Colloids and Interfaces in Life Sciences
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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.
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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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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
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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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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
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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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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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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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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 :
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Some Thermodynamic Principles and Relations
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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
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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
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For an ideal gas ni ¼
