Colloid and Surface Chemistry E.D. Shchukin, A.V. Pertsov, E.A. Amelina and A.S. Zelenew

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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.

Throughout the book the presentation of fundamental theoretical and experimental approaches and results is combined with discussion of general scientific basis of their role in nature and applications in various technological processes.

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Colloid and Surface Chemistry E.D. Shchukin, A.V. Pertsov, E.A. Amelina and A.S. Zelenew

  1. 1. STUDIES IN INTERFACE SCIENCE Colloid and Surface Chemistry
  2. 2. STUDIES IN INTERFACE SCIENCE SERIES EDITORS D. M6bius and R. Miller 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 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. 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 ColloidParticles and Proteins to Interfaces and Formation of Two- Dimensional Arrays by P.A. Kralchevsky and K. Nagayama Vol. 11 Novel Methods to Study Interfacial Layers by D. M6bius and R. Miller Vol. lz Colloid and Surface Chemistry by E.D. Shchukin. A.V. Pertsov, E.A. Amelina ans A.S. Zelenev
  3. 3. Colloid and Surface Chemistry Eugene D. Shchukin The Johns Hopkins University, Department of Geography and Environmental Engineering, Baltimore, MD, USA and Moscow State University, Department of Chemistry, Moscow, Russia Alexandr V. Pertsov Moscow State University, Department of Chemistry, Moscow, Russia Elena A. Amelina Moscow State University, Department of Chemistry, Moscow, Russia Andrei S. Zelenev ONDEO Nalco Company, Naperville, IL, USA 2001 ELSEVIER Amsterdam - London - New York- Oxford- Paris - Shannon - Tokyo
  4. 4. ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 92001 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 IDX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WI P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. ISBN: 0 444 50045 6 ISSN: 1383 7303 0 The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.
  5. 5. 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
  6. 6. 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
  7. 7. 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, nano- cluster 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.
  8. 8. 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 structure- rheological 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.
  9. 9. 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 ofphysical-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
  10. 10. 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 (Max- Planck 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
  11. 11. 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.
  12. 12. viii CONTENTS PREFACE GENERAL INTRODUCTION i xii I. SURFACE PHENOMENA AND THE STRUCTURE OF INTERFACES IN ONE-COMPONENT SYSTEMS 1 I. 1. Introduction to the Thermodynamics of the Discontinuity Surface in a Single Component System 2 1.2. The Surface Energy and Intermolecular Interactions in Condensed Phases 13 1.3. The Effect of the Interfacial Curvature on the Equilibrium in a Single Component System 31 1.3.1 The Laplace Law 31 1.3.2. The Thomson (Kelvin) Law 40 1.4. Methods Used for the Determination of the Specific Surface Free Energy 44 References 59 List of Symbols 61 II. THE ADSORPTION PHENOMENA. STRUCTURE AND PROPERTIES OF ADSORPTION LAYERS AT THE LIQUID-GAS INTERFACE 64 II. 1. Principles of Adsorption Thermodynamics. The Gibbs Equation 65 II.2. Structure and Properties of the Adsorption Layers at the Air-Water Interface 84 II.2.1. The Dilute Adsorption Layers 84 II.2.2. Langmuir and Szyszkowski Equations. Accounting for the Adsorbed Molecules Own Size (Mutual Repulsion) 97 II.2.3. Structure and Properties of Saturated Adsorption Layers 112 II.3. Classification of Surface Active Substances. The Assortment of Synthetic Surfactants 131 II.4. Analytical Chemistry of Surfactants 144 References 160 List of Symbols 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 III.3. Adsorption of Ions. The Electrical Double Layer (EDL) III.3.1. Basic Theoretical Concepts of the Structure of Electrical Double Layer III.3.2. Ion Exchange III.3.3. Electrocapillary Phenomena III.4. Wetting and Spreading III.5. Controlling Wetting and Selective Wetting by Surfactants III.6. Flotation 176 193 194 214 220 225 244 250
  13. 13. ix References 255 List of Symbols 257 IV. THE FORMATION OF DISPERSE SYSTEMS 260 IV. 1. Thermodynamics of Disperse Systems: the Basics 261 IV.2. Thermodynamic Principles of the Formation of New Phase Nuclei 273 IV.2.1. General Principles of Homogeneous Nucleation According to Gibbs and Volmer 273 IV.2.2. Condensation of the Supersaturated Vapor 279 IV.2.3. Crystallization (Condensation) from Solution 280 IV.2.4. Boiling and Cavitation 280 IV.2.5. Crystallization from Melt 282 IV.2.6. Heterogeneous Formation of a New Phase 284 IV.3. Kinetics of Nucleation in a Metastable System 289 IV.4. The Growth Rate of Particles of a New Phase 295 IV.5. The Formation of Disperse Systems by Condensation 300 IV.6. Ultradisperse Systems. Supramolecular Chemistry 311 IV.7. Dispersion Processes in Nature and Technology 313 References 316 List of Symbols 318 V. TRANSFER PROCESSES IN DISPERSE SYSTEMS 320 V. 1. Concepts of Non-Equilibrium Thermodynamics as Applied to Transfer Processes in Disperse Systems. General Principles of the Theory of Percolations 321 V.2. The Molecular-Kinetic Properties of Disperse Systems 327 V.2.1. Sedimentation in Disperse Systems 329 V.2.2. Diffusion in Colloidal Systems 329 V.2.3. Equilibrium Between Sedimentation and Diffusion 333 V.2.4. Brownian Motion and Fluctuations in the Concentration of Disperse Phase Particles 337 V.3. General Description of Electrokinetic Phenomena 349 V.4. Transfer Processes in Free Disperse Systems 361 V.5. Transfer Processes in Structured Disperse Systems (in Porous Diaphragms and Membranes) 373 V.6. Optical Properties of Disperse Systems: Transfer of Radiation 390 V.6.1. Light Scattering by Small Particles (Rayleigh Scattering) 390 V.6.2. Optical Properties of Disperse Systems Containing Larger Particles 402 V.7. Transfer of Ultrasonic Waves in Disperse Systems. Acoustic and Electroacoustic Phenomena 408 V.7.1. Theoretical Principles of Ultrasound Propagation Through Disperse Systems (Acoustics) 409 V.7.2. Electroacoustic Phenomena 417 V.8. Methods of Particle Size Analysis 421
  14. 14. V.8.1. Sedimentation Analysis 426 V.8.2. Sedimentation Analysis in the Centrifugal Force Field 431 V.8.3. Nephelometry. Ultramicroscopy 435 V.8.4. Light Scattering by Concentration Fluctuations 438 V.8.5. Photon Correlation Spectroscopy (Dynamic Light Scattering) 442 V.8.6. Particle Size Analysis by Acoustic Spectroscopy 452 References 454 List of Symbols 456 VI. LYOPHILIC COLLOIDAL SYSTEMS 461 VI. 1. The Conditions of Formation and Thermodynamic Stability of Lyophilic Colloidal Systems 462 VI.2. Critical Emulsions as Lyophilic Colloidal Systems 468 VI.3. Micellization in Surfactant Solutions 472 VI.3.1. Thermodynamics of Micellization 476 VI.3.2. Concentrated Dispersions of Micelle-Forming Surfactants 483 VI.3.3. Formation of Micelles in Non-Aqueous Systems 486 VI.4. Solubilization in Solutions of Micelle-Forming Surfactants. Microemulsions VI.5. Lyophilic Colloidal Systems in Polymer Dispersions References List of Symbols 487 498 502 504 VII. GENERAL CAUSES FOR DEGRADATION AND RELATIVE STABILITY OF LYOPHOBIC COLLOIDAL SYSTEMS 506 VII. 1. The Stability of Disperse Systems with Respect to Sedimentation and Aggregation. Role of Brownian Motion 507 VII.2. Molecular Interactions in Disperse Systems 521 VII.3. Factors Governing the Colloid Stability 536 VII.4. Electrostatic Component of Disjoining Pressure and its Role in Colloid Stability. Principles of DLVO Theory 543 VII.5. Structural-Mechanical Barrier 556 VII.6. Coagulation Kinetics 561 VII.7. The Influence of Isothermal Mass Transfer (Ostwald Ripening) on the Decrease in Degree of Dispersion 571 References 577 List of Symbols 580 VIII. STRUCTURE, STABILITY AND DEGRADATION OF VARIOUS LYOPHOBIC DISPERSE SYSTEMS 583 VIII. 1. Aerosols 584 VIII.2. Foams and Foam Films 596 VIII.3. Emulsions and Emulsion Films 607 VIII.4. Suspensions and Sols 624 VIII.5. Coagulation of Hydrophobic Sols by Electrolytes 629
  15. 15. xi VIII.6. Detergency. Microencapsulation VIII.7. Systems with Solid Dispersion Medium References List of Symbols 636 641 642 646 IX. PRINCIPLES OF PHYSICAL-CHEMICAL MECHANICS 649 IX. 1. Description of Mechanical Properties of Solids and Liquids 651 IX.2. Structure Formation in Disperse Systems 665 IX.3. Rheological Properties of Disperse Systems 689 IX.4. Physico-Chemical Phenomena in Processes of Deformation and Fracture of Solids. The Rehbinder Effect 702 IX.4.1. The Role of Chemical Nature of the Solid and the Medium in the Adsorption-Caused Decrease of Material Strength 705 IX.4.2. The Role of External Conditions and the Structure of Solid in the Effects of Adsorption Action on Mechanical Properties of Solids 715 IX.4.3. The Application of Rehbinder's Effect 723 References 728 List of Symbols 731 SUBJECT INDEX 733
  16. 16. 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
  17. 17. 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
  18. 18. 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 ofparticles 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. Systems with a liquid dispersion medium 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)
  19. 19. TABLE. Different types of disperse systems XV ~ Medium Solid Dispersed Phase Solid Sl/S 2 Liquid Gas S/L S/G Liquid L/S L1/L2 L/G Gas G/S G/L and gels (in the case of structured systems), coarsely dispersed low- concentrated suspensions, and concentrated pastes. Dispersions with a liquid dispersed phase (L~/L2systems) 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 dispersion 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 dispersion medium are represented by rocks, minerals, a variety of construction materials. Most such systems are of the S~/S2types. 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
  20. 20. 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
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. 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
  25. 25. 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 ofchemical 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
  26. 26. 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
  27. 27. 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
  28. 28. 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 ofparticles; 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
  29. 29. XXV the leading scientific schools ofthe 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 ofmonographs, including Surfaceand 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 [6- 28]. 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
  30. 30. 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, Addison- Wesley, 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, Wiley- Interscience, 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,6thed., 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 ofInterface 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 Physical- Chemical Mechanics, Nauka, Moscow, 1992 (in Russian). Hiemenz, P.C., and Rajagopalan, R., Principles of Colloid & Surface Chemistry, 3rded., 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
  31. 31. 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.
  32. 32. 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~r- s-s - - dS where cyis 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 Ags - 2cyAld(the numerical coefficient is due to the film having two sides). From these considerations it follows that the force F 2acting on the wire and due to the film is given by
  33. 33. F2 - = 2cyd Al The case when F~=Fz=2~dcorresponds to an equilibrium between these two forces. Consequently, cycan 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 surfacetension,and is expressed in mN/m or mJ/m2, 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 . , , , - . _ . _ -< Fig. I-1. A schematic drawing of A.Dupr6's experiment I/ o o I- o o %1 Fig. I-2. The action of the surface tension For fluids the surface tension values are numerically equal to those of the corresponding specific surface energies, while for solids one also has to
  34. 34. 4 consider a tensor quantity related to mechanical stresses that are present at the interfacial layer. The existence of the free surface energy can be explained by the presence of unsaturated bonds between the molecules at the interface. The formation of a new interface requires work to be performed, in order to bring molecules to the interface from the bulk. The intermolecular interactions at the interface and in the bulk of a phase are substantially different. In the vicinity of an interface, and at distances comparable with molecular dimensions, the composition and properties of individual phases are no longer continuous. This means that a nonuniform layer exists between the phases, within which a transition from properties characteristic of one phase to those characteristic of another occurs. Such a nonuniform transition layer is referred to as the physical interface of discontinuity, or simply the discontinuity surface, according to Gibbs [1-3]. The thermodynamics of the discontinuity surface can be examined by analyzing how the density offree energy fchanges upon transition from one phase to another. From thermodynamics one can establish the relationship between the free energy, G-, the isobaric-isothermal potential, ~o, and the chemical potential, ~t, for a single component system: pV- pV- p)V , where p is pressure, V is volume, N is a number of moles, and c=N/V is concentration. The density of free energyfis thus given by" f - ~tc- p (I.1)
  35. 35. The phases separated by a flat interface have the same equilibrium values of g and p. Therefore, under such conditions the free energy densities of individual phases differ solely due to the difference in substance concentration. It is hence evident that the free energy density in a vapor is considerably smaller than that in a liquid (Fig. I-3). S f // ! -6 V # / / Fig. I-3.Changesinthe freeenergydensitywithinthe discontinuitysurface Following the original treatment developed by Gibbs, let us define the free energy excess for a two-phase single component system, taking liquid and vapor as an example. Let us choose an imaginary geometrical interface (further referred to as the dividingsurface) somewhere within the physical discontinuity surface. Let an arbitrary prism, drawn in a direction perpendicular to the dividing surface, include volumes V'and VHatthe sides of the liquid and vapor phases respectively (see Fig. I-3). Let us also introduce some characteristic distance,-8', counted from a chosen geometric interface,
  36. 36. 6 below which the free energy density has approximately the same value as in the liquid bulk (f ~f'= const), and a distance +6" above whichf ~f"= const, where f~ is the free energy density in the vapor. The physical surface of discontinuity is, therefore, simply a layer of width 6' + 6"(Fig. I-3). The presence of the discontinuity surface causes the free energy of a real system gto be higher than the quantity ~ + ~r" =f'V'+f~': The latter represents the free energy of an idealized system in which the free energy densities of each phaseSandfHare constant within the entire phase volume. The excess of free energy in a real system, as compared to that in a described idealized one, is given by - (f'v' + f"v") = where c~is the free energy excess per unit interfacial area S. Let us examine trends in c~as the free energy density changes within the interfacial layer. The free energy of an idealized system, assuming that the dividing interface lies in the z = 0 plane, is: V' V" while the free energy of a real system is given by g--S ff (z) dz V',V" The excess of free energy per unit interfacial area is therefore
  37. 37. - S = [f(z)- f'] dz + [f(z)- f"] dz (I.2) V' V" The integration limits in eq. (I.2) can be set as -6' and +6",respectively, since free energy densities are identical to their bulk values outside of the discontinuity surface. Equation (I.2) can thus be written as 0 +8" -8' 0 which is numerically equal to the shaded part of the area under the curveJ(z), as shown in Fig. I-3. From this figure it is also clear that the utilized approach yields a value of ~ which is dependent on the position of the dividing surface. The surface tension is, however, a quantity accessible directly through experimental measurements, and thus should not depend on the type of approach used to model the interface. This contradiction indicates that treatment used is by no means general: equation (I.2) indeed yields cyonly in the case of a particularly positioned dividing surface, corresponding to the position of the so-called equimolecularsurface (see Section 2.1). In the case when other positions of the dividing surface are chosen, the right-hand side of eq. 0.2) yields a quantity which, along with the mechanical work required to form a new interface also includes a term describing chemical work, dependent on the gc(z) function profile, as well as on the placement of dividing surface.
  38. 38. The definition ofG invariant with respect to positioning of the dividing surface can be worked out, if one analyzes trends in the f(z)-gc(z) function within the discontinuity surface. The specified quantity has the same value in the bulk of both phases, equal to the negative external pressure (Fig. I-4). Within the discontinuity surface, pressure p has a tensor nature, making Pascal's law invalid. Meanwhile, the concentration and pressure dependence of the surface energy density, f, given by eq. (I. 1), is valid only in the regions where Pascal's law holds, i.e., where pressure is a scalar quantity (direct summation of a scalar and a tensor within the same equation is not permitted). P f- gc'- -pw ..~ J~._P~=P Fig. I-4. Profile of theJ(z) - gc(z) functionwithinthe discontinuitysurface It is now clear that the quantity Pv - - (f- gc) has units of pressure, and is indeed equal to the pressure in the bulk. It is, however, important to remember thatpv is not equal to the pressure at the interface. The generalized expression for the surface free energy, c~,can be written by analogy with eq. (I.2):
  39. 39. o- (i' V' - gc')} dz + The expressions in parenthesis in both of the above integrals are identical, and equal to -p, while those in square brackets can be replaced by a function of vertical coordinate, pT(z).Consequently, the equation for ~ reads: ey- I[p-pT(z)ldz V',V" The above expression is known as the Bakker equation [4,5]. The quantitypT can be regarded as the "tangential pressure", acting in a plane parallel to the interface and tending to decrease an interfacial area. Taking into account that the difference between PT and p is significant only within the discontinuity surface, the Bakker equation can be written as For temperatures significantly below the critical point, the thickness ey- I[p-pT(z)ldz of the discontinuity surface, 8' + 8" N109 m, which is on the order of molecular dimensions. Since values of the surface tension ~ customarily lie within the range between 10 and 103mJ/m2(mN/m), the average values ofp- - PT - ~ / (8'+8") ~107 to 109 Pa (100 to 10000 atm). In other words, the
  40. 40. 10 tangential pressure within the discontinuity surface is negative and has a very high value, as compared to the bulk hydrostatic pressure p. The negative sign of the tangential pressure characterizes a tendency of an interface to decrease its area. It is now evident that the surface tension cy, which is a macroscopic measure of a tendency of a surface to decrease its area, is indeed an integral characteristic of specific forces acting within the interracial layer. The magnitude of such a tangential force is numerically equal to the shaded area under the curve shown in Fig. I-4, and it does not depend on the position of the dividing surface. The dividing surface can thus be chosen arbitrarily. This feature of the approach will be utilized in Chapter II in deriving the Gibbs equation. It is noteworthy that the above treatment is only valid for fiat interfaces. Things get more complicated if one deals with curved surfaces, for which it is necessary to consider a pressure gradient existing between two phases in contact. In such a case the surface tension becomes dependent on the position of the dividing surface. A position of the dividing surface that yields a minimum value of~ is referred to as the position of the "surface of tension", according to Gibbs. The excess (per unit area) of internal energy, e, and entropy, 11,within the interracial layer can be introduced by analogy with the excess of free energy [6]. These quantities are also dependent on the position of the dividing surface. One can verify that the equations relating cy,~, and ri are very similar to those derived in conventional three-dimensional thermodynamics, i.e."
  41. 41. rI - - dcffdT, ~:= cy - T(do/d T) 11 (I.3) (1.4) Equation (I.4) is analogous to the Gibbs-Helmholtz equation. The results of experimental studies, presented in Figure I-5, indicate that for most unassociated single-component liquids the surface tension is a linear function of temperature: = a(r-ro), 0.5) where % is the surface tension at some reference temperature, T0, and a is an empirical constant. It is understood here that Toexceeds the substance melting point. A direct comparison of eq. (I.5) with eq. (I.3) indicates that the empirical constant, a, is indeed equivalent to the entropy excess, q, within the interfacial layer, which is also practically independent of temperature. The experimentally determined values of a=rl, given in Table I. 1, show that the entropy excess depends little on nature of the substance and for many substances is close to 0.1 mJ/m 2 K. An interfacial layer contains about 10~9 molecules/m 2, assuming that the molecular size b is about 0.3 nm, and hence the entropy excess per molecule (or, so to say, per degree of freedom) equals 0.1 mJ m-2 K-~ / 10~9 m-2 , which is close to the Boltzmann constant, k=1.6x 10.23 J/K. Such an increase in entropy within the interfacial layer of a pure liquid can be explained by the higher mobility of molecules at the interface as compared to that in the bulk.
  42. 42. 12 A direct comparison of equations (I.4) and (I.5) yields - cyo - or(T- To) + aT- cyo + otTo - const, meaning that the excess of internal energy within the interfacial layer is independent of temperature for a broad temperature range (Fig. I-5). TABLE I. 1. The energy characteristics of condensed phases at the liquid-air interface (~, rl, e), and in the bulk (Sr{), [7]. Substance T, K cy rI e 5~ 1/4 ~ / Vm23N A1/3 mJ/m2 mJ/m2K mJ/m2 J/mol mJ/m2 H2 14.7 2.9 0.14 5 9.1xl02 2.8 N 2 70 10.5 0.19 24 5.7x103 16 NH3 284 23 0.14 63 2.1 • 104 70 Octane 293 21.8 0.06 39 Benzene 293 28.9 0.13 67 2.3 x 104 35 HzO 293 72.7 0.16 119 4.5x104 190 NaC1 1096 114 0.07 180 Hg 273 480 0.22 540 5.0x 104 300 Zn 750 753 0.4 1050 Pt 2273 1820 Consequently, the excess of internal energy can be regarded as a universal characteristic of the interfacial layer of a liquid (see Table 1). A constant value of e is an indication of zero heat capacity excess Cs=de,/dT within the interfacial layer of a single-component liquid, meaning that the interface does not provide any additional degrees of freedom associated with the motion of molecules. The finite positive rl reflects the higher entropy of the
  43. 43. 13 existing degrees of freedom, corresponding to molecules oscillating "more freely" in a direction perpendicular to the interface. An increase in the hidden heat of interface formation, tiT, is in line with a reduction in surface tension with increasing temperature, corresponding to e=const. e, o, rl tiT 0 T~ T Fig. I-5. Temperature dependence of the excess per unit area of free energy, ~, internal energy, ~, entropy,q, and hiddenheatof interfaceformationqTwithin the interfacial layer [6] At temperatures close to the critical point, T~, the compositions of neighboring phases become similar, and thus the excess of all thermodynamic parameters vanishes under these conditions. Near the critical point e and decrease drastically within a range of just a few degrees, and the G(T) dependence is no longer linear (Fig. I-5), [6]. 1.2. The Surface Energy and Intermolecular Interactions in Condensed Phases In the previous section a macroscopic definition of the free and total surface energy as the energy excess within the interface was introduced. An alternative way of addressing this matter is an approximate evaluation of the
  44. 44. 14 interaction energy between atoms, molecules, or ions. All of them for simplicity will further be referred to as the "molecules". Let us not consider the contribution of the entropical factor (i.e. the temperature dependence of the surface tension) and assume that o ~ e. To begin with, let us consider a rarefied gas" its condensation into a liquid, or into a solid crystal leads to a decrease in the system energy due to the saturation of interaction forces between molecules in the condensed phase. Such a decrease taken per mole of a substance, is equivalent to the heat of evaporation (or heat of sublimation, taken with the opposite sign) and can be expressed as 1 ~ - AU ~ ---ZN A b/ll , 2 where Z is the coordination number (a number of neighbors closest to a molecule under consideration) in the bulk of a condensed phase; Na is Avogadro's number, and U~l < 0 is the energy with which the adjacent molecules are bound to each other (Fig. I-6). For the molecules located at the surface the coordination numbers, Zs, are smaller than for those in the bulk, and hence the interactions between molecules present within the surface layer are not saturated. Due to these factors the decrease in energy upon condensation is smaller within the surface layer, as compared to that in the bulk by the amount (yS(where S is the surface area). This essentially means that the energy level of molecules at the surface is higher than that of those in the bulk by the amount (yS (Fig. I-6). In other words, the excess of energy within the surface layer can be regarded as
  45. 45. 15 "incomplete lowering" of the system energy upon the establishment of intermolecular bonds. The surface energy can be related to the interaction energy of molecules in the bulk. To show this, let us introduce the work (or energy) of cohesion, W~. This quantity can be defined as the work of the isothermal process required to separate a column of matter having a unit cross-sectional area O . . . . . . . . . . 1 _ _ t Fig. I-6. The schematic illustration of energy lowering (taken per mole of a substance), occurring when molecules are transported fromthe gaseous phase intothe bulk volume and to the surface of a condensed phase into two parts. Since such a process leads to the formation of two new interfaces, having a unit area each, the work of cohesion is simply twice the surface tension: W~- 2c~. If there were, say, n~ molecules per unit area, and each of them, prior to separation, was interacting with the Zs neighboring molecules from the other part of the body, then the work of cohesion, W~ nsZs]u~[. The surface energy can thus be written as 1 1 cr=--Wc ~ nsZslu1 ] (I.6) 2 2- 1 The molecular density at the surface is related to molar volume, Vm,
  46. 46. 16 -2/3 and the volume per molecule, VM= Vm/ NA, as ns ~VM surface tension is thus given by - 2/3 = (g m / NA) . The cy ,~ ~3 ' (I.7) V2m/3N Z where the Zs/ Z ratio is on the order of fractions of a unit, e.g. 1/4. Consequently, according to eq. (I.7), the specific surface energy is proportional to the heat of evaporation (heat of sublimation), and inversely proportional to the molar volume to the power 2/3. Such a correlation between cyand ~ is commonly known as the Stefan rule. The data summarized in Table 1 are in good agreement with this rule. Indeed, changes in the heat of evaporation by three orders of magnitude correspond to a similar increase in the specific surface energy, as one moves along the table from noble gases and molecular crystals to covalent and ionic compounds, and to metals. Since for solids cyis difficult to estimate (see Chapter 1,4 for details), eq. (I.7) can be used to obtain approximate estimates for the surface energy in such systems. The values of the evaporation and sublimation heats are usually quite close to each other, as well as the densities of solid substances and their melts, measured at the melting point. Consequently, the values of the surface energy at the liquid-vapor, Gcv, and at the solid-vapor, CYsv,interfaces are nearly identical. Oppositely, the interfacial energy CYSLat the interface between the solid phase and its melt is usually low" C~scvalues normally do not exceed 1/10 of surface tension values of melt (note that the heats of melting are also on the order of ~ 10% of those of evaporation).
  47. 47. 17 Following the method established originally by Rehbinder, let us relate the surface energy to the internal pressure. The latter is the other quantity used to characterize the intermolecular interactions. To make things simpler, let us assume that the liquid phase is non-volatile (f"~f "), and that the free energy density,f, changes linearly from the bulk value f" to some valuefm within the entire discontinuity surface of thickness 8' = 8 (see Fig. I-7). Let us also treat the surface tension ~ as the work, required to bring molecules contained in the volume of 1 m 2 x ~i m = 8 m 3 (i.e. 8 n molecules, provided that n is a number of molecules per 1 m 3 ) from the bulk to the surface. Such treatment allows us to write cy- k1(fm - f')8, where k~ = 89in the present approach. -6' f // fm f (z) Fig. I-7.A schematicf(z) dependence for a non-volatile liquid-vaporsystem
  48. 48. 18 The quantity 9U,,given by 9U=--= kl (fm - f' ), 8 is an average density of the energy excess (or the deficiency of binding energy within the surface layer), and has the same order of magnitude as the density of intermolecular energy in the bulk. This quantity estimates the "jamming" between molecules in the fluid bulk and is thus close to the internal (molecular) pressure, which is responsible for molecules in liquids and solids being held together [6]. For ideal gases ~ 0, while for real ones it is given by the virial coefficient in the van der Waals equation, describing the intermolecular attraction. In condensed phases the internal pressure is rather high: considering that the surface layer thickness, 6, has molecular dimensions (5~b), and that the values of c~are normally within a broad range between units and thousands mJ/m2, the values of 5U are as high as 107 - 10~~ / m2, i.e. approach many thousands of atmospheres. It is thus clear that the internal pressure, o~Y,is indeed the total of all of the forces per unit area that one has to overcome to bring molecules from bulk to the interface. In other words, the formation of a new interface requires work to be performed against the cohesion forces. Such work in the isothermic process is accumulated within the surface layer in the form of the energy excess, with density o%r ~fm-f ", j/m3. The interpretation of surface energy as a deficiency of intermolecular interaction energy within the surface layer is of great importance, since it closely relates the experimentally assessable macroscopic quantity, cy,to the internal pressure, ~,, not measurable directly. The internal pressure can in fact
  49. 49. 19 be regarded as the "primary" characteristic of intermolecular interactions in the bulk. Quantities that have dimensions and magnitudes similar to those of (note that 1 N/m2 = 1 J/m3 ) describe other properties of condensed phases that are related to the work against cohesion forces. Two examples of such quantities include the modulus of elasticity, E, and the so-called theoretical strength of an ideal crystal, P~d. The former is the force per unit area during an elastic deformation of a solid (assuming a 100% elongation), while the latter has the meaning of the force per unit cross-sectional area that causes a simultaneous cleavage of all bonds within a cross-section to which it is applied. Since V m "~NAb3, and the Stefan rule can also be written as 597 V m ,v / b, the heat of sublimation, 5r{ Consequently, one may write , is also of the same nature as 9g'. ~-- E ~ Pid ,,IPT[ V b All of the above quantities are the macroscopic characteristics of intermolecular interactions. Moreover, they all have the same origin, which arises from interactions between effective electric charges of the same magnitudes as the elementary charge e separated by distances b, comparable to those between atoms. The quantity e2/b (or e2/4Zceob ~ 10-18 J, if SI units are used) has the same order of magnitude as the interaction energy between the neighboring atoms or molecules. The force of such interactions (and thus the bond strength) is given by e2/4~eob2 ~ 10.9 N. An approximate estimation for the energy of cohesion can therefore be obtained by multiplying the first of these
  50. 50. 20 values by the number of atoms per 1 m2 of cross-sectional area, ns = l/b2: Wc - 2or ~ e 2 /4rt~;o b3 . Wc is on the order of magnitude of several thousands mJ/m 2. The estimation of the force, therefore, reads" ~" "~ Pid ~E~ o~" e2 V 4rtg0b4 10~~N / m 2 Changes in the effective charge from several e to fractions of e, and variations of b within a range of few angstroms yield a broad spectrum of 0 (mJ/m 2) values" from units (noble gases) and tenths (common liquids) of mJ/m 2, to thousands of mJ/m 2 for metals and compounds with high melting points. A more precise free energy estimate can be obtained by various methods, depending on the nature of the condensed phases and on the types of intermolecular interactions within them. For instance, the intermolecular distance b can be determined by considering the intermolecular attractive forces along with the Born repulsion. The latter is a repulsion between electron shells of molecules that have been brought into a close contact. The equilibrium distance R ~ b (Fig. I-8) corresponds to a minimum of the interaction potential. An interaction potential can generally be written as al bl u = i (I.8) R" R m The first term in the above expression corresponds to the attractive interaction between molecules, while the second one describes intermolecular repulsion.
  51. 51. 21 The value of m is usually 10-12, while that of n depends on the nature of attractive forces. A steep increase in the Born repulsion energy is observed as the molecules closely approach each other (Fig. I-8). As a result, the potential well depth, u~, for small n values (corresponding to the Coulombic interaction of ions) is mainly determined by the attractive energy of molecules, corresponding to the equilibrium separation distance. The properties of ionic crystals, in which the attraction between oppositely charged ions is Coulombic (i.e. n = 1), are best described by the dependence of the macroscopic characteristics of solids, such as 9U, 72 b 0 Fig. I-8. The potential energy of interaction between two molecules as a function of a separation distance P~d,E, 5rg'/Vm,on the values of e and b. One, however, has to also account for the influence (attractive or repulsive) of ions located further away in the other coordination shells. To do so, it is necessary to carry out a pairwise summation of interactions between all ions at both sides of a future interface (Fig. I-9). The formation of an interface upon the separation of a single crystal into parts causes partial relaxation within the surface layer, which has also to
  52. 52. 22 be accounted for in the calculations. The result of the described summation of interaction energies can be represented by a numerical coefficient with a value around 1. The slight deviation of this result from the one obtained with the simplified method is readily understood, since the closest neighbors are the ones that contribute to the surface energy and work of cohesion most, while the attractive and repulsive interactions between ions in other coordination shells make no significant contribution to the latter, as they approximately cancel each other out. @@@@(9@@ @@(9@@@@ e|174 . | "Q'-@,'c D'| | | | | Fig. I-9. Schematic representation of the summation of interactions within the ionic lattice The van der Waals-type interactions between uncharged species can be approximately described by the Lennard-Jones 6,12 potential" b/- _ a~ b~ R 6 + R12 The coefficient al, characteristic of the intermolecular attraction, describes the contributions from three types of interactions, namely [6,8,9]" 1) the dipole/dipole orientational interaction involving two permanent dipoles, the contribution of which to a~ is proportional to the fourth power of
  53. 53. 23 the dipole moment, ~d ; 2) the permanent dipole/induced dipole interaction, which is the interaction between a dipole and a non-polar molecule of polarizability aM;the z contribution of this interaction to a~ is proportional to gd0tM, 3) the induced dipole/induced dipole (or dispersion, according to London) interaction between two non-polar molecules, the contribution ac of which to a~ is given by 3 2 a L ---hv0c~ M , 4 where h is Plank's constant; v0 is the characteristic frequency of the charge oscillation; hv0 is the minimal energy of a mutual molecular excitation (may correspond to the IR, visible, or UV region in the absorption spectrum). The oscillation frequency, v0, is directly related to the interactions between molecules. The origin of the dispersion interactions arises from the attraction between the fluctuation- induced dipole of one molecule and the dipole of another molecule induced by it. The dipole/dipole interaction can contribute to the total interaction energy from 0 (non-polar molecules) to 50% and more (molecules having a high dipole moment, e.g. water), while the contribution from a dipole/induced dipole interaction usually does not exceed 5-10%. The dispersion interaction, in contrast, may in certain cases account for as much as half of the attraction energy, and even for all of it in the case of the interaction between non-polar
  54. 54. 24 hydrocarbons. A significant feature of dispersion interactions is their additivity" for two different volumes of condensed phase separated by a gap, the summation of attraction energies of individual molecules is valid (even though the value of a~ in the condensed state might be different from that in a vacuum, due to the mutual influence of molecules on each other). Dispersion interactions are especially important when molecules of a condensed phase are separated by distances significantly larger than molecular dimensions. The net dipole moment of macroscopic phases is usually zero" the spatial orientation of their constituent permanent dipoles is such that the dipole electric fields compensate each other. On the contrary, each molecule inside a given phase is polarized by fluctuating dipoles of the other phase, and thus interacts with them. Therefore, the interactions between molecules of different condensed phases at large separation distances are due solely to the dispersion interactions. This case is of primary importance for the investigation of interactions between colloidal particles separated by small gaps filled with dispersion medium (see Chapter VII). The work of cohesion can be estimated using the microscopictheory of Hamaker and De Boer [10,11]. Their model is based on a simple summation of the dispersion interactions between the molecules contained in two semi-infinite volumes of condensed phases, separated by a gap of thickness h (Fig. I-10). The interaction energy per unit interfacial area between two phases, Umo~, is equal to the energy of interaction of all molecules contained above the plane O~within the infinitely long cylinder of unit cross- sectional area S with those contained within the entire volume below plane
  55. 55. 25 02. Such a summation can be well approximated by the integration with respect to four coordinates: z~, z2, R2 , and q0 (Fig. I-10). The result of such integration yields Umo~" All Umo 1 -- - ~ (I.9) 12zth 2 ' where A~=Tt2n2aLis the Hamaker constant having the units of energy (J). O3 E 9 > Z 1 S dzl ZI O1 s f R~2 ('-4 E 9 > Z2 Fig. 1-10. Summation of dispersion interactions according to the method established by Hamaker and De Boer The symmetry of the problem suggests that the cylindrical coordinates zx, z2 (choosing the positive direction ofz~ to be above the plane O~,and that of,z 2 - below the plane O2) , R 2 and q~(Fig. I-10) are the most convenient to use. It is assumed that all molecules contained in the volume element dV~= Sdz~interact in the sameway with all molecules from
  56. 56. 26 the volume element d V2,located at distance Rl2 from d V~.Consequently, one may write 1 gmol- -n2al If lf Ri62 Z 1 Z 2 R2q) dzldZ2 dR2 R2 dq~ In the above expression a~=aL=3/4hv0~2M, since dispersion interactions are the only ones considered; n is the concentration of molecules in volumes 1 and 2. Since all elements of the ring for which Zz=COnStand Rz=const are located at the same distance from d V~, and the ring volume is given by 2gRzdzzdR2, the integration with respect to q0results in All Umo 1 = - 2 7t R 2 IffR62 dz1 dz2 dR2 Z1 2 2 R2 From geometry it follows that RI22=R22+ (zI + z2+h)2. The integration with respect to R2, yields the interaction energy between molecules in volume 1 with those contained between planes z2 and Zz+dZ2,. The result of this integration reads oo All! d(R~)dzldz2 _ R =o[R2+(Zl+Z2+h)2] 3 - All 2~ (z 1+ z2 + h) 4 dzldZ2 The third integration with respect to zI yields All dz 2 6rt(z2 + h) 3 Finally, the fourth integration with respect to z2yields the value of Umo~(in units of energy per unit interfacial area): UmoI = - ~ All 12~h 2
  57. 57. 27 where the "-" sign corresponds to attraction. Another (more strict) way of calculating the energy of dispersion interactions between the two volumes is based on Lifshitz' s macroscopictheory and is briefly summarized in Chapter VII,2. The work of cohesion in a condensed phase containing molecular species can be understood as the value of gmoI in the h-~b limit. In this case h = h0 = b, and hence 1 1 All Wc - (Y -- - -- Umo 1(b) ~ . (I. 1O) 2 24~b 2 At distances comparable with molecular dimensions the summation can no longer be replaced by integration, as was done above. For such cases only some approximate values on the order of intermolecular dimensions can be assigned to b. In organic substances containing polar groups in addition to the dispersion interactions there are also the so-called non-dispersioninteractions, related to the presence of permanent dipoles and multipoles, and especially to the hydrogen bonding. These interactions are the most effective between the closest neighbors, and are not additive at large distances in the bulk. Consequently, one can distinguish (after Fowkes) the dispersion (yd and non- dispersion oncomponents of the free energy, i.e. the net surface energy is o = (yd -t- O n [12,13]. The contribution of each component to the total surface energy is strongly dependent on the nature of the interacting phases. For example, in non-polar media (saturated hydrocarbons), there are only dispersion forces acting between the molecules, yielding one0, and o=od=20
  58. 58. 28 mJ/m 2. In polar liquids, such as water, the dipole/dipole interaction (and especially hydrogen bonding) contribute up to ~ 70% of the total interaction energy, while the contribution of the dispersion interactions does not exceed 30% of that. For water Gnu50 mJ/m2and cyd~20mJ/m2. The value of the ~d component of the surface energy of ionic, metallic, or covalent compounds is usually different from that of organic non-polar substances. This difference is comparable in magnitude to the difference in corresponding densities. The surface tension, cy, is usually high for the compounds in which non-dispersion (high energy) interactions contribute most to cohesion. For such compounds the values of c~are often ~ 103 mJ/m 2, or higher, and the contribution of ~d to the surface tension is not as significant as in the case of hydrocarbons. It is, however, noteworthy that even in such cases the long-range attractive forces are responsible for the destabilization of colloidal systems (see Chapter VII). These forces, because of their additivity, contribute most into interactions between the particles, large as compared to molecules. The input of dispersion and non-dispersion interactions into the surface tension is similar to that into the work of cohesion. The Wc=2cy dependence is valid for any liquid phases, regardless of their polarity. Indeed, two volumes having a unit cross-section merge as they are brought together at a distance h ~ b. Consequently, the two interfaces with the total energy of 2c~vanish completely under these conditions. On the other hand, the relationship 2c~= - Umol(b) is valid for non-polar liquids only, where the intermolecular interactions are governed by the dispersion forces, and (~ ~(~d.
  59. 59. 29 In contrast to liquids, two different volumes of a solid phase can not be merged together upon contact. Since the mobility of molecules within solid phases is low, the differences in the bulk and surface structure of these volumes can not disappear spontaneously. Thus, even at the closest contact possible, the real physical interface having its own characteristic value of the specific surface free energy (y*is present between the two solid phases. For the two solid crystals, u* is referred to as the specific surface energy of the grain boundary, ~gb " For nonpolar solids -1/2 Umo~(b) is less than the surface energy, cy, i.e.-Umol (b)= = 2cy-o*. The interface between grains in a single component polycrystalline substance serves as a specific dividing interface between the two volumes of a solid phase. The structure and the free surface energy, ~gb, of the grain boundary are primarily determined by the degree of disorientation between the individual grains. Weak mutual disorientation between the neighboring areas (blocks) within a crystal corresponds to a small value of Ggb, linearly dependent on the disorientation angle, 0. A simple type of such low angle disorientation is schematically shown in Fig. I-11, a. The edges of incomplete atomic planes (Fig. I-11, a) can be regarded as a special type of linear defects within the solid phases. These defects are also referred to as the edge dislocations (See Chapter IX). The regions of an amorphous material are formed in the vicinity of grain boundaries in the systems consisting of strongly disoriented grains. These regions can be as large as several intermolecular distances in size. The energy of such high-angle grain boundaries is not strongly dependent on the disorientation angle. It is, however, noteworthy that drastic
  60. 60. a minima in the grain boundary energy may appear at certain disorientation angles (Fig. I-11, b). The highest possible values of ~gb are generally dependent on the nature of the solid phases. These values can reach about 1/3 of the interfacial energy at the solid-vapor interface of metals and about half of that value at the same interface of ionic crystals. 30 Ogb "- 0 0 Fig. I-11. Schematicdrawingof a grain boundarycorrespondingto a small disorientation angle 0 (a); specificfree energy,%, as a functionof the disorientationangle0 (b) Increased energy in the vicinity of grain boundaries and areas of other structural defects explains high chemical activity of solid materials in which such imperfections are present at the surface. This energy excess can significantly influence various chemical processes occurring between solids and other phases surrounding them. Two examples of such processes that are of an extreme importance include corrosion and catalysis. The investigation of the influence of structural defects on the reactivity of various solid materials is the primary subject of modern solid-state chemistry.
  61. 61. 31 1.3. The Effect of the Interfacial Curvature on the Equilibrium in a Single Component System Up to now we have considered interfacial phenomena in systems where the interfacial boundaries separating coexisting phases were essentially flat (i.e., with large radius of curvature). The interfacial curvature changes the thermodynamic properties of systems and is responsible for a number of important phenomena, such as capillary effects. The large interfacial curvature is typical of finely dispersed systems, and hence one has to take into account its effects on the thermodynamic properties of such systems. 1.3.1. The Laplace Law Let us consider the equilibrium between a drop of radius r and a large volume of surrounding vapor, at constant temperature and pressure in each phase. Let us assume that near the equilibrium small amount of vapor condenses into liquid, causing an increase in drop radius equal to 8r. Changes in pressure and, therefore, in chemical potential due to this process are negligible and thus these two quantities remain essentially constant. At the equilibrium the thermodynamic potential, ~o, reaches its minimum and therefore under these conditions its first variation 8 G= 0, i.e: ~5,~'=- Ap8 V + 8 (cyS) - - Ap8 V + c~8S+ SScy- 0, where kp-p'-p"; p' is the pressure in a drop, and p"- that in a vapor; Vand S are the volume and the surface area of the drop, respectively.
  62. 62. 32 According to Gibbs, it is possible to chose a position of the dividing surface such that 6cy= O, the so-called surface of tension, for which one can write 6S Ap = cy-- (I.11) 8V For spherical particles 6S- 8nr6r and 6V = 4nr2~)r. Substitution of these expressions into eq. (I. 11) readily yields the Laplace law: --Pc)" z 2cy where r is the radius of a drop. The pressure difference Ap = po between the neighboring phases separated by a curved interface is referred to as the capillary pressure. In the previous case of a liquid drop surrounded by its vapor, the pressure inside the drop is higher than that in the vapor by the amount 2(y/r, while for the opposite case, i.e. a vapor bubble in a liquid, the vapor pressure is greater than that in the liquid by the same amount. The capillary pressure may be viewed as an additive to the internal molecular pressure, which, depending on whether it is positive or negative, can either increase or decrease the internal molecular pressure 5U, as compared to its value 5U0. established for the flat interface: 5U(r) - 5U0 + + Ipol. For a water drop with a modal radius of 1 gin, the capillary pressure po- 2o/r ~ 1.5x 105 Pa (1.5 atm.), which constitutes ~0.1% of the molecular
  63. 63. 33 pressure 9U-~/b -2x 108 Pa (2000 atm.), while for a 10 nm drop the capillary pressure is already -10% of 5~. In agreement with the Laplace equation, the action of the stress field of the curved interface on phases in contact is analogous to the action of an elastic film with tension u located at the surface of tension. It is, however, important to realize that the properties of the interfacial layer are significantly different from those of a film. Namely, the surface tension o is independent of the surface area S, while the tension of the elastic film increases with increasing deformation ~. Due to the existing interfacial curvature of interfaces between individual phases, the corresponding dividing surfaces are no longer equivalent. It is not only the value of • that is of interest to us, but also the curvature radius r of the dividing interface, which depends on the choice of position of the latter. The dividing surface position corresponding to the values of~ and r, characteristic of the real interfacial layer, was referred by Gibbs to as "the surface oftension". When the curvature radii are large, (also taking into account that the discontinuity surface is thin) the difference in the curvature radii of the interface of tension, as compared to those corresponding to the other possible positions of the dividing surface, (e.g. the equimolecular surface, see Chapter II) is negligible. The Laplace equation represents the basic law in the theory of capillary action. The generalized expression of the Laplace equation applied to non- spherical surfaces can be written as In solutions the surface tension may depend on the interfacial area due to the Gibbs effect (see Chapter VII, 3)
  64. 64. 34 p~ - (5" + , where r~ and r2are the principal curvature radii of the surface. In the simplest case, corresponding to a liquid drop in the absence of gravity, both principal curvature radii are identical and constant along the entire interface. In the gravity field the surfaces of small liquid drops and bubbles are still nearly spherical, ifp~=2cy/r>>r(9'- p')g, i.e. r 2 <<a 2 = 2 cy/(9'- -p")g, where 9' and 9" are the densities of the liquid and gaseous phases, respectively, g is the acceleration of gravity, and a is the capillary constant. When the above restriction is not valid, the surface shape deviates from that of a sphere. The drop, however, maintains its vertical axis of symmetry, and has the shape of body of rotation. The capillary pressure in such a drop varies with height: the difference in capillary pressure Ap, corresponding to the height difference Az, is Ap~, - (p' - p") gAz. As established in analytical geometry, the principal curvature radii and the axis of rotation Oz are located in the same plane ( plane xOz in Fig. 1-12). These radii are related to the shape of cross-section of body of rotation by the plane xOz as rl "- r2 ~ [1 + (dz/dx) 2]~ d2z/dx 2 [1 + (dz/dx)2 ]~ dz/dx and
  65. 65. 35 The substitution of the above equation into the generalized Laplace equation (with the dependence of capillary pressure on the z coordinate accounted for) yields the Laplace equation in the differential form, the numerical integration of which leads to the exact mathematical description of the drop or bubble surface shape in the gravitational field [6,14]. The exact description of the equilibrium surface shape is of importance in the evaluation of surface tension from the experimental data at interfaces with high mobility, such as liquid- gas and liquid-liquid ones (See Chapter I, 4). H~4 alma• X X Fig. 1-12. The equilibrium shape of a drop (bubble) placed on a solid support Let us now proceed with the discussion of several characteristic examples of capillary phenomena occurring on the contact of liquids with solid surfaces of various shapes. An important quantity that describes the solid-liquid interfaces is the contact angle ofwetting, 0 (Fig. I-12), which is the angle between the surfaces of liquid and solid phases in contact. The nature and importance of contact angles will be addressed in more detail in Chapter III and let us just state for now that the contact angle, 0, reflects the similarity in the nature of the contacting phases. The smaller this angle, the better the wetting, and therefore the more similar are the solid surface and liquid phase. If the liquid is water,
  66. 66. 36 the surface is referred to as hydrophilic in the case of a good wetting (0<90~ and hydrophobic in the case of a poor one (0>90~ Let us consider the behavior of a liquid in a thin capillary tube immersed into a vessel containing the same liquid. Let us also assume that the radius of this capillary tube is sufficiently small so that the meniscus is spherical. If the wetting of capillary walls with the liquid is good (0<90~ the curvature of fluid surface is negative (r<0), and the meniscus is concave (Fig. I-13). The pressure under such meniscus is lower (as compared to what it would be under the fiat surface) by the amount 2c~/r,and thus, the fluid will rise inside the capillary, until it reaches the level at which the capillary pressure is balanced by the hydrostatic one, i.e p~ - H(p'- p")g, where 9' and P" are the densities of fluid and its saturated vapor (or air), respectively; g is the acceleration of gravity, and H is the height of the fluid rise. The curvature of the fluid surface inside the capillary is determined by the degree of wetting, i.e. by the value of contact angle, 0. The radius of a meniscus curvature, r, is related to the radius of a thin capillary, r0, as r = - r0 / cos 0. The height of capillary rise is then p~ 2~ tacos 0 H - - . (I. 12) (p'- p")g r,,(p'- p")g
  67. 67. 37 i - - , . . . - m F0 Fig. I-13.Fluidrise in a capillarywithwell-wettedwalls The better the liquid wets the capillary walls, the higher its rise at a given value of CYLG.In the case of a non-wetting liquid (0>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
  68. 68. 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 r0 has a shape of surface of rotation, and can be characterized at each point by the two curvature radii r~ and r2(in Fig. I-14, a these radii are of opposite sign, i.e. r~>0 and r2<0), which are related to each other as 1/r~ + 1/r2 = const. If r 1<<r0, both r~ and r2 may be considered to be constant. For the case of perfect wetting (0=0~ the capillary attractive force F that one needs to overcome in order to separate the particles consists of two components. One of them is the capillary pressure force, F~: while the other force component, F 2, is the constituent of the surface tension, acting along the perimeter of wetting: F 2 = 2rtqcy, and the net force, F, is therefore
  69. 69. 1 a 39 Fig. 1-14.The shape of the meniscus is indicative of the strength of the capillaryattractive force, F, between two wettedparticles F-F~+ F2- 7vrl2 oil- 7-2). The value ofF depends significantly on the amount of liquid in the meniscus. As the volume of liquid decreases, e.g. during drying, the attractive force increases and reaches its maximum value, when r~ --'0 (the "vanishing"

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