Colloid and Surface Chemistry (Studies in Interface Science)

S T U D I E S IN I N T E R F A C E S C I E N C E

Colloid and Surface C h e m i s t r y

STUDIES

IN I N T E R F A C E

SCIENCE

SERIES EDITORS D. M6bius

Vol. I Dynamics of Adsorption at Liquid Interfaces

Theory, Experiment, Application by S.S. Dukhin, G. Kretzschmar and R. Miller Vol. z An Introduction to Dynamics of Colloids by J.K.G. Dhont

Vol. 3 Interfacial Tensiometry by A.I. Rusanov and V.A. Prokhorov Vol. 4 New Developments in Construction and Functions of Organic Thin Films edited by T. Kajiyama and M. Aizawa

and

R. Miller

Vol. 8 Dynamic Surface Tensiometry in Medicine by V.N. Kazakov, O.V. Sinyachenko, V.B. Fainerman, U. Pison and R. Miller Vol. 9 Hydrophile-Lilophile Balance of Surfactants and Solid Particles

Physicochemical Aspects and Applications by P. M. Kruglyakov Vol. lo Particles at Fluid Interfaces and Membranes

Attachment of Colloid Particles and Proteins to Interfaces and Formation of TwoDimensional Arrays by P.A. Kralchevsky and K. Nagayama

Vol. 5 Foam and Foam Films by D. Exerowa and P.M. Kruglyakov Vol. 6 Drops and Bubbles in Interfacial Research edited by D. M6bius and R. Miller Vol. 7 Proteins at Liquid Interfaces edited by D. M6bius and R. Miller

Vol. 11 Novel Methods to Study Interfacial Layers by D. M6bius and R. Miller Vol. l z

Colloid and Surface Chemistry by E.D. Shchukin. A.V. Pertsov, E.A. Amelina ans A.S. Zelenev

Colloid and Surface Chemistry

E u g e n e D. S h c h u k i n The Johns Hopkins University, Department of Geography and Environmental Engineering, Baltimore, MD, USA and Moscow State University, Department of Chemistry, Moscow, Russia

Alexandr

V. P e r t s o v

Moscow State University, Department of Chemistry, Moscow, Russia

E l e n a A. A m e l i n a Moscow State University, Department of Chemistry, Moscow, Russia

A n d r e i S. Z e l e n e v ONDEO Nalco Company, Naperville, IL, USA

2001 ELSEVIER Amsterdam - London - New Y o r k - Oxford- Paris - Shannon - Tokyo

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ISBN: ISSN:

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PREFACE

This book covers major areas of modern Colloid and Surface Science (in some countries also referred to as Colloid Chemistry) which is a broad area at the intersection of Chemistry, Physics, Biology and Material Science investigating the disperse state of matter and surface phenomena in disperse systems. The book arises of and summarizes the progress made at the Colloid Chemistry Division of the Chemistry Department of Lomonosov Moscow State University (MSU) over many years of scientific, pedagogical and methodological work. The development of colloid science at Moscow State University and elsewhere in Russia was greatly influenced by the fundamental contributions to its major areas ([ 1-4] in the General Introduction) made by Professor Peter Aleksandrovich Rehbinder (1898 - 1972), Academician of the USSR Academy of Sciences, who chaired and led the Colloid Chemistry Division for more than 30 years. Rehbinder was a great enthusiast of colloid science and an excellent lecturer. The synopsis of his lecture course (published by Moscow State University in 1950) was for a long time used as a textbook by generations of students and still now serves as an example of the most clear, logical and broad coverage of the subject. From 1973 to 1994, the Colloid Chemistry Division was chaired by Rehbinder's closest collaborator and successor, Eugene D. Shchukin, Academician of the Russian, the US and the Swedish Academies of Engineering. Professor Shchukin designed a general lecture course in colloid chemistry, which he taught for many years at the Chemistry Department of

ii MSU, and continues to teach now at John's Hopkins University (JHU, Baltimore, MD, USA). The course includes all major areas of colloid science, covering the basic principles, certain quantitative details, and applications. From year to year the course content has undergone continuous changes in line with the latest developments in the field. The materials of this lecture course were worked up by the faculty of the Colloid Chemistry Division, Professor Rehbinder' s former students Professor Alexandr V. Pertsov and Docent Elena A. Amelina, and, with additional contributions written by them, formed the basis of the textbook entitled "Colloid Chemistry", the second edition of which was published in 1992 (see [5] in General Introduction). That book also included materials from a number of specialized courses designed by the authors at different times. The book became the major text used by students at educational institutions throughout Russia, where colloid chemistry is the mandatory part of the core curriculum in chemistry. On-going progress in colloid and surface science and new approaches in teaching, implemented in the courses taught at MSU, and in the course that by E.D. Shchukin currently teaches at JHU, inspired this new book. The preparation of the manuscript took place simultaneously in two languages: in English and Russian. The text written in Russian by Eugene D. Shchukin, Alexandr V. Pertsov and Elena A. Amelina was simultaneously translated into English by Dr. Andrei S. Zelenev (a former graduate student of Professor Egon Matijevi6), who made significant and substantial contributions to the content of the book. The topics written by Dr. Zelenev include the sections on analytical chemistry of surfactants, transfer of sound in disperse systems (acoustics, electroacoustics and their applications), photon correlation

iii spectroscopy, dynamic tensiometry, monodisperse colloidal systems, and other principal subjects. Among significant innovations in the presentation of material, the authors would like to emphasize the following. In contrast to the traditional separation of electrokinetics as

"specific" colloidal phenomena, and

molecular-kinetic and optical phenomena as "non-specific" ones, Prof. Pertsov combined these in a single chapter (Chapter V) based on the fact that all of these phenomena are examples of different transfer processes taking place in disperse systems. The same chapter includes a description of the scattering of light, as well as different methods of particle size distribution analysis based on transfer processes. The description of electrophoresis and other electrokinetic phenomena can also be found in Chapter V, while the theory of the electrical double layer is discussed much earlier, in Chapter III, which covers the adsorption phenomena. Special emphasis has been put on the description of phase equilibria in surfactant solutions and the investigation of properties of adsorption layers. The coverage of lyophilic colloidal systems, micelle formation, microemulsions, the structure of adsorption layers, structure and properties of emulsions and foams has been expanded. The concepts of the theory of percolations, fractals, molecular dynamics, nanocluster and supramolecular chemistry were introduced. Dr. E. Amelina has completely changed the description of the interactions between dispersed particles, the measurements of these interactions, and the discussion of sedimentation analysis. The application of molecular dynamics and computer modeling to the description of characteristic colloidal phenomena has been illustrated.

iv

Professor Shchukin also performed general editing of the manuscript utilizing his experience in lecturing this course and paying special attention to the presentation of the concepts and applications of physical-chemical mechanics of disperse systems and materials, properties of the structurerheological barrier as a factor of strong stabilization, some features of lyophilic colloidal systems and other research areas, explored by Russian scientific schools and less known abroad. Although this book significantly differs from the earlier "Colloid Chemistry" textbook, it nevertheless focuses on the specifics of educational and research work carried out at the Colloid Chemistry Division at the Chemistry Department of MSU. Many results presented in this book represent the art developed in the laboratories of the Colloid Chemistry Division, in the Laboratory of Physical-Chemical Mechanics (headed by E.D. Shchukin since 1967) of the Institute of Physical Chemistry of the Russian Academy of Science, and in other research institutions and industrial laboratories under the guidance of the authors and with their direct participation. Special attention is devoted in the book to the broad capabilities that the use of surfactants offers for controlling the properties and behavior of disperse systems and various materials due to the specific physico-chemical interactions taking place at interfaces. At the same time the authors made every effort to avoid duplication of material traditionally covered in textbooks on

physical

chemistry, electrochemistry, polymer chemistry, etc. These include adsorption from the gas phase on solid surfaces (by microporous adsorbents), the structure of the dense part of the electrical double layer, electrocapillary phenomena, specific properties of polymer colloids, and some other areas.

Material related to these subjects is presented only to the extent consistent with its relevance to colloid chemistry. The authors made every effort to ensure the proper subdivision of the principal material and additional information. The main principles are discussed mostly on a semi-quantitative and in some cases even qualitative levels. This material is presented using the regular base font. Detailed quantitative derivations and other more cumbersome issues are given in fine print. Newly introduced terms are usually given in italic, while words and phrases of special importance are given with larger letter spacing. Because of the interdisciplinary nature of colloid science and the close links between different topics, references to preceding and subsequent chapters are given throughout the book. The authors believe that this helps in emphasizing the interconnectedness between different topics. In correspondence with the detrimental role that interfacial phenomena play in the formation and stability of disperse systems, the book starts with the description of phenomena at interfaces separating phases that differ by their phase state (Chapters I-III). Then the formation (Chapter IV), properties (Chapters V-VI), and stability (Chapters VII-VIII) of disperse systems are covered. The last chapter (Chapter IX) in the book is devoted to the principles of physical-chemical mechanics, the part of colloid science in the development of which the scientific school established by Rehbinder and Shchukin played the leading role. The current literature in Colloid and Surface Science is broadly represented by the art developed by many well-known scientific schools and published in various journals, series of monographs and books listed in the

vi general introduction. These materials may serve as good sources of additional information on both the details related to particular topics and the course content as a whole. If used as a textbook, this book is primarily suitable for university students majoring in Chemistry and Chemical Engineering who take courses in colloid and surface science. The authors believe that the book will also be useful to graduate students, engineers, technologists, and academic and industrial scientists working in the areas that deal with the applications related to disperse systems and interfacial phenomena. The authors are grateful to Professor Boris D. Summ, the head of the Colloid Chemistry Division of the Chemistry Department at MSU, Professor Victoria N. Izmailova, and to all faculty and colleagues at MSU and in the Department of Geography and Environmental Engineering at JHU for their valuable comments related to the content and teaching of the course in Colloid Chemistry. The authors would also like to thank Professors Reinhard Miller (MaxPlanck Institute, Potsdam/Golm, Germany), Egon Matijevid, Larry Eno (Clarkson University, Potsdam, NY, USA), Dr. Niels Ryde (Elan Pharmaceutical, Inc., King of Prussia, PA, USA), and Dr. Andrei Dukhin (Dispersion Technology, Inc., Mt. Kisco, NY, USA) for valuable comments, suggestions and discussions. The authors are especially indebted to Mr. Harald Hille for his commitment, patience and professional help in editing and proofreading the manuscript. His participation was truly critical, since none of the authors are the native speakers of English. The authors express their most sincere

vii appreciation to Ms. Kristina Kitiachvili (University of Chicago, Chicago, IL, USA) for her help in preparing camera-ready manuscript. Help of Mr. Alexei Zelenev and Dr. Peter Skudarnov is also appreciated.

viii CONTENTS PREFACE

GENERAL INTRODUCTION I. SURFACE P H E N O M E N A AND THE STRUCTURE OF INTERFACES IN O N E - C O M P O N E N T SYSTEMS

I. 1. Introduction to the Thermodynamics of the Discontinuity Surface in a Single Component System 1.2. The Surface Energy and Intermolecular Interactions in Condensed Phases 1.3. The Effect of the Interfacial Curvature on the Equilibrium in a Single Component System 1.3.1 The Laplace Law 1.3.2. The Thomson (Kelvin) Law 1.4. Methods Used for the Determination of the Specific Surface Free Energy References List of Symbols II. THE ADSORPTION PHENOMENA. STRUCTURE AND PROPERTIES OF A D S O R P T I O N LAYERS AT THE LIQUID-GAS INTERFACE

II. 1. Principles of Adsorption Thermodynamics. The Gibbs Equation II.2. Structure and Properties of the Adsorption Layers at the Air-Water Interface II.2.1. The Dilute Adsorption Layers II.2.2. Langmuir and Szyszkowski Equations. Accounting for the Adsorbed Molecules Own Size (Mutual Repulsion) II.2.3. Structure and Properties of Saturated Adsorption Layers II.3. Classification of Surface Active Substances. The Assortment of Synthetic Surfactants II.4. Analytical Chemistry of Surfactants References List of Symbols

i

xii

1

2 13 31 31 40 44 59 61

64 65 84 84 97 112 131 144 160 162

III. INTERFACES BETWEEN CONDENSED PHASES. WETTING 165 III. 1. The Interfaces Between Condensed Phases in Two-component Systems 166 III.2. Adsorption at Interfaces Between Condensed Phases 176 III.3. Adsorption of Ions. The Electrical Double Layer (EDL) 193 III.3.1. Basic Theoretical Concepts of the Structure of Electrical Double Layer 194 III.3.2. Ion Exchange 214 III.3.3. Electrocapillary Phenomena 220 III.4. Wetting and Spreading 225 III.5. Controlling Wetting and Selective Wetting by Surfactants 244 III.6. Flotation 250

ix References List of Symbols IV. THE FORMATION OF DISPERSE SYSTEMS

IV. 1. Thermodynamics of Disperse Systems: the Basics IV.2. Thermodynamic Principles of the Formation of New Phase Nuclei IV.2.1. General Principles of Homogeneous Nucleation According to Gibbs and Volmer IV.2.2. Condensation of the Supersaturated Vapor IV.2.3. Crystallization (Condensation) from Solution IV.2.4. Boiling and Cavitation IV.2.5. Crystallization from Melt IV.2.6. Heterogeneous Formation of a New Phase IV.3. Kinetics of Nucleation in a Metastable System IV.4. The Growth Rate of Particles of a New Phase IV.5. The Formation of Disperse Systems by Condensation IV.6. Ultradisperse Systems. Supramolecular Chemistry IV.7. Dispersion Processes in Nature and Technology References List of Symbols V. TRANSFER PROCESSES IN DISPERSE SYSTEMS

V. 1. Concepts of Non-Equilibrium Thermodynamics as Applied to Transfer Processes in Disperse Systems. General Principles of the Theory of Percolations V.2. The Molecular-Kinetic Properties of Disperse Systems V.2.1. Sedimentation in Disperse Systems V.2.2. Diffusion in Colloidal Systems V.2.3. Equilibrium Between Sedimentation and Diffusion V.2.4. Brownian Motion and Fluctuations in the Concentration of Disperse Phase Particles V.3. General Description of Electrokinetic Phenomena V.4. Transfer Processes in Free Disperse Systems V.5. Transfer Processes in Structured Disperse Systems (in Porous Diaphragms and Membranes) V.6. Optical Properties of Disperse Systems: Transfer of Radiation V.6.1. Light Scattering by Small Particles (Rayleigh Scattering) V.6.2. Optical Properties of Disperse Systems Containing Larger Particles V.7. Transfer of Ultrasonic Waves in Disperse Systems. Acoustic and Electroacoustic Phenomena V.7.1. Theoretical Principles of Ultrasound Propagation Through Disperse Systems (Acoustics) V.7.2. Electroacoustic Phenomena V.8. Methods of Particle Size Analysis

255 257 260 261 273 273 279 280 280 282 284 289 295 300 311 313 316 318 320

321 327 329 329 333 337 349 361 373 390 390 402 408 409 417 421

V.8.1. Sedimentation Analysis V.8.2. Sedimentation Analysis in the Centrifugal Force Field V.8.3. Nephelometry. Ultramicroscopy V.8.4. Light Scattering by Concentration Fluctuations V.8.5. Photon Correlation Spectroscopy (Dynamic Light Scattering) V.8.6. Particle Size Analysis by Acoustic Spectroscopy References List of Symbols VI. LYOPHILIC COLLOIDAL SYSTEMS VI. 1. The Conditions of Formation and Thermodynamic Stability of Lyophilic Colloidal Systems VI.2. Critical Emulsions as Lyophilic Colloidal Systems VI.3. Micellization in Surfactant Solutions VI.3.1. Thermodynamics of Micellization VI.3.2. Concentrated Dispersions of Micelle-Forming Surfactants VI.3.3. Formation of Micelles in Non-Aqueous Systems VI.4. Solubilization in Solutions of Micelle-Forming Surfactants. Microemulsions VI.5. Lyophilic Colloidal Systems in Polymer Dispersions References List of Symbols

426 431 435 438 442 452 454 456 461 462 468 472 476 483 486 487 498 502 504

VII. GENERAL CAUSES FOR DEGRADATION AND

RELATIVE STABILITY OF LYOPHOBIC COLLOIDAL SYSTEMS VII. 1. The Stability of Disperse Systems with Respect to Sedimentation and Aggregation. Role of Brownian Motion VII.2. Molecular Interactions in Disperse Systems VII.3. Factors Governing the Colloid Stability VII.4. Electrostatic Component of Disjoining Pressure and its Role in Colloid Stability. Principles of DLVO Theory VII.5. Structural-Mechanical Barrier VII.6. Coagulation Kinetics VII.7. The Influence of Isothermal Mass Transfer (Ostwald Ripening) on the Decrease in Degree of Dispersion References List of Symbols VIII. STRUCTURE, STABILITY AND DEGRADATION OF VARIOUS LYOPHOBIC DISPERSE SYSTEMS VIII. 1. Aerosols VIII.2. Foams and Foam Films VIII.3. Emulsions and Emulsion Films VIII.4. Suspensions and Sols VIII.5. Coagulation of Hydrophobic Sols by Electrolytes

506 507 521 536 543 556 561 571 577 580

583 584 596 607 624 629

xi VIII.6. Detergency. Microencapsulation VIII.7. Systems with Solid Dispersion Medium References List of Symbols IX. PRINCIPLES OF PHYSICAL-CHEMICAL MECHANICS

IX. 1. Description of Mechanical Properties of Solids and Liquids IX.2. Structure Formation in Disperse Systems IX.3. Rheological Properties of Disperse Systems IX.4. Physico-Chemical Phenomena in Processes of Deformation and Fracture of Solids. The Rehbinder Effect IX.4.1. The Role of Chemical Nature of the Solid and the Medium in the Adsorption-Caused Decrease of Material Strength IX.4.2. The Role of External Conditions and the Structure of Solid in the Effects of Adsorption Action on Mechanical Properties of Solids IX.4.3. The Application of Rehbinder's Effect References List of Symbols SUBJECT INDEX

636 641 642 646 649 651 665 689 702

705

715 723 728 731 733

xii GENERAL INTRODUCTION Colloid Chemistry or, alternatively, Colloid and Surface Science, are the established and traditionally used names of the field of science devoted to the investigation of substances in dispersed state with particular attention to the phenomena taking place at interfaces. Peter A. Rehbinder defined colloid chemistry as the "chemistry, physics, and physical chemistry of disperse systems and interfacial phenomena" [1-6]. The dispersed state and interfacial phenomena can not be separated from each other, as interracial phenomena determine the characteristic properties of disperse systems as well as the means by which one can control such properties. In most chemical disciplines the properties of substances are usually considered within the framework of two "extreme" levels of organization of matter: the macroscopic level, which deals with the properties of continuous homogeneous phases, and the microscopic level, dealing with the structure and properties of individual molecules. In reality, material objects (both natural and man-made products and materials) exist, in nearly all cases, in the

dispersed state, i.e. contain (or consist of) small particles, thin films, membranes and filaments with characteristic interfaces between these microscopic phases. As a rule, the dispersed state is the necessary condition required for the functioning and utilization of real objects. This is especially true for living organisms, the existence of which is governed by the structure of cells and by processes taking place at the cellular interfaces. One of the main objectives of colloid and surface science is the investigation of peculiarities in the structure of systems related to their

xiii dispersed state. Heterogeneous systems (and primarily microheterogeneous systems consisting of two or more phases) in which at least one phase is present in the dispersed state, are referred to as disperse systems. The small particles associated with the dispersed state can still be viewed as phase particles, since they are the carriers of properties close to those of the corresponding macroscopic phases and have characteristic interfaces. Usually the disperse system is characterized as an ensemble of particles of dispersed

phase, surrounded by the dispersion medium. One of the central tasks of colloid science is the investigation of changes in the properties of systems due to changes in their degree of dispersion. If the shape of particles forming the system is more or less close to isometric, the extent of dispersion fineness can be characterized by the particle linear dimension (some effective or mean radius, r), degree of

dispersion (or simply dispersion), D, and the specific surface area, S~. The degree of dispersion is determined as the ratio of the total surface area of particles forming the dispersed phase (at the interface between the dispersed phase and the dispersion medium) to the total volume of these particles. The specific surface area is defined as the ratio of the total surface area of all particles to the total mass of these particles, i.e. S~ - D/9, where 9 is the density of the substance forming the dispersed phase. For the monodisperse system consisting of uniform spherical particles of radius r, one can write that

D = 3/r ; for systems consisting of particles of shapes other than spherical the inverse proportionality between dispersion or specific surface area and the particle size will be maintained with a different numerical coefficient. A more complete description of the dispersion composition of the

xiv disperse system is based on the investigation of the particle size distribution function (for anisometric particles, also the particle shape distribution function). The breadth of the distribution function characterizes the system polydispersity. The range of disperse systems of interest in colloid science is very broad. These include coarse disperse systems consisting of particles with sizes of 1 gm or larger (surface area S < 1 m2/g), and fine disperse systems, including ultramicroheterogeneous colloidal systems with fine particles, down to 1 nm in diameter, and with surface areas reaching 1000 m2/g ("nanosystems"). The fine disperse systems may be both structured (i.e. systems in which particles form a continuous three-dimensional network, referred to as the disperse structure), and free disperse, or unstructured (systems in which particles are separated from each other by the dispersion medium and take part in Brownian motion and diffusion). Based on the aggregate states of the dispersed phase and the dispersion medium one can recognize different kinds of disperse systems, which can be described by the abbreviation of two letters, the first of which characterizes the aggregate state of the dispersed phase, and the second one that of the dispersion medium. In these notations gaseous, liquid and solid states are labeled as G, L, and S, respectively. In the case of two phase systems, one can outline eight different types of disperse systems, as shown in the table below. S y s t e m s with a liquid d i s p e r s i o n m e d i u m represent a broad class of dispersions. The main portion of the book is devoted to these objects, the examples of which include various systems with a solid dispersed phase (S/L type), such as finely dispersed sols (in the case of unstructured systems)

XV

TABLE. Different types of disperse systems

~

Medium

Solid

Liquid

Gas

Solid

Sl/S 2

S/L

S/G

Liquid

L/S

L1/L2

L/G

Gas

G/S

G/L

Dispersed Phase

and gels (in the case of structured systems), coarsely dispersed lowconcentrated suspensions, and concentrated pastes. Dispersions with a liquid dispersed phase (L~/L2 systems) are the emulsions. Dispersions in which the dispersed phase is in a gaseous state include gas emulsions (systems with low dispersed phase concentration) and foams. Systems with a gaseous d i s p e r s i o n medium, known under the common name of aerosols, include smokes, dusts, powders (systems of S/G type) and fogs (L/G type systems). Aerosols containing both solid particles and liquid droplets of dispersed phase are referred to as smogs. Since gases are totally miscible with each other, the formation of disperse systems of G~/G2 type is impossible. Nevertheless, even in the mixtures of different gases one can encounter non-uniformities caused by the fluctuations in density and concentration. Systems with a solid d i s p e r s i o n m e d i u m are represented by rocks, minerals, a variety of construction materials. Most such systems are of the S~/S2 types. Various synthetic and natural porous materials (with closed porosity), such as pumice and solid foams (e.g. styrofoam, bread), belong to the G/S type. The systems of L/S type include natural and synthetic opals and

xvi pearl. One can also classify (rather conditionally) cells and living organisms formed with these cells as L/S-type systems. It is worth outlining here that the subdivision of disperse systems according to dispersed phase and dispersion medium ~ is, strictly speaking, valid only for systems in which the dispersed phase is formed with individual particles. There are, however, a large number of systems in which both phases are continuous and pierce each other. Such systems, referred to as

bicontinuous, include porous solids with open porosity (catalysts, adsorbents, zeolites), various earths and rocks, including oil-containing ones. Gels and jellies forming in polymer solutions, including those that are glue-like (the word "colloid" means "glue-like", from Greek ~:c0kka- glue), are also quite close to bicontinuous systems. The principal peculiarity of fine disperse systems is the presence of highly developed interfaces. These interfaces and the interfacial phenomena occurring at them affect the properties of disperse systems, primarily due to the existence of excessive surface (interfacial) 2 energy associated with interfaces. The excess of interfacial energy reveals its action along the interface in the form of interfacial tension, which tends to decrease interfacial

~In some cases dispersion medium is referred to as the continuous phase 2 The terms "surface" and "interface" are not exactly equivalent. One usually refers to an interface when describing the boundary between condensed phases or between condensed phase and a gas (e.g. solution-air interface), while the term surface is attributed specifically to a border of a condensed phase with either vacuum or gas. However, due to their obvious similarity, these two terms have been used interchangeably. In this book we will continue applying this commonly accepted practice and in many instances will use them as synonyms

xvii area. At the same time, the surface energy is directly related to surfaceforces. The force field of these forces may maintain considerable strength, even at distances from the surface significantly larger than molecular dimensions. The existence of developed surfaces in systems consisting of fine particles results in the need of external energy for the formation of such systems by both the comminution (dispersion) of macroscopic phases and condensation from homogeneous systems. The excessive interfacial energy is the reason for the higher chemical activity of dispersed phases in comparison with macroscopic phases. The result of this higher activity is increased solubility of the dispersed phase in the dispersion medium and an increase in the vapor pressure above the surface of fine particles. The smaller the particle size, the greater the increase in the vapor pressure. The elevated chemical activity and the availability of strongly developed interfaces are the reasons for the high rates of interactions between the dispersed phase and the dispersion medium, and the high rates of mass and energy transfer between them in heterogeneous chemical interactions. The presence of surface forces that lead to changes in the structure and composition of interfaces may have a great influence on these transfer processes. A high free energy excess, particularly in systems with a fine degree of dispersion, is the cause of thermodynamic instability, which is the most important feature of a majority of disperse systems. Thermodynamic instability in turn entails various processes aimed at decreasing the surface energy, which results in the saturation of surface forces. Such processes may occur in a number of ways. For example, in a free disperse system partial saturation of the surface forces may take place in the contact zone between the

xviii particles when the latter approach each other closely, resulting in the formation of aggregates. This phenomenon, referred to as coagulation, corresponds to the transition from a free disperse system to a structured one. A further decrease in the surface energy of disperse system may be caused by a decrease in the interfacial area due to the coalescence of drops and bubbles, or by fusion (sintering) of solid particles, as well as by the dissolution of more active smaller particles with the transfer of substance to less active larger particles. Destabilization due to coagulation, coalescence and diffusional mass transfer leads to changes in the structure and properties of disperse systems. It is important to point out that due to coagulation and bridging of particles, a disperse

system acquires

qualitatively new structural-mechanical

(rheological) properties which entail a conversion of the disperse system into a material. In the end, coalescence may result in the disintegration of a disperse system into constituent macroscopic phases. In a number of applications such degradation of colloidal systems is a desirable goal, as, e.g., in making butter by churning, or dehydration and desalination of crude oil. Along with the classification of disperse systems based on the phase state ofthe dispersed phase and the dispersion medium, and their classification as coarse dispersed or colloidal, structured or unstructured, dilute or concentrated, one can also subdivide disperse systems into lyophilic or lyophobic types. Systems belonging to these principally different classes differ in the nature of colloid stability and in the intensity of interfacial intermolecular interactions. High degree of similarity between the dispersed phase and the dispersion medium, and, consequently, compensation of the

xix interactions at the interface (which usually results in very low values of interfacial free energy) is characteristic of lyophilic disperse systems. These systems, e.g. critical emulsions, may form spontaneously and reveal complete thermodynamic stability with respect to both aggregation into a macrophase and dispersion down to particles of molecular size. In various lyophobic systems (colloidal and coarse disperse), there is a lot less similarity between the dispersed phase and the dispersion medium; here the difference in the structure and properties of contacting phases results in uncompensated interfacial forces (energy excess). Such systems are thermodynamically unstable and require special stabilization. All aerosols, foams, numerous emulsions, sols, etc., are examples of lyophobic systems. Along with typical lyophobic and lyophilic systems, there is a broad range of states which with respect to the nature of their stability can be viewed as intermediate. In controlling the stability of disperse systems, the adsorption of

surface-active substances (surfactants) at the interfaces represents a very important way of decreasing the free energy of the system without decreasing the interfacial area. The adsorption of surfactants results in a partial compensation of unsaturated surface forces. Surface active substances, when introduced into the bulk, spontaneously accumulate at the interface, forming adsorption layers. Adsorption monolayers may radically alter properties of interfaces and the type of acting surface forces. Change in the surface forces may also occur with changes in the electrolyte composition of the dispersion medium due to the effect of electrolyte on the structure of the interfacial

electrical double layer. The use of electrolytes and surfactants allows one to effectively control

XX

the formation and degradation of disperse systems and influence their stability, as well as their structural-mechanical and other properties. Surfactants participate in a variety of microheterogeneous chemical, biochemical and physiological processes, such as micellar catalysis, exchange processes, phenomena involved in membrane permeability, etc. The control of the stability of disperse systems plays a crucial role in many technological applications. It is necessary to point out that finely dispersed state of substance is the primary condition for a high degree of organization of matter. Fine disperse structure is the basis for the strength and durability of materials, such as steel, ceramics and others, and for the strength of tissues in plants and live organisms. Heterogeneous chemical reactions in both industry and living organisms take place only at highly developed interfaces, i.e. in finely dispersed systems. Only fine disperse structure consisting of many tiny cells allows an enormous amount of information to be stored in small physical volumes. This relates to both the human brain and new generations of computers. Since the tendency towards lowering the excess of surface energy in disperse systems may take the form of various types of degradation of such systems, the problem of colloid stability is the central problem, not only in colloid and surface science but in all natural sciences as well. Along with factors responsible for the stabilization of different disperse systems, the conditions necessary for the formation of such systems from macroscopic phases are also part of colloid stability studies. It is clear from everything said so far that colloid and surface science

xxi is a peculiar border area of science that has resulted from interdisciplinary interaction between chemistry, physics, biology and other related areas of science during the gradual process of genesis, separation, differentiation and merging between different areas. This has been very well reflected in the recent book by Evans and Wennerstr6m [7]. Colloid chemistry is closely related to the investigation of the kinetics of interfacial electrochemical processes, microheterogeneity (origination of new phases and structures) in dispersions of natural and synthetic polymers, sorption and ion exchange processes in ultramicroporous systems. It is also closely related to such areas of science as solid state physics and chemistry, molecular physics, material mechanics, rheology, fluid mechanics, etc. All of this determines the fundamental theoretical development and heavy involvement of mathematics in various parts of colloid and surface science, with broad use of the methods of chemical thermodynamics and statistics, the thermodynamics of irreversible processes, electrodynamics, quantum theory, the theory of gaseous and condensed states of substance, structural organic chemistry, the statistics of macromolecular chains, molecular dynamics, methods of various numerical simulation involving high-speed computers, etc. Close interaction between colloid science and other related disciplines helped in the establishment and further enrichment of its experimental basis. Along with classical experimental methods specific to colloid science (determination of the surface tension, ultramicroscopy, dialysis and ultrafiltration, dispersion analysis and porosimetry, surface forces and measurements of particle interactions, studies of the scattering of light, etc.), such methods as various spectroscopic techniques (NMR, ESR, UF and IR

xxii

spectroscopy, luminescence quenching, multiply disrupted total internal reflection, ellipsometry), X-ray methods, radiochemical methods, all types of electron microscopy, are all effectively used in the investigation of disperse systems and interfacial phenomena. The methods of surface studies involving atomic force microscopy, slow electrons, and spectroscopy of secondary ions are also broadly used. The use of these and other methods aids have assisted in solving the main problems of colloid science aimed at the understanding of the nature and mechanisms of interfacial phenomena and processes at the atomic and molecular levels. The specific interdisciplinary nature of colloid science makes it of fundamental importance for such adjoining sciences as biology, soil science, geology and meteorology. Colloid and surface science forms the general physico-chemical basis of modern technology in nearly all areas of industry, including chemical, oil, mining, production of construction, instrumental, and composite materials, pulp and paper, printing, food, pharmaceuticals, paint and numerous other areas. It is very important in agriculture for solving problems related to increasing the soil fertility, application of pesticides and herbicides, etc. Colloid science also plays an important role in handling numerous environmental problems, such as waste water treatment, trapping of aerosols, fighting soil erosion, etc. The close interaction of colloid and surface science with molecular physics and a number of theoretical disciplines has determined its role in the development of natural sciences as a whole. The discovery of the nature of, and the further investigation of Brownian motion, the development of direct

xxiii methods for the determination ofAvogadro' s number, the development of the theory of fluctuations and their studies led to the experimental conformation of the molecular structure of matter and of the limits of applicability of the second law of thermodynamics. Colloid science has established new approaches to the studies of the geological history of the Earth's crust, the origin of life, and mechanisms of vital functions. The work of Thomas Graham (circa 1760) marks the birth of colloid chemistry as an independent branch of science. Like other areas, colloid chemistry has its own long history" some specific colloid-chemical recipes were known to the ancient Egyptians and medieval alchemists. J. Gibbs, W. Thomson (Kelvin), J. Maxwell, A. Einstein, J. Perrin, T. Svedberg, G. Freundlich, I. Langmuir, M. Poliani, S. Brunauer, and other great physicists and chemists took active part in developing understanding and knowledge in various areas of colloid chemistry. The results of their work are reflected throughout this book. In this book the authors acknowledge and pay special attention to the views on general and specific problems of colloid chemistry developed by Russian scientists and the different scientific schools founded by them. Among the great scientists who made significant contributions to the area and are less known to the world scientific community, one should name F.F. Reiss, famous for his discovery of electrokinetic phenomena, A.V. Dumansky, the inventor of a centrifuge and the organizer of the first scientific journal on colloid chemistry, also known for his studies on biopolymers as lyophilic colloidal systems, N.A. Shilov, M.M. Dubinin, A.V. Kiselev (theory of adsorption), I.I. Zhukov (electrosurface phenomena), N.P. Peskov (stability

xxiv and structure ofmicelles ofhydrophobic sols). Another great contributor to the study of adsorption layers, adsorption, and other areas of colloid chemistry was A.N. Frumkin, who also played a pioneering role in the development of modern electrochemistry. B.V. Derjaguin and his associates developed the theory of disjoining pressure and its major components as the principal thermodynamic factor in the stability of colloidal systems. In collaboration with L.D. Landau, B.V. Derjaguin created the modern theory of the stability and coagulation of hydrophobic sols by electrolytes. This theory was independently (and somewhat later) developed by the Dutch scientists W. Vervey and J. Overbeek and is now commonly known as the DLVO theory. P.A. Rehbinder and his scientific school played an important role in developing a number of pioneering ideas of modern colloid and surface science. Among them are the fundamental concepts of different mechanisms of surfactant action at various interfaces, particularly those concerning the formation and properties of structural-mechanical barrier as the factor of strong stabilization of disperse systems; the notion of formation of spatial structures in disperse systems due to the aggregation of particles; the discovery of the influence of the surface-active media on the mechanical properties of solids (Rehbinder's effect). The principal result of the development of Rehbinder's ideas was the creation of Physical-Chemical Mechanics, a new area of colloid chemistry. Chapter IX of this book is devoted specifically to the teachings of Rehbinder and the progress in physical-chemical mechanics achieved by his successors. The current literature in the area of colloid and surface science and interfacial phenomena represents the knowledge and techniques developed in

XXV

the leading scientific schools of the world. Numerous articles regularly appear in such specialized periodicals as the Journal of Colloid and Interface Science, Colloids and Surfaces, Langmuir, Advances in Colloid and Interface Science, Colloid Journal, Journal of Dispersion Science and Technology, Colloid and Polymer Science, Current Opinion in Colloid and Interface Science and others. There are series of monographs, including Surface and Colloid Science (edited by E. Matijevid), Studies in Interface Science (edited by D. M6bius and R. Miller), Surfactants Science Series (founding editor M. Schick), Progress in Colloid and Polymer Science, etc, and many textbooks and monographs [628]. The knowledge published in these books and periodicals will be extensively referenced throughout this book.

References ~

,

,

4. 5. ,

~

Q

Rehbinder, P.A., "Selected Works", vol. 1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Rehbinder, P.A., "Selected Works", vol. 2, Surface Phenomena in Disperse Systems. Physical Chemical Mechanics, Nauka, Moscow, 1979 (in Russian) Shchukin, E.D., Proc. Acad. Sci. USSR, Chem Sci., 10 (1990) 2424 Shchukin, E.D., Colloid J. 61 (1999) 545 Academician Pjotr Aleksandrovich Rehbinder: the Centenary, Moscow, Noviy Vek, 1998 (in Russian) Shchukin, E.D., Pertsov, A.V., Amelina, E.A., Colloid Chemistry, 2nd ed., Vysshaya Shkola, Moscow, 1992 (in Russian) Evans, D.F., Wennerstr6m, H., The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet, 2nded., Wiley-VCH, New York, 1999 Kruyt, H.R. (ed.), Colloid Science, vols.l-2, Elsevier, Amsterdam, 1952

xxvi ,

10. 11. 12.. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

28.

Stauff, J, Colloid Chemistry, Springer Verlag, Berlin, I960 (in German) Sheludko, A., Colloid Chemistry, Elsevier, Amsterdam, 1966 Kerker, M., Surface Chemistry and Colloids, Butterworth, 1975 Sonntag, H., Textbook on Colloid Science, VEB Deutsches Verlag der Wissenschafte, Berlin, 1977 (in German) Mysels, K.J., Introduction to Colloid Chemistry. Krieger, 1978 Voyutsky, S.S., Colloid Chemistry, Translated by N. Bobrov, Mir Publishers, Moscow, 1978 Frolov, Yu.G., A Course in Colloid Chemistry, Khimiya, Moscow, 1982 (in Russian) Vold, R.D., Vold, M.J., Colloid and Interface Chemistry, AddisonWesley, London, 1983 Fridrikhsberg, D.A., A Course in Colloid Chemistry, Translated by G. Leib, Mir Publishers, Moscow, 1986 Ross,S., Morrison, I.D., Colloidal Systems and Interfaces, WileyInterscience, New York, 1988 Everett, D.H., Basic Principles of Colloid Science, Royal Society of Chemistry, 1988 Adamson, A.W., Gast, A.P., Physical Chemistry of Surfaces,6 th ed., Wiley, New York, 1997 Hunter, R.J., Foundations of Colloid Science, vols.l,2, Clarendon Press, Oxford, 1991 Hunter, R.J., Introduction to Modern Colloid Science. Oxford University Press, 1994 Lyklema, J., Fundamentals of Interface and Colloid Science, vols. 1-3, Academic Press, 1991-2000 Mittal, K.L., Surface & Colloid Science in Computer Technology. Perseus Publishing, 1987 Shchukin, E.D. (Editor), Advances in Colloid Chemistry and PhysicalChemical Mechanics, Nauka, Moscow, 1992 (in Russian). Hiemenz, P.C., and Rajagopalan, R., Principles of Colloid & Surface Chemistry, 3rd ed., Dekker, New York, 1997 JGnsson, B., Lindman, B., Holmberg, K., and Kronberg, B., Surfactants and Polymers in Aqueous Solution, Wiley, Chichester, 1998 Borywko, M., (Editor), Computational Methods in Surface & Colloid Science, Marcel Dekker, New York, 2000

I.

SURFACE

PHENOMENA

AND

THE

STRUCTURE

OF

INTERFACES IN ONE-COMPONENT SYSTEMS

The difference in the composition and structure of phases in contact, as well as the nature of the intermolecular interactions in the bulk of these phases, stipulates the presence of a peculiar unsaturated molecular force field at the interface. As a result, within the interfacial layer the density of such thermodynamic functions as free energy, internal energy and entropy is elevated in comparison with the bulk. The large interface present in disperse systems determines the very important role of the surface (interfacial) phenomena taking place in such systems. According to Gibbs [1 ], one can view an interface as a layer of finite thickness within which the composition and thermodynamic characteristics are different from those in the bulk of phases in contact. This approach allows one to describe the properties of interfaces phenomenologically in terms of excesses of the

thermodynamic functions in the interfacial layer in

comparison with the bulk of individual phases. With this approach one does not need to introduce any model considerations regarding the molecular structure of the interfacial layer or utilize particular values of layer thickness.

1.1. Introduction to the Thermodynamics of the Discontinuity Surface in a Single Component System

In a single component system two phases (e.g. liquid and vapor) coexist in equilibrium only if there is a stable interface present between them. Such an interface is formed only if an increase in the surface area results in an increase in the system free energy, i.e. d,9~7dS>0. One may thus introduce the surface free energy, ~z-s ,as the free energy excess, proportional to the interfacial surface area: dS~rs-s

-

dS

-

where cy is the specific surface free energy. This specific surface free energy can be viewed as the work required for a reversible isothermal formation of a unit interface. The existence of a force that tends to decrease the interfacial area can be visualized from an experiment designed by A. Dupr6, schematically illustrated in Fig. I-1. In this experiment a rigid frame of wire with one movable side of length d is dipped into a soap solution, resulting in the formation of a thin film on the wire. Let the force F~ be applied to the sliding wire in the direction shown in Fig. I-1. The displacement of the wire by an amount Al causes an increase in the film area equal to Ald. Therefore, the free energy increases by the amount A g s - 2cyAld (the numerical coefficient is due to the film having two sides). From these considerations it follows that the force F 2 acting on the wire and due to the film is given by

F2 -

The case when

Al

= 2cyd

F~=Fz=2~dcorresponds

to an equilibrium between

these two forces. Consequently, cy can also be defined as the force per unit length of the frame. This force, acting along the interface in a direction perpendicular to the frame, is commonly referred to as the surface tension, and is expressed in mN/m or mJ/m 2 , assuming SI units. The action of the surface tension can be readily understood if we consider a series of forces acting on a film with a circular boundary. In Fig. I-2 these forces are labeled with arrows, and they have the effect of contracting the film towards its center. The length of the arrows corresponds to the magnitude of the forces, while the distance between them represents a unit length. d .

,

, , - . _ . _

I/ o I- o -90~ the meniscus is convex, and the pressure in the fluid under it is increased, as compared to that under the flat interface, resulting in capillary lowering. The capillary phenomena are by all means ubiquitous in nature and our every-day life. The penetration of fluids into thin pores, such as those present in soils, plants and rocks, the impregnation of porous materials and fabrics, the changes in the structure and mechanical properties of soils and grounds upon their wetting, are all due to the capillarity. The action of capillary pressure underlies the mercury porosimetry

method, which

is commonly used for the determination of pore size

distribution in ceramics, adsorbents, catalysts and other porous materials [ 15]. Mercury is known to wet non-metallic surfaces poorly, and thus the capillary pressure, equal to 2c~/r (where r is the pore radius, or the average radius of pores having complex shape), prevents its spontaneous penetration into the pores. The pore size distribution can be established by measuring the volume

38 of mercury forced into the pores of a sample of known weight as a function of applied external pressure. In order to force mercury into nanometer-sized pores, the applied pressure has to be on the order of 108 - 109 Pa (1000 - 10000 atmospheres). An interesting phenomenon based on capillarity is the appearance of a capillary attractive force between particles of moistened solids. As a result of wetting, a meniscus is formed upon the particle contact (Fig. I-14). This meniscus between two contacting particles of radii r 0 has a shape of surface of rotation, and can be characterized at each point by the two curvature radii r~ and r 2 (in Fig. I-14, a these radii are of opposite sign, i.e. r~>0 and

r2>r Fig. I-16. The equilibrium shape of the rotating drop

its axis of revolution, the centrifugal force pulls the drop closer to the axis, causing its transformation into a prolate ellipsoid with the same axes of revolution as the outer tube. Assuming that the ellipsoid can be closely approximated with a cylinder of radius r, and measuring its length, l, and speed of revolution, co, it is possible to evaluate the interfacial tension cyfor a known difference in the densities of liquids"

(3"~

c02(pl - p 2 ) r 3

4 This relationship is known as the Vonnegut expression [24].

47 Somewhat different from the other static methods, the plate balancing

method (also referred to as the Wilhelmyplate method) is commonly used to evaluate the surface tension at the gas-liquid interface. In this method a thin rectangular plate of width d, mounted on an arm of a sensitive recording balance, is immersed into the

liquid under investigation, resulting in the

formation of menisci on both sides (Fig. I-17). It is generally assumed that the liquid wets the plate well. The meniscus shape and the height of liquid rise are determined by the Laplace equation. The weight of liquid lifted by a plate

Fig. I-17. The force balance equilibrium in the Wilhelmy method

(per unit of the plate's perimeter) does not depend on the meniscus shape and at zero contact angle exactly equals the surface tension, cy. Therefore, the force that one needs to apply to balance the plate, F, is the product of the surface tension and the plate perimeter. The surface tension can then be estimated as o = F/2d, provided that the plate is sufficiently thin. No corrections associated with the meniscus shape are required in this method. It is, however,

48 difficult to get the thin edges of the plate smooth, so in reality the perimeter of the plate is a bit greater than double the width. To increase the accuracy of the measurements, the equivalent plate thickness is determined by calibration with liquids of known surface tension. It is also worth mentioning that for nonzero contact angles the surface tension is c~- F/2dcosO, i.e. the Wilhelmy method can also be used to measure contact angles. As compared to static methods, the semi-static methods for surface tension measurement are based on achieving a metastable equilibrium, and focused mainly on investigating the conditions under which the system loses that equilibrium. The threshold of the equilibrium state can generally be reached slowly, and thus the surface tension values obtained by semi-static methods closely resemble those obtained by static ones. The rate of approaching the equilibrium state should be optimized in each system, in order to avoid lengthy measurements and to obtain surface tension values as close to the equilibrium ones as possible. Among the most common semi-static methods are the method of maximum pressure, the du NoiSy ring method and the drop-weight method. The maximum pressure method establishes the maximum value of pressure required to squeeze a bubble (or a drop of another liquid) through the liquid phase [6,25]. When the outside pressure gradient, Ap, is applied across a calibrated capillary immersed into liquid, a gas bubble (or drop of liquid) starts to grow at the capillary tip (Fig. I- 18). As the bubble grows, its curvature radius, r, decreases and finally reaches a minimum value equal to the radius of the capillary, r 0. At this point the bubble surface acquires hemispherical

49

_

-?'W

-

.

_

_

.

.

.

_

Fig. I-18. Change of the curvature radius of the bubble surface that occurs during the determination of surface tension by the maximum pressure method shape. Further increase in the bubble volume results in an increase in the curvature radius ( r > r0 ). At r = r0, the capillary pressure, p~ = 2~/r then reaches its maximum value 2cy/r o. Consequently, at kp < 2~/ro the system is mechanically stable, while at kp > 2~s/r o the capillary pressure is unable to balance the applied pressure, Ap, resulting in rapid bubble growth, followed by its final detachment from the capillary tip. The latter is usually accompanied by a noticeable pressure drop, the registered maximum value of which is

@ m a x --

2cy/ro. From this expression it is evident that

APmaxis directly

related to cy, i.e.

1 cy - -- APmax r 0 . 2 If the capillary diameter is not very small, one has to correct for the non-spherical shape of the bubble in order to enhance the

accuracy of

measurement. Similarly to other methods, one often performs relative measurements, in which the results are compared with the data acquired for other liquids for which the exact values of ~ are known.

50 The force Frequired to detach a well-wetted thin ring of radius rr, from the liquid surface is measured in the du No~y ring detachment method [6,26]. Within the first approximation one can assume that the equation relating the surface tension, o, to the detachment force, F, is analogous to that used in the Wilhelmy plate method, with the exception that the perimeter of the ring is used in place of the plate width, i.e. F -

4grrO. In reality, however, the

curvature of the liquid surface at points of contact with a ring causes the surface tension vectors to be somewhat off the vertical (Fig. I-19). F

I//

9

---

--

6------

I

\\

o'---

' --

0

~-8--m

Fig. I-19. Measurement of the surface tension by the ring detachment method (du Noay) In addition to this, one also has to account for the capillary pressure acting at the ring surface and hampering ring detachment (similar to the attractive capillary force of the menisci). The appropriate correction is achieved by introducing a numerical coefficient into the expression for cy, i.e. F cy-~k, 4xr r

where k is a correction coefficient the value of which depends on the ring

51 geometry and can be evaluated with the help of tables containing the results of the Laplace equation integration. While the du Notiy method is commonly used for measurement of the surface tension at liquid-gas interfaces, it is little used to measure (y at liquid-liquid interfaces, since in the latter case it is difficult to achieve the 0=0 ~ condition. The semi-static method frequently used for the determination of the interfacial tension at the liquid-liquid interface is the drop weight method, based on determining the weight of a liquid drop detaching from a flat capillary tip (Fig. 1-20). Usually a known number of drops is collected, their weight measured, and the average weight of a single drop is estimated from these measurements. This method is also sometimes referred to as the drop volume method.

6 Fig. 1-20. Detachment of a drop from a capillary tip

A rather complex theory of the drop weight method, which makes it possible to tabulate the data required in order to determine the surface tension, has been worked out in some detail [27].

In the first (roughest)

approximation, it can be assumed that, at the moment of detachment the gravity force acting on a drop, P, is balanced by the surface tension forces,

52 equal to the surface tension times the capillary circle length, i.e., P = 2~r0o. However, the detachment of a drop is a more complex process. For example, the fluid "neck" between the drop and portion of liquid that remains attached is of smaller diameter than the capillary tip. Furthermore, when the drop detaches, one or more smaller droplets are usually formed. These factors are accounted for by introducing a correction coefficient k, the tabulated values of which are established from the exact theory of the drop weight method. Thus, the corrected weight of a drop is

p - 2~rocy/k. The use of highly accurate optical drop-counting devices increases the reliability and convenience of the drop weight method, making it a rather popular technique in the lab. Dynamic methods for the determination of surface tension are usually employed in specialized studies of the non-equilibrium states of fluid interfaces, and in the investigation of how fast equilibrium in such systems is reached. A classical example of such methods is the oscillating jet method, which allows one to study the interfacial properties at rather small time intervals. In this method the liquid is ejected from a capillary with an elliptical cross-section, forming a stream with the shape of an elliptical cylinder. The surface tension forces tend to change the shape of the stream into that of a cylinder. These forces acting along with forces of inertia cause the stream to oscillate in a transverse direction, which results in a continuous interchange between the positions of the smaller and larger axes of the ellipse. The theory developed by Rayleigh and later by

Bohr and Sutherland relates the

53 wavelength of the longitudinal stream profile, measured by optical methods, to the surface tension of the fluid. A comparison of the surface tension values obtained with the ones established from static or semi-static measurements allows one to draw conclusions regarding the rate at which the equilibrium surface structure is established, as well as to study the adsorption kinetics. The capillary wave method is based on the generation of harmonic waves on the surface of a bulk volume of liquid [28]. The wavelength of the ripples formed, )~, is a function of the surface tension, which can be evaluated from expressions given by Kelvin: Z,3 p (5"--

2~'c

2

gZ,2 p

4re

2

u 2 _ g ) ~ ~ 2~o 2~

p)~

where p is the density of the liquid, g is the acceleration of gravity, ~) is the velocity of wave propagation, and ~ is the period of the ripples. One can thus determine the surface tension by measuring the wave parameters, which can be done, e.g., by the analysis of the standing waves. Even in the case of standing waves, the solution surface undergoes alternating local expansion and contractions, which may be accompanied by local surface tension changes and the transport of materials between surface layers. The resulting damping is characterized by a damping coefficient, which is another parameter obtained by the capillary wave method [28-31]. The damping coefficient provides information on the exchange of matter and the dilational elasticity of the

54 adsorption layers (see Chapter II). Examples of other dynamic methods based on interfacial relaxation include the oscillating bubble and oscillating drop methods and their variations [32]. In the oscillating bubble method a small air bubble is formed at the tip of a capillary immersed into a solution. The bubble is then forced to undergo harmonic oscillations induced by an oscillating membrane either due to oscillations in a gas volume connected with the capillary or due to oscillations in the pool of solution induced by a piezoelectric driver. By measuring changes in the surface area of the oscillating bubble and the amplitude of the pressure oscillations, one can evaluate surface tension using the appropriate theory [32]. The oscillating drop method is, essentially, a variation of the pendant drop method [32,33]. In this method a system of two interconnected syringes is used, as shown in Fig. I-21. A drop with a definite volume is formed with the help of a precise syringe (syringe 1 in Fig. I-21). Due to the oscillatory motion of the second syringe (with a characteristic frequency), the drop undergoes periodic contraction and expansion. Video images of the oscillating drop are acquired over short time intervals throughout the experiment. The instantaneous values of the surface tension, surface area, and the drop volume are then obtained from the digitized video images using axisymmetric drop-shape analysis [21]. An interesting modification of this method suitable for measurement of the interfacial tension at liquid-liquid interfaces has been proposed by Hsu and Apfel [34]. In the modified method a drop of one liquid is acoustically levitated in another liquid. The drop is then forced to oscillate in the acoustic force field, and the interfacial tension is evaluated from the resonance frequency.

55 motor

syringe 2

1

drop

Fig. 1-21. Oscillating pendant drop [33] As we mentioned above, some methods that we classified as semistatic can be adapted for the measurement of dynamic surface tension, cy(t). These include the drop weight and the maximum bubble pressure methods. In the dynamic drop weight (volume) method the liquid is dosed through a capillary in such a way that a continuous formation of drops takes place. The surface tension is then calculated from the average volume measured for several subsequent drops. Adjusting the liquid dosing rate allows one to age the interface for different periods of time, so that one can carry out measurements at times from less than l s to 30 rain or longer, obtaining the interfacial tension as a function of drop formation time. The dynamic drop weight method allows one to monitor the kinetics of the adsorption of surfactants and proteins. The interpretation of data acquired by the dynamic drop weight method requires the use of rather cumbersome adsorption kinetics theories that take into account such factors as the changing drop surface area

56 during drop growth (the adsorption at a growing surface is slower than at a stationary one), as well as the flow inside the drop [ 18]. These effects are described in detail in [27]. The dynamic maximum pressure method gives one an opportunity to monitor interfacial tension as a function of time in intervals from 1-2 ms to several seconds [18,25,35-40]. The "dynamic regime" of the maximum pressure method is achieved by changing the bubble formation frequency. Rehbinder was the first to alter the bubble growth rate, and hence to change the frequency of bubble formation, in the studies on the surfactant adsorption kinetics [41]. Using a recent design of the maximum bubble pressure instrument described by Miller and co-workers [39,40] one can carry out measurements on a millisecond time frame.

The high resolution of this

method was accomplished by increasing the system volume relative to the detaching bubble volume, and by using electric and acoustic sensors for registering bubble formation frequency. Miller and co-workers also addressed the issue of hydrodynamic effects at short bubble formation times [37]. There are also other dynamic methods that we have not described here. Some of these methods are reviewed in [ 18,25,27,32,42]

The measurement of the surface free energy of solids is a considerably more difficult task than that of liquids. In solids it is usually impossible to reach a thermodynamically reversible increase in the interfacial area, partially due to the high amount of work required for plastic deformation. Nevertheless, a number of methods that allow one to measure (or at least to approximately evaluate) the surface free energy of solids have been developed.

57 For ductile solids, such as metals, the zero-creep method can be employed to measure the surface tension at temperatures close to the melting point. In this method the material of interest is cut into strips of width d, onto which weights of different magnitudes are mounted (Fig. 1-22) [43]. The samples prepared are kept in a thermostat at temperatures somewhat lower than the material's melting point. After a rather long period of time, the change in the strip length A1 is measured. Depending on the magnitude of the applied weight F, the strips either shrink or become elongated due to the action of surface tension. The elongation of the strips is usually a linear function of the applied

b,\\\\\\\\\\\\\\\\\\\\\\\\'q

! lI--ll

0

+AI Fig. 1-22. A schematic representation of the zero-creep method used to determine surface free energy of solids

/

~],

1

Fig. 1-23. Determination of surface tension by single crystal cleavage

force. A point in the AI(F) dependence where AI = 0 (a so-called zero-creep

point) characterizes the force balance between the applied weight and the surface tension acting along the perimeter of the strip. The exact treatment, which accounts for the change in the strip shape at constant volume, shows that an additional numerical coefficient of 89is required, so that the force, F, is

58 F = cyd .

Typical ~ values for various solids determined using the zero-creep method are summarized in Table 1.2. In case of brittle solid materials, especially single crystals with clearly defined layer structure (e.g. mica), it is possible to use the c l e a v a g e m e t h o d developed by Obreimow [44]. In this method the crystal is split along the cleavage plane (Fig. 1-23), and the force that has to be applied to cause further development of the crack, F c, is measured.

TABLE 1.2. The values of the surface free energy, ~, of solids, established by different methods [7] Solid Substance

t, ~

~, mJ/m 2

Method

Ag

909

1140

zero-creep

Au

1040

1350

zero-creep

Co

1350

1970

zero-creep

Cu

900

1750

zero-creep

Ni

1343

1820

zero-creep

Zn

380

830

zero-creep

Zn, (0001) plane

- 195

410

crystal cleavage

Naphthalene

20

60

crystal cleavage

Mica

20

480

crystal cleavage

The relationship between the force, F c , the surface tension, ~, (which in this case represents the work required to form a new interface), crack length, l, thickness, h, width, d, and Young's elasticity modulus, E, of the cleaved layer

59 is given by 6(Fc/) 2 Ed2h 3 Another method that can be used to determine the ~ of solids is based on the investigation of the dependence of solubility on particle size, and involves the use of the Thomson (Kelvin) equation. This method, however, has a significant limitation, owing to the fact that increased solubility of particles obtained by mechanical fragmentation is in part due to numerous defects in the crystal lattice, appearing due to mechanical action. References ~

0

9

~

,

,

~

0

Q

10. 11.

Gibbs, J.W., "The Collected Works of J.W. Gibbs",vol.1, Thermodynamics, Longmans, Green, New York, 1931 Rowlinson, J. S., Widom, B., Molecular Theory of Capillarity, Clarendon Press, Oxford, 1984 Rusanov, A.I., Phasegleichgewichte und Grenzflachenersheinungen, Akademic Verlag, Berlin, 1978 (in German) Bakker, G., in "Wien Harms' Handbuch der Experimental Physik", vol.6, Akademische Verlagsgesellschaft, Leipzig, 1928 (in German) Goodrich, F.C., in "Surface and Colloid Science", vol. 1, E. Matijevid (Editor), Wiley-Interscience, New York, 1969 Padday, J.F., in "Surface and Colloid Science", vol.1, E. Matijevid (Editor), Wiley-Interscience, New York, 1969 Schukin, E.D., Pertsov, A.V., Amelina, E.A., Colloid Chemistry, 2~d ed., Vysshaya Shkola, Moscow, 1992 (in Russian) Derjaguin, B.V., Churaev, N.V., Muller, V.M., Surface Forces, Consultants Bureau, New York, 1987 Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1992 Overbeek, J.Th.G., in "Colloid Science", vol. 1, H.R. Kruyt (Editor), Elsevier, Amsterdam, 1952 De Boer, J.H., and Custers, J.F.H., Z. Phys. Chem., B23 (1934) 225

60 12. 13. 14. 15. 16. 17.

18. 19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Fowkes, F.M., J. Phys. Chem., 66 (1966) 382 Fowkes, F.M., Ind. Eng. Chem., 12 (1964) 40 Gaydos, J., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Winslow, D.N., in "Surface and Colloid Science", vol.13, E. Matijevid, R.J. Good (Editors), Plenum Press, New York, 1984 Adamson, A.W., Gast, A.P., Physical Chemistry of Surfaces, 6th ed., Wiley, New York, 1997 Rusanov, A.I., Prokhorov, V.A., Interfacial Tensiometry, in "Studies in Interface Science", vol.3, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1996 Miller, R., Joos, P., Fainerman, V.B., Adv. Colloid Interface Sci., 49 (1994) 249 Anastasidis, S.H., Chen, J.K., Koberstein, J.T., Siegel, A.F., Sohn, J.E., Emerson, J.A., J. Colloid Interface Sci., 119 (1987) 55 Cheng, P., Li, D., Boruvka, L., Rotenberg, Y., Neumann, A.W., Colloids Surf., 43 (1990) 151 Chen, P., Kwok, D.Y., Prokop, R.M., del Rio, O.I., Susnar, S.S., and Neumann, A.W., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Passerone, A., Ricci, R., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Seifert, A.M., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Vonnegut, B., Rev. Sci. Inst., 13 (1942) 6 Fainerman, V.B., and Miller, R., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Lecomte du No~iy, P., J. Gen. Physiol., 1 (1919) 521 Miller, R., and Fainerman, V.B., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Adin Mann Jr., J., in "Surface and Colloid Science", vol.13, E. Matijevid, R.J. Good (Editors), Plenum Press, New York, 1984 Van den Tempel, M., van de Riet, R.P., J. Chem. Phys., 42(8) (1965) 2769 Lucassen-Reynders, E.H., J. Colloid Interface Sci., 42 (1973) 573 Lucassen-Reynders, E.H., Lucassen, J., Garrett, P.R., Giles, D., and Hollway, F., Adv. Chem. Ser., 144 (1975) 272

61 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

43. 44.

Wantke, K.D., and Fruhner, H., in "Studies in Interface Science", vol.6, D. MObius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Miller, R., Sedev, R., Schano, K.-H., Ng, C., and Neumann, A.W., Colloids Surf. A69 (1993) 209 Hsu, C., Apfel, R.E., J. Colloid Interface Sci., 107 (1985) 467 Kloubek, J., J. Colloid Interface Sci., 41 (1972) 7 Mysels, K.J., Langmuir, 5 (1989) 442 Fainerman, V.B., Makievski, A.V., Miller, R., Colloids Surf., A75 (1993) 229 Fainerman, V.B., Miller, R., and Joos, P., Colloid Polym. Sci., 272(6) (1994) 731 Miller, R., Joos, P., and Fainerman, V.B., Prog. Colloid Polym. Sci., 97 (1994) 188 Mischuk, N.A., Fainerman, V.B., Kovalchuk, V.I., Miller, R., Dukhin, S.S., Colloids Surf. A175 (2000) 207 Rehbinder, P.A., Z. Phys. Chem., 111 (1924) 447 Stebe, K.J., Ferri, J., Datwani, S., Abstracts of 73 ~dACS Colloid and Surface Science Symposium,Massachusets Institute of Technology, Cambridge, MA, 1999 Krotov, V.V., Rusanov, A.I., Physicochemical Hydrodynamics of Capillary Systems, Imperial College Press, London, 1999 Obreimow, I.V., Proc. Royal Soc. London, A127 (1930) 290

List of Symbols

Roman symbols All a

b al, bl aL C

Cs d e

E

Hamaker constant capillary constant distance comparable with molecular dimensions coefficients in Lennard-Jones potential coefficient characterizing dispersion interaction concentration heat capacity excess distance elementary charge modulus of elasticity

62 F force F~, F 2 forces acting on the frame in Dupr6's experiment ~free energy surface free energy free energy density f g acceleration of gravity isobaric-isothermal potential h Plank's constant h thickness of gap between volumes of condensed phase H height of fluid rise in a capillary 5genthalpy, heat of sublimation 5U internal pressure k correction coefficient in drop weight method kl numerical coefficient k Boltzmann constant Al displacement of wire in Dupr6's experiment m, n powers of R in the expression for interaction potential Avogadro's number NA number of moles N n number of molecules per unit volume number of molecules (atoms) per unit area g/s P gravity force (weight force) theoretical strength of ideal crystal Pid pressure P capillary pressure P~ tangential pressure PT R universal gas constant R equilibrium distance R, q0, z cylindrical coordinates distance between two volume elements of a condensed phase R12 r radius of curvature radius of a capillary ro rl, F2 principal curvature radii radius of the du No~y ring rr surface area S absolute temperature T critical point temperature :rc temperature t

63 internal energy interaction energy per unit area interaction energy between neighboring molecules /211 volume V Vm molar volume volume per molecule D velocity of wave propagation We work of cohesion x, y, z Cartesian coordinates Z, Zs coordination number U Umol

Greek symbols

~M

15, 15' q 0

~d V VO

P (~gb ~5n ~SLV (~SV (~SL T CO

empirical constant in eq. (I.5) polarizability small distance specific excess of internal energy specific excess of entropy contact angle wavelength chemical potential dipole moment frequency of radiation oscillation frequency 3.14159... density specific surface free energy; surface tension specific surface energy of grain boundary dispersion component of the specific surface free energy non-dispersion component of the specific surface free energy specific surface energy at liquid/vapor interface specific surface energy at solid/vapor interface specific surface energy at solid/liquid interface period of ripples at the surface of liquid angular speed of revolution

64 II. THE ADSORPTION PHENOMENA. STRUCTURE AND PROPERTIES OF ADSORPTION LAYERS AT THE LIQUID-GAS INTERFACE

A distinctive force field present at the interface may cause changes in the composition of the near-surface layer: different substances, depending on their nature, may either concentrate near the surface, or, alternatively, move into the bulk. This phenomenon, referred to as the adsorption, causes changes in the properties of interfaces, including changes in the interfacial (surface) tension. In disperse systems with liquid dispersion medium, adsorption layers present at the surfaces of dispersed particles may significantly influence the interactions between these particles and hence affect the properties of disperse system as a whole, including its stability. For this reason the investigation of the laws governing the formation, structure and properties of the adsorption layers at different interfaces is of extreme importance, as it allows one to analyze the role such layers play in controlling colloid stability and other properties of disperse systems. The thermodynamics gives a unified description of adsorption at a variety of interfaces of different nature. In contrast to that, some quantitative trends in the adsorption, as well as the methods that one may choose to study the adsorption layers, are very specific to the nature of contacting phases and to the structure of adsorbing molecules. Throughout this chapter, after a brief introduction into the thermodynamics of adsorption phenomena, we will focus on the formation and structure of adsorption layers at liquid-gas interfaces, leaving the discussion of adsorption at interfaces between condensed phases until Chapter III. Among the adsorption phenomena, those taking place at the solid-gas

65 interfaces are peculiar ones. On the one hand, these processes are very well studied with respect to the nature of intermolecular interactions taking place in adsorption layers, while on the other hand the adsorption layers at the solidgas interface can not radically influence interactions between particles, and hence are unable to significantly affect the stability of disperse system with gaseous dispersion medium.

II.1. Principles of Adsorption Thermodynamics. The Gibbs Equation In a two-phase system consisting of two or more components the composition of the discontinuity surface (see Chapter I) may significantly differ from that of a bulk of both phases in contact. Primarily the components that lower the system's free surface energy are expected to accumulate within the discontinuity surface; this spontaneous concentration of substances is referred to as adsorption. The quantitative measure of the adsorption of the ith component, FI, was introduced by Gibbs, and is also referred to as the

adsorption, or the surface excess of the amount of substance. This measure has a meaning of the molar excess of a particular component per unit interfacial area: Fi -

N i - N i - N/" S

where N~ is the total number of moles of the i-th component in the system; N, and N/' are the number of moles of the same component in the bulk of each of the contacting phases if it is implied that the substance concentration is constant at all locations within the phases, up to the geometrical dividing surface of area S.

66 Let us consider a model two-phase two-component system consisting of a solution of hexyl alcohol (component 2) in water (component 1) at equilibrium with their own vapors. A schematic change in the concentration of water c~(z) and that of hexanol c:(z) across the discontinuity surface is shown in Fig. II-1. In the regions below and above the discontinuity surface the concentrations of both components are constant and equal c~' and c 2' in the tl

!!

liquid phase, and c 1 and c 2 in the vapor phase, respectively. Furthermore, due I

II

I

II

to low vapor densities c~ >~c~ and c 2 )) c 2

.

I

z ~__

4

...._

~--~

CI

C2U

,

~__

!

~

,~

I

k

-b'

li-F--

,

-

__

~ C2 t --.---~---.--

Fig. II-1. Changes in the component concentrations within the discontinuity surface Within the discontinuity I

surface the concentration

of water

II

monotonously decreases from c~ to c~, which are the concentrations in the liquid and gas phases respectively, while the hexyl alcohol behaves differently" its concentration increases and substantially exceeds both c 2' and II

C2

9

67 To examine the relationship between the adsorption of a second component and the distribution of the latter within the discontinuity surface, let us draw a prism of cross-sectional area S in the direction perpendicular to the discontinuity surface (Fig. II-1). Let us then compare the amount of substance accumulated within such a prism in the real system and in an idealized one, for which in the z - 0 plane the concentration increase from c2' to c2" has the form of a step function. The adsorption of component 2 (i.e. hexanol) can be estimated as follows: 0

+8"

F 2 - f[c2(z ) - c ; ] d z + -8'

[ [c2(z ) - c2" ]dz,

(II.1)

~3

where - 6' and + 6" are the coordinates of the discontinuity surface, which has a thickness equal to 8 - ~ ' + ~". Geometrically, the adsorption, F2, is represented by a shaded area (Fig. II-1) between the c2 = c2' and c2 - c2" lines and the c2(z) curve. The adsorption of component 1 (i.e. water) can be determined in a similar way. The concentration of water

Cl'(Z) within the part of the

discontinuity surface adjacent to the liquid phase (for which - 6' < z < 0) is smaller than the bulk concentration c( and hence the corresponding integral is negative, as marked in Fig. II-1. The adsorption of water is geometrically equal to the difference in the shaded areas (Fig. II-1), given by the positive term written for the part of discontinuity surface adjacent to the vapor phase: ~"

I 0

> 0,

68 and the negative term for the part of discontinuity surface on the liquid side, respectively: 0

_~'

It is now evident that depending on the choice of the dividing surface position, the adsorption of component 1 can be either positive or negative (corresponding to a deficiency

of a component within the discontinuity

surface), or zero (note that the surface free energy is independent of the dividing surface position, see Chapter I, 1). The dividing surface, the position of which is chosen in such a way that F~ = 0, is referred to as the

equimolecular surface with respect to component 1 (i.e. the solvent). Let us now turn to a more detailed description of the adsorption of component 2 where its concentration within the interfacial layer is significantly higher than that in the bulk. Let us also assume that this component is non-volatile, i.e. that

c2" ~0. To make things simpler, it is

possible to chose such a position of the dividing surface that the second integral in eq. (II. 1) is negligible compared to the first one, and thus the entire physical discontinuity surface is located below the geometrical dividing surface~. The adsorption is then given by

Strictly speaking, such position of the dividing surface differs from that of the equimolecular one (with respect to solvent), however, the difference between the two is too small to cause any significant influence on the results of the present treatment

69 0

c;]dz

_~'

Using the definition of the integral average, the above equation can be written as

F 2 - ( c ~ s) -c2')~ ,

(II.2)

where c~s) is the average concentration of component 2 within the interfacial layer of effective thickness 8. Graphically, the above procedure is equivalent to replacing the "tongue" between the c2(z) curve and the c2' line with a rectangle of equal area, with sides equal to ( c~s)-

c2') and 6 (Fig. II-2).

(s)

C2

|

C2

-5

!

c2 f

Fig. II-2. The evaluation of the adsorption, F, and the surface concentration, c(s)

70 The effective thickness of the adsorption layer differs from that of a surface layer (physical surface of discontinuity) determined from changes in the other parameters, such as the free energy density (see Chapter I, 1). The adsorption, F 2, can, therefore, be viewed as the excess of substance per unit interfacial area within the interfacial layer, as compared to the amount of the same substance within the layer of equivalent thickness located in the bulk. If the substance has a strong tendency to adsorb and its bulk concentration is /

small, then c~s) >>c 2 , and hence F 2 ~ c~s)~5,

(II.3)

i.e., the adsorption approximately equals the amount of substance per unit area within the interfacial layer. This equation, obviously, remains valid when component 2, in addition to being non-volatile, is also insoluble in a liquid ll

I

phase, i.e. when c2 ~ 0 and c2 ~ 0. Under these conditions component 2 is completely concentrated within the surface layer (see Chapter II, 2). The expression (II.3) allows one to calculate the approximate maximum value of substance adsorption, e. g. hexyl alcohol in the present case. If we assume that the thickness of the closely packed adsorbed layer is close to the length of the hexanol molecule (~0.7 nm), and that the alcohol concentration, c~~, is close to its concentration in the liquid phase (~8 kmol m-3), then the adsorption, F, is ---0.6 x 10.5 mol m -2. The relationship between the adsorption (the excess) of a substance and its concentration within the interfacial layer, established by eqs. (II. 1) and (II.3), allows for a better evaluation of the properties ofmonomolecular layers

71 by comparing them with macroscopic phases. The treatment of interfacial layers as individual phases (to which the laws of regular three-dimensional thermodynamics are applied) is the basic concept behind the thermodynamics of finite-thickness layers, developed in the work

of van der Waals,

Guggenheim, and Rusanov [1,2]. The differences in the composition of bulk phases and interfacial layers in multi-component systems result in the re-distribution of the components of individual phases between the bulk volumes and the surface layers when changes in the interfacial area occur. Because of this the increase in the latter requires that chemical work be performed in addition to the mechanical work, o. Both of the terms that constitute the work required to form an interface can be accounted for by introducing the quantity ~, defined as - (Y + 2

~iFi

(II.4)

i

According to the Gibbs phase rule, at constant temperature and volume the binary two-phase model system has only one degree of freedom, meaning that only one variable in eq. (II.4) is independent. It is possible then to replace partial derivatives by full ones. The treatment is simpler for a surface that is equimolecular with respect to solvent, for which F~ = 0; the individual subscripts are no longer needed and can be omitted (i.e., g = g2 and F = F2), and eq. (II.4) becomes ~ - cy+gF. According to Rehbinder, we choose the chemical potential of solute g as an independent variable. The differentiation of the above equation with

72 respect to bt yields d~ d~ dF ~=~+bt~+F, dbt dbt dbt the left-hand side of which can be written as d~F

d/a

d~F dF dF

d/a

The quantity d~/dF, describing the change in the surface free energy with the increase in adsorption, is, by definition, equal to the chemical potential, bt, and hence dF bt~= dbt

do ~+F+ dbt

dF bt~ dbt

Consequently, F = - do / dbt, or do = - F dbt. The above thermodynamic relationship, describing adsorption in a two-component system, was first derived by Gibbs and is known as the Gibbs

equation [3]. It follows from the Gibbs equation that the excess of component within the interfacial layer determines how abrupt the decrease in the surface tension is with correspondingly increasing chemical potential of the adsorbed substance. The Gibbs equation reflects the equilibrium conditions at constant pressure and temperature between the surface layer and the bulk, i.e. the conditions corresponding the system's free energy minimum. The latter becomes more evident if the equation is written in its variational form, i.e." 8 ~ - 8c~ + F d p - 0.

73 It is thus possible to say that at a given value of adsorption, F, the balance between "mechanical" forces, 6G, and "chemical" forces, Fdg, corresponds to the minimum of the system free energy per unit interfacial area. In the other words, there is a balance between the tendency of a system to decrease its surface energy by concentrating some of the species within the surface layer on the one hand, and the disadvantage of such accumulation due to the increase in chemical potential on the other hand. It was shown by Gibbs that for multi-component systems the fundamental adsorption equation can be written as "

17

d~ - - 2

Fidgi '

i=2

where the summation is carried out over all components with the exception of component 1 (the solvent). When the system is at thermodynamic equilibrium, the chemical potential of any component ( including the adsorbed one) is the same in all phases in contact, as well as within the interfacial layer. If g is the chemical potential of solute in the bulk, one can write dg = RTdln(czc), where a is the activity coefficient, and c is the solution concentration. If the solution studied is not too different from the ideal one, the activity coefficient, a -- 1, and the Gibbs equation for two-component system is

74 written

as 2

c d~ r = - ~ ~ . RT dc

(II.5)

It is known that for solutions containing molecular species the condition of ~ - 1 is valid for concentrations up to - 0.1 mol dm -3, and thus the use of the simplified Gibbs equation (II.5) is justified for sufficiently dilute solutions only. On the contrary, the magnitude of substance concentration in the interfacial layer, c~s) - c (s), does not impose any restrictions on the use of eq. (II.5). If the adsorption, F, is expressed in terms of the surface concentration, c (~), and the thickness of adsorption layer, 8, given by eq. (II.2), the Gibbs equation can be written in the following form: c (s) - c

des = R 7 " 6 ~ . dc c

(II.6)

Experimental studies on the surface tension of various solutions showed that the latter can both increase and decrease with increasing solution concentration, depending on the nature of solvents and solutes. Different solutes affect, however, the surface tension of the solvent, %, in different ways" some solutes, when present at extremely small concentrations, can cause a significant decrease in the surface tension, while the others can only insignificantly increase it (Fig. II-3).

2 If the adsorbing species are of the ionic nature, a numerical coefficient may be introduced into the Gibbs equation (II.5). For example, this coefficient equals 1/2,if the substances dissociate into ions of the two types

[4]

75 (J O0 d(~

--- > 1/~:

As we have seen, the structure of the diffuse part of the EDL is determined by the ratio of the potential energy of the electrostatic attraction between counter-ions and the charged surface to the thermal energy of ions. This ratio is given by a dimensionless function, ze%/4kT (or zeq~d/4kT). When the potential energy of interaction between ions and the charged interface is small (zeq)o/4kT < 1), the potential decays exponentially with increasing distance, and its value at any reference point in the diffuse layer is proportional to the surface potential, %. Conversely, if the potential energy of attraction between the ions and the interface exceeds the kinetic energy of their Brownian motion (ze% / 4kT> 1), the surface charge is majorly compensated in the direct vicinity of the charged surface, i.e., the counter-ions present at short distances from the surface effectively screen its cbrge. It is important for one to remember that if the surface potential is high, at short separation distances eq. (III.17) should be replaced by the more accurate eq. (III.18), which takes into account the structure of the dense portion of counter-ion layer, as well as the individual size of the counter-ions. It can be verified that the asymptotic eq. (III. 15) can be readily obtained by

212 extending the integration limit of the Poisson-Boltzmann equation to the interface, i.e. to x - 0. This means that the centers of the ions can be located directly at the interface. The latter has no significant effect on the distribution of potential at large distances from the surface, especially in the situations

ze%~/ 4kT >

when the adsorption layer potential, q~d, is sufficiently high and

> 1. In some cases these distant diffuse layers of counter-ions are the ones that determine colloid stability (see Chapters VII, VIII). The presence of a diffuse layer with elevated concentration of counterions and lowered concentration of co-ions in the vicinity of an interface gives rise to many electric and filtration phenomena taking place in disperse systems. It is of significance that the diffuse layer has an increased total concentration of current carriers (Fig. III-13). For the simplest case of a symmetric electrolyte in agreement with eq. (III.8) one can write that

n

-4-

+n

- no

exp

-

+ exp

kT

=

kT

(III.20)

=2n~176 "kT1 The above treatment is valid for flat interfacial double layer. In disperse systems the EDL can be treated as flat when the size of dispersed particles is substantially larger than the thickness of ion atmosphere. If this is not true, one has to write the Poisson-Boltzmann equation in its complete form, namely: d2____~_~+ d2q)4 d2q) _ dx2 dY 2 dz2 _

~

~

m

.

9v_2zenosinhlZe~p(x)1 ~

~:gO

m

~

~3~;0

,

kT

This equation cannot be solved in quadratures even for the simplest model of

213 spherical or cylindrical particles. The results of the numerical integration of this equation are available for different geometries and cover a broad range of surface potentials and values of ion atmosphere thickness. P.Debye and E. Htickel offered an approximate solution of the above equation for the system consisting of weakly charged spherical particles of radius r, when zeq)o / k T < 1, and sinh (ze% / kT) ~zeq~o / kT. The Poisson-Boltzmann equation, written in spherical coordinates, in this case appears as:

e(R)

R 2 dR

where R is the distance from the center of a particle. The solution of this equation reads r

q~ (R) - q~o ~ - e x p [ - K (R - r ) ] .

This result reflects both the common decay of potential as the distance from the center of a charged sphere (r/R term) increases, and a more rapid exponential decay due to the presence of the diffuse layer (exponential term). Consequently, the potential of a sphere surrounded by diffuse layer decays with distance faster than the potential near the particle in a dielectric medium, or the potential in the vicinity of a flat interface with a diffuse layer. One can say that the distant regions with low potentials are mostly "proliferated" around the charged particle, while those with high potentials occupy a small volume in direct vicinity of the particle surface. For potentials at large distances from the surface of strongly charged particles one can use an expression similar to eq. (III. 19)"

q~(R) -

4kT r ze

exp [ - ~ : ( R - r ) ] .

R

In later sections (in particular in those devoted to colloid stability) we will limit our discussion by considering the flat double layers only.

214 III.3.2. Ion Exchange

Changes in the electrolyte composition of the dispersion medium electrolyte cause some particular changes in the structure of the electrical double layer (EDL), and are followed by ion exchange, during which some of the newly introduced ions enter the double layer, while some of the ions previously located in the EDL return to the solution bulk [19-20]. The nature of the changes to the EDL is determined by the ability of the introduced coions and counter-ions to enter the solid phase, their tendency to become specifically adsorbed at the interface, and the ratio of their charge to that of the ions forming the EDL (the latter is mostly related to counter-ions). One can identify two extreme cases: the indifferent electrolytes, which do not affect the surface potential, r

and

non-indifferent

electrolytes, which are capable of

changing %. The electrolytes of the latter type usually contain ions that are able to enter the crystalline lattice of solids, for instance through isomorphic substitution with ions forming the lattice. The ion exchange process in solutions of indifferent electrolytes can be described in the most general way by the Nikolsky equation, which for rather concentrated (non-ideal) solutions can be written as

c~/z~ 1/z2 C2

a ll/zl -

-

k12

1/z 2 " a2

In this equation a, c, and z are the activities, concentrations, and charges of ions of type 1 and 2, respectively. The ion exchange constant, kl2, is related to the corresponding adsorption potentials, ~ and q)2, by

215

k12-exp((I)l - cI)2./ " kT Depending on the nature of the introduced electrolyte, the ion exchange can affect different regions of the EDL: the diffuse and adsorption regions, and even the layer of potential-determining ions (in which case it is, however, more appropriate for one to talk about the build-up of the crystal lattice of the solid phase with the constituent ions of introduced electrolyte). The diffuse layers of counter-ions are the ones that undergo exchange most easily. Disperse systems consisting of positively charged particles or macromolecules that are surrounded by diffuse layers consisting of exchangeable anions, are referred to as anionites, while systems consisting of negatively charged particles (macromolecules) that are capable of exchanging cations, are referred to as cationites [21 ]. In finely dispersed systems changes in the ionic content of layers containing potential determining ions or counterions may cause a significant change in the composition of the colloidal particles. For example, a particle with a diameter d ~ 10 nm contains (d/d) 3 (30) 3 ~ 3 x 104 ions (assuming that the average ionic diameter d; = 0.3 nm). Out of that many ions

4%dZ/%di 2 ~

4x 103 ions (more than 10%) are located at

the surface, i.e., changes in the ion content of the surface layer may affect a significant portion of the matter making up the particle. The ability of disperse systems to participate in ion exchange is characterized by the exchange capacity, equal to the number of gramequivalents of ions taken up by one kilogram of a substance. Since the ion exchange ability is strongly dependent on the pH, concentrations and

216 composition of the medium, the exchange capacity is usually determined under certain standard conditions, i.e., one uses the conditional ion exchange capacity. For instance, in soil science the exchange capacity is usually determined at pH 6.5, using Ba 2+ as exchangeable ions at an electrolyte concentration of 0.1 N (usually BaC12.which is normally not present in soil). Ion exchange processes play an important role in nature and technology. For example, clay m i n e r a l s reveal a strong ion exchange ability. These minerals are alumosilicates with a lamellar structure (the interlayer distance is -~

nm). The potential determining ions in these

materials are silicic acid surface groups, and the cations play the role of exchangeable counter-ions. Depending on medium composition, the counterions may be Na +(Na-clays), Ca 2+or others. Ion exchange in clays plays an important role in the formation of the so-called s e c o n d a r y ore depo sits: hydrothermal waters containing ions of heavy metals enter the strata rich in clay minerals, where they undergo ion exchange, leaving the heavy metal ions behind, and washing the light ones out. The influence of the adsorption activity of ions on their geochemical fate can be clearly followed by looking at the distribution of potassium and sodium in nature. These elements have approximately equal abundance in the Earth's crust (2.4 and 2.35%, respectively), while sea water contains mostly sodium (there are about 10.8 g of sodium and only 0.4 g of potassium in 1 kg of sea water). Ion exchange taking place in clay deposits at the sea bottom is the reason for such an enrichment: sodium originally present in clays becomes nearly completely displaced by potassium. The ability of soils to actively participate in ion exchange determines

217 their functioning and fertility [22]. Soils are complex disperse systems containing finely dispersed insoluble polysilicic acids and clays and mineral organic substances formed due to the decomposition of organic matter (the so-called Gedroiz soil absorption complex) [22]. Soil composition, productivity and ability to participate in ion exchange are to a large extent dependent on the climate. The weathering of rocks leads to the formation of various clay minerals with ion exchange capacities up to 1 mol kg -1. In regions with high humidity and low content of organic matter (insufficient heat), the erosion of basic oxides (those of alkali and earth alkali elements), and humic acids, as well as the peptization oftrivalent metals (due to weak binding by organic substances) take place. These phenomena lead to soils impoverished in organic substances and valuable ions and containing increased amounts ofpolysilicic acids. Such coils are, consequently, enriched with clays in which metal cations are replaced with hydrogen ions. All of these factors cause soils (especially podsols) to be acidic and have poor productivity. The exchange capacity of podsols falls within a range of 0.05 - 0.2 mol kg -~. Chernozems are formed in regions with a moderate amount of precipitation and sufficient amount of heat. These soils contain a significant amount of organic matter, most of which is present in the form of poorly soluble humates of divalent metals (as calcium and magnesium salts ofhumic acids). Colloidal particles of humates may undergo heterocoagulation with alumosilicates and silicates (see Chapter VIII), forming finely dispersed highly porous structures with exchange capacities reaching 0.6 - 0.8 mol kg -1. These structures are rich in valuable cations and various nutritious substances. They

218 are able to entrap water due to capillary forces and at the same time are airpermeable. Air permeability is of extreme importance for the life of various microorganisms that improve the structure and productivity of soil. Peat soils, having a content of organic matter comparable to that in chernozems, are usually formed in regions with high humidity, which results in the erosion of cations and their replacement with hydrogen ions. Peat soils are thus acidic. The acidic nature of these soils makes difficult the development of plants that would be able to release the hydrogen ions during growth. Binding of these hydrogen ions released by plants (primarily due to ion exchange) is one of the main functions of a productive soil. The use of peat as an organic fertilizer in acidic soils is practical only if simultaneous exchange of protons with other more valuable ions takes place. The latter is achieved by the addition of either calcium carbonate, which causes the replacement of H § ions with Ca 2- ions, or ammonia aqueous solution, which at the same time plays the role of a valuable fertilizer. Ion exchange processes have enormous importance in various technological applications. Softening and de-ionization of water are the two characteristic examples of processes based on ion exchange [19,23]. Water

softening, or the exchange of Ca 2+ ions with Na § can be carried out using highly porous zeolite-type alumosilicate minerals of the general formula A1203.mSiO2.nH20 [24]. In these materials part of the H § ions can be replaced by metal ions. Both natural and synthetic (permutite) minerals are used. Schematically representing a single exchanging group of the Na form of permutite, Na20"A1203"3SiO2"2H20, as NaP, one can write the ion exchange reaction as

219 2NAP+ Ca 2§ Ca(P)2 + 2Na § . Subsequent treatment of the calcium form of permutate with concentrated NaC1 solution results in the regeneration of its sodium form. Another important practical application of ion exchange is the complete removal of ions from water, widely utilized for the preparation of deionized water, and conversion of sea water to fresh water (i.e. desalination). Highly effective ion exchange resins with exchange capacities reaching 10 mol kg -~ are used for ion removal. Ion exchange resins consist of crosslinked polyelectrolytes that form a three-dimensional network [21 ]. Such structure provides ion exchange granules and membranes with high mechanical strength. In aqueous media the resins swell, allowing all ionic groups within the granules to be available for exchange with the dissolved ions. C a t i o n i c resins usually contain sulfonic groups,-SO3, carboxylic _

groups,-COO , or phenolic groups,

C6H40 ,

the exchange capacity of which

increases with increasing pH. The interaction of the resin H-form with an electrolyte solution results in the exchange of electrolyte cations with H§ ions until a certain pH, determined by the strength of the ionic group, is reached. Cationic resins can be regenerated (i.e., converted back into the H-form) by treatment with acid. Ani o ni c resins contain various aminogroups (-NH3; =NH2; =NH) or quaternary substituted ammonium. The exchange capacity of anionic resins increases as the pH is lowered. These resins allow one to remove dissolved _

anions by ion exchange with OH ions. Anionic resins can be regenerated by treatment with alkalis. Apparently, de-ionized water is produced by the

220 sequential ion exchange of water on cationic and anionic resins. In some cases amphoteric ion exchangers (e.g. activated carbon) are used. According to Frumkin, when activated carbon is saturated with hydrogen, it acts as a cationic ion exchanger, but if saturated with oxygen, it turns into an anionic one. The removal of heavy metals from wastewater is another area in which ion exchange resins are used [21]. The ion exchange method allows one to remove such metals as copper, silver, chromium, and radioactive substances. Ion exchange methods of hydrometallurgy in combination with the use of microorganisms capable of converting the heavy metals present in poor ores into soluble compounds constitute a promising direction in the development of mineral and ore processing.

111.3.3. Electrocapillary Phenomena Information regarding the structure of EDL and the nature of some colloidal phenomena resulting from the interactions between ions and the interface can be obtained from the studies of

electrocapillary phenomena,

focusing at how the interfacial charge influences the surface tension. A complete

description

of electrocapillarity

is

given

in

courses

in

electrochemistry. Here we will only discuss the basic laws governing these phenomena that are important for understanding such colloidal phenomena as the adsorption of anionic and cationic surfactants, nucleation (see Chapter IV, 1), and the Rehbinder effect at charged surfaces (see Chapter IX, 4) The repulsion between charges of the same sign in the interfacial double layer should make an increase in the surface area easier, i.e., it should

221 decrease the interfacial tension u. It is well known from electrostatics that the work Wq required to supply a charge q to a spherical surface of radius r at a potential difference of cp=q/4xeeo r is given by 2

Wq

-

q

87~ggor

=2=georq)

2

One may expect that the specific (per unit area) work of charging is exactly the value of the work "already accumulated" by the interface that is needed to ease the increase in the interfacial area. In other words, the specific work of charging is equal to the potential energy lowering: Wq G O - o((p) - 4~r2

q2

ego(p2

32r~2~;~or3

2r

Differentiation of the above equation with respect to q~yields the L i p p m ann e q u a t i o n , which is the main relationship of electrocapillarity: dcy d(p

~oq~ r

q = p~, 4=r 2

(III.21)

where P~,.is the surface charge density. The investigation of the effect made by the applied potential difference on the interfacial tension can be most conveniently carried out on the ideally polarizable surface of liquid metal (most commonly mercury) in aqueous electrolyte solution. It is important that in these experiments one be able to simultaneously measure the potential difference between phases (with respect to some standard electrolyte) and the interfacial tension. The latter is usually

222 done by measuring the highest level reached by mercury, which is retained in the capillary by the surface tension. At the same time one can also determine the double layer charge density from the current carried by the mercury drops of known area. In agreement with the Lippmann equation, in the absence of surfactants the curve showing the surface tension as a function of the potential difference between phases (the electrocapillary curve) contains a maximum at some particular value of q~ (Fig. III-17). This potential, which corresponds to the

,,,rf,ct,,,t / X ,,,rf,ct,,,t

Fig. III-17. The shift in point of zero cha~'geposition due to adsorbed ionic surfactants maximum in the electrocapillary curve (i.e., to ps=0), is referred to as thepoint

of zero charge. The position of the point of zero charge is determined by the adsorption activity of ions present in solution and by the dipole moment of solvent molecules. In the absence of an externally applied potential the prevailing adsorption of Hg 2+ ions occurs at the mercury surface. These ions, present in the solution that is at equilibrium with the mercury, cause the

223 surface to become positively charged. To balance this charge one has to apply a negative potential, q~A has the same shape as the function showing the change in the potential as a function of distance (see Figs. V-7 and V-8). It is worth remembering here that an increase in electrolyte concentration results in compression of the diffuse counter-ion atmosphere, and the greater "portion" of a decay is attributed to the immobilized layer of the dispersion medium, i.e. at x_>1, using eqs. (V.26) and (V.29) one obtains

~,v - )~0 + 4~

8280rK~ 1"1

n.

Experimental studies by Dukhin et al [14] showed that the specific electric conductance of disperse system depends on the frequency of applied field. These findings can be explained by changes in polarization effects at high frequencies. The presence of dispersed particles may significantly affect the value of d i e l e c t r i c c o n s t a n t of disperse system. In some cases, e.g. in nonaggregated (non-flocculated) inverse emulsions (Chapter VIII,3), the dielectric constant is related to the volume fraction of droplets in the emulsion, V~e~, by the Bruggerman relationship

371

~;v = (1 _ Vrel)3

'

where e is the dielectric constant of dispersion medium. Dukhin has shown that the flocculation (aggregation) of emulsion droplets results in an increase in the dielectric constant to values determined by the volume fraction of flocs as a whole, i.e. together with incorporated dispersion medium. In aqueous systems in which particles are surrounded by a welldeveloped double layer, such as in sols and emulsions, sharp increase in dielectric constant is observed at particular frequencies of external field. The observed unusually high values of dielectric constants typical for such systems (Fig. V- 12) are due to the fact that particles move relatively to the surrounding ionic atmosphere as charges of high magnitude. At high frequencies of external field such motion becomes impossible, and dielectric constant assumes its "normal" values. The studies of such trends in dielectric constant are in the basis of dielectric spectroscopy, which is an effective method for investigation of disperse systems, and in particular of emulsions [ 15].

2000

1000 0

I

0.1

_

I

|

1

10

.,

I

100 ~, kHz

Fig. V-12. Dielectric constant of sols and emulsions as a function of frequency of the outer electric field

372 Now we would like to briefly describe other transfer processes that may occur in free disperse systems. It was already stated at the beginning of this chapter that directed motion of particles may be caused by the action of forces other than those originated from the applied electric field. For example, the existing temperature gradient results in the motion of colloidal particles referred to as the

thermophoresis. In aerosols thermophoresis occurs due to a

higher average momentum of molecules striking the particle on a warmer side as compared to that of those hitting it on a cooler side. The net effect is that the particles translate towards the region containing cooler air. This phenomenon explains the deposition of dust on walls near the cold air outlets. Another phenomenon that may have the same nature is photophoresis: the particles may move due to the action of luminous flux which heats up their surface. This process is different from the one taking place in outer space, where the motion of interstellar dust particles can be caused by a direct action of the light pressure. The gradient in the concentration of substance dissolved in dispersion medium may lead to

diffusiophoresis of dispersed colloidal particles. The

theory of diffusiophoresis was developed by B.V. Derjaguin and his collaborators [ 16]. According to the concepts discussed in their studies, there are two major causes for diffusiophoresis. First, the presence of diffusion adsorption layer (containing ions and uncharged molecules) in a vicinity of the surface and the existence of external concentration gradient of solute result is a complex osmotic pressure distribution near the surface, which causes particle motion. In electrolyte solutions the particle velocity due to diffusiophoresis is proportional to the square of ~-potential. Second, changes

373 in the structure of the electric double layer along the particle surface (EDL polarization) result in the generation of a potential difference, Aqo. In this case the rate of diffusiophoresis is proportional to the first power of (-potential. Diffusiophoresis plays a role in life of microorganisms, allowing them to move towards the sources of substances that are vital for their existence.

V.5. Transfer Processes in Structured Disperse Systems (in Porous Diaphragms and Membranes)

In structured disperse systems, where particles of the dispersed phase form united spatial networks, as well as in porous media with open porosity, the existence of double layers at

interfacial boundaries results in some

peculiarities in the processes of substance transfer and electric current transport. We will devote most of our attention to the discussion of transfer phenomena in an individual capillary, which is the simplest element of any structured disperse system, and then only qualitatively address the peculiarities related to complex structure of porous medium. During filtration the laminar flow of dispersion medium with viscosity 11through a capillary of radius r and length l under the pressure gradient, Ap, is described by the P o i s e u i 11e e q u at i o n: gr 4 Ap

Op- 8n 1 where

Qpis a volume of liquid passing through capillary per unit time; Ap/l

is the pressure gradient in capillary. Under these conditions the flow profile,

374 i.e. the cross-sectional distribution of fluid velocities, is parabolic, as shown in Fig. V-13, a. ~NN\\\\\\\\\\\\\\\\\\\

a

2

~\\\\\\X\\\\\\\\\~,

,\xz

~\NN\\\\\\\\\\\\4\\\\\

b

_

~

\

1/4

\

~

\

\

\

\

1uo ~ \ N N N \ \ N N \ X

"k \ \ \ " '4 . ~ \ \ - < / \_\ \ \ \ \ \ \ \

\

C

Fig. V- 13. The fluid velocity distribution profile in a capillary: a - during filtration, b - during electroosmotic transfer, c - during electroosmotic rise Another example of a direct transfer process is the generation of electric current, ID between two ends of the capillary under the applied potential difference, A~. In this case the strength of the outer electric field in the capillary, E = -grad 9 =

AU?/l, while the magnitude of current, I E, is

determined by the capillary cross-sectional area, g r 2, and by the average electric conductivity of the medium in it, X" At high electrolyte concentration and rather large radius of a capillary, when

I E - g r 2 ~ AttJ

(v.32)

l m

>> 1, the value of X is essentially the same as X0, the conductivity of the dispersion medium. If this condition is invalid one must also account for the current transfer by ions of electrical double layer, where the net ion

375 concentration is higher than in the bulk (see Chapter III, Fig III-12). This contribution of electrical double layer may be taken into account if one introduces the correction for the surface conductivity, )~s, which is excessive electric conductivity of the near-surface layers of the dispersion medium. The average electrical conductivity of the dispersion medium in the capillary can be written as -

9~ -

2 9~0 + - - ) ~ s ,

F

where 2/r is the surface to volume ratio of the capillary. As we turn to the discussion of the cross-processes, it would be worth pointing out that when r,r >>1, the mutual displacement of dispersion medium layers occurs only within a thin layer of liquid in a direct vicinity to the wall. Consequently, the velocity distribution in the medium inside the capillary has the profile shown in Fig. V-13, b. The electroosmotic flux ofthe medium, QE, is thus equal to the product between the capillary cross-section and the net electrioosmotic phase displacement velocity, v0, described by the HelmholtzSmoluchowski equation (V.26), i.e.: 2 EEOC AtIJ ~ E -- 1rr2 D 0 -- 71:r

q

l

(V.33)

In agreement with the Onsager reciprocity relationship, the streaming

current, Ip, generated in the capillary due to external pressure drop, Ap, is given by I p - rtr 2 ggO~ A p .

q

376 The flow of medium leads to the appearance of difference in fluid levels in vessels attached to the capillary. The resulting pressure drop,

Ap=ggAH, causes the counter-flow of dispersion medium, and the flow profile in the capillary is such as that shown in Fig. (V-13, c), i.e., near the walls and in the center of a capillary the medium moves in opposite directions. Under the steady-state conditions, when the net flux of medium is zero (QE + Qp=0), the height of electroosmotic rise, HE, is given by

9g An inverse phenomenon, i.e., the appearance of a steady-state potential difference, A~ E, due to the action of the pressure gradient, Ap, (the streaming

potential) is described by the condition IE + Ip - O, and consequently A ~ E _ ~ 0 ~ Ap nX

(V.34)

In a transition from an individual capillary to a real structured disperse system (membrane or diaphragm), one faces complications related to the actual structure of porous medium, in which the transfer of substance and electric current take place. In such systems all previously described basic relationships remain valid, but the radius and length of a single capillary are replaced with coefficients having particular dimensions, referred to as the "structure parameters". In general, the determination of these "structure parameters" is a rather difficult task, but one may expect that in the description of electroosmotic transfer and the electric conductivity of the structured disperse systems these parameters are included in an identical way, similar to the identical dependence of I E and Q~ on r and l, as shown in eqs (V.32) and

377 (V.33). This allows one to determine the electrokinetic potential of disperse system with an unknown structure. By determining the electroosmotic flux and current passing through the investigated system (provided that the additional amount of electrolyte is added to satisfy the 9~~)~0condition) at some particular value of the potential difference, Aqj, one may estimate the electrokinetic potential from the equation rlL0 QE ee0 IE

Many directions of practical use of structured disperse systems (such as of porous diaphragms and membranes) are related to peculiarities of substance transfer through these systems. In addition to the appearance of streaming currents and potentials, generated during filtration, the c h an g e in the c o m p o s i t i o n of the d i s p e r s i o n m e d i u m occurs. Indeed, since the concentration of co-ions in thin channels is substantially lowered, their transport through these channels is impeded. As the flowing fluid tends to restore its electroneutrality, the counterions also become trapped by these fine porous membranes. The process of removing electrolyte from dispersion medium by filtration through membranes with fine pore size is referred to as the reverse osmosis, or ultrafiltration [ 17,18]. This process is used to remove dissolved salts from water and to purify liquids from impurities, such as, heavy metal salts. To facilitate sufficiently high rate of reverse osmosis, one needs to apply a large pressure gradient to the membranes, which requires the use of highly durable membranes.

378 The reverse osmosis takes place during ultrafiltration of sols - the process of the separation of dispersion medium on the fine porous filter under the applied pressure gradient. The resultant ultrafiltrate may have a substantially different composition from that of initial dispersion medium. Interesting peculiarities of mass transfer processes are observed in fine membranes permeable to ions but impermeable to colloidal particles (semipermeable membranes, e.g. collodium film). If such a membrane separates colloidal system or polyelectrolyte solution from pure dispersion medium, some ions pass through the membrane into the dispersion medium. Under the steady-state conditions the so-called D o n n a n e q u i l i b r i u m is established. By repeatedly replacing the dispersion medium behind the membrane, one can remove electrolytes from a disperse system. This method of purifying disperse systems and polymer solutions from

dissolved

electrolytes is referred to as the dialysis. Let us now look at what happens when the unit volume of disperse system containing n charged particles (or n /NA moles of particles) and c moles of electrolyte (e.g., NaC1) 5 is separated from a unit volume of pure distilled water by semipermeable membrane (Fig. V-14). If the effective charge of the particle is q~ (let's assume that q~>0), the diffuse layers of counterions contain ql n / eN A moles of anions (C1 ions in the present example).

5 It is implied that the concentration c is that in the bulk of a solution at distances R significantly greater than the diffuse double layer thickness, 6=1/~:, i.e. at R )) 5=1/~

379 ,~ ~

n

+

-~-~--AqI +

,e.,

nql

CI" eN A

.-'."

-.-

xNa

+

.,.

,..-

".'" xCI" ~o e. :-5

cNaCl

:..'. ,~ , ~

Fig. V-14. Transport of electrolyte through the membrane

The necessary condition for an equilibrium in the system that is close to an ideal solution is that product of concentrations of ions capable of passing through the membrane has to be the same for solutions on both sides of the membrane (in the case of concentrated solutions one has to account for the activity coefficients of ions). For this equilibrium to be reached, x moles of NaC1 have to diffuse through the membrane into pure dispersion medium. The value of x is thus determined by

c + qan - x l ( c - x ) - x 2, eN a from which it follows that x -

c + [ql n / (eN A )] 2c + [q l n / (eN A )]

c.

(v.35)

When the electrolyte concentration, c, is low, while the concentration of colloidal particles, n, and their effective charge are high, i.e. when c
>rp, i.e., when Y -~0, which corresponds to the low-frequency limit, eq. (V-46) becomes

_ ~012 2 Pm Otvisc--9 rP7

/ 02 OP -

pm

((3visc > > r p )

,

and, consequently, when ~visc > ;5th m ,t'p > > and in the case of high frequencies (Xp>>1, Xm))l)'

~Ph ) ,

415

~ / P m C ? l ; mPpCPl: p ~th -- 2x/2-rp

cTPm m _

PmC?

p

Pp Cp

m + ppC %p (rp < < S mth ,rp tmax.

Pnllx f

0

t~in

dP t- tant9 - t - ~

tmx

Fig. V-32. The sediment accumulation curve in polydisperse system

430 The treatment of data acquired in sedimentation analysis usually involves graphical differentiating of the sediment accumulation curve. This method of obtaining particle size distribution is based on the Svedberg - Oden equation" P=q+t~,

dP dt

in which q stands for the weight of particles with sizes greater than r, = r(t), which complete their settling by the time t, i.e. the particles of all fractions that settled by the time t. This equation has a simple physical meaning, since at any given moment, t, the sediment weight increases with the rate dP/dt due to settling of particles with sizes smaller than r, = r(t). Since prior to time t the sedimentation occurred with constant rate, the product t (dP/dt) represents the weight of particles with size r < r, that have settled onto the sedimentation pan by the time t. The value of q - P- t(dP / dt) gives the weight of larger particles that have already completed their settling. The value of q is given by an intercept of the line tangent to the P(t) curve, (Fig. V-33). By plotting such tangent lines and determining the corresponding values of q and the size of particles, r(t), that complete sedimentation by time t, one obtains an integral distribution curve, q(r) / q

Pmax

-Pmax"

,f I ! ! I

i 0

rmin

q

f

/ rms x

r

Fig. V-33. Integral and differential particle size distribution curves

431

f(r)

The differentiation of this curve yields differential distribution curve, dq(r) / Pmax dr , also shown in Fig. V-33. The values ofrmi . and rmax

are determined from the times/max and train, respectively (see Fig. V-32). Sedimentation analysis can be successfully used in systems containing particles with radii in the range between 1 and 100 ~tm. When larger particles settle in a low viscosity medium, such as water, one has to account for the deviations from the Stokes equation due to turbulent flow of medium around the particles, and introduce correction factors accounting for the acceleration of particles at the beginning of sedimentation. Sedimentation of particles with sizes on the order of fractions of a micron and those of smaller sizes is influenced by the diffusion phenomena to a significant extent (see Chapter V, 2.3).

V.8.2. Sedimentation Analysis in the Centrifugal Force Field When particle with radius r settles in centrifugal force field, its velocity,

dR/dt,

is determined by centrifugal acceleration, co2R,where o3 is the

angular velocity of the centrifuge rotor and R is the distance between the particle and the axis of revolution. The particle velocity is given by dR

~3 ~:P3 ( P - Po) r176

dt

B

where B is the friction coefficient. Consequently, one can write ln(R / R 0 )

4/3np3 ( p - P o )

Atr 2

B

m(1-9~ B O

= S,

(V.52)

432 where R 0 and R are the distances between the particle and the axis of revolution at the beginning of sedimentation and after the period of time, At, had elapsed, respectively; m is the mass of particle. The quantity S is referred as the sedimentation coefficient, or the sedimentation constant. If AR = R - R 0 T~ the usual continuous transition between two homogeneous solutions takes place.

r, oc

TCr

90

88

86

f

1

70

75

80

85

90

oxyquinoline, tool %

Fig. VI-5. The temperature - composition phase diagram of the tricosane - oxyquinoline system [11] Particle size analysis in the critical emulsions is a rather complex task, in part due to the high particle concentration. However, such studies were carried out and yielded the size of microdroplets on the order of tens of nm. Similar treatment can also be applied to three-component systems, in which two of the three components are immiscible with each other, but each of these components is infinitely miscible with the third one. The phase diagrams in such three-component systems contain a so-called line of the critical states, which shows the critical composition as a function of temperature. In such systems the critical state can be approached from the side of a two-phase system by both changing the temperature and altering the composition.

472 VI.3. Micellization in Surfactant Solutions

Micellar dispersions, which contain micelles along with individual surfactant molecules, are the typical examples of lyophilic colloidal systems. Micelles are the associates of surfactant molecules with the degree of association, represented by aggregation number, i.e. the number of molecules in associate, of 20 to 100 and even more [1,13,14]. When such micelles are formed in a polar solvent (e.g. water), the hydrocarbon chains of surfactant molecules combine into a compact hydrocarbon core, while the hydrated polar groups facing aqueous phase make the hydrophilic

shell. Due to the

hydrophilic nature of the outer shell that screens hydrocarbon core from contact with water, the surface tension at the micelle - dispersion medium interface is lowered to the values c~_/m) that maintain spherical symmetry is thermodynamically unfavorable, since it has to involve the inclusion of polar groups into the body of a micelle. For this reason the degree of association of molecules in micelles increases not due to the growth of spherical micelles, but due to changes in their shape, i.e. due to the transition to asymmetric structures. The formation of colloidal particles (surfactant micelles) in the disperse system either as a result of spontaneous dispersion of macroscopic phase, or by spontaneous association (condensation) of individual molecules upon the increase of surfactant concentration, corresponds to a qualitative change in the system. The latter undergoes transformation

from

macroheterogeneous or homogeneous state into microheterogeneous colloidal dispersion. Such qualitative change causes an abrupt experimentally observable change in physico-chemical properties, which in most cases represented by a characteristic break on the curves showing various physicochemical parameters as a function of surfactant concentration. As the surfactant concentration in solution increases above some critical concentration, Ccr,one can observe a noticeable increase in the intensity of scattered light, which is characteristic of the formation of a novel dispersed phase. Instead of their usual smooth behavior, described by the Szyszkowski

475 equation, the surface tension isotherms experience a break at c - c~. Further increase in surfactant concentration above c~ results in essentially constant values ofcy (Fig. VI-7). Similarly, the break at c = C~rappears also in the curves showing specific and equivalent (A) conductivities as a function of concentration of an ionic surfactant (Fig. VI-8). The surfactant concentration, c~, above which micellization begins (some experimentally detectable number of micelles form) is referred to as the critical micellization concentration (CMC). Abrupt changes in the properties of surfactant- water system that occur in the vicinity of the CMC, allow one to determine the latter with high precision from the break point in the curves showing various properties as a function of surfactant concentration. In the discussion of micellization we will primarily focus on features of this process that are common for both ionic and non-ionic surfactants. The ability of ionic surfactants to undergo ionization in aqueous solutions results in the generation of charge at the micellar surface, which stipulates some specific features of systems containing such surfactants. A

I

I

!

CMC

c

Fig. VI-7. The surface tension isotherm of aqueous solutions containing micelleforming surfactants

0

. . . . .

CMC

Fig. VI-8. The equivalent electric conductivity of aqueous solutions of ionic surfactants as a function of surfactant concentration

476

VI.3.1. Thermodynamics of Mieellization The equilibrium between dispersed phase (i.e., micelles) and molecular solution o f a surfactant (or the macroscopic phase, in case of saturation) exists in thermodynamically stable systems containing micelle-forming surfactants. One can, to a certain degree of approximation, describe the equilibrium between micelles consisting of m surfactant molecules and molecularly dissolved surfactant as a chemical reaction, namely [ 15,16] m[S] a (S)m, where S stands for surfactant molecules. In agreement with the law of mass action, one can write nmic/NA Kmic

-"

m

CM

where n mic is the number of micelles per 1 m3; cM is the concentration of molecularly dissolved surfactant in kmol m -3, and K mic is the equilibrium constant of micellization.

In systems containing ionic surfactants, the molecules of which undergo dissociation into ions with monovalent counterions, it is more proper to describe the formation ofmicelles with aggregation number m and effective charge q as nmic/NA Kmic -

m

~ q / e

(VI.4)

C M t; i

where ci is the concentration of counterions and e is the charge of electron. Since the CMC corresponds to some particular value of nm~c,determined by the precision of available experimental methods, for electrolyte that contains an ion identical to the one present in a surfactant molecule, eq. (VI.4) yields Ccr = CMC as a function of

477 electrolyte concentration, i.e 9

log C M C -

k~- k 2 log c~,

where k~ = (l/m) log (n ~c/NA Kmic),and k 2 = q / ( m e ) is the degree of dissociation of the ionic groups in a micelle. One can also obtain this expression by examining the work of micelle charging.

The experimental studies indicate that aggregation numbers of surfactant molecules in micelles increase from 20 to 100 or higher, as the surfactant hydrocarbon chain length grows longer. Consequently, the dependence of nmic on the total surfactant concentration in the system, Co, can be represented by a high order parabola, and may be viewed as the curve with inflection point corresponding to the CMC (Fig. VI-9). At low net surfactant concentrations, i.e. when c0 CMC can be readily understood, since the value of ~ is determined by the concentration of molecularly dissolved surfactant. Indeed, in agreement with the Gibbs equation, dc~= Fdg, the condition ofc~- const corresponds to an independence of chemical potential of concentration at Co> CMC, i.e., dg - 0. One can thus

478 say that the formation of micelles causes a characteristic non-ideality of solution above the CMC. f/mic

Cm

C'mi c

I I J

J CMC l

r

,,, ,

o Co Fig. VI-9. The micelle number concentration, r/mic,as a function of the total surfactant concentration in the system, Co

CMC

Co

Fig. VI-10. Changes in the surfactant content in the molecularily dissolved and micellar states as a function of increase in the total surfactant concentration, Co

The amount of substance present in the micellar state, Cmic mnmic /NA, -

-

may exceed the concentration of it in the molecular solution by several orders of magnitude. The micelles thus play a role of a "reservoir" (a depot) which allows one to keep the surfactant concentration (and chemical potential) in solution constant, in cases when surfactant is consumed, e.g. in the processes of sol, emulsion and suspension stabilization in detergent formulations, etc. (see Chapter VIII). A combination of high surface activity with the possibility for one to prepare micellar surfactant solutions with high substance content (despite the low true solubility of surfactants) allows for a the broad use of micelle-forming surfactants in various applications. Important information regarding the nature of the micellization process can be obtained from the studies on the temperature dependence of the CMC. It is worth reminding here that CMC corresponds to the state of

479 thermodynamic equilibrium between micelles and individual surfactant molecules. CMC is the concentration of true solution, C~r,at which a particular, experimentally detectable number ofmicelles per unit volume, nmic,is formed. If one assumes that this measurable micelle concentration, nm~, and the aggregation number of molecules in micelles, m, in the vicinity of the CMC remain constant within some temperatur range, and that the activity coefficient of molecular solution is 1, the thermodynamic expression for the enthalpy of micellization can be written as d In Ccr A ~rVrmic = - RT 2 m ~ dT Numerous experimental studies on micellization in various surfactant solutions indicated that the values of A J~mic are usually very small and often positive [1,15]. Since a spontaneous processes is accompanied by a decrease in the system free energy, small and, moreover, positive values of A ~ mic indicate that e n t r o p i c a l c h a n g e s play a significant role in spontaneous micellization process. Such changes are primarily related to the specific features in the structure of water as a solvent (see Chapter II,2). The driving force for the association of hydrocarbon chains into a micellar core is an increase in the entropy of the system, which occurs primarily due to destruction of iceberg structure present in water. Such structures are present around the hydrocarbon chains of dissolved surfactant molecules. The studies performed with aqueous dispersions of micelle-forming surfactants have shown that the micelle formation by both association of individual molecules and dispersion of macroscopic phase may occur only

480 above certain temperature, referred to as the Krafftpoint, TKr(Fig. VI- 11) [ 13]. Below the Kraffl point the surfactant solubility is small and its concentration is lower than the CMC. The equilibrium between the surfactant crystals and true surfactant solution (the concentration of which rises as the temperature increases) exists in this temperature range. Thus, in surfactant solutions, for which the Kraffl point is in the range of elevated temperatures, the formation of micelles does not occur under the normal conditions. I

I Micelles Crystals

+

+

solution

solution

CMC ution

TK~

T

Fig. VI-11. Phase diagram of a micelle-forming surfactant- water system

Due to micelle formation the total surfactant concentration undergoes an abrupt increase. Since true (molecular) solubility ofsurfactants, determined by the CMC, remains essentially constant, an increased surfactant concentration in solution is caused by an increase in a number of formed micelles. Micellar solubility increases with increase in temperature, and thus a continuous transition from pure solvent and true solution to micellar solution, and further to different liquid crystalline systems and swollen surfactant crystals (see below), may take place in the vicinity of the Krafft point.

481 The molecular solubility and the surface activity of micelle-forming surfactants, respectively, decrease and increase by a factor o f ~ 3 to 3.5 when the hydrocarbon chain is extended by one CH2 group (see Chapter II,2). Since in the vicinity of the Krafft point the value of the C M C differs little from the molecular solubility, C M C within the same homologous series also decreases by a factor of-- 3 to 3.5 upon the transition to each subsequent member. The highest possible lowering of the surface tension at the air - surfactant solution interface ofmicelle-forming surfactants, as well as of"regular" surfactants, is essentially constant within a given homologous series.

The CMC's of all micelle-forming surfactants are usually low (about 10-s to 10-2 kmol m-3), i.e. low concentrations of molecular solutions correspond to the micelles=solution equilibrium. This means that the existence of particles with sizes d, different from the size of micelles, dm, is thermodynamically unfavorable (see Chapter VI,1). The transition from particles of size dm to those with smaller sizes, hence, results in the increase in free energy of the system, and the A J-(log d) curve contains a minimum in the colloidal range at d=dm (see Fig. VI- 1). The increase if A J-occurring at d>h = R - 2r (Fig. VII-5), the major contribution into Usphcomes from the first term in parenthesis in eq. VII-11, which can be written as

2r 2

2r 2

2r 2

r

R2 -4r 2

( R - 2 r ) ( R + 2r)

h(h + 4r)

2h

Consequently,

Ar b/sph

12h

Fig. VII-5. Two particles separated by a thin gap

The energy of molecular interaction (attraction) between two particles each or radius r separated by a thin gap filled with dispersion medium is given by the expression

r2)

In 1 - 4 - ~

r2

r4

~-4-R--T-8- ~

64

r6 R6

~

2r 2 R2 _ 4 r 2

r2

r4

6

~ 2 - ~ + 8 7 - a - + 3 2 Rr 6

9

529

A*r Usph ~

12h = =hrUmo 1

,

(VII.12)

in which eq.(VII.9) is taken into consideration in the right hale

The

interparticle interaction force in this case is given by

A*r

F ~

12h 2 9

(VII. 13)

According to eqs.(VII. 12) and (VII. 13), the molecular attraction force between two identical spherical particles may be written as F(h)

-

71;rUmoI (h).

Derj aguin obtained an analogous expression valid for any interaction potential, U(h) between curved interfaces of various shape 4 [15]" F ( h ) - rckU(h) - ~kA ,~-ff(h).

(VII.14)

In this equation k is the linear parameter related to geometry and determined by the curvature of surfaces in contact. For two spherical particles of different radii, r' and r", k = 2r'r"/(r' + r"); for two cylindrical surfaces positioned at a right angle with respect to each other, k - 2 (r' r") ~/2. For two p l a n e - p a r a l l e l s u r f a c e s separated by an equilibrium distance h0, there is a minimum in specific free energy of interaction (free energy of film), U(ho) = A g f (h0). This minimum (Fig. VII-6) is referred to as the near potential energy minimum, or simply the primary minimum. The

4

Derjaguin's expression is valid only for the surfaces of second order

530 values of h 0 are approximately equal to intermolecular distance in the bulk of condensed phase (or the size of dispersion medium molecules in the residual adsorption layer in the gap). The attraction forces between the surfaces (to make things simpler, we will further consider dispersion interactions only) predominantly act when h > h0, while Born repulsion becomes significant at h 10-2, one can apply the relationships established in the molecular kinetic theory of gases, according to which the resistance to the particle motion is proportional to the cross-sectional area of particles, and the velocity of their motion, 1), due to the applied force, F, is given by F

Ar

F

m

2mM UM

2~r2mMUM NAc

In the above expression mMis the mass of a molecule of gas; ~M=(8kT/71;mM)1/2 is the average velocity of gas molecules; Ar is the mean free path of particle, defined

as

A r -

1/~r2NAc, where c is the concentration of gas. Retardation in

586 particle motion occurs because the rate of collision of molecules with the front side of particles is higher than with the rear side. For a majority of most important aerosol systems the Knudsen number has intermediate values, i.e 10-2 x* ), the object acquires strain ~' -

, which results in the G

accumulation of energy by elastic element. If also ~ < 2~*, then due to the action of the dry friction element, the object maintains residual ("frozen") stress after the applied stress has been removed. This residual stress is equal to ~ - ,~* , and has the sign opposite to that of initial stress. Apparently, the absolute value of residual stress can not exceed ~*. G

Fig. IX-12. The model of internal residual stresses

4. The B i n g h a m m o d e 1 involves a parallel combination of a viscous Newtonian element and plasticity element, as shown in Fig. IX- 13. This model is widely used in description of colloidal structures, such as aqueous suspensions of clay minerals. rib

Fig. IX-13. The Bingham model

664 Since elements in Bingham' s model are parallel to each other, their strains are identical, and the stresses are additive. The shear stress in Coulomb's element can not exceed the critical value, ~*. Consequently, the strain rate generated by viscous element, should be proportional to the difference between the acting stress and the critical shear stress, namely" "~--'C ~

~

o

riB When ~ < ~*, deformation does not occur (Fig. IX-14). Since the parameter ofBingham's model, riB, defines the derivative, d~/d~,- riB, this constant value is referred to as the differential viscosity, in contrast to a variable effective

viscosity, "c/4[- qef('Y )" 7

q0 =

rib

~qo 0

r*

r

Fig. IX-14. The viscoplastic behavior

In order to describe rheological behavior of real systems, in particular under broadly varying conditions (time, stress), one often uses more complex combinations that include above described simplest rheological models. For instance, in order to adequately describe the system, one may need several relaxation times (not just one), or even a broad spectrum of them. Rheological models thus become more complex, and mathematical description of such models becomes cumbersome as well.

665 One of the approaches used to simplify the solution of such problems is based on the use of the so-called electro-mechanical analogies, it. This approach involves modeling of rheological properties with the help of electric circuits and is based on mathematically identical form of laws that describe the transport of electric current and stresses in solid and liquid objects. For instance, the expression for energy stored by a spring,

G72/2, is identical

to the expression for energy of charged capacitor, q2/2C;the expression for energy dissipation in viscous element, fly 2, is identical to the expression for heat release at ohmic resistance, given by R/2. This allows one to describe and model relaxation of mechanical stresses in Maxwell's model as the voltage decrease that occurs upon discharging a capacitor over the resistor in a chain with the time constant equal to t = RC = rl/G. At the same time it is often impossible to describe successfully real systems even by complex models consisting of elements with constant parameters G, q, r* that remain unchanged in the course of deformation. In such cases one needs to introduce models with variable parameters, which, for instance, include elements of non-linear elasticity, G = G(y), non-linear viscosity, 11___ 11(~'),and variable yield stress, i.e., work hardening, z*

__

~*( 3' ).

IX.2. Structure Formation in Disperse Systems

Structure formation that takes place in disperse systems is the result of spontaneous thermodynamically favorable processes of particle aggregation that lead to a decrease in free energy of the system. These processes include coagulation of dispersed phase or condensation of substance in the zones of direct particle contact. The development of spacial networks (disperse structures) of different kinds defines the ability of disperse system to be converted into a material with particular mechanical properties. Such system becomes qualitatively different from its initial, unstructured state. Material strength, Pc, expressed in N m -2, is an important mechanical characteristic, which determines the ability of a material to withstand external

666 stresses without becoming fractured. Let us address disperse structures of g l o b ul ar type, for many of which contributions into Pc value come from a combination of adhesive forces that act between particles at points of their contacts, i.e., from the strength of individual contacts between particles, p~ (expressed in the units of force, N), and the number of such contacts per unit area of a fracture surface, Z, expressed in m -2. Within the limits of additivity approximation, one can write that Pc ~ 7~Pl. The values ofp~ and Z can be both theoretically evaluated and experimentally assessed [ 16-19].

The value of 7~is defined by geometry of the system, primarily by the size, r, and the packing density of particles. The latter is characterized by porosity of the structure, H, which may be defined as the ratio of pore volume, Vp,to the total volume of porous structure, V, i.e., H = Vp/V. One can evaluate ?( = )~(r, H) dependence from particle dispersity and sample

porosity data by employing particular models of disperse structures. In the simplest case of porous monodisperse structures consisting of spherical particles that form crossing chains containing, on average, g particles from node to node (Fig. IX-15), one can describe this dependence for porosities, 1-I >_48%, using the following expressions [20,21]"

Z-

1 ( 2 r ) 2 ~2

;

II - 1 -

(3~-2). 6~ 3

For structures that are not too porous, within the first approximation one may write that

667 1

Z~

(2r)2 '

which allows one to obtain rough estimates for possible values of Z in real systems. For particles with a diameter 2r = 1O0 btm, Z ~ 103 - 104 contacts per 1 c m 2", for particles with 2r ~ 1 gm

X;~ 107- 10 8 cm 2", if 2r ~ 10 nm, ~ ~ 10 ~

to 10 ~2contacts per 1 c m 2. If one accounts for polydispersity and/or anisometric shape of particles, these values will change accordingly.

2r

n = 1.5 Fig. IX-15. A model of globular disperse structure This geometry is pre-defined by a combination ofphysico-chemical processes of the particle formation during dispersion or condensation. The physicochemical and chemical factors are represented to even greater extent and in greater variety in the strength of individual contacts, p~, characteristic that describes adhesive forces between individual particles. This description corresponds to the case of disperse structures of globular type in which the strength originates from a continuous skeleton that forms due to adhesion of individual particles upon the conversion of free disperse system into structured disperse system. There are, however, other types of structures, such as, e.g., cellular structures (in solidified foams and emulsions), in which the skeleton consists of continuous films of solid-like dispersion medium. Such structures, typical for some polymeric systems, may

668 form during the formation of a new phase in a mixture of polymers by condensation. Separate approach is also needed for the description of structures consisting of anisometric particles, fibers. At the same time, in addition to porous structures, one also encounters various compact microheterogeneous systems, which include ceramics, composite materials and natural "construction materials", such as wood, animal bones, etc. Depending on the nature of forces responsible for the adhesion of particles to each other, one can classify the contacts as coagulation and phase [ 16-22]. In the case of coagulation contacts interactions between particles are limited to a simple contact between them, either via a gap filled with the dispersion medium (Fig. IX-16, a), or directly (Fig. IX-16, b). Such contacts may form in cases when the DLVO potential barrier in the system is either surmounted or totally absent. Thus, coagulation contacts correspond to particles in the primary potential energy minimum (see Chapter VIII, 5). Disperse systems containing such contacts may be characterized by weak strength and by mechanical reversibility, which reflects their ability to restore an original structure upon mechanical destruction

(thixotropy).

~

--

I i I I ! I I 1 ,

|

|

w

u

!

|

I

a b c Fig. IX-16. Different contacts between particles: a and b - coagulation contacts; c - phase contact The strength of a coagulation contact, i.e., adhesive force between particles, is determined primarily by molecular forces. For spherical particles, in agreement with eq. (VII. 13), this force is given by

669 A*?"

Pl-

IF(h0)l ~

12h g '

where A* is the Hamaker constant ( with dispersion medium accounted for); h0 is the equilibrium gap width between particles, and r is the curvature radius of particles at the point of their contact. The energy of particle adhesion in a contact in agreement with eq. (VII. 15) is given by

A*r blc

12h o For example, let the coagulation contact be formed by two spherical particles with radii r ~ 1 gin, or by non-spherical particles touching each other with curvatures of the same radii. For h0 between a few tenths of a nanometer and 1 nm, and A*

10 -19 J, one hasp~ ~ 10 .7 - 10 .8 N. The adhesion energy in

a contact, u c, in this case falls within a range between 10 -16 and 10 -~7 J, which significantly exceeds the value of kT (at T ~ 293 K, kT ~ 4x 10 -2j J). For colloidal particles with radii r ~ 10 nm under the same conditions one obtains p~ ~ 10 .9 - 10-~~ determined

and Uc ~ 10 -~9 J, which is on the border of what can be

experimentally

for an individual

contact.

Oppositely,

for

macroscopic molecularly smooth spherical particles with r ~ 1 mm, one finds that p~ ~ 10 .5

-

1 0 -4

N and u c

~

1 0 -14 J ,

which is within

reach by direct

experimental measurements. It is important for one to realize that all of the above numbers are given for l y o p h o b i c

systems in which the interfacial

tension, cy, is on the order of 10 mJ m 2 or higher, and the Hamaker constant, A* ~24~h2c~ (see Chapter III, 1), reaches the value o f ~ 1 0 19 J. In this case in the contact corresponding to a primary potential energy minimum the adhered

670 particles can not be separated by Brownian motion. At the same time, within the same approximation, at low values of interfacial tension, ey, such as tenths and hundredths of mJ m -2, i.e., when A% 10 2~ - 10 .22 J, for particles of r ~ 1 ~tm, one obtains p~ _40 )

25