, membranes is the cytoplasmatic proteins located at the inner surface. These mainly rod-shaped proteins are connected to each other and are ﬁrmly anchored to intrinsic membrane proteins. They thus form a strong, dynamic ﬁbrous network, the socalled cytoskeleton, that controls the lateral diffusion of the intrinsic proteins. The third class is that of the peripheral or extrinsic membrane proteins. They reside entirely in the (aqueous) extracellular or intracellular space. They are physically bound to the membrane by speciﬁc or nonspeciﬁc interactions. Most membrane-linked enzymes are such extrinsic proteins. Biological membranes are dynamic structures. They must be able to respond and conform to changing environmental conditions. To comply with that requirement the membrane has a high degree of ﬂuidity. The lipid bilayer should therefore be in the liquid-crystalline state rather than in the gel-crystalline state (see Section 11.6, Figure 11.16). The liquid crystalline state allows for a high lateral mobility of the protein and lipid molecules whereas the movements normal to the membrane surface are severely restricted. This ascertains the integrity of the membrane implying that asymmetry, for instance, with respect to protein arrangements, is maintained. The ﬂuidity of the bilayer primarily depends on the nature of the aliphatic fatty acid chains in the lipids. The ﬂuidity increases with increasing chain length beyond, say C10 . Furthermore, and more strongly, the order in the bilayer is sensitive to the degree of saturation of the chemical bonds in the chains. The smaller the number of (especially cis) double bonds the more ﬂuid the bilayer is. For a given composition the temperature determines the membrane ﬂuidity. It explains why the degree of saturation of the fatty acids in membranes of plants,

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micro-organisms, and cold-blooded animals tends to decrease with decreasing temperature of the surroundings. For the sake of convenience and to facilitate unambiguous interpretation of experimental results, fundamental research concerning biomembranes is often carried out using model systems. Such models may be monolayers (Chapter 7), Langmuir–Blodgett ﬁlms (Chapter 7), and self-assembled structures such as planar lipid bilayers or liposomes (Chapter 11). These basic structures may be upgraded by inserting or attaching functional molecules to mimic the biological membrane more closely.

19.2 ELECTROCHEMICAL PROPERTIES OF BIOMEMBRANES Biological membranes carry electric charges of different origin. Because of the low polarity and hence low dielectric constant in the core of the lipid bilayer the charges try to avoid the inner membrane zone. Therefore, essentially all the charged groups reside in the peripheral regions at both sides of the membrane. Part of the charge ﬁxed to the membrane originates from dissociation of carboxyl groups of sialic acids that belong to the glycocalyx and that are positioned in the outer part of the glycoproteins. The pK-value for the dissociation of these groups is around 2.6. Hence, at most physiological pH-values the carboxyls are negatively charged. Note that these charges may be located rather far out on the bilayer surface. Other negative charges stem from several phospholipids. Proteins also, with their cationic and anionic groups, contribute to the charge on the membrane. Protons are the charge determining ions for all these groups and the membrane charge density is therefore directly dependent on the pH. At essentially all physiological pH values the net membrane charge is negative, reaching values of a few to a few tens of mC cm2 . Due to different degrees of mobility, entropy, electrostatic effects, and other types of interaction the charged groups are, as a rule, nonuniformly distributed over the peripheral regions of the membrane. Moreover, counterions more or less penetrate the external regions of the membrane structure. For these reasons the models of the electrical double layer that assume a planar impenetrable surface are only poorly applicable to biological membranes. The best approximation may be the model invoking a porous surface that is penetrable for counterions [see Section 9.4, Figure 9.9(b)]. It should also be realized that the porosity (i.e., the depth over which the surface charge and penetrated countercharge is distributed into the membrane) is strongly sensitive to the ionic strength of the medium. The peripheral regions of the membrane, in particular, the glycocalyx, have a polyelectrolyte character (see Section 12.5) and therefore these regions shrink

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388

Chapter 19

with increasing ionic strength. In spite of these complications the charge on the membrane gives rise to an electrical double layer having a thickness that depends on the ionic strength (see Section 9.4.2). In aqueous systems of low ionic strength, such as rivers and lakes, the electrical double layer extends over distances on the order of ten nm, whereas in more saline environments, such as most body ﬂuids, fruit juices, and, above all, sea water, the compensating countercharge extends less than a (few) nm from the membrane surface. Assuming a certain charge distribution in and near the membrane, Poisson’s law, Eq. (9.15), allows for the calculation of the potential proﬁle. The charge density in the inner aliphatic core of the membrane is zero and, consequently, the potential varies linearly in that region. In addition to accommodating the charged groups of the membrane, the peripheral regions have a strong dipolar character. This originates from the organization of dipolar groups and molecules, primarily ester linkages of the phospholipids and water molecules, in such a way that the aliphatic side is positive relative to the exterior side. Taking a constant charge density the potential proﬁle in the peripheral regions is parabolic. Across these regions the potential may decay over several hundred mV. Because of the asymmetric composition of the membrane the potential drops across the inner and outer peripheral regions may be quite different. At either side of the membrane the charge is usually thought to be diffusely distributed so that the potential falls off exponentially as described by Eq. (9.27) or (9.28). It is noted that at the extracellular side of the bilayer the space charge density rðxÞ and, hence, the potential proﬁle cðxÞ is not only determined by countercharge but also by charged groups ﬁxed at the extended parts of the glycocalyx. At low ionic strength the glycocalyx is packed more or less loosely with a corresponding effect on rðxÞ and hence cðxÞ. It implies that the outer structure of the membrane is controlled by a subtle interplay between electrochemical properties of the glycocalyx and the ionic strength of the medium. The permeability of the biological membrane is different for different ions and this gives rise to an electric potential difference between the bulk phases at the inner and outer sides of the membrane, respectively. The origin of this potential difference, referred to as the (trans)membrane potential cm , is discussed more extensively in Section 19.4. As a rule, the potential at the inner side is lower implying a negative value of cm . In biological membranes cm is typically in the range of a few tens of milliVolts. Based on these considerations the overall electric potential in and at a biological membrane is qualitatively shown in Figure 19.3. The whole membrane, surrounded by an electrolyte solution at both sides, may be considered as a condenser with a capacitance C given by s ee ð19:1Þ C ¼ 0; cm d

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Physicochemical Properties of Biological Membranes

389

Figure 19.3 Schematic representation of the electrostatic potential across a biomembrane.

where s is the charge difference (per unit surface area) between the two sides of the membrane required to generate the membrane potential cm , d is the thickness, and e the dielectric constant of the membrane. The capacitance of the membrane is relatively constant because e and d are essentially invariant. The capacitance can be experimentally assessed; a typical value for the capacitance of a cell membrane is 10 mF m2 . Taking 10 nm for the membrane thickness results in a value of 9 for the dielectric constant. This value is much higher than that for the dielectric constant of pure lipids, which is about 3.5. The reason for the difference is probably due to the polar head groups of the lipids and the heterogeneity of the membrane, notably its (glyco)proteins and water content. It follows therefore that the strength of the electric ﬁeld in a biological membrane, especially in both peripheral regions, may reach values in the range of 107 to 109 V m. Such strong ﬁelds are likely to affect the orientation and arrangement of charged molecules in the membrane and, thereby, the membrane structure and mechanical properties. Finally, it should be realized that we discussed the membrane’s electrochemical properties as if the core and the peripheral regions were homogeneous in which the charges were smeared out. However, each of these zones is heterogeneous in itself and in reality the local microconditions may deviate drastically from the average picture presented in Figure 19.3.

19.3 TRANSPORT IN BIOLOGICAL MEMBRANES The biological membrane is a barrier for the motion of molecules. Hydrophilic molecules are obstructed by the core of the lipid bilayer but their mobility at the

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390

Chapter 19

polar peripheral regions of the membrane may be enhanced. Conversely, crossing the polar peripheries by hydrophobic substances is strongly impeded so that these substances may be trapped in the hydrophobic core, where they have considerable freedom to move laterally. Thus the bilayer forms an obstacle for both hydrophilic and hydrophobic molecules. However, to allow for life processes to occur various compounds have to be transported in and out of the cells and subcellular organelles and therefore they have to traverse the membrane. This transport is rarely, if ever, governed by the lipid bilayer properties. Instead, special regulatory mechanisms control membrane permeation. Figure 19.4 illustrates different types of transport through a membrane. Diffusion without the aid of a transport mediator, as depicted in Figure 19.4(a), may occur to some extent under the inﬂuence of a very high transmembrane concentration difference and, in the case of charged species, a high transmembrane potential. Of course, such diffusion processes are more likely to occur where the lipid bilayer packing is perturbed, that is, where it is less dense. Such packing defects could be induced by incorporating certain compounds such as alcohols in the bilayer. Membranes may, furthermore, become leaky due to the penetration of antibiotics, toxins, and so on. Because of their speciﬁcity these biomolecules may induce selective permeability. Membrane permeability is low, in particular for ionic compounds and for low molecular weight ions even more so than for the larger organic ions that are less polar. The transport of the small inorganic electrolytes relies essentially fully on carriers [Figures 19.4(b) and (c)] and on transmembrane channels [Figure 19.4(d)]. Experiments indicate that in the case of carrier-mediated transport the hopping mode is often more realistic than the traveling of a carrier-ion complex across the membrane. The last-mentioned mechanism would require relatively large perturbations in the membrane, whereas jumping of an ion from one site to another could result from local periodic oscillations. Whatever the transport mechanism is, it should be realized that biological membranes are dynamic structures that respond to ambient changes. Thus a membrane often responds to solute–membrane interactions through a feedback mechanism.

(a)

(b)

(c)

(d)

Figure 19.4 Types of ion transport across a biomembrane. (a) Diffusion without transport mediator; carrier-mediated transport using (b) the traveling or (c) the hopping mode; (d) transport through a transmembrane channel.

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Physicochemical Properties of Biological Membranes

391

Ion transport may occur in simultaneous ﬂuxes of two or more ions in a constant stochiometric ratio. Such coupled ﬂuxes are referred to as cotransports. There are two types of cotransport: ﬂuxes of cations and anions in the same direction (symport) and oppositely directed ﬂuxes of ions of the same charge sign (antiport). If in a cotransport the two ﬂuxes involve equal numbers of charges the process is electroneutral; it is not inﬂuenced by the presence of an electric ﬁeld. However, if the stochiometry between the ﬂuxes is such that a net amount of charge is transported, the process is rheogenic which means that it generates an electric current. Rheogenic processes manifest themselves by electric conductivity of the medium in which they occur. Rheogenic transport is sensitive to electric ﬁeld conditions. With respect to the driving force for membrane permeation we can distinguish between active and passive transport. Transport of a component i is called passive if, at constant temperature and pressure, its ﬂux is driven by a gradient of the (electro)chemical potential: dðmi þ zi FcÞ=dx 6¼ 0. The ﬂux of i is towards the region where its (electro)chemical potential has a less positive (or more negative) value. Active membrane transport uses a source of Gibbs energy other than the (electro)chemical potential gradient of the transported compound. In this way ions or uncharged molecules can be transported against their own (electro)chemical potential gradients. Hence, active transport is required to maintain (electro)chemical potential gradients across the membrane. The Gibbs energy change during this ‘‘uphill’’ transport is delivered by metabolic reactions, in most cases the hydrolysis of ATP into ADP. The active transport mechanism causes, for instance, a nonequilibrium distribution of Naþ and > cextracellular and Kþ across almost all cell membranes; that is, ccytoplasma Kþ Kþ cytoplasma extracellular cNaþ < cNaþ . Now, we proceed to discuss a simple model for the kinetics of ion permeation through a membrane that may be used to deduct ion permeability from the experimentally accessible electric conductivity. Transport of a solute from one side (1) of the membrane to the other side (2) may be considered as the net result of adsorption and desorption at both sides of the membrane, combined with a permeation step. Permeation of ion i involves the passage of a Gibbs energy barrier that is determined by the proﬁle of dðmi þ zi FcÞ=dx. In our model the barrier is the highest in the apolar center of the lipid bilayer, as shown in Figure 19.5. From the laws of conservation for the permeating ion i it follows that dGi;1 dGi;1 dGi;1 ¼ k~Gi;1 þ kGi;2 ð19:2Þ dt dt þ dt ~

and

dGi;2 dGi;2 dGi;2 ¼ þ k~Gi;1 kGi;2 ; dt dt þ dt ~

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Chapter 19

µ i + zi F ∆ ψ

392

α zi F ∆ ψ m

zi F ∆ ψm

Figure 19.5 Electrochemical potential proﬁle for an ion in a lipid bilayer membrane (—— cm ¼ 0; – – – cm 6¼ 0).

in which Gi is the adsorbed amount of i at the side of the membrane indicated by 1 or 2, and where dGi dGi and dt dt þ

~

represent adsorption and desorption rates, respectively. The constants k~ and k are the rate constants for transfer through the membrane from 1 to 2, and reversed. The rates of adsorption and desorption are given by dGi ¼ ka ci ð19:3Þ dt þ and

dGi ¼ kd G i : dt

ð19:4Þ

The separation of i and, hence, the charge separation between 1 and 2 is given by the net ﬂux Ji of i across the membrane: ~

Ji;1!2 ð¼ Ji;2!1 Þ ¼ k~Gi;1 kGi;2 :

ð19:5Þ ~ In the absence of a transmembrane potential (i.e., cm ¼ 0) k ¼ k k0 , which is the so-called standard rate constant for permeation. However, in general, cm has a ﬁnite value (see Section 19.4) and this causes a difference between the values of k~ and k , azi Fcm ð19:6Þ k~ ¼ k0 exp RT ~

~

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Physicochemical Properties of Biological Membranes

and ~

k ¼ k0 exp

ð1 aÞzi Fcm : RT

393

ð19:7Þ

Equations (19.6) and (19.7) are based on the concept that a ﬁnite value of cm lowers the height of the Gibbs energy barrier with an amount of azi Fcm , as is indicated in Figure 19.5. Therefore, ion transfer from 1 to 2 is accelerated by a factor exp½azi Fcm =RT and for the reversed direction retarded by exp½ð1 aÞzi Fcm =RT . The ion permeability of a biological membrane may be approximated by measuring the electric conductivity Km across a unilamellar lipid bilayer. Such a bilayer immersed in a solution of the desired electrolyte represents a symmetrical situation; that is, ci;1 ¼ ci;2 ð ci Þ; Gi;1 ¼ Gi;2 ð Gi Þ and cm ¼ 0. Applying a small a.c. ﬁeld ðca:c: Þ across the symmetrical bilayer leads to an electric conductivity given by zi FJi;1!2 : Km ¼ ð19:8Þ ca:c: (Note that a d.c. ﬁeld would cause polarization across the bilayer and hence disturb the symmetrical conditions.) It follows from (19.5) through (19.7) that for a symmetrical bilayer, for which a ¼ 0:5, z Fca:c: z Fca:c: zF 1þ i Ji;1!2 ¼ k0 Gi 1 i ¼ i ca:c: k0 Gi : RT 2RT 2RT ð19:9Þ (In deriving (19.9) the approximation exp½x ¼ 1 þ x, for small values of x, has been used.) Combining (19.8) and (19.9) gives z2i F 2 k G: ð19:10Þ RT 0 i For low ion concentration the adsorbed amount scales linearly with the concentration, Gi ¼ ðka =kd Þci [cf. Section 14.2.2, Eq. (14.8) and Section 14.3.2, Eq. (14.25)], so that Km ¼

z2i F 2 Pc; ð19:11Þ RT i i where Pi is the effective permeability of the bilayer for the ion under consideration, deﬁned as Km ¼

P k0 ðka =kd Þ:

ð19:12Þ

Equation (19.11) allows for the calculation of the effective permeability of ion i through the bilayer. Ionic permeabilities, thus established, are tabulated in the literature. For instance, the effective permeability of bilayers of lecithin (Figure 7.1) is 9:5 1015 m s1 for Naþ , 7:6 1013 m s1 for Cl , and 105 m s1 for Hþ .

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conductivity × 1011 / Ω –1 cm–2

394

Chapter 19 5 3 Km, min

4 3

2 Km, min

2 Km, min

1

0

10

20

30

40

50

60

70 time / s

80

90

Figure 19.6 Fluctuation of ion transport through a lipid bilayer containing a small amount of gramicidin A. The distinct steps in the conductivity reﬂect the opening and closing of individual transmembrane channels. (From D. A. Haydon and S. B. Hladky. Q. Rev. Biophys. 5: 187, 1972.)

In biological membranes ions are almost exclusively transported through dynamic channels. A well-known helically structured channel-forming compound is the polypeptide gramicidin A. Measuring the electric conductivity of a bilayer in which a very small amount of gramicidin A is incorporated may reveal the contributions of the individual channels. Monitoring Km in realtime gives spikes of a height of n Km;min , with n ¼ 1; 2; 3; . . . and Km;min the conductivity of one channel. Figure 19.6 shows that in a 0.5 M NaCl solution the conductivity of one channel for Naþ is on the order of 1011 O1 and that the lifetime of a channel is in the range of a second. The Naþ conductivity of one gramicidin A channel may be compared with that of a lipid bilayer. Thus for an a.c. ﬁeld of 102 V a conductance of 1011 O1 corresponds to an electric current of 1013 C s1 which means an Naþ ﬂux of about 1018 mole s1 . The inner radius of the helical gramicidin A channel is estimated to be 0.2 nm so that the Naþ ﬂux per unit cross-sectional area of the channel amounts to 2 mole s1 m2 . Based on a lipid bilayer permeability of 9:5 1015 m s1 for Naþ (see above) a ﬂux of almost 2 1012 mole s1 m2 is calculated when applying a potential of 102 V. It reveals that the resistance for Naþ permeation across the lipid membrane is 1012 times higher than for transport through a gramicidin A channel.

19.4 THE TRANSMEMBRANE POTENTIAL In Section 19.2 we already mentioned the existence of an electric potential difference between the two sides, 1 and 2, of the membrane: the transmembrane

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Physicochemical Properties of Biological Membranes

395

potential cm ¼ c2 c1 . For biological membranes the cytoplasmic side is usually denoted as side 2 and the extracellular side as side 1. As a rule cm attains negative values of a few tens of milliVolts. The transmembrane potential results from different permeabilities for different ions and from different ion concentrations at the two sides of the membrane. Consider the simple case of a membrane-enclosed solution of a (low molecular weight) electrolyte Mzþ Xz immersed in its solvent. If the membrane is permeable for only one type of ion of the electrolyte, charge separation across the membrane results, giving rise to a transmembrane potential that opposes further ion transport. The equilibrium potential cm follows directly from the equilibrium condition for the permeating ion i: mi;1 þ zi Fc1 ¼ mi;2 þ zi Fc2 and with (3.27) relating mi to ci this gives

cm ¼

RT ci;1 ln : zi F ci;2

ð19:13Þ

Membrane potentials as expressed by (19.13) that result solely from semipermeability of the membrane are called Nernst potentials. In Section 9.5 we discussed the Donnan effect, that is, the expulsion of electrolyte from an electrical double layer around a charged surface or a polyelectrolyte. It has been shown that the Donnan effect leads to an unequal distribution of ions between the double layer and the bulk solution and, hence, to an electric potential difference, the Donnan potential. Thus the Donnan potential is a special case of the Nernst potential. A more general situation is one in which various ions are present for which the membrane has different permeabilities. For each ion i the ﬂux across the membrane may be described by the Nernst–Planck equation (19.14) that accounts for one-dimensional diffusion [ﬁrst term on the right-hand side in (19.14)] and electric conductivity [second term on the right-hand side in (19.14)]:

Ji ¼ Di

dci ðxÞ jzi j dcðxÞ ; u c ðxÞ dx zi i i dx

ð19:14Þ

where Di is the diffusion coefﬁcient of i in the membrane, dci ðxÞ=dx and dcðxÞ=dx are the concentration gradient of i and the potential gradient across the membrane, respectively, and where ui is the electric mobility of i. Di and ui are related through ui ¼ zi FDi =RT :

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ð19:15Þ

396

Chapter 19

At steady-state condition with respect to charge separation, that is, for a constant value of cm , the electric currents due to the individual ion ﬂuxes cancel P

i zi

Ji ¼ 0:

ð19:16Þ

Combining (19.14) through (19.16) gives P jzi j dci ðxÞ u dcðxÞ RT i zi i dx P : ¼ dx F i jzi jui ci ðxÞ

ð19:17Þ

Integration of (19.17) requires knowledge of either cðxÞ or ci ðxÞ in the membrane. It may be understood from the foregoing discussion that these functionalities are rather complicated. Two very crude, but frequently used, approximations are (1) a linear concentration gradient ci ðxÞ ¼

ci;2 ci;1 x þ ci;1 d

and (2) a linear potential decay cðxÞ ¼

cm x þ c1 d

across the membrane of thickness d. Inserting a linear concentration proﬁle in (19.17) and subsequent integration from x ¼ 0 to x ¼ d yields P jzi j u ðc ci;1 Þ P jz ju c RT i zi i i;2 P cm ¼ ln P i i i i;2 jz ju ðc c Þ jz F i;1 i i i i;2 i i jui ci;1 :

ð19:18Þ

Based on the assumption of a linear potential drop (which seems to be more realistic along an ion-transporting channel than along other cross-sections of the membrane) Eq. (19.17) can be developed into cm ¼

P Pþ uc þ uc RT ln Pi þ i i;1 Pi i i;2 ; F i ui ci;2 þ i ui ci;1

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ð19:19Þ

Physicochemical Properties of Biological Membranes

397

P P where þ and are summations over all cations and anions, respectively. The equations (19.18) and (19.19) show that cm arises from differences in ion mobilities and concentrations. Because of the heterogeneous structure of the biological membrane, diffusion coefﬁcients and, hence, ion mobilities vary with the location in the membrane. It is therefore more practical to replace ui by the effective permeabilities Pi for the ions crossing the membrane under the inﬂuence of an electric ﬁeld. Then (19.19) becomes P Pþ Pc RT i Pi ci;1 þ Pi i i;2 ; ln P þ cm ¼ F i Pi ci;2 þ i Pi ci;1

ð19:20Þ

which is known as the Goldman–Hodgkin–Katz equation. Equation (19.20) illustrates that, at given ion concentrations at sides 1 and 2, cm is primarily determined by the most permeable ions. For the limiting case of a semipermeable membrane enclosing the electrolyte Mzþ Xz with, for instance, PMzþ PXz (19.20) reduces to cm ¼

RT cMzþ ;1 ln cMz ;2 F

which is the Nernst potential as given in (19.13).

EXERCISES 19.1

Comment on the following statements. (a) Transport of a type of ion across a biological membrane occurs until there are equal concentrations of this ion at both sides of the membrane or zero membrane potential. (b) At physiological conditions biological membranes are in the gelcrystalline state in order to maintain their asymmetry. (c) The hydrocarbon chains of the phospholipids in the membranes of cold-blooded animals show a higher degree of saturation than in warm-blooded animals. (d) The extension of the glycocalyx decreases with increasing ionic strength. (e) The existence of a membrane potential lowers the Gibbs energy barrier for ions to traverse the membrane.

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398

19.2

19.3

Chapter 19

(a)

Give a qualitative sketch of the structure and composition of a biological membrane. (b) Mention three modes of ion transport across a biomembrane. (c) Explain the difference between active and passive ion transport across a biomembrane. Vesicles of a certain phospholipid are introduced in pure water up to a vesicle volume fraction of 5% Initially, the vesicles contain a 0.4 M aqueous solution of LiCl. At t ¼ 0 gramicidin A is added to the dispersion and thereafter Liþ is released from the vesicles. The concentration of Liþ in the continuous phase is monitored. Results are given below.

t=s

0

20

40

60

Liþ =M

0

1:0 103

2:0 103

3:8 103

(d) What is the role of gramicidin A? (e) Calculate the rate constant for the permeation of Liþ through the vesicle membrane. The net transport of Liþ terminates when the Liþ concentration in the continuous phase reaches 102 M. (f) What is the reason that the Liþ transport does not continue until equal Liþ concentrations inside and outside the vesicles are reached? (g) Calculate the membrane potential at equilibrium. At which side of the membrane does the potential attain the highest value? Consider an artiﬁcial lipid bilayer immersed in an aqueous solution of a polyacid of pH 4. The temperature is 20 C. An a.c. ﬁeld of 10 mV across the bilayer results in an electric current of 103 C s1 m2 . (a) Which ions cause the electric current across the bilayer? Calculate the effective permeability for those ions. Which assumption(s) did you have to make for the calculation? Next, at side (1) of the membrane the solution is replaced by a 0.01 M HCl=0.03 M KCl solution and at side (2) by a 0.05 M HCl=0.01 M KCl solution. The membrane potential cm ð¼ c2 c1 Þ reaches 41 mV. (b) What do you conclude about the effective permeabilities for the different ions in the system? Subsequent addition of valinomycin, a carrier for monovalent cations (but not for Hþ ), causes a shift of cm from 41 mV to 25 mV. (c) Explain the change of cm . Calculate the effective permeability for Kþ .

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Physicochemical Properties of Biological Membranes

399

SUGGESTIONS FOR FURTHER READING A. Baszkin, W. Norde (eds.). Physical Chemistry of Biological Interfaces, New York: Marcel Dekker, 2000, Chapters 7–10. M. Bender (ed.). Interfacial Phenomena in Biological Systems, New York: Marcel Dekker, 1991, Chapters 3, 8, 10, 14. M. Blank (ed.). Electrical Double Layers in Biology, New York: Plenum, 1986. M. Bloom, E. Evans, O. G. Mouritsen. Physical properties of the ﬂuid lipid-bilayer component of cell membranes: A perspective, Quart. Rev. Biophys. 24: 293–397, 1991. H. Ti Tien. Bilayer Lipid Membranes. Theory and Practice, New York: Marcel Dekker, 1974.

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Implant Infection

Bioﬁlms may be readily formed on the surfaces of implanted artiﬁcial organs, synthetic blood vessels, joint replacements, and indwelling catheters. The picture shows a cardiac-assist device. Infection may occur via the prosthetic conduits. Because of the high resistance against antibiotics and chemotherapy bioﬁlmassociated infections usually lead to recurrent inﬂammation. Reoperation may be necessary, but inﬂammation could also result in osteomyelitis, amputation, or even death. The development of longer-term performing synthetic implants, showing reduced infectious complications, requires full understanding of bacterial adhesion.

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20 Bioadhesion

One of the most relevant, if not the most relevant, colloidal and interfacial phenomenon in life sciences is bioadhesion, that is, the joining together of surfaces of which at least one is of biological nature. Usually, bioadhesion involves the association of a biological cell (including bacterial cells) with the surface of a living or an inanimate substratum. In the special case where the adhesion is between particles of comparable size it may be referred to as aggregation, and adhesion at a gas–liquid interface is also called ﬂotation. Bioadhesion is a common occurrence in biological systems where it is often of vital importance. Perhaps the most well-known example of cell adhesion is that of blood platelets to injured blood vessels where they stop bleeding and defend the organism against wound infection. Blood platelets also adhere when they contact inanimate surfaces but then they induce an adverse effect: a thrombus may develop. Cell adhesion also plays an important role in the recognition and elimination of foreign particles from the body. This process involves ingestion of the foreign particles by special single cells, the monocytes and granulocytes. Furthermore, there are indications that cell growth in tissue culture is stimulated by attachment to surfaces. It is not clear whether this effect is due to the mere attachment to the surface or to the stretching of the cell upon adhesion. Most of the cell adhesion concerns bacteria. In natural environments about 99% of bacterial mass exists at surfaces which seems to imply that the association with surfaces is beneﬁcial. Indeed, bacteria may adhere to survive. To mention a few examples: (1) in aquatic systems nutrients tend to accumulate at surfaces and this is a good reason for micro-organisms to adhere there as well; (2) bacteria may adhere to surfaces to avoid transport by ﬂow to a hostile environment, for instance, from the oral cavity into the gastrointestinal tract; (3) in the adhered

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402

Chapter 20

state the bacteria are less susceptible to environmental attacks, such as by antibiotics and chemicals. In some technological and medical applications cell adhesion is advantageous but in others it is detrimental. In bioreactors cell adhesion is stimulated to form a bioﬁlm for favorable production conditions. In environmental biotechnology micro-organisms immobilized to soils and sediments are exploited for the degradation and detoxiﬁcation of natural and anthropogenic compounds. In contrast, bioﬁlm formation may cause contamination problems in water puriﬁcation systems, in food processing equipment, and on kitchen tools. Similarly, bacterial adhesion on synthetic biomaterials used, for example, for artiﬁcial organs, vascular prostheses, voice prostheses, joint replacements, and so on, as well as for extracorporeal devices such as contact lenses, hemodialysis membranes, catheters, and blood bags, may cause severe infections. Furthermore, bioﬁlms on heat exchangers, ﬁlters, separation membranes, and pipelines, and also on ship hulls oppose heat and mass transfer and increase frictional resistance. These consequences clearly result in decreased production rates and increased costs. Another recent development suggests utilizing adhesion and subsequent proliferation and differentiation of autologous stem cells on biomaterials with the intention of improving the biocompatibility of intracorporeal prostheses and artiﬁcial organs. Thus cell adhesion occurs for various reasons and in different appearances. As surfaces of living systems are involved, speciﬁc recognition mechanisms undoubtedly play crucial roles. Nevertheless, since cell adhesion is a rather general phenomenon it is likely that these speciﬁc mechanisms are superimposed on a basic, generic interaction mechanism. Bioadhesion is very complicated from a physical–chemical point of view. Interfacial tensions, wetting, and electrical properties of the surfaces are prominently involved. Because the (aqueous) medium from which the cells adhere usually contains surface active molecules, notably proteins, the cells adhere as a rule onto an adsorbed proteinaceous layer. The preformed adsorbed layer will therefore largely determine the subsequent cell adhesion process. Furthermore, biological cells often carry polymeric substances at their surfaces. These components may inﬂuence the interaction with a substratum surface in various ways, as explained in Chapter 16. Understanding bioadhesion therefore requires a thorough knowledge of various aspects of colloid and interface science.

20.1 A QUALITATIVE DESCRIPTION OF BIOFILM FORMATION Although bioﬁlms formed in different systems may have different appearances, their formation follows a common sequence of events. In Figure 20.1 the

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Bioadhesion

403

subsequent steps in bioﬁlm formation are schematically depicted. Very quickly, say within seconds, after exposing a substratum to a bioﬂuid a layer of adsorbed organic molecules, particularly proteins, is observed on its surface [Figure 20.1(a)]. In the race for the surface, protein adsorption occurs well before cells arrive. The protein molecules are transported to the substratum at a much higher rate simply because they are much smaller and occur in a much higher number concentration. When the cells arrive [Figure 20.1(b)] they see essentially an adsorbed protein layer rather than the pristine substratum surface. Biological cells respond speciﬁcally to proteins. A pronounced example is the adhesion of blood platelets induced by adsorbed ﬁbrinogen whereas ﬁbrinogen in solution does not bind to platelets. As discussed in Chapter 15 the three-dimensional structure of a protein molecule is more often than not perturbed by adsorption and the mode and extent of perturbation depends on surface characteristics such as electrical charge density, wettability (hydrophobicity), and steric effects. Moreover, in systems containing mixtures of proteins the composition of the adsorbed layer will depend on the type of sorbent surface. The cellular response depends on the structure and composition of the adsorbed protein layer and, hence, indirectly, on the substra-

(a)

(b)

(c)

(d)

Figure 20.1 Subsequent steps in bioﬁlm formation: (a) adsorption of organic (proteinaceous) molecules leading to a conditioning ﬁlm; (b) initial cell adhesion; (c) anchoring of adhered cells through (polymeric) exudates; (d) microbial colonization of substratum surface.

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tum surface. Thus the preformed protein layer conditions the substratum surface with respect to the subsequent events. The cells may approach the substratum by various transport mechanisms, that is, diffusion, convection, sedimentation, active transport (locomotion), or a combination thereof. Then, for the actual attachment of the cell at the surface, two stages may be distinguished, the ﬁrst one determined by generic physical–chemical interactions operating over a relatively long separation distance (a few to a few tens of nm) between cell and substratum whereafter more speciﬁc short-range interactions take over. These short-range interactions, acting on a subnanometer scale, are of a physical–chemical nature as well (e.g., hydrogen bonding, ion pairing, and hydrophobic interaction). They are called ‘‘speciﬁc’’ because they originate from strongly localized groups in speciﬁc surface architectures of the cell and the substratum that are stereochemically complementary and thus allow speciﬁc recognition. The long-range interactions in bioadhesion may be more or less successfully predicted by applying concepts from colloid and interface science. Describing the short-range interactions requires detailed knowledge of the stereochemistry for each individual case and such subtle information is usually not available. After attachment the biological cells may form extrusions or pseudopods and bacteria may excrete (polymeric) exudates allowing them to ﬁrmly anchor at the surface [Figure 20.1(c)]. Eventually, adhered cells start growing and proliferate to colonize the substratum surface [Figure 20.1(d)]. A bioﬁlm has formed. In physical–chemical theories the adhering cells are considered as inert particles; that is, adhesion-induced responses and stimuli, characteristic of living cells, are neglected. Moreover, as mentioned before, prediction of the strength of short-range speciﬁc recognition interactions suffers from a lack of information on the stereochemical structure. Therefore, in this chapter we restrict ourselves primarily to the initial stage of cell adhesion that is governed by long-range forces, thereby realizing that control of the initial stage may be decisive for subsequent stages in bioﬁlm formation. Because of the complexity of the bioﬂuid and the nature of the surfaces involved application of physicochemical concepts to these living systems is limited and provides us with qualitative or, at best, semiquantitative answers.

20.2 BIOLOGICAL SURFACES Particle deposition and adhesion may be approximated by applying theories for the stability of lyophobic colloids (Chapter 16) and for interfacial tension (Chapter 5) and wetting (Chapter 8). These theories assume nonbiological, well-deﬁned particles and substrata. They may fail to quantitatively describe adhesion in living systems. Notably, the following aspects should be realized.

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405

As a rule, particles and substrata are not smooth. In Chapter 19 the biological membrane, constituting the cell wall, was described as a dynamic structure composed of a bilayer of phospholipids to which intrinsic and extrinsic proteinaceous components are associated. The outer surface of the membrane, the glycocalix, contains various polysaccharide structures that protrude in the solution over signiﬁcant distances. With respect to the cell wall composition of bacteria two classes are distinguished, indicated as gram-positive and gramnegative (based on Gram staining). The cell wall of gram-positive bacteria is made up of a biological membrane enveloping the cytoplasm and around that membrane a rigid, 15 to 80 nm thick, layer of peptidoglycans. Polymeric material of different nature, mainly proteins and=or polysaccharides, may be attached to the surface and extend into the solution. Gram-negative bacteria have a much thinner rigid peptidoglycan layer, 1 to 2 nm, sandwiched between two membranes. Hence, the outer surface of gram-negative bacteria is similar to that of other biological, nonbacterial, cells. Bacterial cell wall structures are schematically depicted in Figure 20.2. The complex, heterogeneous, and uneven structure of the surfaces implies that in the adhered state the area of contact comprises various bonds of different strengths. Therefore adhesion is sensitive to the detailed geometry of the surfaces in the contact zone and it is often impossible to quantify the different types of interactions that are involved.

gram-positive

gram-negative

glycocalyx

peptidoglycan layer

outer membrane

cytoplasmic membrane

Figure 20.2 Schematic illustration of the structures of the walls of gram-positive and gram-negative bacteria.

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2.

3.

4.

5.

Animal cells are more or less deformable and they may undergo spreading after deposition at the substratum surface. Because of their peptidoglycan layer bacteria are rather rigid and are not likely to deform substantially after contact with a substratum. Biological cell walls are porous; they are more or less penetrable for water and small solute molecules, such as ions. Notably, the peptidoglycan layer in bacterial cell walls is extremely porous as its density is about 20% (w=v). Various kinds of bacteria have thin microﬁlaments on their surface. They protrude outward in the surrounding solution. These protrusions appear to be actively involved in adhesion processes. Based on their morphology two types of such adhesion-mediating surface appendages may be distinguished: ﬁmbriae and ﬁbrils. Fimbriae are 0.2 to 2.0 mm in length and 2 to 10 nm in diameter; their composition is proteinaceous containing a high fraction of hydrophobic amino acid residues. Fimbriae can be either rigid or ﬂexible and the number of ﬁmbriae on a cell may vary from a few to several hundreds. Fibrils are much shorter than ﬁmbriae, usually less than 0.2 mm and their diameter is too thin to be practically measurable. Fibrils on one and the same bacterial cell may have different lengths and they may be densely or sparsely distributed all over the bacterial surface, often homogeneously but sometimes in lateral or polar tufts. Schematic illustrations of cells carrying ﬁmbriae and ﬁbrils are shown in Figure 20.3. As ﬁmbriae and ﬁbrils facilitate adhesion, they are often found on bacteria that have to adhere for survival. Low molecular weight surfactants and, especially, proteinaceous molecules usually have formed an adsorbed conditioning ﬁlm on the substratum surface before the cells or bacteria arrive. These low molecular and polymeric surfactants may be excreted by the (bacterial) cells themselves. In most biological systems the structure and composition of the conditioning ﬁlm is not well known.

Figure 20.3 Cartoons of cells carrying different types of surface appendages (ﬁmbriae and ﬁbrils) in various patterns.

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Now, having indicated the major obscurities in deﬁning biological surfaces we may be able to judge the value of applying physicochemical theories to bacterial adhesion.

20.3 PHYSICOCHEMICAL MODELS FOR CELL DEPOSITION AND ADHESION Cell adhesion may be portrayed either from a biological or a physical–chemical point of view. The biological approach recognizes that the surfaces are part of nonequilibrium living systems that respond to signals and stimuli. It deals with biological recognition in which various kinds of ligands and receptors play their roles. In the physical–chemical approach the surfaces are considered to be biologically inert; only generic interactions are accounted for. These assumptions allow the development of models to which physical–chemical concepts are applied. It has already been mentioned in Section 20.1 that, because of insufﬁcient knowledge of the biological surface structures on a subnanometer scale, short-range interactions between stereochemically complementary groups are difﬁcult to assess. The generic interactions are often the prelude to the biologically directed speciﬁc interactions between the cell and the substratum. In the following a few theories from colloid and interface science, presented in previous chapters, are called upon to describe cell adhesion.

20.3.1 Capillarity The phenomenon of capillary pressure, or, as it is otherwise called, Laplace pressure, was discussed in Chapter 6. Capillary pressure arises on the close approach of two surfaces with a wetting ﬁlm between them. See Figure 20.4. In the case of well-wettable surfaces the wetting ﬁlm forms a meniscus of radius R and surface tension g. The capillary pressure is inversely proportional to the curvature and, hence, for a

Figure 20.4 Capillary force resulting from the wetting ﬁlm between a disc-like particle (left), a spherical particle (right), and a ﬂat substratum surface.

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given ﬁlm thickness it increases with increasing wettability. When the surfaces are completely wetted, that is, when the contact angle between the liquid and the two surfaces is zero, the meniscus attains maximum curvature and, as a result, so does the capillary pressure. For a ﬂat disc of radius a near a planar surface the capillary pressure equals g=R and the total force f pulling the disc towards the surface is f ¼

pa2 g ; R

ð20:1Þ

showing that the maximum force is reached at maximum curvature, that is, at minimum ﬁlm thickness. For a sphere of radius a at a planar surface the capillary pressure is gð1=R 1=a0 Þ and for the total force f, 1 1 02 ð20:2Þ f ¼ pa g 0 ; R a where a0 is the radius of the circle described by the liquid line along the particle surface. Note that R and a0 are interdependent. Thinning the ﬁlm has an increasing effect on the capillary pressure, but at the same time the area subject to the capillary pressure becomes smaller. Hence, the force goes through a maximum for a certain R; a0 combination. It is obvious that adhesion by capillary forces is only effective for particles that are not completely immersed in a suspending liquid medium. Thus, it may well occur at the inner wall of a container (ﬂask, syringe, blood bag, etc.) above or beyond the suspension surface, as well as on surfaces of various utensils after washing or rinsing.

20.3.2 Stability of Lyophobic Colloids Bacteria and other biological cells, having dimensions in the mm range, belong to the colloidal domain. Depending on such conditions as pH and ionic strength they have electrically charged groups on their surfaces. In most cases substratum surfaces are charged as well. The surfaces of many biological cells are covered with polymeric material and the surroundings may contain dissolved polymers. It is therefore often assumed that the initial stage in bioadhesion, the deposition of the cell on the substratum surface, in a suspending medium is primarily determined by the same types of interaction that govern the stability against aggregation of lyophobic colloids. These are 1. 2. 3.

dispersion interaction; interaction between electrical double layers; and polymer-mediated interactions (bridging, steric repulsion, and depletion aggregation).

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In Chapter 16 it was explained how all of these interactions contribute to the total Gibbs energy of adhesion adh G as a function of the separation distance h between the particle and the substratum. The contributions from dispersion interactions and electrical double layer interactions are combined in the DLVO theory to give adh GðhÞ ¼ Gdisp ðhÞ þ Gedl ðhÞ:

ð20:3Þ

In Section 16.1 expressions for Gdisp ðhÞ and Gedl ðhÞ are given for spherical and planar geometries. Here, we add expressions for a spherical particle interacting with a planar surface: Gdisp ðhÞ ¼

A123 a 6h

ð20:4Þ

with A123 the Hamaker constant for interaction between (1) the particle and (2) the surface across (3) the medium, and ( 2c13 c23 1 þ expðkhÞ 2 2 Gedl ðhÞ ¼ pee0 aðc13 þ c23 Þ 2 ln 1 expðkhÞ c13 þ c223 ) þ ln½1 expð2khÞ ;

ð20:5Þ

where ee0 is the dielectric permittivity of the medium, k the reciprocal Debye length as deﬁned by (9.29), and c the potential at the interface indicated by the index. Equation (20.5) is applicable if the surface potentials are not too high, say, jcj < 50 mV, which is usually the case in biological environments. It further includes the assumption that the electric potentials at the surfaces remain constant while the surfaces approach each other. The argument for choosing the potentials, rather than the charge densities, constant is the slowness by which the relatively large particles encounter the substratum, allowing complete relaxation of the surface ionizable groups. Applying (20.4) and (20.5) requires knowledge of various system parameters. For biological systems these are often far from well known. For instance, the value of the Hamaker constant A123 is in most cases rather uncertain. In aqueous environments dispersion interactions between biological cells and solid substrates are essentially always attractive and for A123 values in the range between 1020 and 1022 J are taken. Under natural conditions most biological cells and most substrates are negatively charged with ensuing negative values for c13 and c23, which causes repulsive electrical double layer interactions. The potentials c13 and c23 are commonly approximated by the electrokinetic potentials z at the respective surfaces. However, for biological surfaces, assessment of the value of z is not straightforward. The walls of biological cells are usually penetrable to solvent and low molecular weight electrolyte and ﬂuid may ﬂow within the outer surface layer

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∆adh G(h)

composed of loosely structured polymeric molecules and surface appendages. Hence surface conductance and hydrodynamic softness complicates derivation of the z-potential from experimentally observed electrokinetic phenomena (cf. Chapter 10). Despite these uncertainties the DLVO approach may give a qualitative insight in adh GðhÞ. The general shapes of the curves of adh GðhÞ are shown in Figure 20.5. These curves are similar to those for int GðhÞ in Figure 16.5. At low ionic strength adh GðhÞ forms a barrier for deposition at a separation between the particle and the substratum surface of at least several tens of nm. Mainly because the biological cells are typically in the mm range (which is rather large on the colloidal scale), the maximum height of the barrier may easily reach tens or even hundreds of kB T -units and is therefore practically insurmountable. It prevents direct contact between the cell and the substratum in the primary minimum. At intermediate ionic strength, ranging, say, between 0.05 and 0.15 M, which occurs in most bioﬂuids, the barrier is less high and situated at a separation of only a few nm. At a somewhat larger distance adh GðhÞ shows a shallow minimum, the secondary minimum. Again, due to the relatively large dimensions of the particle the depth of the secondary minimum may exceed a few kB T which is sufﬁciently deep to capture the particles. Cells adhering in the secondary minimum are laterally mobile and therefore easily rinsed off the substratum surface. Furthermore, they may be released by lowering the ionic strength of the medium. At high ionic strength electrical double layer repulsion is strongly suppressed so that there is a net attraction between the particle and the substratum at all separation distances. The cells deposit in the primary minimum.

1

2 3

h

Figure 20.5 Gibbs energy of adhesion of a particle at a ﬂat substratum surface according to the DLVO theory: (1) low ionic strength; (2) intermediate ionic strength; (3) high ionic strength.

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Cell surface appendages and protrusions are known to play an active role in bioadhesion. In addition to speciﬁc recognition mechanisms they may facilitate attachment to the substratum by reducing the overall repulsion. First, these protuberances are more strongly curved (smaller value of a) than the overall shape of the cell and this causes a proportional reduction of the barrier in adh GðhÞ. Second, a deviating value of the local electric potential may result in less repulsion, or even attraction, between a cell surface appendix and the substratum surface. The inﬂuence of polymers on aggregation and adhesion is discussed at some length in Section 16.3. Here, we summarize the main features in qualitative terms. Polymers that are dissolved in the surrounding medium but that do not adsorb on either one of the surfaces drive the surfaces together. This is caused by an osmotic force arising from a zone adjacent to the surfaces that is depleted of polymer molecules. When the polymer molecules adsorb at one or at both surfaces different effects could result. When the surfaces are only partly covered with polymer, the same molecule may attach to both surfaces when they approach, thereby linking the particle to the substratum. Clearly, in the case of electrical double layer repulsion such bridging can only occur if the loops and tails of the adsorbed polymer molecules reach out from the surface over a distance farther than the position of the barrier for deposition calculated from the DLVO theory and indicated in Figure 20.5. Hence, to allow for bridging the polymers should be longer the lower the ionic strength of the medium is. For the cases where both surfaces are fully covered with a polymer layer or where one surface is covered whereas the polymer does not adsorb at the other surface, close encounter of the two surfaces leads to steric repulsion. See Figures 16.10 and 16.11. To prevent adhesion by steric repulsion the polymer layer should be sufﬁciently thick that the onset of the repulsive interaction is at a separation where the attractive dispersion interaction is too small to let the particles adhere in the secondary minimum. Figure 16.10 illustrates this requirement. It was explained in Chapter 15 that preadsorbed polymers do not readily desorb. It could therefore well be that upon approach of the surfaces the adsorbed polymer layers are not fully relaxed; that is, they are not in thermodynamic equilibrium. The contributions from steric effects may, in a ﬁrst approximation, be added to those of dispersion and electrical double layer interactions resulting in adh GðhÞ proﬁles as shown for int GðhÞ in Figures 16.9 and 16.10. However, when the polymers are polyelectrolytes the contributions from steric and electrical double layer interactions are not additive. This is because the conformation of the adsorbed polyelectrolyte molecules responds to changing electrical conditions. From the foregoing it follows that in media of low ionic strength, such as drinking water, rain water, rivers, and lakes, adhesion of most bacteria and other biological cells is obstructed by electrical double layer repulsion. Only bacteria

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equipped with ﬁmbriae or ﬁbrils may bridge the cell–substratum separation in such systems. In an environment of intermediate salinity such as, for instance, saliva, waste water, and various food products, both DLVO and steric interactions have to be considered for an assessment of cell deposition. In more salty systems, such as urine, blood, lachrymal ﬂuid, and especially sea water, electrical double layer interactions are strongly suppressed and the initial stage of cell adhesion is dominated by dispersion and steric interactions. Thus bioadhesion is sensitive to factors of electrical and compositional nature. This offers various ways to manipulate cell adhesion, for example, by changing the pH and=or ionic strength of the medium and by adding adsorbing or nonadsorbing polymer molecules of different sizes. These possibilities can be exploited for immobilization of biological cells in bioreactors, for bioremediation of soils and sediments, and for many other applications.

20.3.3 Balance of Interfacial Tensions; The Wetting Approach Describing bioadhesion in terms of wetting assumes the creation of a cell– substratum interface at the expense of a cell–liquid and a substratum–liquid interface. Consider a spherical particle P, immersed in a liquid L, having an interfacial tension gPL . The particle approaches a substratum S with an interfacial tension gSL . On adhesion the particle will undergo some deformation to give a particle–substratum contact area of pr2 with an interfacial tension gPS . This process is depicted in Figure 20.6. The Gibbs energy of adhesion per particle (neglecting the contribution from deformation) is given by adh G ¼ ðgPS gPL gSL Þpr2 :

ð20:6Þ

Adhesion occurs for adh G < 0. Unfortunately gPL ; gSL, and particularly gPS cannot be assessed unambiguously. The problems involved in determining interfacial tensions of solid surfaces were explained in Chapter 5. Values may

P

P S

S

Figure 20.6 Deformation of a spherical particle contacting a ﬂat solid surface.

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be derived from contact angle data but that requires the use of controversial theories. Another serious shortcoming of the wetting approach is the assumption of close contact between the particle and the substratum forming a P=S interface. Consequently, the wetting approach ignores the particle–substratum separation dependence of adh G. Unlike the DLVO types of interaction and the steric interactions the wetting approach does not reveal the occurrence of a barrier that obstructs deposition of the particle in a Gibbs energy minimum at close separation, nor does it show the existence of a secondary minimum in which the particles may be arrested. Furthermore, because appendages and hydrated polymers are often present at biological surfaces, close contact between the cell and the substratum is not a reasonable assumption. Despite these inadequacies the wetting approach may be useful to indicate qualitatively the tendency of a deformable cell to spread over a substratum surface. As the critical surface tension of wetting gc of a solid is deﬁned as the surface tension of a hypothetical liquid that would just wet that surface with zero contact angle (cf. Section 8.4), the driving force for cell spreading increases with increasing value of gc . Hence, gc values may be used to rank surfaces with respect to inducing cell spreading. The DLVO theory, the principles of polymer-mediated steric interactions, and the wetting approach are all generic and they should therefore be interconnected. However, the relation between DLVO and steric interactions on the one hand and the interfacial tensions will not be appreciated from the foregoing discussion. In Section 20.4 the apparent disparate approaches are united in a general thermodynamic analysis of particle adhesion.

20.4 GENERAL THERMODYNAMIC ANALYSIS OF PARTICLE ADHESION Based on thermodynamic arguments we show that forces between approaching interfaces are related to variations in adsorption at these interfaces and, hence, to variations in their interfacial tension g. Consider two ﬂat, rigid, and smooth plates immersed in a multicomponent solution at constant temperature and pressure. Components from the solution may adsorb onto these plates. For the sake of simplicity we restrict the analysis to the situation of two identical plates, each having a surface area A, and to nonelectrolyte components in the solution. The same arguments apply but the reasoning becomes more complicated when the interacting interfaces are different and when electrolytes are in the solution. In a system as represented in Figure 20.7 the interacting plates are positioned at a separation h by an external force f. (Note that f is positive if the plates repel each other and negative when their

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p V, T, n1 ...nj A

f

h

Figure 20.7 Two parallel plates of area A immersed in a j-component mixture and interacting across a separation distance h.

interaction is attractive.) By reversibly moving the plates so that the separation changes with dh work of f dh is performed on the system. The differential Gibbs energy of the system relative to a reference system, which is identical except for the presence of the plates, is given by dðG G Þ ¼ ðS S ÞdT þ

j P i¼1

mi dðni n i Þ þ 2gdA f dh;

ð20:7Þ

where the superscript ‘‘’’ refers to the reference system and G; S; T ; m; n, and g have their usual meaning. The difference ðni n i Þ is the number of moles of species i that are adsorbed on the two plates, ns . Integration at constant T ; m; g, and h yields G G ¼

j P

mi nsi þ 2gA:

ð20:8Þ

i¼1

Redifferentiation of (20.8), subsequent subtraction from (20.7), and dividing by A, we obtain, at constant T, 2dg ¼ sdh þ 2

j P

Gi dmi ;

ð20:9Þ

i¼1

where s is the force per unit area, s f =A, and Gi nsi =2A. By deﬁning G1 0 according to the concept of the Gibbs dividing plane (see Section 3.9), it follows that 2dg ¼ sdh þ 2

j P

Gi dmi ;

i¼2

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ð20:10Þ

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415

where Gi is the adsorption of component i relative to component 1 (for which, for convenience, the solvent is usually taken). Cross-differentiation in (20.10) gives @s @Gi ¼2 ð20:11Þ @h mi6¼1 @mi h which reveals that the interaction force to keep the plates at separation h varies with the composition between the plates if the adsorption changes with separation, at constant chemical potential. Assuming an ideal solution, so that dmi ¼ RT d ln ci (3.27), we obtain after integration with respect to ci , ð ci @Gi d ln ci ; ð20:12Þ sj sji ¼ 2RT ci¼0 @h ci6¼1 where sj is the interplate force in the j-component mixture and sji the force in the mixture from which component i has been eliminated. Applying (20.12) ð j 1Þ times to eliminate subsequently all the solution components (except the solvent) and by adding the resulting ð j 1Þ expressions for ðsj sji Þ we obtain ð ci j P @Gi sj s1 ¼ 2RT d ln ci ð20:13Þ i¼2 ci¼0 @h ci6¼1 with s1 being the force between the plates in the pure solvent. Equation (20.13) states that the interplate force in a mixture is inﬂuenced by Pj the separation dependence of the adsorption of any component i ¼ 6 1. When i¼2 Gi increases P during closer approach of the plates P d ji¼2 Gi =dh < 0, the separation dependent adsorption is attractive, but when d ji¼2 Gi =dh > 0 it is repulsive. The derivation of the preceding equations requires the system to be in equilibrium; that is, it must be fully relaxed. It implies that adsorption of the various components i 6¼ 1 should reach their equilibrium values at any distance when the plates change their separation. This requirement is usually not met for adsorbed polymers and, hence, (20.13) is not applicable to steric stabilization by polymers that are ﬁxed at the approaching surfaces. The usefulness of the thermodynamic analysis may be illustrated with some examples. 1.

When two nonpolarizable charged surfaces approach during which their electrical double layers are fully relaxed, they do so at constant surface potentials (see Section 16.1.2). It then follows directly from Le Chatelier’s principle that the surface charge densities decrease, which implies a reduction of the surface excess of charge determining ions. Thus the net adsorption decreases upon approach which requires the force to be repulsive. This is the electrical double layer repulsion.

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2.

3.

The presence of nonadsorbing polymer molecules in the solution gives rise to a layer adjacent to the plates that is depleted of these polymer molecules; in other words, the polymer is negatively adsorbed. When the plates approach down to a separation distance where the depletion layers overlap, the negative adsorption decreases: dG=dh < 0, which implies an attractive force leading to so-called depletion adhesion (cf. Section 16.3.1). Consider two plates that are separated by a solution of (ﬂexible) polymers which adsorb reversibly at both plate surfaces. Decreasing the plate separation leads to increased adsorption which manifests itself in attractive polymer bridging between the two surfaces. On closer approach the polymers desorb due to conformational restrictions, causing a repulsive interaction force.

In these examples the contributions of the solution components to the interplate interaction force are in principle accessible by determining the adsorption isotherms of these components as a function of plate separation. However, in practice this is often not or hardly realizable. In the case of two different surfaces a and b the interfacial tensions of both surfaces enter the equations and the adsorption of component i has to be considered for each plate separately. Thus, 2g has to be replaced by ðga þ gb Þ and 2Gi by ðGi;a þ Gi;b Þ. Another useful relation derived from (20.10) is @g s ¼ 2 ; ð20:14Þ @h mi6¼1 which shows that the interfacial tension of the interacting plates varies with separation distance. Allowing for the separation dependence of g in the wetting approach of (bio)adhesion, gPS as it appears in (20.6) should be replaced by gPL ðhÞ þ gSL ðhÞ with h being the separation between the particle and the substratum. Then (20.6) is modiﬁed to become adh GðhÞ ¼ ðgPL ðhÞ gPL Þ þ ðgSL ðhÞ gSL Þ pr2 : ð20:15Þ ðgPL ðhÞ gPL Þ and ðgSL ðhÞ gSL Þ are obtained from (20.14) by integrating sðhÞ from inﬁnite separation to h. In this way the wetting approach provides the particle substratum interaction as a function of their separation distance. Then, it is compatible with the DLVO and steric interaction models (Section 20.3.2) or with any other adhesion model describing adh G as a function of h. In bioadhesion the interacting surfaces include outer regions of biological membranes or bacterial cell walls. These are far less deﬁned than the model surfaces in the foregoing thermodynamic analysis. Nevertheless, the interdependence of the forces and the compositions in the contact regions remain qualitatively correct.

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Complex bioﬂuids contain several diverse components that adsorb and desorb in the course of the bioadhesion process. Any physical–chemical model that does not take into account all these adsorption changes is, in principle, insufﬁcient and should be judged as not better than a crude approximation.

EXERCISES 20.1

20.2 20.3 20.4

Comment on the following statements. (a) The ‘‘conditioning ﬁlm’’ formed at a surface preceding adhesion of biological cells contains mainly low molecular mass components rather than polymeric ones. (b) Capillary pressure is irrelevant for the adhesion of a particle that is fully immersed in a liquid medium. (c) Adhesion of smooth particles at an electrically repelling, smooth surface is, for given values of the surface potentials, enhanced by decreasing the size of the particles. (d) Adhesion in the so-called ‘‘secondary minimum’’ is reversible with respect to variation in the ionic strength. (e) The afﬁnity of cell adhesion at a substratum decreases with increasing critical surface tension of wetting of that substratum. (f) The interfacial tensions of two attracting interfaces decrease during their approach. See Exercise 6.4. See Exercise 8.4, but now for a spherical biological cell B instead of a protein molecule P. The radius of the cell is 500 nm. Two uncharged ﬂat plates are immersed in an aqueous solution containing a nonadsorbing polysaccharide. The molar mass of the polysaccharide is 25,000 D and the radius of gyration of its molecules is 25 nm. The polymer solution behaves ideally. (a) Give a graph for this system for sj s1 as a function of c, based on Eq. (20.13). (b) Calculate the Gibbs energy of adhesion adh G (per m2 interacting interfacial area) as a function of the separation distance h between the plates for a 1% (mass=volume) polysaccharide solution and a temperature of 37 C. Dispersion interactions may be neglected. What is the interaction force at these conditions? (c) Assume that water is a good solvent, so that the polysaccharide molecules adopt a swollen coil conformation. Derive how, for a given mass concentration of the polysaccharide, the adhesion force (per unit area) depends on the molar mass of the polysaccharide.

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20.5

Consider negatively charged bacteria (B) with radius 500 nm and a tenuous layer of highly solvated ﬂexible uncharged polymers attached to their surfaces. The bacteria are suspended in a 102 M aqueous NaCl solution at 25 C. The surface polymer layer is penetrable without constraint for water molecules, Naþ and Cl ions. The bacteria adhere irreversibly to teﬂon (T) but reversibly to glass (G). The surface potentials at the glass and the teﬂon are 30 mV and at the bacterial cell surface (underneath the polymer layer) 20 mV. The Hamaker constants for the B–G and the B–T interactions across the aqueous medium are 5 1021 J and 1021 J, respectively. (a) Explain the different afﬁnities for adhesion at the glass and the teﬂon. (b) When the NaCl concentration is reduced to 103 M the bacteria spontaneously detach from the glass surface. The surface potential at the bacteria and at the substrata are independent of the NaCl concentration. What is the mechanism of the detachment from the glass surface? Use this experimental ﬁnding to estimate the thickness of the polymer layer. (c) Sandy soils contain mainly silicium oxide particles. The physical– chemical properties of the surfaces of such particles are similar to those of glass. Suppose that cells of B could be used to decontaminate sandy soils. How would you manage to direct transport of B through the soil from one contaminated area to the other? (d) How would you change conditions to prevent the bacteria from adhering to teﬂon?

SUGGESTIONS FOR FURTHER READING Y. H. An, R. J. Friedman (eds.). Handbook of Bacterial Adhesion, Totowa, NJ: Humana, 2000. R. C. W. Berkeley, J. M. Lynch, J. Melling, P. R. Rutter, B. Vincent (eds.). Microbial Adhesion to Surfaces, Chichester, UK: Ellis Horwood, 1980. P. Bongrand, P. M. Claesson, A. S. G. Curtis (eds.). Studying Cell Adhesion, Heidelberg: Springer-Verlag, 1994. W. G. Characklis, K. C. Marshall. Bioﬁlms, New York: Wiley-Interscience, 1990. M. A. Hjortoso, J. W. Roos (eds.). Cell Adhesion, New York: Marcel Dekker, 1995. D. C. Savage, M. Fletcher (eds.). Bacterial Adhesion, New York: Plenum, 1985.

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