S T U D I E S IN I N T E R F A C E S C I E N C E
Colloid and Surface C h e m i s t r y
STUDIES
IN I N T E R F A C E
SCIENCE
SERIES EDITORS D. M6bius
Vol. I Dynamics of Adsorption at Liquid Interfaces
Theory, Experiment, Application by S.S. Dukhin, G. Kretzschmar and R. Miller Vol. z An Introduction to Dynamics of Colloids by J.K.G. Dhont
Vol. 3 Interfacial Tensiometry by A.I. Rusanov and V.A. Prokhorov Vol. 4 New Developments in Construction and Functions of Organic Thin Films edited by T. Kajiyama and M. Aizawa
and
R. Miller
Vol. 8 Dynamic Surface Tensiometry in Medicine by V.N. Kazakov, O.V. Sinyachenko, V.B. Fainerman, U. Pison and R. Miller Vol. 9 Hydrophile-Lilophile Balance of Surfactants and Solid Particles
Physicochemical Aspects and Applications by P. M. Kruglyakov Vol. lo Particles at Fluid Interfaces and Membranes
Attachment of Colloid Particles and Proteins to Interfaces and Formation of TwoDimensional Arrays by P.A. Kralchevsky and K. Nagayama
Vol. 5 Foam and Foam Films by D. Exerowa and P.M. Kruglyakov Vol. 6 Drops and Bubbles in Interfacial Research edited by D. M6bius and R. Miller Vol. 7 Proteins at Liquid Interfaces edited by D. M6bius and R. Miller
Vol. 11 Novel Methods to Study Interfacial Layers by D. M6bius and R. Miller Vol. l z
Colloid and Surface Chemistry by E.D. Shchukin. A.V. Pertsov, E.A. Amelina ans A.S. Zelenev
Colloid and Surface Chemistry
E u g e n e D. S h c h u k i n The Johns Hopkins University, Department of Geography and Environmental Engineering, Baltimore, MD, USA and Moscow State University, Department of Chemistry, Moscow, Russia
Alexandr
V. P e r t s o v
Moscow State University, Department of Chemistry, Moscow, Russia
E l e n a A. A m e l i n a Moscow State University, Department of Chemistry, Moscow, Russia
A n d r e i S. Z e l e n e v ONDEO Nalco Company, Naperville, IL, USA
2001 ELSEVIER Amsterdam - London - New Y o r k - Oxford- Paris - Shannon - Tokyo
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PREFACE
This book covers major areas of modern Colloid and Surface Science (in some countries also referred to as Colloid Chemistry) which is a broad area at the intersection of Chemistry, Physics, Biology and Material Science investigating the disperse state of matter and surface phenomena in disperse systems. The book arises of and summarizes the progress made at the Colloid Chemistry Division of the Chemistry Department of Lomonosov Moscow State University (MSU) over many years of scientific, pedagogical and methodological work. The development of colloid science at Moscow State University and elsewhere in Russia was greatly influenced by the fundamental contributions to its major areas ([ 1-4] in the General Introduction) made by Professor Peter Aleksandrovich Rehbinder (1898 - 1972), Academician of the USSR Academy of Sciences, who chaired and led the Colloid Chemistry Division for more than 30 years. Rehbinder was a great enthusiast of colloid science and an excellent lecturer. The synopsis of his lecture course (published by Moscow State University in 1950) was for a long time used as a textbook by generations of students and still now serves as an example of the most clear, logical and broad coverage of the subject. From 1973 to 1994, the Colloid Chemistry Division was chaired by Rehbinder's closest collaborator and successor, Eugene D. Shchukin, Academician of the Russian, the US and the Swedish Academies of Engineering. Professor Shchukin designed a general lecture course in colloid chemistry, which he taught for many years at the Chemistry Department of
ii MSU, and continues to teach now at John's Hopkins University (JHU, Baltimore, MD, USA). The course includes all major areas of colloid science, covering the basic principles, certain quantitative details, and applications. From year to year the course content has undergone continuous changes in line with the latest developments in the field. The materials of this lecture course were worked up by the faculty of the Colloid Chemistry Division, Professor Rehbinder' s former students Professor Alexandr V. Pertsov and Docent Elena A. Amelina, and, with additional contributions written by them, formed the basis of the textbook entitled "Colloid Chemistry", the second edition of which was published in 1992 (see [5] in General Introduction). That book also included materials from a number of specialized courses designed by the authors at different times. The book became the major text used by students at educational institutions throughout Russia, where colloid chemistry is the mandatory part of the core curriculum in chemistry. On-going progress in colloid and surface science and new approaches in teaching, implemented in the courses taught at MSU, and in the course that by E.D. Shchukin currently teaches at JHU, inspired this new book. The preparation of the manuscript took place simultaneously in two languages: in English and Russian. The text written in Russian by Eugene D. Shchukin, Alexandr V. Pertsov and Elena A. Amelina was simultaneously translated into English by Dr. Andrei S. Zelenev (a former graduate student of Professor Egon Matijevi6), who made significant and substantial contributions to the content of the book. The topics written by Dr. Zelenev include the sections on analytical chemistry of surfactants, transfer of sound in disperse systems (acoustics, electroacoustics and their applications), photon correlation
iii spectroscopy, dynamic tensiometry, monodisperse colloidal systems, and other principal subjects. Among significant innovations in the presentation of material, the authors would like to emphasize the following. In contrast to the traditional separation of electrokinetics as
"specific" colloidal phenomena, and
molecular-kinetic and optical phenomena as "non-specific" ones, Prof. Pertsov combined these in a single chapter (Chapter V) based on the fact that all of these phenomena are examples of different transfer processes taking place in disperse systems. The same chapter includes a description of the scattering of light, as well as different methods of particle size distribution analysis based on transfer processes. The description of electrophoresis and other electrokinetic phenomena can also be found in Chapter V, while the theory of the electrical double layer is discussed much earlier, in Chapter III, which covers the adsorption phenomena. Special emphasis has been put on the description of phase equilibria in surfactant solutions and the investigation of properties of adsorption layers. The coverage of lyophilic colloidal systems, micelle formation, microemulsions, the structure of adsorption layers, structure and properties of emulsions and foams has been expanded. The concepts of the theory of percolations, fractals, molecular dynamics, nanocluster and supramolecular chemistry were introduced. Dr. E. Amelina has completely changed the description of the interactions between dispersed particles, the measurements of these interactions, and the discussion of sedimentation analysis. The application of molecular dynamics and computer modeling to the description of characteristic colloidal phenomena has been illustrated.
iv
Professor Shchukin also performed general editing of the manuscript utilizing his experience in lecturing this course and paying special attention to the presentation of the concepts and applications of physical-chemical mechanics of disperse systems and materials, properties of the structurerheological barrier as a factor of strong stabilization, some features of lyophilic colloidal systems and other research areas, explored by Russian scientific schools and less known abroad. Although this book significantly differs from the earlier "Colloid Chemistry" textbook, it nevertheless focuses on the specifics of educational and research work carried out at the Colloid Chemistry Division at the Chemistry Department of MSU. Many results presented in this book represent the art developed in the laboratories of the Colloid Chemistry Division, in the Laboratory of Physical-Chemical Mechanics (headed by E.D. Shchukin since 1967) of the Institute of Physical Chemistry of the Russian Academy of Science, and in other research institutions and industrial laboratories under the guidance of the authors and with their direct participation. Special attention is devoted in the book to the broad capabilities that the use of surfactants offers for controlling the properties and behavior of disperse systems and various materials due to the specific physico-chemical interactions taking place at interfaces. At the same time the authors made every effort to avoid duplication of material traditionally covered in textbooks on
physical
chemistry, electrochemistry, polymer chemistry, etc. These include adsorption from the gas phase on solid surfaces (by microporous adsorbents), the structure of the dense part of the electrical double layer, electrocapillary phenomena, specific properties of polymer colloids, and some other areas.
Material related to these subjects is presented only to the extent consistent with its relevance to colloid chemistry. The authors made every effort to ensure the proper subdivision of the principal material and additional information. The main principles are discussed mostly on a semi-quantitative and in some cases even qualitative levels. This material is presented using the regular base font. Detailed quantitative derivations and other more cumbersome issues are given in fine print. Newly introduced terms are usually given in italic, while words and phrases of special importance are given with larger letter spacing. Because of the interdisciplinary nature of colloid science and the close links between different topics, references to preceding and subsequent chapters are given throughout the book. The authors believe that this helps in emphasizing the interconnectedness between different topics. In correspondence with the detrimental role that interfacial phenomena play in the formation and stability of disperse systems, the book starts with the description of phenomena at interfaces separating phases that differ by their phase state (Chapters I-III). Then the formation (Chapter IV), properties (Chapters V-VI), and stability (Chapters VII-VIII) of disperse systems are covered. The last chapter (Chapter IX) in the book is devoted to the principles of physical-chemical mechanics, the part of colloid science in the development of which the scientific school established by Rehbinder and Shchukin played the leading role. The current literature in Colloid and Surface Science is broadly represented by the art developed by many well-known scientific schools and published in various journals, series of monographs and books listed in the
vi general introduction. These materials may serve as good sources of additional information on both the details related to particular topics and the course content as a whole. If used as a textbook, this book is primarily suitable for university students majoring in Chemistry and Chemical Engineering who take courses in colloid and surface science. The authors believe that the book will also be useful to graduate students, engineers, technologists, and academic and industrial scientists working in the areas that deal with the applications related to disperse systems and interfacial phenomena. The authors are grateful to Professor Boris D. Summ, the head of the Colloid Chemistry Division of the Chemistry Department at MSU, Professor Victoria N. Izmailova, and to all faculty and colleagues at MSU and in the Department of Geography and Environmental Engineering at JHU for their valuable comments related to the content and teaching of the course in Colloid Chemistry. The authors would also like to thank Professors Reinhard Miller (MaxPlanck Institute, Potsdam/Golm, Germany), Egon Matijevid, Larry Eno (Clarkson University, Potsdam, NY, USA), Dr. Niels Ryde (Elan Pharmaceutical, Inc., King of Prussia, PA, USA), and Dr. Andrei Dukhin (Dispersion Technology, Inc., Mt. Kisco, NY, USA) for valuable comments, suggestions and discussions. The authors are especially indebted to Mr. Harald Hille for his commitment, patience and professional help in editing and proofreading the manuscript. His participation was truly critical, since none of the authors are the native speakers of English. The authors express their most sincere
vii appreciation to Ms. Kristina Kitiachvili (University of Chicago, Chicago, IL, USA) for her help in preparing camera-ready manuscript. Help of Mr. Alexei Zelenev and Dr. Peter Skudarnov is also appreciated.
viii CONTENTS PREFACE
GENERAL INTRODUCTION I. SURFACE P H E N O M E N A AND THE STRUCTURE OF INTERFACES IN O N E - C O M P O N E N T SYSTEMS
I. 1. Introduction to the Thermodynamics of the Discontinuity Surface in a Single Component System 1.2. The Surface Energy and Intermolecular Interactions in Condensed Phases 1.3. The Effect of the Interfacial Curvature on the Equilibrium in a Single Component System 1.3.1 The Laplace Law 1.3.2. The Thomson (Kelvin) Law 1.4. Methods Used for the Determination of the Specific Surface Free Energy References List of Symbols II. THE ADSORPTION PHENOMENA. STRUCTURE AND PROPERTIES OF A D S O R P T I O N LAYERS AT THE LIQUID-GAS INTERFACE
II. 1. Principles of Adsorption Thermodynamics. The Gibbs Equation II.2. Structure and Properties of the Adsorption Layers at the Air-Water Interface II.2.1. The Dilute Adsorption Layers II.2.2. Langmuir and Szyszkowski Equations. Accounting for the Adsorbed Molecules Own Size (Mutual Repulsion) II.2.3. Structure and Properties of Saturated Adsorption Layers II.3. Classification of Surface Active Substances. The Assortment of Synthetic Surfactants II.4. Analytical Chemistry of Surfactants References List of Symbols
i
xii
1
2 13 31 31 40 44 59 61
64 65 84 84 97 112 131 144 160 162
III. INTERFACES BETWEEN CONDENSED PHASES. WETTING 165 III. 1. The Interfaces Between Condensed Phases in Two-component Systems 166 III.2. Adsorption at Interfaces Between Condensed Phases 176 III.3. Adsorption of Ions. The Electrical Double Layer (EDL) 193 III.3.1. Basic Theoretical Concepts of the Structure of Electrical Double Layer 194 III.3.2. Ion Exchange 214 III.3.3. Electrocapillary Phenomena 220 III.4. Wetting and Spreading 225 III.5. Controlling Wetting and Selective Wetting by Surfactants 244 III.6. Flotation 250
ix References List of Symbols IV. THE FORMATION OF DISPERSE SYSTEMS
IV. 1. Thermodynamics of Disperse Systems: the Basics IV.2. Thermodynamic Principles of the Formation of New Phase Nuclei IV.2.1. General Principles of Homogeneous Nucleation According to Gibbs and Volmer IV.2.2. Condensation of the Supersaturated Vapor IV.2.3. Crystallization (Condensation) from Solution IV.2.4. Boiling and Cavitation IV.2.5. Crystallization from Melt IV.2.6. Heterogeneous Formation of a New Phase IV.3. Kinetics of Nucleation in a Metastable System IV.4. The Growth Rate of Particles of a New Phase IV.5. The Formation of Disperse Systems by Condensation IV.6. Ultradisperse Systems. Supramolecular Chemistry IV.7. Dispersion Processes in Nature and Technology References List of Symbols V. TRANSFER PROCESSES IN DISPERSE SYSTEMS
V. 1. Concepts of Non-Equilibrium Thermodynamics as Applied to Transfer Processes in Disperse Systems. General Principles of the Theory of Percolations V.2. The Molecular-Kinetic Properties of Disperse Systems V.2.1. Sedimentation in Disperse Systems V.2.2. Diffusion in Colloidal Systems V.2.3. Equilibrium Between Sedimentation and Diffusion V.2.4. Brownian Motion and Fluctuations in the Concentration of Disperse Phase Particles V.3. General Description of Electrokinetic Phenomena V.4. Transfer Processes in Free Disperse Systems V.5. Transfer Processes in Structured Disperse Systems (in Porous Diaphragms and Membranes) V.6. Optical Properties of Disperse Systems: Transfer of Radiation V.6.1. Light Scattering by Small Particles (Rayleigh Scattering) V.6.2. Optical Properties of Disperse Systems Containing Larger Particles V.7. Transfer of Ultrasonic Waves in Disperse Systems. Acoustic and Electroacoustic Phenomena V.7.1. Theoretical Principles of Ultrasound Propagation Through Disperse Systems (Acoustics) V.7.2. Electroacoustic Phenomena V.8. Methods of Particle Size Analysis
255 257 260 261 273 273 279 280 280 282 284 289 295 300 311 313 316 318 320
321 327 329 329 333 337 349 361 373 390 390 402 408 409 417 421
V.8.1. Sedimentation Analysis V.8.2. Sedimentation Analysis in the Centrifugal Force Field V.8.3. Nephelometry. Ultramicroscopy V.8.4. Light Scattering by Concentration Fluctuations V.8.5. Photon Correlation Spectroscopy (Dynamic Light Scattering) V.8.6. Particle Size Analysis by Acoustic Spectroscopy References List of Symbols VI. LYOPHILIC COLLOIDAL SYSTEMS VI. 1. The Conditions of Formation and Thermodynamic Stability of Lyophilic Colloidal Systems VI.2. Critical Emulsions as Lyophilic Colloidal Systems VI.3. Micellization in Surfactant Solutions VI.3.1. Thermodynamics of Micellization VI.3.2. Concentrated Dispersions of Micelle-Forming Surfactants VI.3.3. Formation of Micelles in Non-Aqueous Systems VI.4. Solubilization in Solutions of Micelle-Forming Surfactants. Microemulsions VI.5. Lyophilic Colloidal Systems in Polymer Dispersions References List of Symbols
426 431 435 438 442 452 454 456 461 462 468 472 476 483 486 487 498 502 504
VII. GENERAL CAUSES FOR DEGRADATION AND
RELATIVE STABILITY OF LYOPHOBIC COLLOIDAL SYSTEMS VII. 1. The Stability of Disperse Systems with Respect to Sedimentation and Aggregation. Role of Brownian Motion VII.2. Molecular Interactions in Disperse Systems VII.3. Factors Governing the Colloid Stability VII.4. Electrostatic Component of Disjoining Pressure and its Role in Colloid Stability. Principles of DLVO Theory VII.5. Structural-Mechanical Barrier VII.6. Coagulation Kinetics VII.7. The Influence of Isothermal Mass Transfer (Ostwald Ripening) on the Decrease in Degree of Dispersion References List of Symbols VIII. STRUCTURE, STABILITY AND DEGRADATION OF VARIOUS LYOPHOBIC DISPERSE SYSTEMS VIII. 1. Aerosols VIII.2. Foams and Foam Films VIII.3. Emulsions and Emulsion Films VIII.4. Suspensions and Sols VIII.5. Coagulation of Hydrophobic Sols by Electrolytes
506 507 521 536 543 556 561 571 577 580
583 584 596 607 624 629
xi VIII.6. Detergency. Microencapsulation VIII.7. Systems with Solid Dispersion Medium References List of Symbols IX. PRINCIPLES OF PHYSICAL-CHEMICAL MECHANICS
IX. 1. Description of Mechanical Properties of Solids and Liquids IX.2. Structure Formation in Disperse Systems IX.3. Rheological Properties of Disperse Systems IX.4. Physico-Chemical Phenomena in Processes of Deformation and Fracture of Solids. The Rehbinder Effect IX.4.1. The Role of Chemical Nature of the Solid and the Medium in the Adsorption-Caused Decrease of Material Strength IX.4.2. The Role of External Conditions and the Structure of Solid in the Effects of Adsorption Action on Mechanical Properties of Solids IX.4.3. The Application of Rehbinder's Effect References List of Symbols SUBJECT INDEX
636 641 642 646 649 651 665 689 702
705
715 723 728 731 733
xii GENERAL INTRODUCTION Colloid Chemistry or, alternatively, Colloid and Surface Science, are the established and traditionally used names of the field of science devoted to the investigation of substances in dispersed state with particular attention to the phenomena taking place at interfaces. Peter A. Rehbinder defined colloid chemistry as the "chemistry, physics, and physical chemistry of disperse systems and interfacial phenomena" [1-6]. The dispersed state and interfacial phenomena can not be separated from each other, as interracial phenomena determine the characteristic properties of disperse systems as well as the means by which one can control such properties. In most chemical disciplines the properties of substances are usually considered within the framework of two "extreme" levels of organization of matter: the macroscopic level, which deals with the properties of continuous homogeneous phases, and the microscopic level, dealing with the structure and properties of individual molecules. In reality, material objects (both natural and man-made products and materials) exist, in nearly all cases, in the
dispersed state, i.e. contain (or consist of) small particles, thin films, membranes and filaments with characteristic interfaces between these microscopic phases. As a rule, the dispersed state is the necessary condition required for the functioning and utilization of real objects. This is especially true for living organisms, the existence of which is governed by the structure of cells and by processes taking place at the cellular interfaces. One of the main objectives of colloid and surface science is the investigation of peculiarities in the structure of systems related to their
xiii dispersed state. Heterogeneous systems (and primarily microheterogeneous systems consisting of two or more phases) in which at least one phase is present in the dispersed state, are referred to as disperse systems. The small particles associated with the dispersed state can still be viewed as phase particles, since they are the carriers of properties close to those of the corresponding macroscopic phases and have characteristic interfaces. Usually the disperse system is characterized as an ensemble of particles of dispersed
phase, surrounded by the dispersion medium. One of the central tasks of colloid science is the investigation of changes in the properties of systems due to changes in their degree of dispersion. If the shape of particles forming the system is more or less close to isometric, the extent of dispersion fineness can be characterized by the particle linear dimension (some effective or mean radius, r), degree of
dispersion (or simply dispersion), D, and the specific surface area, S~. The degree of dispersion is determined as the ratio of the total surface area of particles forming the dispersed phase (at the interface between the dispersed phase and the dispersion medium) to the total volume of these particles. The specific surface area is defined as the ratio of the total surface area of all particles to the total mass of these particles, i.e. S~ - D/9, where 9 is the density of the substance forming the dispersed phase. For the monodisperse system consisting of uniform spherical particles of radius r, one can write that
D = 3/r ; for systems consisting of particles of shapes other than spherical the inverse proportionality between dispersion or specific surface area and the particle size will be maintained with a different numerical coefficient. A more complete description of the dispersion composition of the
xiv disperse system is based on the investigation of the particle size distribution function (for anisometric particles, also the particle shape distribution function). The breadth of the distribution function characterizes the system polydispersity. The range of disperse systems of interest in colloid science is very broad. These include coarse disperse systems consisting of particles with sizes of 1 gm or larger (surface area S < 1 m2/g), and fine disperse systems, including ultramicroheterogeneous colloidal systems with fine particles, down to 1 nm in diameter, and with surface areas reaching 1000 m2/g ("nanosystems"). The fine disperse systems may be both structured (i.e. systems in which particles form a continuous three-dimensional network, referred to as the disperse structure), and free disperse, or unstructured (systems in which particles are separated from each other by the dispersion medium and take part in Brownian motion and diffusion). Based on the aggregate states of the dispersed phase and the dispersion medium one can recognize different kinds of disperse systems, which can be described by the abbreviation of two letters, the first of which characterizes the aggregate state of the dispersed phase, and the second one that of the dispersion medium. In these notations gaseous, liquid and solid states are labeled as G, L, and S, respectively. In the case of two phase systems, one can outline eight different types of disperse systems, as shown in the table below. S y s t e m s with a liquid d i s p e r s i o n m e d i u m represent a broad class of dispersions. The main portion of the book is devoted to these objects, the examples of which include various systems with a solid dispersed phase (S/L type), such as finely dispersed sols (in the case of unstructured systems)
XV
TABLE. Different types of disperse systems
~
Medium
Solid
Liquid
Gas
Solid
Sl/S 2
S/L
S/G
Liquid
L/S
L1/L2
L/G
Gas
G/S
G/L
Dispersed Phase
and gels (in the case of structured systems), coarsely dispersed lowconcentrated suspensions, and concentrated pastes. Dispersions with a liquid dispersed phase (L~/L2 systems) are the emulsions. Dispersions in which the dispersed phase is in a gaseous state include gas emulsions (systems with low dispersed phase concentration) and foams. Systems with a gaseous d i s p e r s i o n medium, known under the common name of aerosols, include smokes, dusts, powders (systems of S/G type) and fogs (L/G type systems). Aerosols containing both solid particles and liquid droplets of dispersed phase are referred to as smogs. Since gases are totally miscible with each other, the formation of disperse systems of G~/G2 type is impossible. Nevertheless, even in the mixtures of different gases one can encounter non-uniformities caused by the fluctuations in density and concentration. Systems with a solid d i s p e r s i o n m e d i u m are represented by rocks, minerals, a variety of construction materials. Most such systems are of the S~/S2 types. Various synthetic and natural porous materials (with closed porosity), such as pumice and solid foams (e.g. styrofoam, bread), belong to the G/S type. The systems of L/S type include natural and synthetic opals and
xvi pearl. One can also classify (rather conditionally) cells and living organisms formed with these cells as L/S-type systems. It is worth outlining here that the subdivision of disperse systems according to dispersed phase and dispersion medium ~ is, strictly speaking, valid only for systems in which the dispersed phase is formed with individual particles. There are, however, a large number of systems in which both phases are continuous and pierce each other. Such systems, referred to as
bicontinuous, include porous solids with open porosity (catalysts, adsorbents, zeolites), various earths and rocks, including oil-containing ones. Gels and jellies forming in polymer solutions, including those that are glue-like (the word "colloid" means "glue-like", from Greek ~:c0kka- glue), are also quite close to bicontinuous systems. The principal peculiarity of fine disperse systems is the presence of highly developed interfaces. These interfaces and the interfacial phenomena occurring at them affect the properties of disperse systems, primarily due to the existence of excessive surface (interfacial) 2 energy associated with interfaces. The excess of interfacial energy reveals its action along the interface in the form of interfacial tension, which tends to decrease interfacial
~In some cases dispersion medium is referred to as the continuous phase 2 The terms "surface" and "interface" are not exactly equivalent. One usually refers to an interface when describing the boundary between condensed phases or between condensed phase and a gas (e.g. solution-air interface), while the term surface is attributed specifically to a border of a condensed phase with either vacuum or gas. However, due to their obvious similarity, these two terms have been used interchangeably. In this book we will continue applying this commonly accepted practice and in many instances will use them as synonyms
xvii area. At the same time, the surface energy is directly related to surfaceforces. The force field of these forces may maintain considerable strength, even at distances from the surface significantly larger than molecular dimensions. The existence of developed surfaces in systems consisting of fine particles results in the need of external energy for the formation of such systems by both the comminution (dispersion) of macroscopic phases and condensation from homogeneous systems. The excessive interfacial energy is the reason for the higher chemical activity of dispersed phases in comparison with macroscopic phases. The result of this higher activity is increased solubility of the dispersed phase in the dispersion medium and an increase in the vapor pressure above the surface of fine particles. The smaller the particle size, the greater the increase in the vapor pressure. The elevated chemical activity and the availability of strongly developed interfaces are the reasons for the high rates of interactions between the dispersed phase and the dispersion medium, and the high rates of mass and energy transfer between them in heterogeneous chemical interactions. The presence of surface forces that lead to changes in the structure and composition of interfaces may have a great influence on these transfer processes. A high free energy excess, particularly in systems with a fine degree of dispersion, is the cause of thermodynamic instability, which is the most important feature of a majority of disperse systems. Thermodynamic instability in turn entails various processes aimed at decreasing the surface energy, which results in the saturation of surface forces. Such processes may occur in a number of ways. For example, in a free disperse system partial saturation of the surface forces may take place in the contact zone between the
xviii particles when the latter approach each other closely, resulting in the formation of aggregates. This phenomenon, referred to as coagulation, corresponds to the transition from a free disperse system to a structured one. A further decrease in the surface energy of disperse system may be caused by a decrease in the interfacial area due to the coalescence of drops and bubbles, or by fusion (sintering) of solid particles, as well as by the dissolution of more active smaller particles with the transfer of substance to less active larger particles. Destabilization due to coagulation, coalescence and diffusional mass transfer leads to changes in the structure and properties of disperse systems. It is important to point out that due to coagulation and bridging of particles, a disperse
system acquires
qualitatively new structural-mechanical
(rheological) properties which entail a conversion of the disperse system into a material. In the end, coalescence may result in the disintegration of a disperse system into constituent macroscopic phases. In a number of applications such degradation of colloidal systems is a desirable goal, as, e.g., in making butter by churning, or dehydration and desalination of crude oil. Along with the classification of disperse systems based on the phase state ofthe dispersed phase and the dispersion medium, and their classification as coarse dispersed or colloidal, structured or unstructured, dilute or concentrated, one can also subdivide disperse systems into lyophilic or lyophobic types. Systems belonging to these principally different classes differ in the nature of colloid stability and in the intensity of interfacial intermolecular interactions. High degree of similarity between the dispersed phase and the dispersion medium, and, consequently, compensation of the
xix interactions at the interface (which usually results in very low values of interfacial free energy) is characteristic of lyophilic disperse systems. These systems, e.g. critical emulsions, may form spontaneously and reveal complete thermodynamic stability with respect to both aggregation into a macrophase and dispersion down to particles of molecular size. In various lyophobic systems (colloidal and coarse disperse), there is a lot less similarity between the dispersed phase and the dispersion medium; here the difference in the structure and properties of contacting phases results in uncompensated interfacial forces (energy excess). Such systems are thermodynamically unstable and require special stabilization. All aerosols, foams, numerous emulsions, sols, etc., are examples of lyophobic systems. Along with typical lyophobic and lyophilic systems, there is a broad range of states which with respect to the nature of their stability can be viewed as intermediate. In controlling the stability of disperse systems, the adsorption of
surface-active substances (surfactants) at the interfaces represents a very important way of decreasing the free energy of the system without decreasing the interfacial area. The adsorption of surfactants results in a partial compensation of unsaturated surface forces. Surface active substances, when introduced into the bulk, spontaneously accumulate at the interface, forming adsorption layers. Adsorption monolayers may radically alter properties of interfaces and the type of acting surface forces. Change in the surface forces may also occur with changes in the electrolyte composition of the dispersion medium due to the effect of electrolyte on the structure of the interfacial
electrical double layer. The use of electrolytes and surfactants allows one to effectively control
XX
the formation and degradation of disperse systems and influence their stability, as well as their structural-mechanical and other properties. Surfactants participate in a variety of microheterogeneous chemical, biochemical and physiological processes, such as micellar catalysis, exchange processes, phenomena involved in membrane permeability, etc. The control of the stability of disperse systems plays a crucial role in many technological applications. It is necessary to point out that finely dispersed state of substance is the primary condition for a high degree of organization of matter. Fine disperse structure is the basis for the strength and durability of materials, such as steel, ceramics and others, and for the strength of tissues in plants and live organisms. Heterogeneous chemical reactions in both industry and living organisms take place only at highly developed interfaces, i.e. in finely dispersed systems. Only fine disperse structure consisting of many tiny cells allows an enormous amount of information to be stored in small physical volumes. This relates to both the human brain and new generations of computers. Since the tendency towards lowering the excess of surface energy in disperse systems may take the form of various types of degradation of such systems, the problem of colloid stability is the central problem, not only in colloid and surface science but in all natural sciences as well. Along with factors responsible for the stabilization of different disperse systems, the conditions necessary for the formation of such systems from macroscopic phases are also part of colloid stability studies. It is clear from everything said so far that colloid and surface science
xxi is a peculiar border area of science that has resulted from interdisciplinary interaction between chemistry, physics, biology and other related areas of science during the gradual process of genesis, separation, differentiation and merging between different areas. This has been very well reflected in the recent book by Evans and Wennerstr6m [7]. Colloid chemistry is closely related to the investigation of the kinetics of interfacial electrochemical processes, microheterogeneity (origination of new phases and structures) in dispersions of natural and synthetic polymers, sorption and ion exchange processes in ultramicroporous systems. It is also closely related to such areas of science as solid state physics and chemistry, molecular physics, material mechanics, rheology, fluid mechanics, etc. All of this determines the fundamental theoretical development and heavy involvement of mathematics in various parts of colloid and surface science, with broad use of the methods of chemical thermodynamics and statistics, the thermodynamics of irreversible processes, electrodynamics, quantum theory, the theory of gaseous and condensed states of substance, structural organic chemistry, the statistics of macromolecular chains, molecular dynamics, methods of various numerical simulation involving high-speed computers, etc. Close interaction between colloid science and other related disciplines helped in the establishment and further enrichment of its experimental basis. Along with classical experimental methods specific to colloid science (determination of the surface tension, ultramicroscopy, dialysis and ultrafiltration, dispersion analysis and porosimetry, surface forces and measurements of particle interactions, studies of the scattering of light, etc.), such methods as various spectroscopic techniques (NMR, ESR, UF and IR
xxii
spectroscopy, luminescence quenching, multiply disrupted total internal reflection, ellipsometry), X-ray methods, radiochemical methods, all types of electron microscopy, are all effectively used in the investigation of disperse systems and interfacial phenomena. The methods of surface studies involving atomic force microscopy, slow electrons, and spectroscopy of secondary ions are also broadly used. The use of these and other methods aids have assisted in solving the main problems of colloid science aimed at the understanding of the nature and mechanisms of interfacial phenomena and processes at the atomic and molecular levels. The specific interdisciplinary nature of colloid science makes it of fundamental importance for such adjoining sciences as biology, soil science, geology and meteorology. Colloid and surface science forms the general physico-chemical basis of modern technology in nearly all areas of industry, including chemical, oil, mining, production of construction, instrumental, and composite materials, pulp and paper, printing, food, pharmaceuticals, paint and numerous other areas. It is very important in agriculture for solving problems related to increasing the soil fertility, application of pesticides and herbicides, etc. Colloid science also plays an important role in handling numerous environmental problems, such as waste water treatment, trapping of aerosols, fighting soil erosion, etc. The close interaction of colloid and surface science with molecular physics and a number of theoretical disciplines has determined its role in the development of natural sciences as a whole. The discovery of the nature of, and the further investigation of Brownian motion, the development of direct
xxiii methods for the determination ofAvogadro' s number, the development of the theory of fluctuations and their studies led to the experimental conformation of the molecular structure of matter and of the limits of applicability of the second law of thermodynamics. Colloid science has established new approaches to the studies of the geological history of the Earth's crust, the origin of life, and mechanisms of vital functions. The work of Thomas Graham (circa 1760) marks the birth of colloid chemistry as an independent branch of science. Like other areas, colloid chemistry has its own long history" some specific colloid-chemical recipes were known to the ancient Egyptians and medieval alchemists. J. Gibbs, W. Thomson (Kelvin), J. Maxwell, A. Einstein, J. Perrin, T. Svedberg, G. Freundlich, I. Langmuir, M. Poliani, S. Brunauer, and other great physicists and chemists took active part in developing understanding and knowledge in various areas of colloid chemistry. The results of their work are reflected throughout this book. In this book the authors acknowledge and pay special attention to the views on general and specific problems of colloid chemistry developed by Russian scientists and the different scientific schools founded by them. Among the great scientists who made significant contributions to the area and are less known to the world scientific community, one should name F.F. Reiss, famous for his discovery of electrokinetic phenomena, A.V. Dumansky, the inventor of a centrifuge and the organizer of the first scientific journal on colloid chemistry, also known for his studies on biopolymers as lyophilic colloidal systems, N.A. Shilov, M.M. Dubinin, A.V. Kiselev (theory of adsorption), I.I. Zhukov (electrosurface phenomena), N.P. Peskov (stability
xxiv and structure ofmicelles ofhydrophobic sols). Another great contributor to the study of adsorption layers, adsorption, and other areas of colloid chemistry was A.N. Frumkin, who also played a pioneering role in the development of modern electrochemistry. B.V. Derjaguin and his associates developed the theory of disjoining pressure and its major components as the principal thermodynamic factor in the stability of colloidal systems. In collaboration with L.D. Landau, B.V. Derjaguin created the modern theory of the stability and coagulation of hydrophobic sols by electrolytes. This theory was independently (and somewhat later) developed by the Dutch scientists W. Vervey and J. Overbeek and is now commonly known as the DLVO theory. P.A. Rehbinder and his scientific school played an important role in developing a number of pioneering ideas of modern colloid and surface science. Among them are the fundamental concepts of different mechanisms of surfactant action at various interfaces, particularly those concerning the formation and properties of structural-mechanical barrier as the factor of strong stabilization of disperse systems; the notion of formation of spatial structures in disperse systems due to the aggregation of particles; the discovery of the influence of the surface-active media on the mechanical properties of solids (Rehbinder's effect). The principal result of the development of Rehbinder's ideas was the creation of Physical-Chemical Mechanics, a new area of colloid chemistry. Chapter IX of this book is devoted specifically to the teachings of Rehbinder and the progress in physical-chemical mechanics achieved by his successors. The current literature in the area of colloid and surface science and interfacial phenomena represents the knowledge and techniques developed in
XXV
the leading scientific schools of the world. Numerous articles regularly appear in such specialized periodicals as the Journal of Colloid and Interface Science, Colloids and Surfaces, Langmuir, Advances in Colloid and Interface Science, Colloid Journal, Journal of Dispersion Science and Technology, Colloid and Polymer Science, Current Opinion in Colloid and Interface Science and others. There are series of monographs, including Surface and Colloid Science (edited by E. Matijevid), Studies in Interface Science (edited by D. M6bius and R. Miller), Surfactants Science Series (founding editor M. Schick), Progress in Colloid and Polymer Science, etc, and many textbooks and monographs [628]. The knowledge published in these books and periodicals will be extensively referenced throughout this book.
References ~
,
,
4. 5. ,
~
Q
Rehbinder, P.A., "Selected Works", vol. 1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Rehbinder, P.A., "Selected Works", vol. 2, Surface Phenomena in Disperse Systems. Physical Chemical Mechanics, Nauka, Moscow, 1979 (in Russian) Shchukin, E.D., Proc. Acad. Sci. USSR, Chem Sci., 10 (1990) 2424 Shchukin, E.D., Colloid J. 61 (1999) 545 Academician Pjotr Aleksandrovich Rehbinder: the Centenary, Moscow, Noviy Vek, 1998 (in Russian) Shchukin, E.D., Pertsov, A.V., Amelina, E.A., Colloid Chemistry, 2nd ed., Vysshaya Shkola, Moscow, 1992 (in Russian) Evans, D.F., Wennerstr6m, H., The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet, 2nded., Wiley-VCH, New York, 1999 Kruyt, H.R. (ed.), Colloid Science, vols.l-2, Elsevier, Amsterdam, 1952
xxvi ,
10. 11. 12.. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
28.
Stauff, J, Colloid Chemistry, Springer Verlag, Berlin, I960 (in German) Sheludko, A., Colloid Chemistry, Elsevier, Amsterdam, 1966 Kerker, M., Surface Chemistry and Colloids, Butterworth, 1975 Sonntag, H., Textbook on Colloid Science, VEB Deutsches Verlag der Wissenschafte, Berlin, 1977 (in German) Mysels, K.J., Introduction to Colloid Chemistry. Krieger, 1978 Voyutsky, S.S., Colloid Chemistry, Translated by N. Bobrov, Mir Publishers, Moscow, 1978 Frolov, Yu.G., A Course in Colloid Chemistry, Khimiya, Moscow, 1982 (in Russian) Vold, R.D., Vold, M.J., Colloid and Interface Chemistry, AddisonWesley, London, 1983 Fridrikhsberg, D.A., A Course in Colloid Chemistry, Translated by G. Leib, Mir Publishers, Moscow, 1986 Ross,S., Morrison, I.D., Colloidal Systems and Interfaces, WileyInterscience, New York, 1988 Everett, D.H., Basic Principles of Colloid Science, Royal Society of Chemistry, 1988 Adamson, A.W., Gast, A.P., Physical Chemistry of Surfaces,6 th ed., Wiley, New York, 1997 Hunter, R.J., Foundations of Colloid Science, vols.l,2, Clarendon Press, Oxford, 1991 Hunter, R.J., Introduction to Modern Colloid Science. Oxford University Press, 1994 Lyklema, J., Fundamentals of Interface and Colloid Science, vols. 1-3, Academic Press, 1991-2000 Mittal, K.L., Surface & Colloid Science in Computer Technology. Perseus Publishing, 1987 Shchukin, E.D. (Editor), Advances in Colloid Chemistry and PhysicalChemical Mechanics, Nauka, Moscow, 1992 (in Russian). Hiemenz, P.C., and Rajagopalan, R., Principles of Colloid & Surface Chemistry, 3rd ed., Dekker, New York, 1997 JGnsson, B., Lindman, B., Holmberg, K., and Kronberg, B., Surfactants and Polymers in Aqueous Solution, Wiley, Chichester, 1998 Borywko, M., (Editor), Computational Methods in Surface & Colloid Science, Marcel Dekker, New York, 2000
I.
SURFACE
PHENOMENA
AND
THE
STRUCTURE
OF
INTERFACES IN ONE-COMPONENT SYSTEMS
The difference in the composition and structure of phases in contact, as well as the nature of the intermolecular interactions in the bulk of these phases, stipulates the presence of a peculiar unsaturated molecular force field at the interface. As a result, within the interfacial layer the density of such thermodynamic functions as free energy, internal energy and entropy is elevated in comparison with the bulk. The large interface present in disperse systems determines the very important role of the surface (interfacial) phenomena taking place in such systems. According to Gibbs [1 ], one can view an interface as a layer of finite thickness within which the composition and thermodynamic characteristics are different from those in the bulk of phases in contact. This approach allows one to describe the properties of interfaces phenomenologically in terms of excesses of the
thermodynamic functions in the interfacial layer in
comparison with the bulk of individual phases. With this approach one does not need to introduce any model considerations regarding the molecular structure of the interfacial layer or utilize particular values of layer thickness.
1.1. Introduction to the Thermodynamics of the Discontinuity Surface in a Single Component System
In a single component system two phases (e.g. liquid and vapor) coexist in equilibrium only if there is a stable interface present between them. Such an interface is formed only if an increase in the surface area results in an increase in the system free energy, i.e. d,9~7dS>0. One may thus introduce the surface free energy, ~z-s ,as the free energy excess, proportional to the interfacial surface area: dS~rs-s
-
dS
-
where cy is the specific surface free energy. This specific surface free energy can be viewed as the work required for a reversible isothermal formation of a unit interface. The existence of a force that tends to decrease the interfacial area can be visualized from an experiment designed by A. Dupr6, schematically illustrated in Fig. I-1. In this experiment a rigid frame of wire with one movable side of length d is dipped into a soap solution, resulting in the formation of a thin film on the wire. Let the force F~ be applied to the sliding wire in the direction shown in Fig. I-1. The displacement of the wire by an amount Al causes an increase in the film area equal to Ald. Therefore, the free energy increases by the amount A g s - 2cyAld (the numerical coefficient is due to the film having two sides). From these considerations it follows that the force F 2 acting on the wire and due to the film is given by
F2 -
The case when
Al
= 2cyd
F~=Fz=2~dcorresponds
to an equilibrium between
these two forces. Consequently, cy can also be defined as the force per unit length of the frame. This force, acting along the interface in a direction perpendicular to the frame, is commonly referred to as the surface tension, and is expressed in mN/m or mJ/m 2 , assuming SI units. The action of the surface tension can be readily understood if we consider a series of forces acting on a film with a circular boundary. In Fig. I-2 these forces are labeled with arrows, and they have the effect of contracting the film towards its center. The length of the arrows corresponds to the magnitude of the forces, while the distance between them represents a unit length. d .
,
, , - . _ . _
I/ o I- o -<
Fig. I-1. A schematic drawing of A.Dupr6's experiment
\
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
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, according to Gibbs
or simply the discontinuity surface,
[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)
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. Changes in the free energy density within the discontinuity surface 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
dividing surface) somewhere
within the physical
discontinuity surface. Let an arbitrary prism, drawn in a direction perpendicular to the dividing surface, include volumes V'and VHat the 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,
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 g t o 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
-
S
=
[ f ( z ) - f ' ] dz + V'
[ f ( z ) - f " ] dz
(I.2)
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 cy only in the case of a particularly positioned dividing surface, corresponding to the position of the so-called
equimolecular surface (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 dividing surface.
gc(z) function profile, as well as on the placement of
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)
function within the discontinuity surface
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):
o-
(i'
-
gc')} dz +
V'
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 103 mJ/m 2 (mN/m), the average values o f p - PT - ~ / (8'+8") ~ 107 to 109 Pa (100 to 10000 atm). In other words, the
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."
11 (I.3) r I - - dcffdT, ~: = cy - T ( d o / d T)
(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 To exceeds 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.
12 A direct c o m p a r i s o n o f equations (I.4) and (I.5) yields
- cy o - o r ( T -
To) + a T -
cy o + otTo - c o n s t ,
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 mJ/m2
rI mJ/m2K
e mJ/m 2
5~ J/mol
1/4 ~ / Vm23 N A 1/3 mJ/m 2
H2
14.7
2.9
0.14
5
9.1xl02
2.8
N2
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 5.0x 104
300
Hg
273
480
0.22
540
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
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 hidden heat of interface formation qTwithin 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
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 2
A b/ll
,
where Z is the coordination number (a number of neighbors closest to a molecule under consideration) in the bulk of a condensed phase; N a 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, Z s, 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
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 from the gaseous phase into the 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 Z s 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 2
c r = - - Wc ~
1 nsZslu 1 ] 21
(I.6)
The molecular density at the surface is related to molar volume, Vm,
16 -2/3
and the volume per molecule, VM= Vm/ NA, as n s ~ VM
= (g m /
NA)
- 2/3
. The
surface tension is thus given by
cy ,~
V2m/3N ~3 Z '
(I.7)
where the Z s / 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 cy and ~ 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 cy is 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).
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
number of molecules per 1 m
3 )
m 3
(i.e. 8 n molecules, provided that n is a
from the bulk to the surface. Such treatment
allows us to write c y - k 1 (fm - f ' ) 8 , where k~ = 89in the present approach.
f
//
fm f (z) -6'
Fig. I-7. A schematicf(z) dependence for a non-volatile liquid-vapor system
18 The quantity 9U,, given by
9U=--= 8
kl ( f m - f ' ),
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/m 2, the values of 5U are as high as 10 7 - 10~~ / m 2, 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
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/m 2 = 1 J/m 3 ) 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
Vm
"~NA b 3, and the Stefan rule can also be written as 597
V m ,v
/ b, the heat of sublimation, 5r{ , is also of the same nature as 9g'. Consequently, one may write
~--
E ~ Pid
V
b
,,IPT[
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 e 2/4Zceob ~ 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
20 values by the number of atoms per 1 m 2 of cross-sectional area, n s = 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
4rtg0 b4
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 u=
al R"
i
bl Rm
(I.8)
The first term in the above expression corresponds to the attractive interaction between molecules, while the second one describes intermolecular repulsion.
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
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" a~ b/- _
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
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 a L ---hv0c~
4
2 M
,
where h is Plank's constant; v 0 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, v 0, 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
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 microscopic theory 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 crosssectional area S with those contained within the entire volume below plane
25 02. Such a summation can be well approximated by the integration with respect to four coordinates: z~, z 2, R 2 , and q0 (Fig. I-10). The result of such integration yields Umo~"
All Umo 1 -- - ~
12zth 2 '
(I.9)
where A~=Tt2n2aL is the Hamaker constant having the units of energy (J). Z1
S
O3
E
dzl
9
>
ZI
O1
s
R~2 f
('-4
E
Z2
9
>
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 o f , 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 d V~= Sdz~ interact in the same way with all molecules from
26 the volume element d V2, located at distance Rl2 from d V~. Consequently, one may write
1
gmol- -n2al I f l f Ri62dzldZ2 dR2 R2 dq~ R2q) Z1 Z2
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 q0 results in
All Umo 1 = - 2
7t
R2
IffR62 Z1 22
dz1 dz2 dR2
R2
From geometry it follows that RI22=R22 + (z I + 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
The result of this integration reads
Zz+dZ2,.
oo
All!
d(R~)dzldz2 R =o[R2+(Zl+Z2+h)2]
_
All 2~ (z 1 + z 2 + h) 4 dzldZ2
3-
The third integration with respect to z I 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): All Umo I = - ~ 12~h 2
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 gmo I in the h -~b limit. In this case h = h0 = b, and hence
1
1 2
Wc - (Y -- - - - Umo 1(b) ~
All 24~b 2
.
(I.
1O)
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-dispersion interactions, 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 o n components 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
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/m 2 and cyd~20 mJ/m 2. 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.
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, o f 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
30 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.
Ogb
"-
0
0
a
Fig. I-11. Schematic drawing of a grain boundary corresponding to a small disorientation angle 0 (a); specific free energy, % , as a function of the disorientation angle 0 (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.
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;
S are the volume and the surface area of the drop, respectively.
V and
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 = c y - 8V
(I.11)
For spherical particles 6 S - 8nr6r and 6 V = 4nr2~)r. Substitution of these expressions into eq. (I. 11) readily yields the Laplace law: 2cy --Pc)"
z
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
p o - 2o/r ~ 1.5x 105 Pa (1.5 atm.), which constitutes ~0.1% of the molecular
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
of tension". 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 nonspherical 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)
34
p~ - (5"
+
,
where r~ and r 2 are 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 <
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 x O z as
[1 + (dz/dx) 2]~ rl "-
d2z/dx 2
[1 + (dz/dx)2 ] ~ r2 ~
dz/dx
and
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 liquidgas 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 of wetting, 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,
36 the surface is referred to as and
hydrophobic in the
hydrophilic in the case
of a good wetting (0<90~
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 ' where 9' and
P" are
p")g,
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 = - r 0 / cos 0. The height of capillary rise is then
H -
p~ (p'- p")g
-
2~ tacos 0 r, , (p' - p " ) g
.
(I. 12)
37
F0
i
- - , . . . -
m
Fig. I-13. Fluid rise in a capillary with well-wetted walls 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
38 of mercury forced into the pores of a sample of known weight as a function of applied external pressure. In order to force mercury into nanometer-sized pores, the applied pressure has to be on the order of 108 - 109 Pa (1000 - 10000 atmospheres). An interesting phenomenon based on capillarity is the appearance of a capillary attractive force between particles of moistened solids. As a result of wetting, a meniscus is formed upon the particle contact (Fig. I-14). This meniscus between two contacting particles of radii r 0 has a shape of surface of rotation, and can be characterized at each point by the two curvature radii r~ and r 2 (in Fig. I-14, a these radii are of opposite sign, i.e. r~>0 and
r2<0),
which are related to each other as 1/r~ + 1/r 2 = const. If r 1<
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
39 a
1
Fig. 1-14. The shape of the meniscus is indicative of the strength of the capillary attractive force, F, between two wetted particles
F-F~+
F 2 - 7vrl2
oil- 7-2).
The value o f F 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"
40 meniscus 2 ). In this case r 2
-
-r12/2ro, and F = 2~r0o , as can be easily verified
from Fig. I-14, a. When the amount of liquid increases to the extent that a cylindrical meniscus with r 2
~
oo
and r~ ~ r 0 is formed (Fig. 1-14, b), the
attractive force decreases to the value F =~r0o. Furthermore, the attractive force vanishes completely (F =0), when r~-r2=2r0 (Fig. I-4, c). The capillary attractive force
is responsible for the commonly
observed strong adhesion between slightly moistened sand grains, as well as for much the weaker attachment between them at high water content. Capillary phenomena play an important role in determining the extent of particle adhesion and thus are critical for the stability and mechanical properties (see Chapter IX) of various clays, pastes, and powders.
1.3.2. The Thomson (Kelvin) Law According to the Laplace law, the pressure in a liquid drop increases with increasing interracial curvature. The increase in pressure causes an increase in the chemical potential of the liquid equal to A~t', which, assuming that the liquid is incompressible, can be written as 20 A~' = ~ V m , r
(I.13)
where Vmis the molar volume of liquid phase, r is the curvature radius of the
2 It is important to realize that this discussion implies the existence of a continuous phase. The expression may no longer be valid when the meniscus radius becomes comparable with molecular dimensions b; the Laplace equation was derived assuming r>>b
41 drop (of the surface of tension, to be more exact). For the liquid to remain in equilibrium with its vapor, the chemical potential g" of the latter should experience an increase by exactly the same increment, i.e. Ag"=AIa', which means that the equilibrium vapor pressurep(r) over the curved interface should be higher than that over the flat interface P0. If the vapor follows the ideal gas law, its chemical potential increment can be written as Ag" - RTln p ( r ) . P0
(I. 14)
Setting the expressions for Ag' and Ag" equal to each other, one obtains the Thomson (Kelvin) equation, which describes the equilibrium between liquid and vapor, separated by a curved interface"
P(r) - p~ exp ( 2c~ Vm '
(I.15)
which can be approximated with
p(r)~po
1+
2~ Vm'] p(r) - Po 2cy Vm ) i.e. ~---rRT ' Po r RT
The Thompson (Kelvin) equation clearly indicates that the smaller the drop radius, the higher the equilibrium vapor pressure above it. The value of R T / V mis close to that of the molecular pressure o~'and consequently, the ratio 2CYVm/ r R T has the same order of magnitude as the ratio of the capillary pressure p, to the molecular pressure 5U,, i.e.
42
2Vm
_+P~ m
rRT
For drops having a radius of 1 gm this ratio does not exceed 10-3; the Kelvin equation is applicable to all systems with the exception of those in which the drop size is comparable to molecular dimensions. It is noteworthy
that
the
Gibbs-Freundlich-Ostwald
equation,
describing the size dependence of the solubility c(r) of drops or solid crystals is similar to eq.(I. 15):
c ( r ) - c 0 exp / 2cYVm/ rRT '
where Co is the solubility of a macroscopic phase.
The increase in the chemical potential of a substance in the dispersed state is formally related to the surface curvature of the particles; but in fact, according to eq. (I. 11), the chemical potential increase is due to the increase in the surface area fraction (and therefore in the surface free energy) per unit volume with decreasing particle size. The increase in the number of surface atoms as the particle volume decreases is also typical of crystals, the surface of which consists of flat faces. Applying treatment similar to that described for the spherical particles, one can establish that for crystals the change in the chemical potential with increasing degree of dispersion is described by the equation, similar to eq. (I. 13): A~t=
2 c~i Vm
hi
in which the distance hi between the i-th face and the center of crystal, the specific surface free energy, % of the i-th face replace the values of the drop radius, r, and specific surface
43 free energy, o, respectively. The equilibrium state for various faces of crystal is determined by the condition of Ag=const., which yields the Curie-Wulff expression, stating that the ratio of the free surface energy of a particular face to its distance from the crystal center is constant for all faces in equilibrium state, i.e.: 13"1
0"2
0"i
h1
h2
hi
= const.
The above relationship also follows from the Gibbs's condition for the minimum of the surface free energy of the equilibrium crystal"
~ ~ S -- ~ E
(YiSi - 0 .
In agreement with both the Curie-Wulffexpression and the above equation, the faces that bear the lowest energy have the largest area, and are the closest to the crystal center (Fig. I- 15). Conversely, the faces that are further away from the center have higher surface energy and are, therefore, less developed. 9
,.
,,,
T 02
ol< 0 2 i
-h~ ~"l
I
h,
" !
0" 1 hi
02 ha
\o, . . . . . " Z / " Fig. I-15. The shape of an equilibrium crystal
The Thomson (Kelvin) law is the basis for the description of such phenomena as capillary condensation, nucleation (Chapter IV) and the isothermal mass transfer of substances (see Chapter VII). Capillary condensation is the process of vapor condensation in the fine
44 pores of solid adsorbent that occurs at pressures lower than the one existing over the flat surface (it is implied that the wetting of
adsorbent by a
condensing liquid is good). In agreement with the Thomson (Kelvin) law, the smaller the pore size, the lower the pressure at which condensation takes place. The latter is utilized in the industrial recycling of various volatile solvents, as well as in the analysis of adsorbent pore geometry.
1.4. Methods Used for the Determination of the Specific Surface Free Energy
The specific surface free energy and the surface tension, cy,numerically equal to it can be accurately determined at the gas-liquid and liquid-liquid interfaces. In this section we describe the general principles on which various methods for the determination of surface tension are based. In general, these methods can be classified as static, semi-static and dynamic. This classification, however, may be further extended with the "dynamic versions" of some methods listed below as static or semi-static exist. Extensive review of the methods used to determine the surface tension is given in [6,16-18]. The static methods are based on
studies of stable equilibrium
spontaneously reached by the system. These techniques yield truly equilibrium values of the surface tension, essential for the investigation of properties of solutions. Examples of the static methods include the capillary rise method, the pendant and sessile drop (or bubble) methods, the spinning (rotating) drop method, and the Wilhelmy plate method. The capillary rise method in its simplest formulation is based on the
45 use of eq. (I-12). The use of thin capillaries allow for the formation of spherical menisci. It is also advantageous to use capillaries that are well wetted by the fluid (0=0~ since it allows one to avoid complications associated with measurement of the contact angle. More precise results can be obtained if one corrects for the fluid volume above the lower edge of the meniscus. If the meniscus is spherical, this correction equals the difference between the volume of a cylinder, whose height equals its radius, and the volume of a hemisphere of the same radius, i.e.
rTrr 2 -
2 / a g r 3 - 1/aTl;r 3.
The precision of surface tension measurements using the capillary rise method can be further increased if the deviation of the meniscus shape from the spherical is taken into account. This correction is especially important when capillaries of large radii are used. Corrections for non-spherical meniscus curvature are based on tabulated numerical solutions of the differential Laplace equation [6]. The capillary rise method yields rs values with a precision of up to hundredths of mN/m. The sessile and pendant drop (or bubble) methods are based on the investigation of the shapes of drops and bubbles in the gravity field, and require the results of Laplace equation integration. In these methods the surface tension is established through the measurement of parameters that describe the deviation of the drop shape from the spherical. The necessary parameters can be evaluated from the digitized video images using the
axisymmetric drop shape analysis [19-21 ]. For the sessile drop shown in Fig. 1-12, these parameters are the maximum drop width alma x and the distance H* between the drop's top edge and the section of maximum width, dmax . A
46 comparison of the measured alma x and H* values with the ones evaluated numerically by the integration of Laplace equation allows one to estimate the surface tension. These methods become especially advantageous in measuring the surface tension at high temperatures [22]. In order to establish the values of the required parameters, the drops are photographed, using either optical devices with a long focal distance, or X-rays photography. The spinning (rotating) drop method allows one to measure very small values of the interfacial tension at the liquid-liquid interface [23]. Let us consider a tube filled with liquid into which a drop of another liquid of lower density is introduced (Fig. I-16). Upon rotation of the tube around oq
l l>>r Fig. I-16. The equilibrium shape of the rotating drop
its axis of revolution, the centrifugal force pulls the drop closer to the axis, causing its transformation into a prolate ellipsoid with the same axes of revolution as the outer tube. Assuming that the ellipsoid can be closely approximated with a cylinder of radius r, and measuring its length, l, and speed of revolution, co, it is possible to evaluate the interfacial tension cyfor a known difference in the densities of liquids"
(3"~
c02(pl - p 2 ) r 3
4 This relationship is known as the Vonnegut expression [24].
47 Somewhat different from the other static methods, the plate balancing
method (also referred to as the Wilhelmyplate method) is commonly used to evaluate the surface tension at the gas-liquid interface. In this method a thin rectangular plate of width d, mounted on an arm of a sensitive recording balance, is immersed into the
liquid under investigation, resulting in the
formation of menisci on both sides (Fig. I-17). It is generally assumed that the liquid wets the plate well. The meniscus shape and the height of liquid rise are determined by the Laplace equation. The weight of liquid lifted by a plate
Fig. I-17. The force balance equilibrium in the Wilhelmy method
(per unit of the plate's perimeter) does not depend on the meniscus shape and at zero contact angle exactly equals the surface tension, cy. Therefore, the force that one needs to apply to balance the plate, F, is the product of the surface tension and the plate perimeter. The surface tension can then be estimated as o = F/2d, provided that the plate is sufficiently thin. No corrections associated with the meniscus shape are required in this method. It is, however,
48 difficult to get the thin edges of the plate smooth, so in reality the perimeter of the plate is a bit greater than double the width. To increase the accuracy of the measurements, the equivalent plate thickness is determined by calibration with liquids of known surface tension. It is also worth mentioning that for nonzero contact angles the surface tension is c~- F/2dcosO, i.e. the Wilhelmy method can also be used to measure contact angles. As compared to static methods, the semi-static methods for surface tension measurement are based on achieving a metastable equilibrium, and focused mainly on investigating the conditions under which the system loses that equilibrium. The threshold of the equilibrium state can generally be reached slowly, and thus the surface tension values obtained by semi-static methods closely resemble those obtained by static ones. The rate of approaching the equilibrium state should be optimized in each system, in order to avoid lengthy measurements and to obtain surface tension values as close to the equilibrium ones as possible. Among the most common semi-static methods are the method of maximum pressure, the du NoiSy ring method and the drop-weight method. The maximum pressure method establishes the maximum value of pressure required to squeeze a bubble (or a drop of another liquid) through the liquid phase [6,25]. When the outside pressure gradient, Ap, is applied across a calibrated capillary immersed into liquid, a gas bubble (or drop of liquid) starts to grow at the capillary tip (Fig. I- 18). As the bubble grows, its curvature radius, r, decreases and finally reaches a minimum value equal to the radius of the capillary, r 0. At this point the bubble surface acquires hemispherical
49
_
-?'W
-
.
_
_
.
.
.
_
Fig. I-18. Change of the curvature radius of the bubble surface that occurs during the determination of surface tension by the maximum pressure method shape. Further increase in the bubble volume results in an increase in the curvature radius ( r > r0 ). At r = r0, the capillary pressure, p~ = 2~/r then reaches its maximum value 2cy/r o. Consequently, at kp < 2~/ro the system is mechanically stable, while at kp > 2~s/r o the capillary pressure is unable to balance the applied pressure, Ap, resulting in rapid bubble growth, followed by its final detachment from the capillary tip. The latter is usually accompanied by a noticeable pressure drop, the registered maximum value of which is
@ m a x --
2cy/ro. From this expression it is evident that
APmaxis directly
related to cy, i.e.
1 cy - -- APmax r 0 . 2 If the capillary diameter is not very small, one has to correct for the non-spherical shape of the bubble in order to enhance the
accuracy of
measurement. Similarly to other methods, one often performs relative measurements, in which the results are compared with the data acquired for other liquids for which the exact values of ~ are known.
50 The force Frequired to detach a well-wetted thin ring of radius rr, from the liquid surface is measured in the du No~y ring detachment method [6,26]. Within the first approximation one can assume that the equation relating the surface tension, o, to the detachment force, F, is analogous to that used in the Wilhelmy plate method, with the exception that the perimeter of the ring is used in place of the plate width, i.e. F -
4grrO. In reality, however, the
curvature of the liquid surface at points of contact with a ring causes the surface tension vectors to be somewhat off the vertical (Fig. I-19). F
I//
9
---
--
6------
I
\\
o'---
' --
0
~-8--m
Fig. I-19. Measurement of the surface tension by the ring detachment method (du Noay) In addition to this, one also has to account for the capillary pressure acting at the ring surface and hampering ring detachment (similar to the attractive capillary force of the menisci). The appropriate correction is achieved by introducing a numerical coefficient into the expression for cy, i.e. F cy-~k, 4xr r
where k is a correction coefficient the value of which depends on the ring
51 geometry and can be evaluated with the help of tables containing the results of the Laplace equation integration. While the du Notiy method is commonly used for measurement of the surface tension at liquid-gas interfaces, it is little used to measure (y at liquid-liquid interfaces, since in the latter case it is difficult to achieve the 0=0 ~ condition. The semi-static method frequently used for the determination of the interfacial tension at the liquid-liquid interface is the drop weight method, based on determining the weight of a liquid drop detaching from a flat capillary tip (Fig. 1-20). Usually a known number of drops is collected, their weight measured, and the average weight of a single drop is estimated from these measurements. This method is also sometimes referred to as the drop volume method.
6 Fig. 1-20. Detachment of a drop from a capillary tip
A rather complex theory of the drop weight method, which makes it possible to tabulate the data required in order to determine the surface tension, has been worked out in some detail [27].
In the first (roughest)
approximation, it can be assumed that, at the moment of detachment the gravity force acting on a drop, P, is balanced by the surface tension forces,
52 equal to the surface tension times the capillary circle length, i.e., P = 2~r0o. However, the detachment of a drop is a more complex process. For example, the fluid "neck" between the drop and portion of liquid that remains attached is of smaller diameter than the capillary tip. Furthermore, when the drop detaches, one or more smaller droplets are usually formed. These factors are accounted for by introducing a correction coefficient k, the tabulated values of which are established from the exact theory of the drop weight method. Thus, the corrected weight of a drop is
p - 2~rocy/k. The use of highly accurate optical drop-counting devices increases the reliability and convenience of the drop weight method, making it a rather popular technique in the lab. Dynamic methods for the determination of surface tension are usually employed in specialized studies of the non-equilibrium states of fluid interfaces, and in the investigation of how fast equilibrium in such systems is reached. A classical example of such methods is the oscillating jet method, which allows one to study the interfacial properties at rather small time intervals. In this method the liquid is ejected from a capillary with an elliptical cross-section, forming a stream with the shape of an elliptical cylinder. The surface tension forces tend to change the shape of the stream into that of a cylinder. These forces acting along with forces of inertia cause the stream to oscillate in a transverse direction, which results in a continuous interchange between the positions of the smaller and larger axes of the ellipse. The theory developed by Rayleigh and later by
Bohr and Sutherland relates the
53 wavelength of the longitudinal stream profile, measured by optical methods, to the surface tension of the fluid. A comparison of the surface tension values obtained with the ones established from static or semi-static measurements allows one to draw conclusions regarding the rate at which the equilibrium surface structure is established, as well as to study the adsorption kinetics. The capillary wave method is based on the generation of harmonic waves on the surface of a bulk volume of liquid [28]. The wavelength of the ripples formed, )~, is a function of the surface tension, which can be evaluated from expressions given by Kelvin: Z,3 p (5"--
2~'c
2
gZ,2 p
4re
2
u 2 _ g ) ~ ~ 2~o 2~
p)~
where p is the density of the liquid, g is the acceleration of gravity, ~) is the velocity of wave propagation, and ~ is the period of the ripples. One can thus determine the surface tension by measuring the wave parameters, which can be done, e.g., by the analysis of the standing waves. Even in the case of standing waves, the solution surface undergoes alternating local expansion and contractions, which may be accompanied by local surface tension changes and the transport of materials between surface layers. The resulting damping is characterized by a damping coefficient, which is another parameter obtained by the capillary wave method [28-31]. The damping coefficient provides information on the exchange of matter and the dilational elasticity of the
54 adsorption layers (see Chapter II). Examples of other dynamic methods based on interfacial relaxation include the oscillating bubble and oscillating drop methods and their variations [32]. In the oscillating bubble method a small air bubble is formed at the tip of a capillary immersed into a solution. The bubble is then forced to undergo harmonic oscillations induced by an oscillating membrane either due to oscillations in a gas volume connected with the capillary or due to oscillations in the pool of solution induced by a piezoelectric driver. By measuring changes in the surface area of the oscillating bubble and the amplitude of the pressure oscillations, one can evaluate surface tension using the appropriate theory [32]. The oscillating drop method is, essentially, a variation of the pendant drop method [32,33]. In this method a system of two interconnected syringes is used, as shown in Fig. I-21. A drop with a definite volume is formed with the help of a precise syringe (syringe 1 in Fig. I-21). Due to the oscillatory motion of the second syringe (with a characteristic frequency), the drop undergoes periodic contraction and expansion. Video images of the oscillating drop are acquired over short time intervals throughout the experiment. The instantaneous values of the surface tension, surface area, and the drop volume are then obtained from the digitized video images using axisymmetric drop-shape analysis [21]. An interesting modification of this method suitable for measurement of the interfacial tension at liquid-liquid interfaces has been proposed by Hsu and Apfel [34]. In the modified method a drop of one liquid is acoustically levitated in another liquid. The drop is then forced to oscillate in the acoustic force field, and the interfacial tension is evaluated from the resonance frequency.
55 motor
syringe 2
1
drop
Fig. 1-21. Oscillating pendant drop [33] As we mentioned above, some methods that we classified as semistatic can be adapted for the measurement of dynamic surface tension, cy(t). These include the drop weight and the maximum bubble pressure methods. In the dynamic drop weight (volume) method the liquid is dosed through a capillary in such a way that a continuous formation of drops takes place. The surface tension is then calculated from the average volume measured for several subsequent drops. Adjusting the liquid dosing rate allows one to age the interface for different periods of time, so that one can carry out measurements at times from less than l s to 30 rain or longer, obtaining the interfacial tension as a function of drop formation time. The dynamic drop weight method allows one to monitor the kinetics of the adsorption of surfactants and proteins. The interpretation of data acquired by the dynamic drop weight method requires the use of rather cumbersome adsorption kinetics theories that take into account such factors as the changing drop surface area
56 during drop growth (the adsorption at a growing surface is slower than at a stationary one), as well as the flow inside the drop [ 18]. These effects are described in detail in [27]. The dynamic maximum pressure method gives one an opportunity to monitor interfacial tension as a function of time in intervals from 1-2 ms to several seconds [18,25,35-40]. The "dynamic regime" of the maximum pressure method is achieved by changing the bubble formation frequency. Rehbinder was the first to alter the bubble growth rate, and hence to change the frequency of bubble formation, in the studies on the surfactant adsorption kinetics [41]. Using a recent design of the maximum bubble pressure instrument described by Miller and co-workers [39,40] one can carry out measurements on a millisecond time frame.
The high resolution of this
method was accomplished by increasing the system volume relative to the detaching bubble volume, and by using electric and acoustic sensors for registering bubble formation frequency. Miller and co-workers also addressed the issue of hydrodynamic effects at short bubble formation times [37]. There are also other dynamic methods that we have not described here. Some of these methods are reviewed in [ 18,25,27,32,42]
The measurement of the surface free energy of solids is a considerably more difficult task than that of liquids. In solids it is usually impossible to reach a thermodynamically reversible increase in the interfacial area, partially due to the high amount of work required for plastic deformation. Nevertheless, a number of methods that allow one to measure (or at least to approximately evaluate) the surface free energy of solids have been developed.
57 For ductile solids, such as metals, the zero-creep method can be employed to measure the surface tension at temperatures close to the melting point. In this method the material of interest is cut into strips of width d, onto which weights of different magnitudes are mounted (Fig. 1-22) [43]. The samples prepared are kept in a thermostat at temperatures somewhat lower than the material's melting point. After a rather long period of time, the change in the strip length A1 is measured. Depending on the magnitude of the applied weight F, the strips either shrink or become elongated due to the action of surface tension. The elongation of the strips is usually a linear function of the applied
b,\\\\\\\\\\\\\\\\\\\\\\\\'q
! lI--ll
0
+AI Fig. 1-22. A schematic representation of the zero-creep method used to determine surface free energy of solids
/
~],
1
Fig. 1-23. Determination of surface tension by single crystal cleavage
force. A point in the AI(F) dependence where AI = 0 (a so-called zero-creep
point) characterizes the force balance between the applied weight and the surface tension acting along the perimeter of the strip. The exact treatment, which accounts for the change in the strip shape at constant volume, shows that an additional numerical coefficient of 89is required, so that the force, F, is
58 F = cyd .
Typical ~ values for various solids determined using the zero-creep method are summarized in Table 1.2. In case of brittle solid materials, especially single crystals with clearly defined layer structure (e.g. mica), it is possible to use the c l e a v a g e m e t h o d developed by Obreimow [44]. In this method the crystal is split along the cleavage plane (Fig. 1-23), and the force that has to be applied to cause further development of the crack, F c, is measured.
TABLE 1.2. The values of the surface free energy, ~, of solids, established by different methods [7] Solid Substance
t, ~
~, mJ/m 2
Method
Ag
909
1140
zero-creep
Au
1040
1350
zero-creep
Co
1350
1970
zero-creep
Cu
900
1750
zero-creep
Ni
1343
1820
zero-creep
Zn
380
830
zero-creep
Zn, (0001) plane
- 195
410
crystal cleavage
Naphthalene
20
60
crystal cleavage
Mica
20
480
crystal cleavage
The relationship between the force, F c , the surface tension, ~, (which in this case represents the work required to form a new interface), crack length, l, thickness, h, width, d, and Young's elasticity modulus, E, of the cleaved layer
59 is given by 6(Fc/) 2 Ed2h 3 Another method that can be used to determine the ~ of solids is based on the investigation of the dependence of solubility on particle size, and involves the use of the Thomson (Kelvin) equation. This method, however, has a significant limitation, owing to the fact that increased solubility of particles obtained by mechanical fragmentation is in part due to numerous defects in the crystal lattice, appearing due to mechanical action. References ~
0
9
~
,
,
~
0
Q
10. 11.
Gibbs, J.W., "The Collected Works of J.W. Gibbs",vol.1, Thermodynamics, Longmans, Green, New York, 1931 Rowlinson, J. S., Widom, B., Molecular Theory of Capillarity, Clarendon Press, Oxford, 1984 Rusanov, A.I., Phasegleichgewichte und Grenzflachenersheinungen, Akademic Verlag, Berlin, 1978 (in German) Bakker, G., in "Wien Harms' Handbuch der Experimental Physik", vol.6, Akademische Verlagsgesellschaft, Leipzig, 1928 (in German) Goodrich, F.C., in "Surface and Colloid Science", vol. 1, E. Matijevid (Editor), Wiley-Interscience, New York, 1969 Padday, J.F., in "Surface and Colloid Science", vol.1, E. Matijevid (Editor), Wiley-Interscience, New York, 1969 Schukin, E.D., Pertsov, A.V., Amelina, E.A., Colloid Chemistry, 2~d ed., Vysshaya Shkola, Moscow, 1992 (in Russian) Derjaguin, B.V., Churaev, N.V., Muller, V.M., Surface Forces, Consultants Bureau, New York, 1987 Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1992 Overbeek, J.Th.G., in "Colloid Science", vol. 1, H.R. Kruyt (Editor), Elsevier, Amsterdam, 1952 De Boer, J.H., and Custers, J.F.H., Z. Phys. Chem., B23 (1934) 225
60 12. 13. 14. 15. 16. 17.
18. 19. 20. 21.
22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
Fowkes, F.M., J. Phys. Chem., 66 (1966) 382 Fowkes, F.M., Ind. Eng. Chem., 12 (1964) 40 Gaydos, J., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Winslow, D.N., in "Surface and Colloid Science", vol.13, E. Matijevid, R.J. Good (Editors), Plenum Press, New York, 1984 Adamson, A.W., Gast, A.P., Physical Chemistry of Surfaces, 6th ed., Wiley, New York, 1997 Rusanov, A.I., Prokhorov, V.A., Interfacial Tensiometry, in "Studies in Interface Science", vol.3, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1996 Miller, R., Joos, P., Fainerman, V.B., Adv. Colloid Interface Sci., 49 (1994) 249 Anastasidis, S.H., Chen, J.K., Koberstein, J.T., Siegel, A.F., Sohn, J.E., Emerson, J.A., J. Colloid Interface Sci., 119 (1987) 55 Cheng, P., Li, D., Boruvka, L., Rotenberg, Y., Neumann, A.W., Colloids Surf., 43 (1990) 151 Chen, P., Kwok, D.Y., Prokop, R.M., del Rio, O.I., Susnar, S.S., and Neumann, A.W., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Passerone, A., Ricci, R., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Seifert, A.M., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Vonnegut, B., Rev. Sci. Inst., 13 (1942) 6 Fainerman, V.B., and Miller, R., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Lecomte du No~iy, P., J. Gen. Physiol., 1 (1919) 521 Miller, R., and Fainerman, V.B., in "Studies in Interface Science", vol.6, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Adin Mann Jr., J., in "Surface and Colloid Science", vol.13, E. Matijevid, R.J. Good (Editors), Plenum Press, New York, 1984 Van den Tempel, M., van de Riet, R.P., J. Chem. Phys., 42(8) (1965) 2769 Lucassen-Reynders, E.H., J. Colloid Interface Sci., 42 (1973) 573 Lucassen-Reynders, E.H., Lucassen, J., Garrett, P.R., Giles, D., and Hollway, F., Adv. Chem. Ser., 144 (1975) 272
61 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
43. 44.
Wantke, K.D., and Fruhner, H., in "Studies in Interface Science", vol.6, D. MObius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Miller, R., Sedev, R., Schano, K.-H., Ng, C., and Neumann, A.W., Colloids Surf. A69 (1993) 209 Hsu, C., Apfel, R.E., J. Colloid Interface Sci., 107 (1985) 467 Kloubek, J., J. Colloid Interface Sci., 41 (1972) 7 Mysels, K.J., Langmuir, 5 (1989) 442 Fainerman, V.B., Makievski, A.V., Miller, R., Colloids Surf., A75 (1993) 229 Fainerman, V.B., Miller, R., and Joos, P., Colloid Polym. Sci., 272(6) (1994) 731 Miller, R., Joos, P., and Fainerman, V.B., Prog. Colloid Polym. Sci., 97 (1994) 188 Mischuk, N.A., Fainerman, V.B., Kovalchuk, V.I., Miller, R., Dukhin, S.S., Colloids Surf. A175 (2000) 207 Rehbinder, P.A., Z. Phys. Chem., 111 (1924) 447 Stebe, K.J., Ferri, J., Datwani, S., Abstracts of 73 ~dACS Colloid and Surface Science Symposium,Massachusets Institute of Technology, Cambridge, MA, 1999 Krotov, V.V., Rusanov, A.I., Physicochemical Hydrodynamics of Capillary Systems, Imperial College Press, London, 1999 Obreimow, I.V., Proc. Royal Soc. London, A127 (1930) 290
List of Symbols
Roman symbols All a
b al, bl aL C
Cs d e
E
Hamaker constant capillary constant distance comparable with molecular dimensions coefficients in Lennard-Jones potential coefficient characterizing dispersion interaction concentration heat capacity excess distance elementary charge modulus of elasticity
62 F force F~, F 2 forces acting on the frame in Dupr6's experiment ~free energy surface free energy free energy density f g acceleration of gravity isobaric-isothermal potential h Plank's constant h thickness of gap between volumes of condensed phase H height of fluid rise in a capillary 5genthalpy, heat of sublimation 5U internal pressure k correction coefficient in drop weight method kl numerical coefficient k Boltzmann constant Al displacement of wire in Dupr6's experiment m, n powers of R in the expression for interaction potential Avogadro's number NA number of moles N n number of molecules per unit volume number of molecules (atoms) per unit area g/s P gravity force (weight force) theoretical strength of ideal crystal Pid pressure P capillary pressure P~ tangential pressure PT R universal gas constant R equilibrium distance R, q0, z cylindrical coordinates distance between two volume elements of a condensed phase R12 r radius of curvature radius of a capillary ro rl, F2 principal curvature radii radius of the du No~y ring rr surface area S absolute temperature T critical point temperature :rc temperature t
63 internal energy interaction energy per unit area interaction energy between neighboring molecules /211 volume V Vm molar volume volume per molecule D velocity of wave propagation We work of cohesion x, y, z Cartesian coordinates Z, Zs coordination number U Umol
Greek symbols
~M
15, 15' q 0
~d V VO
P (~gb ~5n ~SLV (~SV (~SL T CO
empirical constant in eq. (I.5) polarizability small distance specific excess of internal energy specific excess of entropy contact angle wavelength chemical potential dipole moment frequency of radiation oscillation frequency 3.14159... density specific surface free energy; surface tension specific surface energy of grain boundary dispersion component of the specific surface free energy non-dispersion component of the specific surface free energy specific surface energy at liquid/vapor interface specific surface energy at solid/vapor interface specific surface energy at solid/liquid interface period of ripples at the surface of liquid angular speed of revolution
64 II. THE ADSORPTION PHENOMENA. STRUCTURE AND PROPERTIES OF ADSORPTION LAYERS AT THE LIQUID-GAS INTERFACE
A distinctive force field present at the interface may cause changes in the composition of the near-surface layer: different substances, depending on their nature, may either concentrate near the surface, or, alternatively, move into the bulk. This phenomenon, referred to as the adsorption, causes changes in the properties of interfaces, including changes in the interfacial (surface) tension. In disperse systems with liquid dispersion medium, adsorption layers present at the surfaces of dispersed particles may significantly influence the interactions between these particles and hence affect the properties of disperse system as a whole, including its stability. For this reason the investigation of the laws governing the formation, structure and properties of the adsorption layers at different interfaces is of extreme importance, as it allows one to analyze the role such layers play in controlling colloid stability and other properties of disperse systems. The thermodynamics gives a unified description of adsorption at a variety of interfaces of different nature. In contrast to that, some quantitative trends in the adsorption, as well as the methods that one may choose to study the adsorption layers, are very specific to the nature of contacting phases and to the structure of adsorbing molecules. Throughout this chapter, after a brief introduction into the thermodynamics of adsorption phenomena, we will focus on the formation and structure of adsorption layers at liquid-gas interfaces, leaving the discussion of adsorption at interfaces between condensed phases until Chapter III. Among the adsorption phenomena, those taking place at the solid-gas
65 interfaces are peculiar ones. On the one hand, these processes are very well studied with respect to the nature of intermolecular interactions taking place in adsorption layers, while on the other hand the adsorption layers at the solidgas interface can not radically influence interactions between particles, and hence are unable to significantly affect the stability of disperse system with gaseous dispersion medium.
II.1. Principles of Adsorption Thermodynamics. The Gibbs Equation In a two-phase system consisting of two or more components the composition of the discontinuity surface (see Chapter I) may significantly differ from that of a bulk of both phases in contact. Primarily the components that lower the system's free surface energy are expected to accumulate within the discontinuity surface; this spontaneous concentration of substances is referred to as adsorption. The quantitative measure of the adsorption of the ith component, FI, was introduced by Gibbs, and is also referred to as the
adsorption, or the surface excess of the amount of substance. This measure has a meaning of the molar excess of a particular component per unit interfacial area: Fi -
N i - N i - N/" S
where N~ is the total number of moles of the i-th component in the system; N, and N/' are the number of moles of the same component in the bulk of each of the contacting phases if it is implied that the substance concentration is constant at all locations within the phases, up to the geometrical dividing surface of area S.
66 Let us consider a model two-phase two-component system consisting of a solution of hexyl alcohol (component 2) in water (component 1) at equilibrium with their own vapors. A schematic change in the concentration of water c~(z) and that of hexanol c:(z) across the discontinuity surface is shown in Fig. II-1. In the regions below and above the discontinuity surface the concentrations of both components are constant and equal c~' and c 2' in the tl
!!
liquid phase, and c 1 and c 2 in the vapor phase, respectively. Furthermore, due I
II
I
II
to low vapor densities c~ >~c~ and c 2 )) c 2
.
I
z ~__
4
...._
~--~
CI
C2U
,
~__
!
~
,~
I
k
-b'
li-F--
,
-
__
~ C2 t --.---~---.--
Fig. II-1. Changes in the component concentrations within the discontinuity surface Within the discontinuity I
surface the concentration
of water
II
monotonously decreases from c~ to c~, which are the concentrations in the liquid and gas phases respectively, while the hexyl alcohol behaves differently" its concentration increases and substantially exceeds both c 2' and II
C2
9
67 To examine the relationship between the adsorption of a second component and the distribution of the latter within the discontinuity surface, let us draw a prism of cross-sectional area S in the direction perpendicular to the discontinuity surface (Fig. II-1). Let us then compare the amount of substance accumulated within such a prism in the real system and in an idealized one, for which in the z - 0 plane the concentration increase from c2' to c2" has the form of a step function. The adsorption of component 2 (i.e. hexanol) can be estimated as follows: 0
+8"
F 2 - f[c2(z ) - c ; ] d z + -8'
[ [c2(z ) - c2" ]dz,
(II.1)
~3
where - 6' and + 6" are the coordinates of the discontinuity surface, which has a thickness equal to 8 - ~ ' + ~". Geometrically, the adsorption, F2, is represented by a shaded area (Fig. II-1) between the c2 = c2' and c2 - c2" lines and the c2(z) curve. The adsorption of component 1 (i.e. water) can be determined in a similar way. The concentration of water
Cl'(Z) within the part of the
discontinuity surface adjacent to the liquid phase (for which - 6' < z < 0) is smaller than the bulk concentration c( and hence the corresponding integral is negative, as marked in Fig. II-1. The adsorption of water is geometrically equal to the difference in the shaded areas (Fig. II-1), given by the positive term written for the part of discontinuity surface adjacent to the vapor phase: ~"
I 0
> 0,
68 and the negative term for the part of discontinuity surface on the liquid side, respectively: 0
_~'
It is now evident that depending on the choice of the dividing surface position, the adsorption of component 1 can be either positive or negative (corresponding to a deficiency
of a component within the discontinuity
surface), or zero (note that the surface free energy is independent of the dividing surface position, see Chapter I, 1). The dividing surface, the position of which is chosen in such a way that F~ = 0, is referred to as the
equimolecular surface with respect to component 1 (i.e. the solvent). Let us now turn to a more detailed description of the adsorption of component 2 where its concentration within the interfacial layer is significantly higher than that in the bulk. Let us also assume that this component is non-volatile, i.e. that
c2" ~0. To make things simpler, it is
possible to chose such a position of the dividing surface that the second integral in eq. (II. 1) is negligible compared to the first one, and thus the entire physical discontinuity surface is located below the geometrical dividing surface~. The adsorption is then given by
Strictly speaking, such position of the dividing surface differs from that of the equimolecular one (with respect to solvent), however, the difference between the two is too small to cause any significant influence on the results of the present treatment
69 0
c;]dz
_~'
Using the definition of the integral average, the above equation can be written as
F 2 - ( c ~ s) -c2')~ ,
(II.2)
where c~s) is the average concentration of component 2 within the interfacial layer of effective thickness 8. Graphically, the above procedure is equivalent to replacing the "tongue" between the c2(z) curve and the c2' line with a rectangle of equal area, with sides equal to ( c~s)-
c2') and 6 (Fig. II-2).
(s)
C2
|
C2
-5
!
c2 f
Fig. II-2. The evaluation of the adsorption, F, and the surface concentration, c(s)
70 The effective thickness of the adsorption layer differs from that of a surface layer (physical surface of discontinuity) determined from changes in the other parameters, such as the free energy density (see Chapter I, 1). The adsorption, F 2, can, therefore, be viewed as the excess of substance per unit interfacial area within the interfacial layer, as compared to the amount of the same substance within the layer of equivalent thickness located in the bulk. If the substance has a strong tendency to adsorb and its bulk concentration is /
small, then c~s) >>c 2 , and hence F 2 ~ c~s)~5,
(II.3)
i.e., the adsorption approximately equals the amount of substance per unit area within the interfacial layer. This equation, obviously, remains valid when component 2, in addition to being non-volatile, is also insoluble in a liquid ll
I
phase, i.e. when c2 ~ 0 and c2 ~ 0. Under these conditions component 2 is completely concentrated within the surface layer (see Chapter II, 2). The expression (II.3) allows one to calculate the approximate maximum value of substance adsorption, e. g. hexyl alcohol in the present case. If we assume that the thickness of the closely packed adsorbed layer is close to the length of the hexanol molecule (~0.7 nm), and that the alcohol concentration, c~~, is close to its concentration in the liquid phase (~8 kmol m-3), then the adsorption, F, is ---0.6 x 10.5 mol m -2. The relationship between the adsorption (the excess) of a substance and its concentration within the interfacial layer, established by eqs. (II. 1) and (II.3), allows for a better evaluation of the properties ofmonomolecular layers
71 by comparing them with macroscopic phases. The treatment of interfacial layers as individual phases (to which the laws of regular three-dimensional thermodynamics are applied) is the basic concept behind the thermodynamics of finite-thickness layers, developed in the work
of van der Waals,
Guggenheim, and Rusanov [1,2]. The differences in the composition of bulk phases and interfacial layers in multi-component systems result in the re-distribution of the components of individual phases between the bulk volumes and the surface layers when changes in the interfacial area occur. Because of this the increase in the latter requires that chemical work be performed in addition to the mechanical work, o. Both of the terms that constitute the work required to form an interface can be accounted for by introducing the quantity ~, defined as - (Y + 2
~iFi
(II.4)
i
According to the Gibbs phase rule, at constant temperature and volume the binary two-phase model system has only one degree of freedom, meaning that only one variable in eq. (II.4) is independent. It is possible then to replace partial derivatives by full ones. The treatment is simpler for a surface that is equimolecular with respect to solvent, for which F~ = 0; the individual subscripts are no longer needed and can be omitted (i.e., g = g2 and F = F2), and eq. (II.4) becomes ~ - cy+gF. According to Rehbinder, we choose the chemical potential of solute g as an independent variable. The differentiation of the above equation with
72 respect to bt yields d~ d~ dF ~=~+bt~+F, dbt dbt dbt the left-hand side of which can be written as d~F
d/a
d~F dF dF
d/a
The quantity d~/dF, describing the change in the surface free energy with the increase in adsorption, is, by definition, equal to the chemical potential, bt, and hence dF bt~= dbt
do ~+F+ dbt
dF bt~ dbt
Consequently, F = - do / dbt, or do = - F dbt. The above thermodynamic relationship, describing adsorption in a two-component system, was first derived by Gibbs and is known as the Gibbs
equation [3]. It follows from the Gibbs equation that the excess of component within the interfacial layer determines how abrupt the decrease in the surface tension is with correspondingly increasing chemical potential of the adsorbed substance. The Gibbs equation reflects the equilibrium conditions at constant pressure and temperature between the surface layer and the bulk, i.e. the conditions corresponding the system's free energy minimum. The latter becomes more evident if the equation is written in its variational form, i.e." 8 ~ - 8c~ + F d p - 0.
73 It is thus possible to say that at a given value of adsorption, F, the balance between "mechanical" forces, 6G, and "chemical" forces, Fdg, corresponds to the minimum of the system free energy per unit interfacial area. In the other words, there is a balance between the tendency of a system to decrease its surface energy by concentrating some of the species within the surface layer on the one hand, and the disadvantage of such accumulation due to the increase in chemical potential on the other hand. It was shown by Gibbs that for multi-component systems the fundamental adsorption equation can be written as "
17
d~ - - 2
Fidgi '
i=2
where the summation is carried out over all components with the exception of component 1 (the solvent). When the system is at thermodynamic equilibrium, the chemical potential of any component ( including the adsorbed one) is the same in all phases in contact, as well as within the interfacial layer. If g is the chemical potential of solute in the bulk, one can write dg = RTdln(czc), where a is the activity coefficient, and c is the solution concentration. If the solution studied is not too different from the ideal one, the activity coefficient, a -- 1, and the Gibbs equation for two-component system is
74 written
as 2
c d~ r = - ~ ~ . RT dc
(II.5)
It is known that for solutions containing molecular species the condition of ~ - 1 is valid for concentrations up to - 0.1 mol dm -3, and thus the use of the simplified Gibbs equation (II.5) is justified for sufficiently dilute solutions only. On the contrary, the magnitude of substance concentration in the interfacial layer, c~s) - c (s), does not impose any restrictions on the use of eq. (II.5). If the adsorption, F, is expressed in terms of the surface concentration, c (~), and the thickness of adsorption layer, 8, given by eq. (II.2), the Gibbs equation can be written in the following form: c (s) - c
des = R 7 " 6 ~ . dc c
(II.6)
Experimental studies on the surface tension of various solutions showed that the latter can both increase and decrease with increasing solution concentration, depending on the nature of solvents and solutes. Different solutes affect, however, the surface tension of the solvent, %, in different ways" some solutes, when present at extremely small concentrations, can cause a significant decrease in the surface tension, while the others can only insignificantly increase it (Fig. II-3).
2 If the adsorbing species are of the ionic nature, a numerical coefficient may be introduced into the Gibbs equation (II.5). For example, this coefficient equals 1/2,if the substances dissociate into ions of the two types
[4]
75 (J O0 d(~
---<_ R75 dc
d~ dc
R~
C
(s)
c
Fig. II-3. The surface tension isotherms for the surface active (1) and surface inactive (2) substances
Substances that lower the surface tension of the system (dcy/dc< 0) are referred to as surface active substances, or surfactants. It follows from the Gibbs equation that the adsorption of such compounds is positive, i.e. their concentration within the surface layer is higher than that in the bulk. For example, at air-water and water-hydrocarbon interfaces the surface active compounds are the ones containing hydrocarbon (non-polar) chain and a polar group ( -OH, -COOH, -NH 2, etc) in their structure. Such an asymmetric (diphilic) structure of surfactant molecules accounts for their similarity to the nature of both contacting phases: a well-hydrated polar group has the strong affinity towards the aqueous phase, while the hydrocarbon chain has the affinity towards the non-polar phase. The surface tension of such surfactants themselves at the interface with air is usually ~25 mJ/m 2, i.e., significantly lower than that of water (72.75 mJ/m2).
76 Rehbinder [5] proposed to use the quantity G as the measure of s u r f a c e activity"
lm/ d)
c---~ 0
which has the units of mJ
m -2 /
kmol m -3. In surfactant solutions there are
no limits on the magnitude of c(S)/c ratio, and the surface activity, given in this case by c (s)
G = RTS~ can be enormously high, i.e., the surface tension, o, can decrease drastically with an increase in solution concentration, c. Such behavior is especially typical for slightly soluble surfactants. Conversely, the dissolved inorganic electrolytes cause only a minor increase in the surface tension of water (Fig. II-3, curve 2), which according to the Gibbs equation corresponds to negative adsorption, i.e., a deficiency of solute within the surface layer as compared to the bulk (c (s) < c). Such a deficiency can be easily explained: it is unfavorable for the hydrated ions to reach the surface closer than the distance equal to their hydration shell radius (the penetration of ions into the surface layer is thermodynamically unfavorable since additional energy is required for dehydration). In the limiting case the solution layer in the vicinity of the surface may contain no ions at all, i.e., (c (s)- c)/c - -1 (see eq. (II.6)), and the increase in the surface tension with increasing concentration is governed by the dG/dc = - S R T relationship, where 8 is the ion hydration layer thickness. The latter has a size on the order of that of water molecules (i.e., 8 does not exceed fractions of a
77 nanometer), and hence for aqueous solution of an electrolyte the maximum value of the slope of the ~ (c) dependence at room temperature equals to 8.3 J mol -~K -~ • 300 K x 4x 10 -~~m = 10 .6 J m mol -~, whichjs equivalent to 1 mJ
m -2 / (kmol m-3). The latter value corresponds to an increase in surface tension of only 1 mJ/m 2 in a 1 mol dm -3 solution, which is very small compared to the surface tension of pure water itself (72.75 mJ/m 2 at room temperature). Substances that increase the surface tension of solvent are referred to as
surface inactive. There are also examples of cases where solutes do not cause any detectable change in the surface tension of solvent, as occurs in, e.g., an aqueous solution of sugar. In his work "Water as a surface active substance" [5] Rehbinder emphasized that the deficiency of solute molecules within the surface layer corresponds to the enrichment of the latter with solvent, meaning that a negative adsorption of solute is equivalent to a positive adsorption of solvent, which acts as a surface active substance. Rehbinder studied the AgTI(NO3)2 -
H20 (Fig. II-4, curve 1) system at 90~ curve 2) system at 100~
and AgNH4(NO3) 2 - H20 (Fig. II-4,
In both cases he was able to measure the surface
tension over the entire concentration range: from pure water to pure melted salt, as shown in Fig. II-4. The same figure also illustrates the surface tension isotherm ofpropionic acid (curve 3) at 90~
given for comparison. The linear
increase in the surface tension in the right portion of the isotherms 1 and 2 is observed up to 30% of salt, which acts as a surface inactive substance under these conditions. On the left side of the isotherm, corresponding to low concentrations of water in a melted salt, the surface tension - concentration profile is different: water, when present at
very low concentrations,
78 substantially reduces the surface tension, and thus behaves like an "ordinary" surfactant. (Jr
,naN na O
120
120
80
- 80
40
4O
!
0
,
!
40:
,1
!
80
1" % H20
Fig. II-4. The surface tension isotherms established at the interface with air in systems with water content changing from 0 to 100%" 1 - AgTI(NO3)2 - H20 system at 90~ 2 AgNH4(NO3) 2 - H20 system at 100~ and 3 - C3H7COOH - H20 system at 90~ [5]
This example clearly illustrates that the concepts of surface activity and inactivity are in fact relative, and do not represent absolute properties of a substance; they both depend on the nature of interface. Indeed, water is surface active with respect to salts, whose surface tension is higher, and surface inactive in the other cases, e.g. at the alcohol air interface. Alcohols and other diphilic substances show strong surface activity with respect to water, and are inactive at the non-polar hydrocarbon air interface. The salts can show high surface activity towards more refractory
79 salts, oxides, and liquid metals; some oxides and low melting point metals are capable of reducing the surface tension of refractory metals, and of covalent compounds. A substance that has lower surface tension is usually a surface active one. If component A is surface active with respect to component B, then the component B is inactive with respect to A, i.e. the surface tension, o(c), isotherms usually exhibit a monotonous behavior. The isotherms with a minimum are rarely encountered. They may be observed in systems consisting of substances whose surface tension values are close, such as, e.g., a carbon disulfide - dichloroethane system.
When the components have limited
solubility in each other, the surface tension isotherms consist of two separate sections separated by a gap, the borders of which correspond to the concentrations of saturated solutions. It is essential to point out that when the mutual solubility of components is limited, an interface with a very low interfacial tension may form between two saturated solutions (see Chapter III,
2). Due to their unique diphilic nature organic surfactants remain surface active at most interfaces (obviously, within the range of their thermal stability). As a rule, these compounds do not lower the surface tension by more than 30 - 50 mN/m. A very strong decrease in the surface tension of high-energy surfaces of refractory oxides, metals, and other compounds is caused by substances of similar molecular nature. The latter relates not only to the liquid-vapor, but also to the solid - vapor and solid - liquid interfaces (see Chapter IX, 4). The
adsorption
of
insoluble
substances
from
vapors
is
80 thermodynamically identical to the described case of the adsorption of nonvolatile components from solutions. At low pressures, the Gibbs equation can approximately be written as F__
p d~s , RT dp
(II.7)
where p is the vapor pressure. This case is especially important in the analysis of gas and vapor adsorption on solid adsorbents. The Gibbs equation contains three independent variables F, o, and g (defined either via concentration or pressure, c or p, respectively), and is a typical thermodynamic relationship. Therefore, it is not possible to retrieve any particular (quantitative) data without having additional information. In order to establish a direct relationship between any two of these three variables, it is necessary to have an independent expression relating them. The latter may be in a form of an empirical relationship, based on experimental ,.
studies of the interfacial phenomena (or the experimental data themselves). In such cases the Gibbs equation allows one to establish the dependencies that are difficult to obtain from experiments by using other experimentally determined relationships. For example, the surface tension is relatively easy to measure at mobile interfaces, such as liquid- gas and liquid- liquid ones (see Chapter I). For water soluble surfactants these measurements yield the surface tension as a function of concentration (i.e., the surface tension
isotherm). The Gibbs equation allows one then to convert the surface tension isotherm to the adsorption isotherm,
F (c), which is difficult to obtain
experimentally. The value of adsorption of practically insoluble non-volatile
81 surfactants can be pre-set to a desired value by placing a very small known amount of substance on a surface of a known area. If the surface onto which the surfactant is placed is on one side limited by a moving barrier, it is possible to continuously adjust the surface area, and hence to control the adsorption (Fig. II-5). .... .......
%
___,.~ ..- .
o
..__--Y__
I
I
.Ng
t_! _ ---~
Fig. ]l-5. Displacement of the barrier due to the action of two-dimensional pressure of the adsorption layers of insoluble surfactants
Such a technique, proposed by Pockels and extensively developed by Langmuir [6] (see Chapter II, 2), allows one to study the surface tension adsorption dependence of insoluble surfactants. If a pure solvent is present on one side of the barrier, and the solvent with the adsorption layer on the other, the forces acting on the barrier from each of these sides differ. Considerations, similar to those used in the description of the Dupr6 experiment (see Chapter I, 1), suggest that a force per unit barrier length, % , is acting in the direction of the pure solvent, and a force cy(F) < % is acting in the opposite direction along the surface covered with the adsorption layer. The net force per unit of barrier length is directed towards the pure solvent (Fig. II-5), and equals the difference in the surface tension between the surface of pure solvent and that surface covered with adsorption layer. This force, equal to % - ~ (F), is referred to as the twodimensional p r e s s u r e , zv, of the adsorption layer"
82
---
- o(r)
(II.8)
- -Ao(r).
This principle was utilized by Langmuir in his instrument designed to measure two-dimensional pressure, i.e. to obtain the Jr(F) dependence. Equation (II.8) is also valid for soluble surfactants as well. However, in the latter case the value of~r can not be directly measured using Langmuir's method, but can be established from surface tension measurements as the drop in the surface tension: ~v = -Ao = % - o(c) (refer to Chapter II, 2 regarding the identity between iv and -Ao). If the F(g) or F(c) dependence is known, the two-dimensional pressure can be obtained by integrating the Gibbs equation, i.e." o
K--
Id(y ~o
~
c
~(r=o)
o
IFdg-RTIF(c)dlnc.
When adsorption takes place at the surface of a highly porous solid adsorbent, the surface excess can be readily measured, e.g. by measuring the increase in the adsorbent weight in the case of adsorption from vapor, or by following the decrease in the adsorbate concentration during adsorption from solutions. Studies of the adsorption dependence on vapor pressure (or solution concentration) reveal F(p) (or F(c)) adsorption isotherms. In both cases the two-dimensional pressure isotherm can be established from the Gibbs equation (see Chapter II, 2, and Chapter VII, 4). Therefore, it is as a rule possible to establish the dependence between the two of three variables present in the Gibbs equation: the surface tension isotherm, o(c), for mobile interfaces and soluble surfactants, the two-dimensional pressure, zr(c), isotherm for insoluble
83 substances, or the F(c) or F(p) adsorption isotherms for the adsorption on porous solid adsorbents and highly disperse powders. It is a more difficult task to independently measure all three quantities found in the Gibbs equation, namely the surface tension, the adsorption, and the chemical potential of the adsorbed component. Such measurements, carried out in a number of studies, have confirmed the validity of the Gibbs equation. For example, McBain [6], along with the measurements of surface tension as a function of solution concentration, also investigated changes in the adsorption of a surface active component. This was done by removing a thin layer (- 0.05 - 0.1 mm) of liquid from solution surface with a fast moving knife. The amount of the surfactant in that "cut" layer was further determined by means of chemical analysis. If the surface activity is high, and the bulk surfactant concentration is low, it can be assumed that the surfactant is present entirely within the adsorption layer. This assumption is reasonable, since the bulk amount of surfactant in such a small volume is negligible, and hence the adsorption, F, can be estimated. These measurements showed that the experimentally measured values of adsorption were in good agreement with those estimated from the Gibbs equation. Similar experiments were also carried out using radioactive tracers. In addition to utilizing experimental data in the integration of the Gibbs equation, one can also employ equations based on the molecular statistical analysis of the adsorption phenomena, e.g.
the relationships
resulting from various models describing the structure of adsorption layers. Alternatively, a comparison of the experimentally established ~(c), Jr(F), etc. values with the ones evaluated from the Gibbs equation, allows one to draw
84 a number of fundamental conclusions regarding the structure of adsorption layers and the behavior of surfactant molecules within such layers.
II.2. Structure and Properties of the Adsorption Layers at the Air-Water Interface
Let us next consider the characteristic properties of interface and adsorption layers, comparing the behavior of water soluble surfactants to that of insoluble ones. We will gradually move from the simplest cases to more complex ones, revealing the nature of intermolecular interactions in the adsorption layers. In doing so, we will analyze the typical relationships that describe the properties of adsorption layers, namely the surface tension isotherm, ~(c), the adsorption isotherm, F(c), the two-dimensional pressure isotherm, ~r(F), etc.
11.2.1. The Dilute Adsorption Layers If the adsorption value is small compared to F ~ 10.5 mol
m -2
(see
Chapter II, 1), the surfactant concentration is low, not only in the bulk, but also within the surface layer. In this case the thermodynamics of ideal solutions can be used to describe both the bulk solution and the surface layer. The chemical potential of the surfactant molecules in the bulk can then be written as P - P0 + R T l n c ,
(II.9)
and a similar expression is also valid for the surface layer, i.e (s) _ g~s) + RTlnc(S).
(II. 10)
85 In both of the above equations the standard parts of the chemical potentials, bt0 and bt~s), are not identical. The surface and bulk solutions are in equilibrium with each other, if their chemical potentials are equal, namely (s) + R T lnc (s) - go + R T lnc. Consequently,
c (s)
= exp !a o
c
_
(II. 11)
RT
i.e.,within the limits of the ideal solutions approximation, the surfactant concentration in the surface layer is proportional to that in the bulk. Equation (II.11) is analogous to the Henry equation describing the partition of a substance between two phases. Thus, the surfactant concentration range within which the dilute adsorption layers exist is often referred to as the Henry
region. In the case when c (s) >>c (strong adsorption), a direct comparison of the Gibbs equation with eq. (II. 11) allows one to write
G -
dcr dc
c,s,
F RT - ~ S R T c c
- 8RTexp
1
- const
RT
(II. 12)
Therefore, the absolute value of the d~s/dc derivative equals the surface activity, G, not only in the limit of zero concentration (c-~ 0), but also within some definite concentration range, in which the properties of the adsorption layers can be described by eq. (II. 10). If the d~/dc derivative is constant, i.e. d ~ / d c - A~/c, the Gibbs
86 equation can easily be integrated to yield a linear dependence of the surface tension, a, and two-dimensional pressure, ~r, on the surfactant concentration (Fig. II.6)"
- -Ao 13
- ~0 - ~(~)
d~ - -c-dc
-
cG.
(II. 13)
7g
I
I
I
Oo
0
[ \n+21 I
I
I
In+21
I I c'+l:c'+2"c"
n+l
I
"~ 10"3.2"1
n
I I
I
c 0 On+2 Cn+l Cn Fig. II-6. The surface tension, c~(c),and two-dimensional pressure, 7r(sM),isotherms of three surfactants, the consecutive members of a homologous series. The concentration range corresponds to the Henry region
At low surfactant concentrations the linear o(c) dependence can indeed be experimentally verified. Thus, a simple model describing adsorption layers as ideal solutions allows one to integrate (within a certain approximation) the Gibbs equation. The experimental studies carried out by Ducleaux and Traube (1884 1886) showed that within the same homologous series the extension of a surfactant chain length by each C H 2 group causes an increase in [d~/dc]by a factor of 3 to 3.5 (Fig. II-6). This observation is known as the
Ducleaux
-
87
Traube rule, more often referred to as the Traube rule. In agreement with eq. (II. 12), the Ducleaux-Traube rule corresponds to a linear relationship between the surfactant chain length and the go - ~t~~)value. The latter may be viewed as the work of adsorption performed under standard conditions. Indeed, let us assume that the work of adsorption, g0 - g~s), is a linear function of the number of carbon atoms in the surfactant hydrocarbon chain, i.e: ~t0 -- ILl'~s)
_ q)l + n q ) C H 2 ,
(II. 14)
where qo~ reflects the change in the interaction energy between the surfactant polar group and water molecules due to the relocation of surfactant from the bulk to the surface layer. The ratio of surface activities of the two subsequent members of homologous series is then determined by the work of transfer, q~cn2, of a single CH2 group into the surface layer"
Gn+l = G~
exp(q~CH2 / R T ) .
The value of q~cn2 that corresponds to the Ducleaux-Traube rule is approximately equal to RT In (3-3.5)~ 3 kJ. A linear drop in the surface tension with increasing surfactant concentration (the Henry region) in agreement with eq. (II. 12) corresponds to a linear increase in adsorption" F-~;
Gc RT
88 i.e. the slopes of F(c) lines increase by a factor of 3 to 3.5 for each successive homologous series member (Fig. II-7). Comparison of the above expression with eq. (II. 13) yields er = FRT. (II. 15) Equation (II.15) shows that at low adsorption values the two dimensional pressure, a:, is proportional to the adsorption, and the proportionality constant equals the product of the universal gas constant, R, and the
Pmax
........
f
/
.g~mP
/
/ /
/
/
f
11---
/
.+2A
0
/
~
Cn+2
n+l
Cn+1
Cn
C
Fig. II-7. Adsorption isotherms of three surfactants, the consecutive members of a homologous series. The solid lines correspond to the Henry region
absolute temperature, T, regardless of the surfactant nature. For surfactants with high adsorption activity at low bulk concentrations, the adsorption, defined by eq. (II.2) as the excessive quantity, is identical to the amount of substance per unit surface area. The quantity Sm-- 1/F is then the area occupied by a mole of adsorbed substance in the adsorption layer. Equation (II. 15) can be written in a form similar to the ideal gas law, i.e." 7ZS m - R T ,
(II. 16)
89 or
7vs M - kT, where SM is the area occupied by one molecule in the adsorption layer. The linear decrease in surface tension at low surface concentrations of adsorbate has a simple molecular-kinetic nature, and can be, approximately, viewed as the action of the pressure of adsorbed molecules on a barrier that separates them from the pure solvent. It is more correct to regard the two-dimensional pressure as analogous to the osmotic pressure. In surfactant solutions, in which F > 0 and c (s) > c, the osmotic pressure is greater at the surface than in the bulk. This causes "pumping" of water into the surface layer, and thus eases the formation of a new surface, which is in fact equivalent to the decrease in the surface tension. The opposite situation, where F < 0 and c (s) < c, is typical with surface inactive substances, when one can say that the energy required for a new surface formation (determined by ~) increases due to the need to perform additional work against the osmotic pressure forces. This work is directed towards the displacement of electrolyte from the surface layer into the bulk. Equation (II. 16) is the state equation o f the ideal two-dimensional gas, which is represented by the two-dimensional pressure isotherm, IV(SM),shown in Fig. II-8. To analyze the behavior of real systems, and to study the origin of deviations of such systems from ideal two-dimensional state, the twodimensional pressure isotherm is plotted in ~rsM- ~rcoordinates. The isotherm of an ideal two-dimensional gas in these coordinates is a straight line, parallel to the x axis (Fig. II-9). If the two-dimensional pressure and area per molecule
90 are expressed in mN/m and nm 2, respectively, the corresponding value of kT is ~4 mN m -~ nlTl 2. 7/" ~SM~
mN (rim) 2 1"11
\ \\
4 \
0
SM
Fig. II-8. The isotherm of an ideal twodimensional gas plotted in 7V-SM coordinates
7C Fig. II-9. Same isotherm as in Fig. II-8 plotted in ~VSM- 7Ccoordinates
The relationships establishing the links between the c, F (or SM), IV values and dc~/dc implied that the surfactants were water soluble. The solubility itself, however, was not directly included in any of the equations, and it is thus reasonable to assume that all conclusions derived are valid for the insoluble surfactants as well. Indeed, the latter was confirmed by direct measurements of the two-dimensional pressure as a function of adsorption, carried out by Langmuir and his successors. In order to obtain this relationship Langmuir constructed an instrument, which measured the two-dimensional pressure in the adsorption layers of insoluble surfactants with
sufficient
accuracy. This instrument, known as Langmuir's film balance, is shown in Fig. II-10. In this device the light, freely-sweeping barrier attached to a dynamometer is mounted on the surface of a rectangular flat cuvette with waxcoated walls. In the modern versions of Langmuir's instrument the cuvettes are made of Teflon, and highly accurate automated force-measuring devices
91 are used as dynamometers. In Langmuir' s instrument the migration of surfactant molecules out of the working zone is prevented by blocking the gap between the cuvette walls and the barrier with thin golden plates or wax-coated threads. The second, adjustable, barrier allows one to change the working zone area of the device. The first barrier is referred to as the measuring barrier and the second one as the subsidiary barrier.
o(13
Fig. II-10. Schematic representation of Langmuir's film balance, the instrument used to measure the two-dimensional pressure in adsorption layers of insoluble surfactants
A small amount of a very dilute surfactant solution in a volatile solvent is placed on the surface of the water confined in the cuvette between the two barriers. For example, the dilution of 1 c m 3 of 10.2 mol dm 3 solution of cetyl alcohol in benzene with 1000 cm 3 of highly pure benzene yields a 10.5 mol dm -3 cetyl alcohol solution. Placing 0.1 cm 3 of this solution, containing 10 .9 mol of surfactant, on an area of 100 cm 2 yields an adsorption of 10.7 mol m 2, which constitutes about 1% of the maximum adsorption value, Fmax. The magnitude of the adsorption can easily be varied by adjusting the position of the subsidiary barrier, and therefore, by varying the area onto which a known amount of a surfactant had been placed. Upon the evaporation of solvent the force acting on a measuring barrier, F, is balanced by a load
92 applied to the dynamometer. The resulting value of F is normalized over the barrier width, yielding the two-dimensional pressure zv. The dependence of the t w o - d i m e n s i o n a l pressure, zv, on the adsorption, F (or on the area per m o l e c u l e in the adsorption layer, SM), can be obtained by repeating the force measurements
at different values of adsorption,
i.e., by carrying out
m e a s u r e m e n t s with different amounts of surfactant solution and at different positions of the subsidiary barrier.
The adsorption layers of insoluble surfactants can be transferred from the airsolution interface onto the solid substrate, forming the so-called Langmuir-Blodgettfilms [7]. The deposition of adsorption layers takes place on a slide that moves perpendicular to the airsolution interface containing the adsorption layer. Film transfer usually occurs at constant and rather high value of the two-dimensional pressure, controlled with a Wilhelmy plate. If the slide is moved by turns in both upward and downward directions, polymolecular films form. In these films the neighboring monolayers alternately come into contact with their hydrocarbon chains and polar groups, as shown in Fig. II-11, b. Ifthe slide always moves into one direction, the "polar" films (with uncompensated dipole moment ofsurfactant molecules in the neighboring monolayers) may be formed. In such layers the polar groups of one monolayer come into contact with hydrocarbon chains of the other. If the slide is hydrophobic, and moves in the downward direction, the X-type films are formed (Fig. II-11, a). The deposition on a hydrophilic slide moving in the upward direction results in the formation of Z-type films (Fig. II-11, c). The Langmuir-Blodgett technique allows one to form structures with the predetermined molecular arrangement, in which the neighboring monolayers have the desired composition. For example, by using water insoluble organic acids with sufficiently long chain length, such as stearic, the composition of deposited layers can be modified by changing the electrolyte content of the substrate solution by introducing polyvalent ions that form insoluble salts with the acids.
93
~
IV.V%I,..~.~ ~ v x . " v ' v
a ~V'..,',....-.~ ~
~
O.'V'v",JXs
VX,-'v-,,.IV ~
,9j - , . - " , , A Z ' 4 l r ~ , , t , , A
V'~"V"Z~
~ O , % . , V , v , .
v'v",/v'~
,j-,.,-,,.,',~
v~...vv,Q
~ v , , . , . v , , z ~
,..'v "v'z'4
,,~,-"v "z~
~
-1
b
r
Fig. II-11. Different types of Langmuir-Blodgett films deposited on a solid surface of a moving slide: a - X-type; b - Y-type, and c - Z-type Among the perspective applications of Langmuir-Blodgett films, we can name their use as monochromators and analyzers of soft (long-wave) X-ray and neutron radiation, and for the synthesis of light transmitting, electrically conducting and superconducting thin films on the surfaces of solids, utilized in novel electronic devices.
Langmuir and his successors (N. Adam, E. Rideal, and others) devoted a lot of attention to the behavior of various lower-molecular weight compounds (higher members of homologous series of acids, alcohols and
94 amines), as well as to the behavior of polymers, including proteins and nucleic acids. Their results are described in some detail in. It was shown that for a variety of substances at the limit of zero adsorption (F-~0) the product of area per molecule in the adsorption layer, SM, and the two-dimensional pressure, ~r, indeed approaches kT, regardless of the surfactant molecular structure. It is now evident that the ideal two-dimensional gas equation of state, eq. (II.16), can be applied to dilute adsorption layers of both soluble and insoluble surfactants. At the same time, the similar behavior of surfactants in the dilute adsorption layers, regardless of the nature of the constituent molecules and their interactions with the solution underneath the adsorption layer, leads to the conclusion that the relationship between the adsorption and the two-dimensional pressure, ~r(F), is the principal characteristic of the adsorption layer, independent of the bulk properties of the surfactant solution. Conversely, the dF/dc value, characterizing the ability of a substance to adsorb, significantly depends on the nature of both the surfactant and the solvent. This value increases sharply as we move from one member of a homologous series to the other. Different abilities of surfactants to adsorb, on the one hand, and the identity of their behavior in the dilute adsorption layer itself, on the other hand, indicate that the increase in the dF/dc value in the homologous series is related to the differences in behavior of the members of homologous series in the solution and not in the adsorption layer. The latter indicates that for dilute adsorption layers the g0 - g~s)value is determined by the state of energy of surfactant molecules in the bulk. In the other words, the standard part of the chemical potential of molecules in the adsorption layer, g~s), (eq. (II. 14)) can be considered to be constant at low adsorption, and thus
95 the increase in go - g~s)within homologous series can be attributed solely to the increase in the standard part of the chemical potential, go, corresponding to the bulk. The above conformities are confirmed by trends in the solubility of surfactant homologs, Co, which decreases with increasing adsorption activity. As the surfactant chain length is extended by each CH2 group, the solubility indeed decreases by a factor of 3 - 3.5. For slightly soluble compounds the molar solubility (measured in mol m -3) is approximately given by the equation
co =
1 exp[-(
0-
Vm
,
where Vm is the molar volume of solute, and g~ is the chemical potential of pure surfactant. A decrease in solubility by a factor of 3 to 3.5 with increasing chain length indicates that g0 is a linear function of the number of carbon atoms in the surfactant hydrocarbon chain, and the proportionality coefficient, q0c.2, is the same as in the equation for the work of adsorption"
f
tU'O - q)l + n q ) c n 2 ,
t
where q~l accounts for interactions between the surfactant polar group and water. The value of presence of
CH 2
qDCH2
reflects the disadvantage associated with the
segments in water, i.e. a tendency of hydrocarbon chains to
get pushed out from the water into the surface layer. In the case of the
96 adsorption of diphilic surfactant molecules at the air-solution interface, the hydrocarbon chains are the reason for the surface activity. Recent studies have shown that the energetics of the adsorption of organic surfactants from aqueous solutions, as well as the energetics of the dissolution of surfactants and hydrocarbons in water, has a complex nature related to the properties of water as a solvent. According to the currently accepted views regarding the structure of water, developed in works by D. Bemal, R. Fowler, L. Pauling, H. Scheraga, J. Samoilov and others [8], at near-room temperatures the directed hydrogen bonds between the water molecules result in the formation of highly ordered structures, resembling those encountered in ice crystals (of course, in the case of water the long range order, typical of solid crystals, does not exist!). It is customary to talk about the existence of ice-like regions in liquid water. The degree of ordering depends on temperature, and is responsible for the commonly known unique o
properties of water, such as the fact that its maximum density is at 4 C, and that it decreases in density on freezing, etc. When molecules of organic compounds that contain hydrocarbon chains are introduced into water, the average degree of ordering increases, causing
the entropy to decrease.
Consequently, a partial disordering due to the displacement of hydrocarbon chains from the bulk into the surface layer (or into the bulk of another nonpolar phase, see Chapter III) results in an increase in entropy. The latter is the cause of interactions, commonly known as hydrophobic interactions [9].
97
II.2.2. Langmuir and Szyszkowski Equations. Accounting for the Adsorbed Molecules Own Size (Mutual Repulsion) The effect of the size of surfactant molecules can in the first approximation be accounted for, if one assumes the existence of some limiting value of the adsorption, Fmax, such that the adsorption no longer changes once it has been reached. Such an assumption allows one to integrate the Gibbs equation in the concentration range where adsorption does not change, i.e" RTF,~•
dcr dlnc
const.
The integration of this expression yields zc--Acy-
RTFmaxlnc + B
(II.17)
where B is the integration constant. An equation similar to the one above, describing the concentration dependence of the surface tension in the region of relatively high surfactant concentrations 3 was offered by Milner (1907). This equation suggests that the surface tension is a linear function of the logarithm of the solution concentration. Szyszkowski (1908-1909) carried out precise measurements of the surface tension of solutions containing various carboxylic acids ranging from butyric to caproic, as well as their isomers. He managed to find an empirical relationship that described with high precision all of his results" :re - ~o - cy - b In( A c + 1),
(II. 18)
3 The surfactant solution ideality condition (II.9) should remain valid within the considered concentration range
98 where the constant b is the same for the entire homologous series, and constant A increases 3 - 3.5 times with increasing the hydrocarbon chain by one C H 2 segment, which agrees with the Ducleaux-Traube rule (Fig.
II-12).
O O0 //
!
0
I
i
I I I
I I I
] .......
I......
Cn+2
n+l
I
e,,
Cn+1
c
Fig II-12. The validity of the Ducleaux- Traube rule for the three surfactants, the neighboring members of the same homologous series
The Szyszkowski equation (II. 18) satisfies both limiting conditions: it is consistent with
the linear dependence of the surface tension on
concentration within the Henry region, and agrees with eq. (II.17) at sufficiently high concentrations. At the same time, the criteria which define "low" and "sufficiently high" concentrations are set. Indeed, at low (as compared to a =
1/A)
concentrations the logarithm can be expanded in series,
yielding ~-
Within this range the derivative
cy o - c~ ~ A b c .
-dcy/dc
is constant and equals
99 dcy dc
:G-Ab-8RTexp(g
_. Rg(0s)t T
(II.19)
Within the Henry region the dependence of the adsorption on concentration is given by
Ab
F=mc. RT
If c )) a, eq. (II. 18) readily yields eq. (II. 17), since under these conditions In
(Ac +
1) ~ In
(Ac). The integration constant B equals RT Fmax In A, and the
empirical constant b in the Szyszkowski equation is interpreted as b - RTFma x .
(II.20)
Therefore, we have established once again that the linear dependence of the adsorption on concentration corresponds to the initial linear region in the surface tension- concentration dependence (see Fig. (II-7)). Since b is constant within the homologous series, it is the value of constant A that determines the steepness of the adsorption increase with increasing concentration. For this reason constant A is referred to as the
activity.By comparing
adsorption
eqs. (II. 19) and (II.20) one establishes that A is related
to the work of adsorption, g0- g~s~,as A- G b
G RTFma x
8 exp ~~ Fmax RT
.
(II.21)
The highest limiting value of the adsorption corresponds to the logarithmic region in the surface tension - concentration dependence. This
100 observation can be explained by the formation of a dense monomolecular layer. At high concentrations the surface tension isotherms plotted in cy - In c coordinates are parallel to each other with slopes of -RT Fmax (Fig. II-13) o o0
~\
r4;
n+2
iN I tanq~ = - b = - RTIT,,,.~,
I
I II
-In(A,,+1)
fl
l
i ............ I
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
~ ......
lnc - In(A,,+2)
- In(A,,)
Fig. II-13 The surface tension isotherms plotted in cy- In (c) for the three surfactants, the neighboring members of a homologous series The decrease in the surface tension at constant adsorption, occurring in agreement with the Gibbs equation, is solely due to the increase in chemical potential of the adsorbed substance caused by the increased concentration of the latter in solution. As is commonly known, the increase in the chemical potential in a stable two-component system always corresponds to the concentration increase. For the present case it translates into the increase of surface concentration, and consequently, of the adsorption. Therefore, in the concentration region where the surface tension linearly depends on the log of concentration, a slow but finite,
increase in adsorption not detected
experimentally should occur. At the same time a sharp increase in the chemical potential of the surfactant molecules in the adsorption layer
101 corresponds to a small increase in adsorption. This allows one to draw an analogy between the properties of adsorption layers at maximum adsorption and those of condensed phases at elevated pressure. In the case of condensed phases a negligibly small increase in density corresponds to a rise in chemical potential. It will be shown further that this analogy has an important physical meaning for the description of properties of adsorption layers formed by insoluble surfactants. The independence of the maximum adsorption (or the lowest possible area per molecule in the dense adsorption layer Sm~n= Sl) of the surfactant chain length can only be explained if we assume that as the adsorption values reach Fmax, the surfactant molecules are closely packed and oriented normal to the surface. Estimates of the limiting values of adsorption, Fma x = b / RT, from the experimental c~(c) dependencies, with the
successive evaluation of
minimum area per molecule, s~ = 1/NA Fmax, for carboxylic acids, are =0.21 nm 2, which agrees with the values established by other methods, e.g. by X- ray diffraction on surfactant crystals. Let us now analyze the general relationship between F and c values over the entire range of concentrations. The derivative of the Szyszkowski equation (II. 18) with respect to concentration reads: dcy
Ab = -
dc
Ac
+ 1
A = -RTF
max A c + 1
Comparison of the above expression with the Gibbs equation (II.5) yields the relationship between the adsorption and the concentration:
102
Ac
F-F
max
c
Ac+ 1
F max ~a ' .+
C
(II.22)
The above adsorption isotherm equation is k n o w n as the L a n g m u i r equation. Originally it was derived by other m e a n s and for another system, n a m e l y for the adsorption from the gas phase onto a solid adsorbent. The surface o f the latter contained fixed sites onto which the a t t a c h m e n t o f m o l e c u l e s o f adsorbing substance occurred (i.e., the localized adsorption took place).
In the theoretical derivation of the Langmuir equation (II.22) which is usually presented in some detail in physical chemistry textbooks, the solid surface is modeled as a chessboard (Fig. II-14), each site of which is able with equal probability to host the adsorbed molecules (no more than one molecule per site is allowed). The treatment is restricted to the case of localized adsorption, i.e. when the exchange between molecules of the gas phase and those adsorbed on the surface is considered, while the possibility of migration of molecules from one site to another is not taken into account. The rates of adsorption and desorption are functions of the fraction of sites occupied, 0a = F / [-'max." If the molecules in the adsorption layer are not interacting with each other, the rate of adsorption, Ua, is proportional to the fraction of unoccupied sites, (1 - 0a), and the vapor pressure p:
Ua--ka(1--0a)P
;
the rate of desorption, Ud, depends only on a fraction of occupied sites, i.e.' O d
where
ka
--
k d Oa ,
and kd are the adsorption and desorption rate constants, respectively. At the initial moment of contact between the adsorbing gas molecules and the bare
103 solid surface, the adsorption rate is the highest, while the desorption rate is equal to zero (Fig. II-15). As the surface is covered with the adsorbed molecules, the rates of adsorption and desorption become equal to each other, and dynamic equilibrium, u, = Ud, is established, i.e. F 0
a
(k a / k d) p -
Fmax
A 'p =
.
l+(k a/kd) p
(II.23)
A'p+l
The adsorption activity, A', thus has the meaning of the ratio between adsorption and desorption rate constants.
L tl n
IE tia--1)d
Q 0 Fig. II-14. A schematic diagram of localized adsorption according to Langmuir
tid 0
t
Fig. II-15. Change in the rates of adsorption, 1)a, and desorption, Ud, as the dynamic equilibrium is established
The experimental data for the adsorption on solid adsorbents from the gaseous phase in the range of moderate vapor pressure values of the adsorbing component often correlate well with the empirical Freundlich isotherm: r = [3p 1/" ,
where [3 and n are constants; n is usually of the order of several units. It is, however, necessary to emphasize that Freundlich isotherm neither has a simple theoretical meaning, nor does it yield the initial linear dependence of the adsorption on concentration, or any finite constant value of limiting adsorption.
The L a n g m u i r equation constituted an era in the theory o f adsorption
104 and chemisorption and in the theory of heterogeneous catalysis based on it. The Langmuir equation can only be used to describe reversible processes and is not applicable to the description ofchemisorption involving chemical bond formation. The transition from the case of a gas with pressure p to that of a solution of concentration c in contact with the solid phase (the adsorbent) does not significantly influence the logic of the described derivation. Therefore, the Langmuir equation can also be used to describe the localized adsorption from solution taking place at the solid interface. The comparison ofthe empirical Szyszkowski equation (II. 18) with the Gibbs equation (II.5) indicates that Langmuir adsorption isotherm (II.22) is well suited also for the description of adsorption at the air - surfactant solution interface. It is interesting to point out that at the gas - solid interface, for which eq. (II.22) was originally derived various deviations from Langmuirian behavior are often observed. The applicability ofeq. (II.22) to a successful description of adsorption from a solution was established by Langmuir himself, when he compared his adsorption isotherm to the Gibbs equation and ended up with the Szyszkowski equation as a result. The transition from localized to non-localized adsorption ( which can be viewed as the transition from fixed adsorption sites to moving ones) does not, therefore, change general trends in the adsorption in the cases described. One should also keep in mind that the liquid interface is more uniform in terms of energy than the solid interface, which contains active sites with different interaction potentials. 4 The latter is probably the reason why
The applicability of the Freundlich equation at intermediate vapor pressures is related to the non-uniformity of the solid surface in energy.
4
105 the Langmuir equation is well suited for the liquid surface. The Langmuir adsorption isotherm (eq. (II.22)) satisfies the limiting conditions described earlier. At low concentrations
c <
1/A
it yields the
asymptote F = FmaxAc,
which corresponds to the linear dependence of adsorption on concentration. The slope of the F(c) line is defined by the value of A. Comparison with the Ducleaux-Traube rule indicates that this slope increases by a factor of 3 to 3.5 with the transition to each subsequent member in the homologous series. When c = a, the adsorption equals half the maximum value, i.e. F In the case when Fma x
c >>ct, A c + 1 ,~ A c ,
= Fma x
/ 2.
which yields the second asymptote F =
(see eq. II.22). The value of a can also be determined from the
intersection point of these two asymptotes (Fig. II-16). The adsorption isotherms plotted in
F /Fmax 1- (I-" /Fmax)
- Ar coordinates are identical for all
F FIIIIlX
F mllx 2
0
~t=l/A
c
Fig. II-16. The determination of the adsorption activity, A substances that obey the Langmuir equation, and yield a single straight line with a unit slope. The surface tension isotherms can be combined into one line
106 in a similar manner by using ~ - A c coordinates. For higher members of the homologous series that have limited solubility such combined isotherms can be obtained up to the concentrations corresponding to saturated solutions. In the case of complete mutual solubility of components, the value of adsorption included in the Gibbs equation, defined by eq. (II.2) as the excess of a component in the surface layer over the bulk volume, should pass through a maximum and then reach zero, corresponding to a pure surface active component. It is important to remember that in the range of high surfactant concentrations, the replacement of the adsorption as the excess quantity with the total amount of surfactant in the adsorption layer is not acceptable. Moreover, it is necessary to account for the bulk activity coefficients of dissolved substances in the solution bulk.
['max and A - 1/a from
In order to determine the values of
experimental F(c) dependence, one may write eq. (II.22) as C
(~
C
F
1-"max
F max
The experimental data plotted in c/F - c coordinates should then yield a straight line, the inverse slope and the intercept of which correspond to the values of Fmax and ~/Fmax, respectively. Expressing the value of A in eq. (II.18) via the work of adsorption using eq (II.22), allows one to write: -1 F F - F m a x (l + l / Ac) -1 - Fma x
1+
max 6c
exp
9 (II.24)
RT
107 Replacing in eq. (II. 18) concentrations with adsorption using the eq. (II.22), one may obtain the relationship between the two-dimensional pressure and the adsorption, namely 2-
cy0 - o - Rgl-'max In
Fmax F max - F
.
(II.25)
In agreement with the above expression, the two-dimensional pressure should increase infinitely, as the adsorption tends to reach its limiting value Fmax . The measurements carried out with insoluble surfactants using
Langmuir's
balance showed that the abrupt increase in two-dimensional pressure may indeed take place when the area per molecule SM decreases to its minimum value s~ (Fig. II-17).
7Cmax
0
s1
SM
Fig. II- 17. The isotherms oftwo-dimensional pressure ~r(SM)in the region of high compression of adsorption layer
Such an increase in two-dimensional pressure is, however, not limitless. It is limited by some value ~max, at which the adsorption layers lose their stability, folds similar to hummocks on ice fields appear (Fig. II-18), and polymolecular adsorption layers form.
108
llll l
I II
Fig. II-18. Schematic representation of the loss of stability of the adsorption layers at 7r= ~max
The limiting value of the area per molecule, Sl, can be viewed as the molecule's own cross-sectional area; for certain types of surfactants this value is approximately the same and is independent of the hydrocarbon chain length, which confirms our previous statement regarding the alignment of molecules normal to the surface in the compact adsorption layer. The independence of the limiting area per molecule from the surfactant hydrocarbon chain length in fact indicates that the former is determined by the cross-sectional area of the hydrocarbon chain. Let us now return to the surfactant concentration range where the adsorption has not yet reached its limiting value, but the area per molecule, SM, is no longer large enough for the similarity between the adsorption layer and the ideal two-dimensional gas to be valid. We can assume that in such a situation the arrangement of hydrocarbon chain segments at the surface takes (on average) the intermediate position between being vertically aligned and horizontally spread. In this case the size of individual molecules, i.e. the intermolecular repulsion and (as will be shown later) the attraction, will have an effect on adsorption. If the attraction between molecules is weak, the dependence of the two-dimensional pressure on the area per molecule (see Fig. II-17) can be
109 described by the expression proposed by Volmer 5
K(S M -- S1 ) - i T
(II.26)
which is analogous to the ideal gas law corrected for the volume of individual molecules. In order to obtain the value of s~ from the experimentally determined two-dimensional pressure - area per molecule dependence, it is convenient to plot data in the ~rsM- a: coordinates. In the absence of any noticeable attraction between molecules, the experimental data plotted in these coordinates fall onto a straight line (Fig II-19), the slope of which yields the area per molecule in the dense adsorption layer, while the intercept corresponds to k T - 4 mN m -~ nm 2. If the molecular weight, M, of the studied substance is unknown, it is not possible to estimate adsorption in moles/m 2 , nor can therefore, the area per molecule, sM, be established from a known sample weight m. In this case, instead of zrsM, the product of the two-dimensional pressure and the macroscopic area S between the barriers in the Langmuir balance (see Fig Ill 0) is plotted against zc, and eq. (II.23) can be rewritten as
r S-
m m s! - ~ N A 7c + - - R T , M
i.e. the experimental data fall onto a straight line in zrS- Jr coordinates (see Fig.
5A similar expression can be obtained if the logarithmic part of eq.(II.25) is written as -ln (1 - F / Fmax) and further expanded into a x-1 series using an approximation that In x -- 2 ~ . x+l
110 II-20). The extrapolation of this line to Jr- 0 yields the molecular weight, M; the area per molecule in the dense adsorption layer, s], can then be evaluated from the slope. :rrSM, m N (nrn)2
4 -~-_,
(pH)2 . y , . .
7CS
a_ . . . . . .
, /- ` /t ( . ~~ n
In
~RT
0
Fig. II-19. The ~S M (~) isotherm with intermolecular repulsion taken into consideration
~'~'_
(pH),
- tt aa nn t pq~ = = ss IIN t,~AA M
..T.TL _ _ _
7c
Fig. II-20. The determination of the molecular weight M, and area per molecule s~ from the zcS0r)isotherm
The described method was used to determine molecular weights of proteins and nucleic acids and to study their structure in the adsorption layers. The method allows one to obtain valuable information regarding the conformation of molecules within the surface layer. The latter determines the area these molecules occupy in a two-dimensional film. During the measurements the pH of the medium was adjusted to values at which molecules are charged due to ionization, so that one did not have to introduce a correction for intermolecular attraction. The protein conformation depends on the pH of the medium, which is crucial for the ionization and hydration of ionogenic groups [10]. With changes in the pH, the slopes of the ~rSM(Jr)lines, i.e. the values of s~, also change (Fig. II-20). Upon the compression of films formed by globular proteins (such as albumine, globuline, hemoglobin, trypsine, and others) up to a pressure of~20 mN m -~ the two-dimensional pressure isotherms are quite reversible. At somewhat higher compression, such that the area per amino-group reaches
111 -0.17 nm 2, the two-dimensional pressure increases abruptly, causing irreversible changes in the film structure. The films may acquire some specific insolubility, as well as some peculiar structural and mechanical (rheological) properties, which are, to a great extent, related to changes in the conformation and structure of the protein molecules [5]. A stronger compression of films to about 0.05 - 0.1 nm 2 per group causes their collapse, resulting in the formation of folds (and, possibly, even polymolecular layers) and final detachment from the surface. It is noteworthy that many proteins in the monolayer state retain their enzymatic activity and are capable of taking part in specific chemical reactions. For this reason the colloid-chemical methods used to investigate the properties of protein films, combined
with other
techniques, represent
valuable tools for the study of the properties of proteins. These methods allow one to examine more closely mechanisms of transport phenomena that take place at cellar interfaces in biological systems. The latter are the interfaces at which the accumulation of surface active substances with biological and physiological activity occur.
These substances, when present at such
interfaces, reveal their important unique properties (e.q. enzymatic activity). The low rate at which equilibrium between the adsorption layer and the bulk is established is typical for high molecular weight surface active substances for which the surface tension gradually decreases with time. The measurement of the surface tension by static and semi-static methods (see Chapter I, 4) as a function of time during the formation of adsorption layers allows one to retrieve information on the kinetics of adsorption phenomena [11,12].
112 11.2.3. Structure and Properties of Saturated Adsorption Layers Let us now discuss the most general case of the behavior of molecules in the adsorption layer, accounting for both intermolecular repulsion (the effect of the size of individual molecules) and attraction. Since the energy of intermolecular attraction between the hydrocarbon chains increases as the chain length increases, it seems obvious that the strongest effects of intermolecular interactions should be observed with the long-chain practically insoluble surfactants. Indeed, the basic concepts regarding the structure and properties of adsorption layers in which the attraction between hydrocarbon chains played a significant role, were established with the Langmuir balance. It turned out that the adsorption layers may not only have states similar to those present in bulk phases, but also some unique ones for which no analogs in the latter exist. At present, there is no commonly accepted classification of adsorption layers that would both include all possible states that occur within the dense adsorption layers, and give substantiative information regarding their nature. Adamson [7] distinguished the following types of surface films. 1. Gaseous G-films, which to a good approximation follow either the two-dimensional ideal gas law (II. 16), or the equation describing gas with molecules of finite dimensions (Fig. II-21). For instance, such films are formed by fatty acids at low two-dimensional pressures or sufficiently high temperatures. Sometimes vapor-type films which exist at temperatures below the condensation point of the adsorption layer are singled out from this class (see below).
113
mI I
0 69
I
0.4 I~.~~\\\ ~VSM---4 S \ 0.2 ~r~
~ ,!
0
10
.... I
20
'
G ,,'I'
30
!
-
sM, (nm) 2
Fig. II-21. The two-dimensional pressure isotherms of some members of the fatty acids homologous sequence: I - lauric (Cj2); 2 - myristic (C~4);3 - pentadecanoic (Czs); 4- palmitic (C16)
2. L i q u i d
expanded
L2 - films, for which the area per molecule
usually stays within the range of 0.4 - 0.5 to 0.22 nm 2 when the twodimensional pressure is raised from very small values (tenths, hundredths of mN m ~ ) to several mN m -~. Figure II-22 shows the two-dimensional pressure isotherms of pentadecanoic acid measured at different temperatures, ~
as
indicated by numbers above the curves. The L 2 films are formed by many substances among which are the acids that have medium long chains at elevated two-dimensional pressures. The formation of liquid expanded films is especially typical with substances having branched structure. In such films the area per molecule significantly exceeds the cross-sectional areas of the hydrocarbon chains. The latter are present in a condensed state, forming film of a thickness that is smaller than the hydrocarbon chain length of the surfactant molecules, which increases with increasing two-dimensional pressure. Such a state, having no bulk phase analogue, may possibly form due to the mutual attraction between hydrocarbon chains with the simultaneous repulsion between polar groups.
114 7g'~ mN (run) 2 ITI
J0 20 10-
L
0
0.1
0.2
0.3
0.4
0.5 SM, (nn~l) 2
Fig. II-22. The two-dimensional pressure isotherms Ir(SM) of adsorption layers of pentadecanoic acid at different temperatures
3.
Liquid
condensed
L1 - films
are characterized by low
compressibility (steep increase in ~r- SM curves). For these films the extrapolation of Z(SM)curve to z=0 yields s~ ~ 0.22 nan2, which is only slightly higher than the cross-sectional area of a surfactant hydrocarbon chain. The L~ films are usually formed from L 2 films at high two-dimensional pressures. At elevated temperatures the higher fatty acids (starting with tridecanoic) yield liquid films directly, without forming the L 2 films first. Surfactants with large polar groups (e.g., phenols) yield liquid films in which the area per molecule exceeds 0.22 nm 2. A transitional region of relatively high compressibility may exist between L~ and L 2 films. This region corresponds to the so-called "intermediate films" (see region Iofthe curves shown in Fig. II-22), the nature of which is not yet fully understood. 4. Solid, or S-films, have even less compressibility than the liquid
films; the limiting value of area per molecule in such films, obtained by extrapolation to zero two-dimensional pressure, is 0.206 nm 2. The most important differences between the S-films and those of other types become
115 evident when their rheological properties are compared. In liquid films the flow already occurs at low shear stresses (see Chapter IX), and the shear rate is a linear function of the shear stress, while the solid films can withstand significant shear stresses without any residual deformation prior to their collapse. The solid and liquid adsorption films can be distinguished from each other by a qualitative test known as the "blow-off method". In this method the surface of liquid that carries the adsorption layer is covered with a layer of fine powder (usually talc), onto which a stream of air is directed at an angle. In the case of liquid surface films the motion of particles can be observed, while in the case of the solid ones the latter does not occur, but sometimes a separation of large "ice floe"- type regions moving as one piece is observed. Let us now turn our attention to a more detailed discussion of transitions that occur between different types of films, and in particular to the way in which these transitions are influenced by the temperature, the structure of surfactant molecules and the composition of the medium. The direct transition from gaseous and vapor films to liquid and solid condensed ones is a two-dimensional first-order phase transition that is quite similar to a threedimensional vapor condensation. A decrease in the area per molecule in the adsorption layer in the region of gaseous films causes a gradual increase in pressure up to the level that corresponds to the condensation pressure of a saturated two-dimensional vapor, ~rc, at an area per molecule equal to s c (see Fig. II-21). The subsequent compression of film is not accompanied by an increase in the two-dimensional pressure" the two-dimensional vapor transforms into the two-dimensional condensed state, which can be either liquid expanded, liquid condensed, or solid, depending on the nature of the
116 surfactant and temperature, and, in some cases, on the composition on the bedding bulk solution (in part on the pH value). The validity of this interpretation of two-dimensional pressure isotherms in the region of constant two-dimensional pressure was confirmed by A. Frumkin in his studies dealing with peculiarities of the surface electric potential in the condensation region.
The following simplified considerations show how a sudden change in the potential near the surface is related to the structure of the adsorption layer. Let us view the surfactant molecules as dipoles with dipole moment, lad, oriented at an angle 7~to the interface, and the film as a whole - as an electric capacitor with an equivalent dielectric constant e (Fig. II-23).
V//////,/,,. P r o b e
2
Fig. II-23. A schematic structure of the surfactant adsorption layer in the region of twodimensional condensation (a schematic representation of Frumkin's experiments) The total specific (i.e., per unit area) dipole moment of the film equals
~d
sinz
=ps 8 ,
SM where Ps is the surface charge density, 8 is the thickness of the film viewed as a flat capacitor with specific capacitance equal to ee 0/8;
l/s M
= FN A is the number of molecules per unit
surface area. The potential difference between the plates of such a capacitor equals
Aq) -
Ps 8 8~;0
=
~d sin Z SMSgO
= ks F sinz,
117 where ks = gd NA/ gg0. Thus, the potential of the adsorption layer (represented by a capacitor) depends on the orientation of the surfactant molecules. Studies of the surface potential as a function of adsorption allow one to obtain information on the orientation of surfactant molecules during the condensation of adsorption layers. A special probe consisting of an electrode with a small amount of radioactive tracer on its surface can be used for the surface potential measurements. The radioactive isotope ionizes air in the vicinity of the electrode surface. This probe allows one to measure the surface potential Aq). Studies carried out by Frumkin showed that the following relationship is valid for fatty acids (Fig. II-24):
( d A q o / d F ~ = sinz________~~L 6. (dAq) / d F ) G
sinzG
If we assume that in the condensed adsorption layer of fatty acid the orientation of surfactant molecules is nearly vertical (sin ZL = 1), then the average angle of orientation of the surfactant molecules in the vapor adsorption layer is - 10~ , which agrees with the nearly horizontal orientation of surfactant molecules in the dilute adsorption layers. When one moves the electrode with a small area ( the probe) along the surface in the condensation region, abrupt
Aq) ~L a n a L - \ dF )L
/
tan~
(dzXq~] c =\-~j G
/
/
'
~
.
G
,
,
F Fig. II-24. The potential difference Aqo as a function of adsorption, F, for the transition from gaseous film to the condensed one. leaps in the measured potential are revealed. These are indicative of the existence of"islands" composed of condensed two-dimensional phase, among which the areas covered by the o
gaseous adsorption layer are present. No abrupt changes in the measured potential are
118 observed during the migration of the probe along the surface in both homogeneous regions of adsorption layers, when the two-dimensionalpressure, 7v,is either smaller or larger than the condensation pressure, ~vc. The
two-dimensional
pressure
isotherm
characterizing
the
condensation ofsurfactant molecules in the adsorption layer (i.e., the transition from gaseous films to condensed ones), has a shape similar to that of a threedimensional real gas given by the van der Waals equation:
p+
(Vm - b y ) - aT.
According to Frumkin, a similar equation can be used to describe the properties of the two-dimensional adsorption layers, i.e.,
I
(sM-Sl)-kr,
(II.27)
where as, like av, characterizes the molecular attraction. The increase in temperature leads to an increase of two-dimensional condensation pressure and to an narrower condensation region. The value of Sl in eq. (II.27) depends on the nature of the condensed state. For liquid expanded films formed by derivatives of aliphatic hydrocarbons, s~ is usually ~0.5 nm 2. If the condensation results in liquid condensed films, s~ is close to 0.22 nm 2, while for solid films it is 0.206 nm 2 (in the case of molecules with linear hydrocarbon chains). An increase in the hydrocarbon chain length of the surfactant molecules (i.e. stronger attraction between the latter) has an effect similar to
119 that caused by lowering the temperature: a decrease in the condensation pressure of vapor films, Jrc, with simultaneous broadening of the area in which the two-phase and intermediate states exist (see Fig. II-21). The ionization of polar groups of the adsorbed molecules, e.g. of acids at alkaline pH of the water on the surface of which the film is formed, on the contrary, increases the repulsion and causes the region corresponding to the gaseous state to expand. Expanded films are often formed by unsaturated compounds (e.g., oleic acid) and compounds with several different polar groups attached to their hydrocarbon chains. Oppositely, molecules of triglycerides form condensed unexpanded films in which the area per molecule (0.66 nm 2) is three times that of surfactants with one aliphatic chain. In order to better follow how the attraction between hydrocarbon chains influences the trends in the JrSM Or) curves, let us write the twodimensional pressure isotherm (II.27) as
~s M
- -
kT
__
-
as -
-~-
+
S1 .
SM
At not too high two-dimensional pressures, and, consequently, not too small areas per molecule in the adsorption layer, the deviation of ~rsMfrom kT is mainly due to the contribution from the negative - as/S a term, which describes the attraction: the JrSM(Tr)curve deviates downward from kT=4 (Fig. II-25). For more compressed (high Jr values) films, when the values of SM become close to those of s~, the last positive term in the above equation, describing the "repulsion", may acquire a rather high value, resulting in the JrsMvalue higher than that of kT.
120 ~6'M
Fig. II-25. rcsM(rr) isoterms in the presence of attraction and repulsion between molecules in adsorption layer Everything stated above is mainly related to the insoluble surfactants. As emphasized earlier, the soluble surfactants,whose polar group interacts weakly with water, contain a short hydrocarbon chain, and the intermolecular interactions between the hydrocarbon chains in such compounds are of the less importance, as a rule. Let us now examine how the shape of the adsorption and two-dimensional pressure isotherms of such surfactants can change, when two-dimensional condensation takes place in the adsorption layer. During condensation with changing adsorption, the constant two-dimensional pressure corresponds to a constant value of chemical potential of the substance, in agreement with the Gibbs equation. This is similar to the three-dimensional case when the chemical potential is independent of the ratio of the liquid-vapor content. Thus, for soluble surfactants that undergo surface condensation, the surface condensation process should occur at some constant bulk concentration, co, i.e. the condensation has the form of an abrupt change in adsorption from some F c = 1/sen A to a value approximately equal to the limiting adsorption, ['max"The dependence of adsorption on concentration is represented by curves 3 and 4 shown in Fig. II-26. The quantity Ac~ relative, or dimensionless, concentration corresponding to twodimensional vapor saturation
increases with
increasing temperature and decreasing
surfactant hydrocarbon chain length. If the formation of liquid expanded adsorption layers takes place, the F(c) dependence is smoother (curve 2 in Fig. II-26). The effect that the two-dimensional condensation of soluble surfactants has on the surface tension isotherm is less pronounced: according to the Gibbs equation, a jump in the
121
1-' Fm"x - -
3' '
2'
4
'
1
i
I,
(3
I I
l
o (AcJ,,+, (Ac&
ac
Fig. II-26. The adsorption, F(c), and surface tension ~ (c) isotherms of soluble surfactants, corresponding to the formation of gaseous (curves 2 and 2') and condensed (curves 3, 3 ~4, and 4) films
d~/dc value should correspond to an abrupt change in adsorption at constant concentration, Cc, i.e, there should be a bend in the ~(c) curve (Fig. II-26). In the event when formation of liquid expanded films is taking place, the adsorption increases gradually, and the ~(c) curve shows a twist instead of a bend (curve 2' ). Similar phenomena were also observed with carboxylic acids having relatively long chains but still rather soluble in water, e.g .with the lauric acid that was investigated by Frumkin.
The solubility of lauric
(C12) acid
is sufficiently high, so that one is
able to carry out the surface tension measurements. Yet at the same time, its solubility and dissolution rate are low enough for one to be able to study the properties of the adsorption layers formed with the Langmuir balance, provided that one works fast. The latter feature allowed Frumkin to compare trends in the 7VSM(TV)curves obtained for the same substance using methods established for both soluble and insoluble surfactants.
The experimental
122 curves shown in Fig. II-27 clearly indicate that the data collected using each of these methods are comparable. /I'SM~
m__N(nm)2 Ill
8 C4
Cls
0
4
8
12
16
20
Jr, mN/m Fig. II-27. The IVSM(TV)isotherms of the adsorption layers in a series of fatty acids established from the two-dimensional pressure measurements by the Langmuir method (C12, C~5 chains), and from the measurements of the surface tension (C4 - C12)
These results agree with those obtained by means of the Langmuir method for acids having hydrocarbon chains longer than C12 and with c~(c) measurements carried out with shorter-chain members of homologous series. Good agreement between these two series of curves can serve as convincing evidence of the fact that the properties of the adsorption layers of both soluble and insoluble surfactants are close, and are not directly related to the solubility of surfactant molecules in the supporting liquid. Similarly to condensed phases, the adsorption layers have some specific
mechanicalproperties characteristic of their phase state. The investigation of mechanical properties of adsorption layers can be performed using the torque pendant technique (Fig. II-28). A disk attached to a thin elastic thread is
123
f
J
Fig. II-28. Schematic drawing of the instrument used to study the mechanical properties of adsorption layers
placed at the surface of water covered with the adsorption layer. The disk is either placed at the center of a cylindrical cuvette, or is surrounded with a ring such that the gap between the latter and the disk is constant. By twirling the limb holding the upper edge of the thread (or by rotating the cuvette) one creates a torque of known value at the edge of the disk. The deformation of the adsorption layer that progresses throughout the gap between the disk and the ring is measured. The measured values can, for instance, be read on the scale illuminated by the lightbeam reflected from the mirror mounted on the disk. Alternatively, one may also rotate the cuvette (outer ring) at a given rate, and determine (from the torsion angle of the thread) the shear stress originating at the periphery of the disk
(see Chapter IX, 1).
Investigation of the mechanical properties of the adsorption layers formed with surfactant molecules of different nature has shown that the condensed adsorption layers can indeed be present in both solid and liquid states. Solid adsorption layers (Fig. II-29, line 1) do not reveal irreversible
124 deformations up to a certain load, ~sc, above which destruction occurs.
3
111
0
i
,
,
1
-
_,
,
....
, .
~sc
.
.
.
.
"~sc
.
.
.
.
"~s
Fig. II-29. The e (z~) dependence corresponding to different mechanical behaviors of the adsorption layers
In the case of liquid adsorption layers, the deformation by shear occurs at any small load (Fig. II-29, line 2), and the shear rate, ~ is proportional to the applied stress, ~ (the torque angle of the thread). The latter allows one to estimate the surface viscosity of the adsorption layer, which is strongly dependent on the nature of surfactants. The adsorption layers may also reveal more complex rheological behavior, intermediate between that of liquids and solids (Fig. II-29, curve 3). It was shown by Izmailova et al that the rheological properties of adsorption layers formed with
high molecular weight surfactants and
biopolymers play an important role in ensuring the stability of disperse systems stabilized by such layers (see Chapters VI-VIII). With the daily growing variety of surfactants (see Chapter II,3) with different and sometimes complex structures, the demand for accurate and precise methods for detailed investigation of the behavior of surfactants at the air/water interface, i.e. in the adsorption layers, has increased. One is usually
125 interested both in determining the surface coverage and in collecting information on the structural properties of adsorption layers, such as information with respect to the arrangement of surfactant molecules within adsorption layers, e.g. the effective surfactant chain length, the tilt angle of surfactant chains, and the area per surfactant molecule. Over the last decades a number of advanced techniques for studying the properties of surfactant layers have emerged. Let us briefly address the principles behind some of these methods, namely neutron and X-ray reflection, radioisotope methods, and optical methods such as ellipsometry, second harmonic and sum frequency generation. An excellent review and analysis of modern instrumental methods for the investigation of adsorption layers can be found in [ 15,16]. The technique that is commonly used nowadays for the investigation of the structure of adsorption layers is the specular neutron reflection. Neutron reflection studies are based on the measurement of the intensity of a reflected beam, generated from a collimated beam of neutrons with the wavelength )~ falling on the air-solution interface at a grazing angle 0. The quantity that is measured is referred to as the reflectance, R, which is given by the ratio of the reflected intensity to the intensity of the incident beam [ 17]. From reflectance measurements one can estimate the adsorption and obtain information on the orientation of molecules in the adsorption layer.
In neutron reflection the surfactant adsorption layer can be viewed as a layer of uniform refractive index over the underlying solution with a different refractive index. The refractive index for neutrons (not to be confused with the refractive index for light) is so close to unity that it is much more convenient to describe reflection in terms of the scattering length
density, 9, that is related to the refractive index, n, as
126
n 2 - 1-~p,
The scattering length density, P, is the sum of scattering length densities, bJ, and number densities,
nj, of atomic species,j, namely f3-E
bjnj , J
For a surfactant monolayer the reflectance is related to the scattering length density and the thickness of adsorption layer, 8, as [15]
K4
p1_P0)2 +,P2-[1,
2 4- 2( Pl - P 0 ) ( P 2 - D 1)COS(K~
,
where indices 1, 0, and 2 are related to the adsorption monolayer, the air and the substrate solution, respectively, and K is the momentum transfer related to the grazing angle of incidence as K = (4re sin0)/)v. The scattering length density is an empirically determined quantity with a very peculiar feature, namely that it is significantly different for hydrogen and deuterium, bn = -3.74x 10.5 A, and bD = 6.67x10 -s A, respectively [15]. It is apparent that since bn and bD are of opposite signs, it is possible to add some D20 to H20 to yield a mixture with zero scattering length density. The D20/H20 mixture having mole fraction of D20 of 0.088 is referred to as
null reflecting water, because there is no reflection at the interface
between such water and air. If deuterium-substituted surfactant is dissolved in this null reflecting water, the neutron reflection occurs only when the surfactant forms an adsorption layer at the airsolution interface. For this case the general expression for reflectance is considerably simplified, namely
K4
2P12COS_K~_ ,
and using the definition of the scattering length density, one can relate R to the scattering
127 length, b~, and number density, n~, of the surfactant monolayer:
R~I~~ -
4bl2n2sin2
-~ --
.
These equations illustrate that the level ofreflectivity is determined by p, while the variation of reflectivity with ~: is determined by the layer thickness, ~5. By fitting the experimental reflectivity values to the monolayer model described by the above two expressions one can obtain the values for p~ and 5. From the established value of the scattering length density and known
b~, one can estimate the number density, nl From these parameters it is then possible
to estimate the adsorption, F, as F-
1 NASM
=
918 NAbl
Since the reflection from a monolayer ofsurfactant over null reflecting water occurs due the presence of deuterium in the surfactant molecules, the value of adsorption estimated from reflectance measurements will be dependent on the particular position within the surfactant molecule in which hydrogen was substituted for deuterium. Consequently, the substitution of hydrogen with deuterium can make reflection sensitive to any part of the layer, meaning that one can study separately the reflection from different portions of the hydrocarbon tails, surfactant polar head, or counter-ion. In each case different values of adsorption will be obtained depending on the position and conformation of the deuterium substituted portion of the surfactant molecule in the adsorption layer. The results of a variety of such measurements will thus yield information on the adsorption layer structure. Also, one can obtain information on the distribution of water molecules within the adsorption layers and the extent to which water penetrates into such layers. The interpretation of these data requires the acceptance of a certain model characterizing the adsorption layer. For example, the model described by Lee [ 18] and Simister [ 19] divides the surfactant layer into an upper portion consisting solely of a fraction, ~, of hydrocarbon chains, and a lower portion consisting of the remainder ("immersed") fraction, 1-~, of hydrocarbon chains, water and all polar heads. Each of these layers can then be described with its own scattering length density, and an
128 expression for reflectivity similarto those shown above can be obtained [15]. Similarly to the reflection of neutrons, the reflection of X-rays can also be used to investigate adsorption layer properties. X-ray reflection is governed by the same principles as neutron reflection. This technique is more useful in cases when the adsorption layers reveal long-range order, i.e. it is of greater use for studying the properties of insoluble adsorption layers. However, even in the absence of long-range order, X-ray reflection studies can yield some useful data due to the fact that X-rays are absorbed by the solution surface much more strongly than are neutrons. This results in lower background scattering from X-rays, and thus much lower values of R that can be measured. The obvious disadvantage, though, of using X-ray reflection is that the contrast between different portions of the adsorption layer can not be as easily manipulated as in neutron reflection. In the ideal situation, one may use both techniques simultaneously, as was done by Lee [20]. The radiotracer technique was introduced quite some time ago [21 ]. The use of the radiotracer method for assaying the total surfactant concentration in solutions is mentioned in Section II.4 of this chapter and is rather straight forward. In studies of adsorption layers at the air-liquid interface the radiotracer method is used for determining the adsorption, F. The species whose adsorption is to be measured are labeled with a [3-emitting tracer, usually tritium. Because of the strong absorption of [3 radiation, the detector is dominated by the signal coming from the surface. Typically measurement involves three steps" measurement of the sample, measurement of a background correcting the radiation emission from the bulk solution and the measurement of a calibration specimen for the purpose of scaling the
129 signal. As compared to neutron reflection, the radiotracer method has much lower sensitivity and does not offer as broad capabilities. Fluorescence and Brewster angle microscopy are two microscopic techniques that are employed in the study of adsorption layers at the airsolution interface. Fluorescence microscopy is based on the analysis of the steady-state fluorescence from fluorescent probes (molecules of flourescent surfactant, fluorescent dyes) incorporated into adsorption layers. The analysis of fluorescence may yield information on the structure of the adsorption layer, as well as on chemical and structural changes taking place in such layers. For example, fluorescence microscopy allows one to directly visualize the coexistence of two phases, such as L1 and L 2 films. In these cases the change in the fluorescence intensity occurs in the same area where surface pressurearea isotherms exhibit transition between films of different types. Change in the fluorescence intensity usually occurs either due to changes in the medium in which the probe is present or due to changes in the geometrical orientation of the probe itself within the adsorption layer. Brewster angle microscopy (BAM) is a technique that allows one to probe the two-dimensional organization of adsorption layers. The main feature behind this technique is the absence of reflection when p-polarized light 6 is impinged at the air-water interface at a Brewster angle, 0Br -- 53.1 ~ When the adsorption layer is present at the surface, the Brewster conditions are altered and the reflection of light impinged at the interface at a Brewster angle is
By convention, in a p-polarized wave the electric vector oscillates in the plane parallel to the plane of incidence. The wave in which the electric vector oscillates in the plane perpendicular to the plane of incidence is referred to as the s-polarized wave. See e.g. [17] 6
130 observed. The reflected light is used for imaging the adsorption layer. Brewster angle microscopy images are usually digitized and filtered in order to reduce diffraction fringes caused by the coherent nature of laser light. Similarly to the case of fluorescence microscopy, there is a good correlation between changes in the shape of two-dimensional pressure isotherms and BAM images. One can clearly distinguish between smooth images of L 2films and images of S films showing high degree of aggregation of molecules and containing many defects and holes [16]. Finally, let us briefly mention three optical techniques that play an important role in studies of the adsorption layers. These are ellipsometry,
second harmonic generation, and sum frequency spectroscopy. The main subject of ellipsometry is the measurement of the thickness and index of refraction of adsorption layers. This technique is based on the difference in reflection of s- and p-polarized light. The monochromatic light from a He-Ne laser is passed through a polarizer, then rotated by passing through a compensator and impinged on the interface. The reflected light is elliptically polarized and is measured by the polarization analyzer. Of interest are the phase shift and the ratio of amplitudes between s- and p- waves in the reflected light. When an adsorption layer is present, these two quantities are related to the complex index of refraction and film thickness.
Second harmonic
generation (SHG) and sum frequency generation spectroscopy (SFG) are techniques based on non-linear optical phenomena that occur in the high light fluxes generated by lasers. In SHG two photons with energy hv are coupled to produce a photon with energy 2hv, and in the SFG two laser beams with two different frequencies are overlapped to produce a beam with a frequency that
131 is the sum of the frequencies of the original beams. The use of SHG and SFG in the study of adsorption layers is based on the fact that their generation is forbidden in media with a center of symmetry, such as the bulk solution, but is allowed at the interface, which is not centrosymmetric. The signals are thus generated from the adsorption layer only. SHG and SFG yield information on the orientation and conformation of molecules in the adsorption layer, and even on the structuring of water molecules underneath the adsorption layer.
II.3. Classification of Surface Active Substances. The Assortment of Synthetic Surfactants
The ability of surfactants to alter the properties of interfaces through adsorption, and, therefore, to influence properties of the disperse systems, is broadly utilized in numerous technological applications. The ways in which surfactants influence interfacial properties depend to a significant extent on the chemical nature and structure of the phases in contact and on that of the surfactant molecules, as well as on the experimental conditions. Following Rehbinder [5], we will classify surfactants according to the physico-chemical
mechanism of their action both on the disperse system as a whole, and at the interface. Four large groups of surfactants can be distinguished; these groups will be briefly characterized below. I. Substances that reveal surface activity only (or mostly) at the airwater interface. These are the middle and higher members of homologous series of aliphatic alcohols and acids and some more complex natural compounds. The surfactants belonging to this group decrease the surface
132 tension at the air-water interface (see Chapter III, 3) and show moderate
wetting action. These substances can be used as foaming agents, particularly to form foams with low stability (e.g. in flotation). Some highly surface active substances that belong to this group (octyl and iso-amyl alcohols, also some other substances) are used as defoamers (see Chapter VIII, 2). II. Substances of various nature that show surface activity at various interfaces (solid-liquid, liquid-liquid). Surfactants under the conditions of a strong decrease in the interfacial free energy favor the development of new interfaces in the processes of destruction, dispersing and processing of solids (see Chapters IV, IX) and during emulsification of liquids. Owing to their action, all of these surfactants can be classified as dispersants. The substances belonging to this surfactant group also allow one to control the selective
wetting of surfaces (see Chapter III,3). A characteristic feature of surfactants belonging to both of these groups is inability to form spatial gel-like networks both at interfaces and in the bulk. III. Surfactants that form gel-type networks (i.e., mostly solid-like, see Chapter VII, 5) within the adsorption layers and in the bulk. These surfactants may not necessarily be highly surface active. The majority of the substances belonging to this group, known as protective colloids, are complex synthetic or natural polymers containing a large number of polar groups (e.g., proteins, glucosides, cellulose derivatives, polyvinyl alcohol, etc.). These substances are typically used as highly effective stabilizers for different moderately concentrated disperse systems, e.g. foams, emulsions, suspensions. The surfactants belonging to this group may act as plasticizers for highly
133 concentrated dispersions (pastes). The mechanism of action of these compounds is discussed in Chapters VII - IX. IV. Detergents are surface active substances that have features of all three surfactant groups described above, and in addition they are able to spontaneously form thermodynamically stable colloidal systems (for
micellization in surfactant solutions please refer to Chapter VI). The particles that are washed away may become incorporated into the nuclei or micelles, i.e.
solubilization (See Chapter VI) takes place. Various anionic, cationic, and nonionic surfactants that are encountered further in this section are typically members of this surfactant group. Based on their chemical structure, the surface active substances can be subdivided into two large classes. On the one hand, there are the organic surfactants containing diphilic molecules. These substances are universally surface active at most interfaces (obviously, below their decomposition temperatures), but their ability to decrease surface tension is relatively small, usually around 30 - 40 mJ/m 2. On the other hand, there are also various (primarily inorganic) substances that show selective, but often very high surface activity at a specific interface (where the lowering of (y can be quite severe). Examples of such substances include low-melting point metals with respect to higher melting point ones, water with respect to some salts, etc. Silicon-organic and other element-organic surface active substances with improved thermal stability and other unique properties, thanks to which they can be used under rough conditions (high temperatures and pressures, aggressive media), represent a special class in the modern assortment of available surfactants. Another important group of surface active substances
134 that deserves special attention are the compounds containing fluorinated and perfluorinated groups. These substances are able to cause stronger decrease in c~as compared to those with hydrophobic hydrocarbon groups. Apparently, one can come up with various means of classifying surfactants, and hence the subdivision of surfactants into four different groups based on the mechanism of their action is by no means universal. Indeed, these groups may merge or overlap with each other. The classification described was developed primarily for "ordinary" organic surfactants, but to a certain extent it can be applied to other substances as well. This is particularly true for the representatives of group II, i.e. for the dispersants (see Chapter IX, 4). The description of properties and applications of various surfactants in industry, agriculture, medicine, etc, as well as information on the synthesis and manufacturing of surfactants can be found in the literature [22-25]. Alkylbenzenesulfonates, alkylsulfates, olefinsulfonates, oxyethylenated surfactants, block co-polymers, and quaternary ammonium salts represent the main classes of synthetic organic surfactants. The main characteristics that determine the level of production of particular surfactants, aside from physico-chemical properties, are cost, availability of raw materials and environmental safety. The latter is characterized primarily by
biodegradability,
which is defined as the time
required to reduce the surfactant concentration in the ambient medium by a given factor [26]. The synthesis of surfactants with good biodegradability is an important task, as the environmental issues has become of a major concern. For example, the accumulation of surfactants within the adsorption layers formed in natural water reservoirs affects the life of various living organisms
135 (e.g. by affecting the transport of oxygen). The formation of stable foams at the water surface and in purifying filters due to surfactant adsorption is a significant hazard to the environment 7 . Surfactants with linear chains usually have the highest biodegradation rates, while those containing aromatic rings and branched chains (especially those with quaternary carbon atoms) are poorly digested by microorganisms. For that reason the use of normal paraffins as raw materials in surfactant production benefits the environment. Based on their chemical nature, organic surfactants with diphilic structures are classified as anionic, cationic, amphyphilic, and nonionic. Anionic surfactants are reasonably inexpensive and fairly universal, so they occupy a predominant place of about 60% in the world surfactant market. The contribution of nonionic surfactants to world production is about 30% and is growing; cationics constitute about 10%, while synthetic amphyphilic surfactants account for only fractions of a percent. Anionic surfactants are organic compounds which upon dissociation in water produce large anions containing hydrocarbon chain. These anions are the "carriers"of the surface activity, while cations are not surface active at the air-solution interface. The description of the most important anionic surfactants is given below [27].
7 The following rather symbolic but at the same time graphic comparison gives one a feeling for the role that the surfactants manufactured worldwide annually can play when they are distributed within the adsorption layers: if distributed evenly, 6 million tons of surfactants with Mr ~ 200 g mol -~ can cover the surface of our planet in 10-15 monolayers
136 Salts of the c a r b o x y l i c acids (soaps) ~, described by the general formula RCOO- - Me +, where R represents the C8 - C~8 hydrocarbon chain, +
and Me + can be Na + (in solid soaps), K + (in liquid soaps), or NH 4. Rather simple
manufacturing
technology,
relatively
low
cost,
and
full
biodegradability are the important characteristics of these surfactants. The soaps of carboxylic acids show good detergency only in alkaline media. In acidic media, due to the formation of poorly soluble fatty acids, and in hard water (due to the formation of insoluble calcium and magnesium salts) these surfactants show rather poor detergency. For a long time the soaps of carboxylic acids were manufactured by saponification of natural fats, which are the esters of glycerol and various fatty acids. A large consumption of food materials promoted the development of commercially manufactured synthetic fatty acids (SFA). As shown further down, the unbranched C~o - C20 synthetic fatty acids can be manufactured by a number of methods and are widely used in surfactant production. The salts of aromatic sulfonic acids, a l k y l a r y l s u l f o n a t e s (most commonly alkylbenzenesulfonates), with the general formula RArSO3 Me +are the least expensive and most widely available synthetic surfactants. These surfactants contribute about 70% of all the manufactured anionics, which include more than 100 species. The anion of a strong acid present in the structure of these surfactants accounts for the good detergency of these
8 Calcium, magnesium and aluminum salts of fatty acids, insoluble in water but soluble in hydrocarbons, are used as components of lubricant oils and greases containing oil as the dispersion medium. The derivatives of more complex organic acids, such as cholic acid found in bile, humic acids present in soils, etc., play an important role in nature
137 substances in both alkaline and acidic media, and in hard water. Alkylbenzenesulfonates
are usually synthesized by the alkylation
of benzene with chloroalkanes or alkenes, which is followed by subsequent sulfonation and neutralization steps. In this synthesis so-called tetrapropylene, consisting of a mixture of isomeric C~0 - C15 alkenes with a large fraction of branched compounds, was a popular aliphatic raw material prior to 1964. The biodegradation rate of sodium tetrapropylbenzenesulfonate made from tetrapropylene is, however, very low (Fig. II-30, curve 1), for which reason the use and production of this compound is banned in most developed countries [28]. Due to improved biodegradation, the production of linear unbranched alkylbenzenesulfonates, which better degrade in biosphere (Fig. II-30, curve 2), had rapidly developed. The major raw materials that are used for the benzene alkylation in this case are the normal paraffins, to a large extent obtained from low boiling point fractions collected during oil refining. Sodium
propyl-
and
butylnaphthalenesulfonates
are
also
representatives of this group of synthetic surfactants. c, % 80 6O 40 20 0
5
10
t, days
Fig. II-30. Kinetics ofsurfactant biodegradation: 1 - sodiumtetrapropylenebenzenesulfonate; 2 - linear alkylbenzenesulfonates; 3 - linear alkylsulfates.
138 A 1ky 1s u 1fat e s are the esters of sulfuric acid with the general formula ROSO~ Me +, where R is usually C~0 - C~8. The use of these surfactants is advantageous from the environmental prospective (Fig. II-30, curve 3), but these substances cost more than alkylarylsulfonates. Alkylsulfates can be synthesized either from higher fatty alcohols (HFA) by the sulfoesterification reaction with subsequent neutralization, or from long chain alkenes by a direct addition reaction followed by neutralization. A l k y l s u l f o n a t e s , RSO3 Me + with R typically being C~0 - C20, are biodegradable, show good detergency action over a broad pH range and in hard water, and are thus widely used. Alkylsulfonates are obtained by the chlorosulfonation or sulfooxydation of higher paraffins with subsequent neutralization. Among other perspective surfactants alkenesulfonates, also known as olefinsulfonates, are worth mentioning. The surfactants with other anionic groups, such as phosphates (salts of phosphoric acid esters), different salts of thiosulfonic acids, xanthates, thiophosphates, etc., are also classified as anionics. Anionic surfactants are widely used as wetting agents, foamers, and constitute major components of detergents. These substances are the main micelle-forming surfactants (see Chapter VI) with the largest assortment and production volume. Anionic surfactants are most actively used in an alkaline medium, but can also be used under acidic conditions, for instance as agents assisting in the removal of oxide films from metal surfaces by acids. Cationic surfactants, when dissociated in water, yield developed organic cations, which are the "carriers" of the surface activity. Primary, secondary and tertiary aliphatic and aromatic amines, quaternary ammonium
139 salts and pyridine derivatives all represent common examples of cationic surfactants [29]. F a t t y a m i n e s can be synthesized by reacting alkylhalides with either ammonia or lower amines, from fatty acids and their derivatives (i.e., amides and ammonium salts), or by ammonolysis of fatty alcohols. Amines dissociate and reveal surface activity primarily under acidic conditions. Higher homologs, such as octadecylamine, are insoluble in water, but soluble in oil. Q u a t e r n a r y a m m o n i u m salts, [RN(R')3] + X-, where R are C~2 C~8; R' is CH3, or C2H5, and X- is usually C1- or Br-, are obtained by reacting higher aliphatic amines with alkylhalides, or by reacting long-chain alkylhalides with lower tertiary amines. P y r i d i n i u m salts, such as + _
X
R
are obtained by reacting pyridine with alkylhalides. The quaternary ammonium and pyridinium salts are soluble in both acidic and alkaline media. Substances with hydrocarbon chain lengths between C~2 and C~8 may reveal strong antibacterial action. Cationic surfactants are used as corrosion inhibitors, flotation aids, biocides, fungicides, and disinfectants. Their use is, however, limited, due to a relatively high cost. Pyridinium salts are employed in the textile industry as dye fixing agents during fabric dyeing, as well as in final fabric finishing. For example, Xelan
140
C 17H35C ~ N H C H 2 N C s H 5
CI-
is used in the treatment of fabrics in order to make them waterproof. In the process of thermal treatment, Xelan decomposes yielding insoluble hydrophobic compounds. Amphyphylic (ampholyte) surfactants are compounds containing both acidic (usually carboxylic) and basic (usually amino-group substituted to different degrees) functional groups in their structure. Depending on the ambient pH, these surfactants behave as cationic (pH < 4) or anionic (pH 9 12). Within the pH range between pH 4 and pH 9, amphophylic surfactants behave as nonionic compounds. Many natural compounds, including all aminoacids and proteins, belong to this group of surfactants. Examples of synthetic analogs of such substances include alkylaminoacids, e.g. cetylaminoacetic acid, C~6H33NH-CH2COOH. These compounds are, however, costly and difficult to prepare, and thus their use as surfactants is not very common. Nonionic surfactants are compounds soluble in both acidic and alkaline media that do not undergo dissociation in aqueous solutions. As a rule, these compounds are the products of ethylene oxide addition to various substances with developed hydrocarbon chains, namely oxyethylenated primary
and
secondary
R(R')CHO(CH2CH20),H, RCOO(CH2CH20),H,
fatty
alcohols,
polyethyleneglycol and
RO(CH2CH20)~H,
esters
oxyethylenated
of
fatty
acids,
alkylphenols,
141
RC6H40(CH2CH20)~H.In all these compounds R is usually C 8 -
C9,
and n is
the average number of ethylene oxide segments. One can also obtain oxyethylenated derivatives of other compounds, such as sulfamides, phosphoric acid esters, etc. The oxyethylenated fatty alcohols are biodegradable,
while
oxyethylenated alkylphenols are not. The latter maintain poor biodegradability even if linear alky! chains are introduced into their structures. Nonionic surfactants have broad industrial applications. For example, in tertiary oil recovery nonionic surfactants are the ingredients of the solutions that are pumped into oil wells during perimeter flooding which favors the movement of oil towards the well opening. (see Chapter Ill, 5). A promising class ofnonionic surfactants are thepluoronics, which are the block co-polymers of ethylene and propylene oxides with molecular weights ranging from 2,000 to 20,000 g mol -~. The solubility and surface activity of these compounds is determined by the ratio of the lengths of polyoxypropylene (carrier of hydrophobicity) to polyoxyethylene chains (carrier of hydrophilicity). Nonionic surfactants are also widely used in detergent formulations. Their detergency action is comparable to that of high quality soaps; nonionic detergents can be used with success in both soft and hard waters, and in alkali, acidic or neutral medium. These compounds also show low foaming action, and can even be used as defoamers. The possibility of controlling the properties ofnonionic surfactants by altering a number of ethylene oxide units, along with low manufacturing costs, have led to wide use and increased production levels of these compounds.
142 Glycerides, glucosides, saccharides, sorbitan derivatives, Tweens and Spans, etc. also belong to the class ofnonionic surfactants. Mono- and diesters of fatty acids and multiatomic alcohols are oil soluble surfactants with low solubility in water. Sulfoesterification of these compounds followed by subsequent neutralization allows one to obtain water soluble surfactants. Many representatives of this group, such as sucrose esters, are non-toxic, tasteless and odorless, which makes them attractive for use in the pharmaceutical, food and perfume industries. Naturally occurring surfactants are represented by various biologically active substances, among which lipids, proteins, and cholic acids are of particular significance for the life cycle of living organisms. L ipids
are the esters of glycerol or sphingosine (long-chain
aminoalcohol) and saturated or unsaturated C~2 - C~8 fatty acids. Most lipids have two hydrophobic chains per molecule; the polar heads may contain various chemical groups, such as esters (mono-, di-, and triglycerides), phosphates (phospholipids), and carbohydrates (in a large group of glycolipids). Structures of some common lipids belonging to different classes are shown in Fig. II-31. In live organisms lipids along with proteins are the main constituents of complex biological structures, such as cell membranes. The surface activity of proteins (as well as many of their other functions) depends on the so-called tertiary structure of protein molecules, which is determined by the spatial arrangement of their polypeptide chains. This tertiary molecular structure depends in turn on the primary protein structure - the aminoacid sequence, which is determined by the genetic code of a cell. The surface of a protein globule is mosaic-like: it contains both polar
143
I ~H3 ~H3 ~H2 ~H2
2
~H3
CH 3
3 CH 3
CH 2 CH2
CH2
I
CH2
C.H2
~H2
CH2
:H2
C~---O C---~O
(:H2 ( ',H 2
!
I
/o
I
CH2--CH
CH~
O
Ie
O~P~O
I
O
!
~CH2--CH
I
/ O
CHz
I e
O=P--O
I I
0
~H z ~)~H2
CH 3
:H2
(;H 2
I
O
H2
(:H 2
CH2
C-~_O
H2
;H2
CHz
C--'O
CH2
~H2
CH 2
[
H2
(:H 2
9
CH
:H2
i
~H3
4 CH3
H2
CH 2 H2
ll3
;H 2 .
II
:Hl
I CH
C--'O
CH
COH NH \CH / I O
Ie
:H 2
!!
C~-O
[
COH NH "~CH /
I
O
i
O---~P---O [
H CH HOC / %O
o
I
CH CHzOH
CH2
COH H
N H 3 COO-
CH3
:H2
CH2
.
Fig. II-31. Examples of some lipids" 1 - dicholinophosphate (lecithin - glycerophospholipid); 2 - phospatydilserin (glycerophospholipid); 3 - sphingomyelin (sphingophospholipid); 4 cerebroside (sphingoglycolipid)
and non-polar regions. Such structure is more or less characteristic for all proteins, including the membrane ones. At interfaces proteins usually adsorb in a globular form; in some cases changes in the conformation of macromolecules in the adsorption layer may take place. Protein adsorption is, to a significant extent, irreversible, which makes it difficult to describe using the Gibbs equation [30].
The manufacture of surfactants (among which micelle forming ones play a leading role) is a heavily developed area of chemical industry. World production of surfactants is on the order of - 10 million tons a year. The assortment of commercially available surfactants
144 consists of more than 500 trade names. According to the available statistical data surfactants are used in nearly one hundred types of industries, and have 3.5 to 4.0 thousand different applications. The fraction of industrial surfactants used in highly developed countries, such as United States and Japan, is on the order of 60% and is continuously growing. Among the major surfactant consumers are such industries as mining, mineral and oil processing, metalworking, textile, construction (additives to concrete and asphalt), transportation (oils, lubricants, lubricating cooling liquids, etc), polymer materials industries, lacquer and paint manufacturing (plasticiser additives and filler activators), pharmaceutical, food, perfumes, printing industry, water treatment, and many others. The use of surfactants for making household and industrial detergents still remains a major area. Let us now briefly address the issue of raw materials for surfactant production. As discussed previously, unbranched paraffins, ethylene oxide, synthetic fatty acids (SFA) and higher fatty alcohols (HFA) are the major raw materials that are utilized. Along with the oxidation of paraffins, the hydrocarboxylation ofolefins (~-olefins and normal olefms), there is a promising method of SFA synthesis. HFAs are produced by organoaluminum synthesis, direct hydration of SFAs, or by oxysynthesis, using linear olefins. The latter are, in turn, obtained by dehydrogenating liquid paraffins and by cracking of the solid ones. Among the methods used to manufacture a-olefins, one can mention high- and low-temperature oligomerization of ethylene. Liquid paraffins may be obtained from the suitable oil fractions using zeolites, or by carbamide deparaffinization.
11.4. Analytical Chemistry of Surfactants 9
Numerous commercial, practical and scientific applications require qualitative, quantitative and structural analysis of surface active substances.
Although this material is not directly related to colloid science, the authors have decided to include it, because in the experimental investigation of various colloidal phenomena discussed in this book one frequently encounters the task of measuring the amount of surfactant in different phases 9
145 These include studies of adsorption and micellization phenomena (Chapters III, VI), the formulation of detergents, investigation of the effect of surfactants on the mechanical properties of various materials (see Chapter IX), control over the biodegradability of surfactants into the environment, etc. Although qualitative and structural analyses of surfactants are certainly important, we will only briefly mention them and will devote most of our attention to the discussion of methods used for the quantitative analysis of surfactants, i.e. methods for determining surfactant concentration in solutions. The subjects of qualitative and structural analysis are extensively covered in the literature and those interested may refer to the relevant monographs and references therein [31-35]. Although in many cases the concentration of surfactant in solution may be determined from the measurements of surface tension at the airsolution interface, there are also a variety of other methods, both physicochemical and chemical, that one can use to assay the surfactant content. Sometimes the use of alternative methods allows one to measure very low surfactant content, not accessible via surface tension measurements. Most of these methods are universal and, with proper modification, can be applied to the analysis of surfactants of all types: anionic, cationic, amphoteric and nonionic. Among the chemical methods gravimetric and volumetric ones are most common. In gravimetry surfactants are quantitatively precipitated upon the addition of a suitable precipitating reagent. The precipitates are then dried to constant mass and weighed. Volumetric methods include either single phase or two-phase titrations. These methods are based on stoichiometric chemical reactions between the surfactant and the titrant. The titration end-points are
146 usually determined using suitable indicators that form colored complexes with either the titrant or the surfactant (or both). The titration is then monitored by observing the change in color due to the transition between the indicatorsurfactant and indicator-titrant complexes. Alternatively, the end-point determination can be based on the differences in color between solutions of pure indicator and solutions containing complexes of the indicator with a surfactant. In the two-phase titration the reaction vessel also contains an organic liquid, immiscible with the aqueous surfactant solution. The end-point determination is based on the difference in solubilities of surfactant-indicator and titrant-indicator complexes and/or reaction products in the aqueous and organic phases. Among the physico-chemical methods, spectroscopic, electrochemical, radiochemical, and chromatographic ones
are used most frequently.
Spectroscopic methods are suitable for the determination ofsurfactants whose molecules contain structural features that absorb electromagnetic radiation of certain wavelengths. If such features are not present, the surfactants may be reacted with suitable substances yielding the absorbing products. Commonly used electrochemical methods are potentiometry, which involves the use of surfactant-selective electrodes, and polarography, based on the reduction or oxidation reactions taking place at the dropping mercury electrode. In radiochemical analysis one is measuring the radiation emitted by surfactant molecules labeled with radioactive isotopes. Chromatographic methods are based on the difference in adsorption-desorption kinetics of different surfactants at the surface of material forming the stationary phase (Chapter III). As a result of multiple adsorption and desorption cycles, the surfactants
147 are carried by the mobile phase along the column (or plate in thin layer chromatography) and become separated from other substances. These methods are in most cases used for the analysis of surfactants in mixtures. Below we will give more detailed description of the analytical techniques, suitable for the quantitative analysis of surfactants, we have just mentioned.
Gravimetric methods can be used for the determination of all classes ofsurfactants:
anionic, cationic, nonionic and amphoteric. Many (but not all) anionic surface active substances form salts with inorganic cations that are sufficiently insoluble to ensure accurate determination by gravimetry [31]. Typical examples include precipitation of sulfates and sulfonates as calcium and barium salts. The precipitates are filtered, dried and ashed to form BaSO4 and CaSO4, respectively. One of the oldest methods of determining anionics in dry detergent mixtures is extraction with hot ethanol with subsequent evaporation of the extract to recover the surfactant [35]. This method allows one.to successfully separate surfactants from inorganic salts, such as sodium sulfate. Oppositely, cationic surfactants, particularly quaternary ammonium salts, can be quantitatively precipitated by various anions. The most common reagents used to precipitate cationic
surfactants
are
sodium
tetraphenylborate,
ammonium
reinecate,
(NH4)+[Cr(NH3)z(SCN)4]- , and various heteropolyacids, such as e.g. tungstophosphoric acid [32]. Heteropolyacids are also used for gravimetric determination of amphoteric surfactants, since in acidic media the latter behave as cationic ones. In some cases it is possible to base gravimetric determination of cationic surfactants on the analysis of the counter-ion, as, for instance, can be done to precipitate chloride with AgNO3 solution. Among nonionic surfactants those containing polyoxyethylene (EO) and polyoxypropelene (PO) units can be analyzed gravimetrically by precipitation with tungstophosphoric and molybdophosphoric acids together with barium ion. Apparently, the presence of quaternary ammonium surfactants interferes with the analysis by precipitation with heteropolyacids. Nonionic EO/PO surfactants can also be determined by precipitation with barium tetraiodobismuthate. In this
148 case one usually performs sedimetry, i.e. measures the volume rather than the weight of precipitate. It is worth mentioning here that even though various insoluble compounds formed by surfactants can be analyzed by means of gravimetry, one often dissolves the precipitates in appropriate solvents and carries out volumetric, colorimetric or potentiometric determination of surfactant derivatives in the solutions
formed. This allows one to
significantly increase the sensitivity of determination and speed up the analysis. Analysis by gravimetry is usually more labor-intensive than other methods but is still used because of high reproducibility, simplicity and low cost of the necessary equipment. The disadvantage of gravimetric analysis is its non-specificity. There are no reagents that precipitate only particular surfactants. Volumetric methods are much faster than gravimetric ones. They are broadly used for accurate assay of surfactant matter. Many anionic and cationic surfactants can be determined by means of an acid-base titration in suitable solvents. These methods are usually not specific for surfactants and can be reliably used only when there are no unknown components
present
in
solutions.
For
example,
acid-base
titration
of
linear
alkylbenzenesulfonates may yield erroneous results due to interference from such buffering substances as sodium silicate and sodium tripolyphosphate. To minimize interference, anionic surfactants are often titrated with cationic ones, and vice versa. Most of these methods have the disadvantage that, due to similarities with the corresponding titrant in each case, small quantities of anionic surfactants interfere with the determination of cationics, while small amounts of cationic surfactants interfere with the analysis of anionic ones. In reality this rarely poses a problem, since anionic and cationic surfactants are in most cases used separately. A typical indicator for the titration of anionic surfactants with cationic ones is thiocyanatocobalt (II). At the titration end-point cationic titrant forms a complex with the indicator, causing a color change from pink to violet. Quaternary amines can also be titrated with polyvinyl sulfate using the cationic dye toluidine blue O as an indicator. In this case the end-point is characterized by color change from purple to blue. A classic method that is commonly used and which deserves special attention is socalled two-phase titration. An extensive review of this volumetric technique with detailed
149 experimental procedures was given by Reid et al [36]. In the case of ionic surfactants, the basic principle utilized in two-phase titration is the same as described above, namely, a stoichiometric reaction between an anionic and a cationic surfactant, which yields a salt that is sparingly soluble in water and lacks surfactant properties:
R- CH2-OSO3Na
+
R I ~ N+/ R2 R3/
_ Cl
RI~ N+/ R2
~ 84
O3SOCH2-R + NaCl
R3/
The principle behind volumetric determination ofnonionics is the already mentioned ability of polyethers to form complexes with large cations, and in particular with Ba 2+, in the presence of large anions, such as tetraphenylborate, tetraiodobismuthate, anions of heteropolyacids, and some others. Usually, either a colorimetric or potentiometric method of end-point detection is used. Two-phase titration is usually less sensitive to interference from other foreign (nonsurfactant) species. When applied to anionic surfactant systems, two-phase titration is primarily used for the analysis of alkyl sulfates and sulfonates. Typical titrants in these applications are solutions of cetylpyridinium bromide, cetyltrimethylammonium bromide, and benzyldimethylalkyl (Cl2 or C14)ammonium chloride. The most accurate results are, however, obtained with p-tert-octyl-phenoxy-ethoxy-ethyl-dimethyl-benzylammonium chloride, also known under the commercial name of Hyamine 1622. The structure of Hyamine 1622 is shown in Fig. II-32,a. The titration end-point is registered with a suitable indicator, which plays an important role in the titration process. A variety of indicators has been described in the literature [31,36]. An example of an effective indicator used in two-phase titration is a mixture of anionic dyestuff, disulfine blue VN, and cationic dyestuf, dimidium bromide, the structures of which are shown in Fig. II-32
(b,c).
Let us describe the mechanism of two-phase titration in some detail, using the titration of sodium dodecylsulfate (SDS) with Hyamine 1622 in the presence of mixed dye indicator as an example. At the beginning of the titration, when only SDS is present in aqueous solution, dimidium bromide reacts with it to form a salt that is insoluble in water, but
150 soluble in chloroform, present as a separate phase at the bottom of the titration flask. This salt colors the chloroform phase in pink. The disulphine blue VN component of the mixed indicator remains dissolved in the aqueous phase, coloring the latter in blue. When a standardized solution of titrant is added to the mixture, a colorless salt is formed due to the I
I
H3C-C-CH2-C-~ ~ I t_\ / CH3 CH3
I_+
OCH2-CH2-OCH2-CH2-~q-CH2
CI-
CH3 a
NH2
- ~ ' ~ - C~N+(C2HS)2 H2
3Br-
NaO3S ~ - ~ SOSq~N +(C2H5)2
b
c
Fig. II-32. Chemical structures of reagents frequently used in two-phase titration; the titrant, Hyamine 1622 (a); components of mixed indicator, disulphine blue VN (b), dimidium bromide (r reaction between Hyamine 1622 and SDS. This salt is insoluble in water but dissolves readily in chloroform. At the equivalence point, dimidium bromide is completely transported from the chloroform phase back into the aqueous phase, and the chloroform layer acquires a light grey color, possibly due to a complete transfer of dimidium bromide into the aqueous phase and the presence of a very small quantity of a salt of disulphine blue with the cationic titrant. The excess of Hyamine 1622 reacts with disulphine blue VN to form a salt that dissolves readily in chloroform, coloring the latter in blue. Thus, the point of color transition between pink and blue is taken as the titration endpoint. It can be detected either visually or colorimetrically, if higher accuracy is desired. It is noteworthy that anionic and cationic dyestuffs can be utilized as indicators individually. In this case one looks for the
151 disappearance of pink color (or appearance of blue color) in the chloroform layer.
The two-phase titration can be also successfully used for the determination of cationic surfactants using standardized solutions of anionic surfactants. In this case the mechanism of titration is the reverse, i.e. the chloroform layer undergoes color change from blue to pink. More often, and especially at higher concentrations, cationic surfactants are determined by two-phase titration using sodium tetraphenylborate titrant and bromophenol blue indicator [32]. Tetraphenylborate ion is also used in the determination of potassium, rubidium and cesium, so the ions of these metals, if present, interfere with the determination of quaternary ammonium surfactants by this method. Many amphoteric surfactants at acidic pH behave like cationic ones, so they can be determined by the standard two-phase titration method using dimidium bromide/disulfine blue VN mixed indicator. Precise analysis can, however, be difficult due to the formation of emulsions in the vicinity of the end-point. These problems can be minimized by reversing the titration, i.e., using amphoteric surfactant as a titrant, and by adding ethanol. The two-phase titration can also be adapted to the determination of nonionic surfactants. This technique has an advantage over titration with tetraphenylborate, as it allows one to eliminate interference from anionic surfactants that form ion pairs with bariumnonionic surfactant complexes, thus competing with tetraphenylborate. Titrants used to titrate nonionics are tetrakis(4-chlorophenyl)borate and tetrakis(4-fluorophenyl)borate [37]. Excess of the titrant results in a change of color of the cationic dye, Victoria blue B, in the organic phase. Two-phase titration of non-ionic surfactants has an important peculiarity: unlike twophase titration of anionic and cationic surfactants, the reaction between nonionic polyethers and tetrakis(4-chlorophenyl)borate titrant does not occur on a mole-per-mole basis. The stoichiometry of this reaction is dependent on the number of EO units per mole of a particular surfactant. Spectroscopic methods are also commonly used for the analysis of surfactants. Among these methods ultraviolet~visible spectrophotometry and infrared~near-infrared
spectroscopy are used for the measurement ofsurfactant concentration, while such techniques as nuclear magnetic resonance (NMR) and mass-spectroscopy (MS) are extensively used for
152 structural analysis and for studies of the association between surfactant molecules in solutions. Ultraviolet/visible (UV/vis) spectrophotometry can be successfully applied to the determination of surfactants of all types, the molecules of which contain structural features that absorb electromagnetic radiation in the UV and visible ranges. Determination by spectrophotometry can be d i r e c t in the case when the surfactant molecule by itself contains the said features, or indirect, when the light-absorbing substance is produced in a reaction between a surfactant and an other reagent. Direct determination by UV spectrophotometry can be performed with surfactants containing benzene rings, such as e.g. arylsulfonates, alkyl(aryl)pyridinium salts, and ethoxylated alkylphenols. As an example, let us examine the UV absorption spectra of sodium poctylbenzenesulfonate (NaOBS), shown in Fig. II-33. These spectra have two characteristic absorption bands. The less-intense band with
~'max-"
260 nm is characteristic of the aromatic
component of the surfactant structure (the fine structure seen in this band indicates that there is no conjugation between the aromatic ring and the sulfonate group [39] ). The band with ~max --
220 nm has higher intensity than the band with
)l,max =
220 nm and is characteristic of
a combined absorption of the aromatic ring and the sulfonate group (the distorted shape of the high-intensity band at high NaOBS concentration is due to the overlap with the instrumental peak with )~max-- 190 nm). It was established that the Beer's law holds over 2.4
2.0
1.6
,-9
~
-- 1 . 4 x 10-___~ 5 mo_.__lldin__-3
= 3.4 x 10 .4 mol dm -3
1.2
0.8
0.4
0.0 200
220
240
260
280
300
320
wavelength (nm) Fig. II-33. Characteristic UV absorption spectra of NaOBS solutions recorded in HC104/NaC104 background electrolyte [38]
153 broad concentration ranges, and either of these bands can be used for the analysis, depending on the surfactant concentration range of interest. The corresponding extinction coefficients are e2oo
=
11873 and ~:26o 387 dm 3 tool -1 cm l [38]. The very high value of e22o offers the =
possibility of determining NaOBS concentrations as low as 5 x 10-v mol dm -3. Apparently, direct spectrophotometric methods are non-selective due to the fact that all species containing aromatic rings interfere with determination. However, these methods can be successfully used in cases when only one surfactant is present or when one is interested in determining the total surfactant content. Indirect spectrophotometric methods are in most cases based on the formation of ion pairs that are extractable into organic solvents. These methods are often used in combination with volumetric and gravimetric methods, as many precipitated surfactant complexes can be dissolved in the appropriate solvents and analyzed colorimetrically. The spectrophotometric determination of the end-point in two-phase titration is often carried out. For anionic surfactants colorimetric methods utilize the formation of an ion pair between the surfactant anion and a cationic dye. Similarly to two-phase titration, colorimetric determination is based on the fact that the ion pair is extractable into organic solvent, while the dye by itself is not. A characteristic example of the analysis of anionic surfactant is the determination ofalkylsulfates and alkyl(aryl)sulfonates as their complex with methylene blue extracted into chloroform [31 ]. The absorbance of chloroform extract is measured at 625 nm versus chloroform background. This methods allows one to analyze alkylsulfates and alkyl(aryl)sulfonates separately. Alkylsulfates, in contrast to sulfonates, are easily hydrolyzed by hydrochloric acid. The products of hydrolysis do not interact with methylene blue and are not transferred into chloroform. Some other cationic dyes, such as dimidium bromide, can also be used. In fact, the use of the latter allows one to achieve much higher sensitivity than that obtained with methylene blue. Like anionic surfactants, cationics can also be determined by formation of ion pairs with colored water-soluble anionic species, extractable into chloroform only in the presence of cationic surfactants. One of the most widely used methods is based on disulfine blue [40]. The absorbance of chloroform extract containing cationic surfactant - disulfine blue complex is measured at 628 nm against chloroform background. Picric acid and sodium p-(2-hydroxy-
154 1-naphthylazo)benzenesulfonate, also known as Orange II, are other reagents that form ion pairs with cationic surfactants (and with amphoteric surfactants at low pH). The corresponding ion pairs are extracted with 1,2-dichlorethane and chloroform in the case of picric acid and Orange II, respectively. The absorbance ofpicric acid derivative is measured at 365 nm, while that of a complex with Orange II at 485 nm. In the case of nonionic surfactants, most spectrophotometric methods are based on the ability ofpolyethers to form complexes with large cations, such as K § and Ba 2+. Nonionic surfactants can be analyzed if they have at least four ethylene oxide units per molecule. The most
commonly
used
method
is
based
on
the
formation
of
a
potassium
tetrathiocynatocobaltate(II) complex with materials containing polyether linkages [31 ]. This complex can be extracted into chloroform and its absorbance can be monitored at 318 nm. Infrared (IR) and near-infrared spectroscopy are attractive and commonly used methods for the qualitative and structural analysis of surfactants. Due to the availability of numerous alternative methods, the use of IR and near-IR spectroscopy for the quantitative analysis of surfactants remains infrequent. Nevertheless, a number of analytical methods utilizing IR and near-IR spectroscopy exist [31 ]. For instance, alkylbenzenesulfonates and alkylsulfates can be assayed by measuring sulfonate absorption at 1172 - 1175 cm -~ or the sulfate absorption at 1206 - 1215 cm -1 [41 ]. Weak absorption bands in 1429 - 1333 cm ~ allow one
to
selectively
determine
relative
amounts
of
linear
and
branched
alkylbenzenesulfonates, which is clearly an example of a situation when IR spectroscopy is advantageous over other techniques. A number of IR and near IR spectroscopic methods have also been established for the analysis ofnonionic surfactants. In the case ofnonionoics, nearIR spectroscopy allows one to monitor the vibration of hydroxyl groups terminating surfactant at 1450 and 2050 nm. For ethoxylated surfactants, IR spectroscopy can be used to assay ethylene oxide content by monitoring the - C H 2- absorption band at 2295 cm ~ and the oxygen band at 2485 cm -1.
Electrochemical methods, namely polarography and potentiometry, can also be used for quantitative analysis of surfactants of all types. Several types of polarographic methods are available to measure the concentration of surfactants in solution via the effects they cause on a dropping mercury electrode at the surface of which they adsorb. Although
155 surfactants themselves do not have a polarographic action, they have effects on one or several types of polarographic currents. Through their adsorption at the surface of the mercury drop, surfactants alter the capacitance of the double layer, and thus decrease the charging current. They also affect the rate at which electroactive species can reach the surface of the electrode by either diffusion or convection, thus decreasing the diffusion or convection currents, respectively. Apparently, polarographic techniques (with the exception of tensammetry, in some cases) can not be used to differentiate between surfactants of different types. Polarographic determination of surfactants based on the measurements of decrease in diffusion and convection currents requires the presence of electroactive species, such as, copper ions or dissolved oxygen. The decrease in the convection current is reflected as a decrease in the height of the maximum on the polarogram. When performing classical polarography, one usually adds surfactant to suppress the polarographic maxima. The decrease in the maximum height is proportional to the surfactant concentration and thus can be used for surfactant quantitative analysis. Methods that utilize the measurement of the charging current and the suppression ofpolarographic maxima are generally sensitive to ~ 0.1 mg/L of surfactant. Polarographic techniques based on the measurement of changes in capacitance do not require the presence of electroactive species. Surfactant molecules when adsorbed at the electrode lower the capacitance proportionally to the concentration of dissolved surfactant. The change in capacitance can be observed in direct current (d.c.) polarography, but is seen much better in the case of alternating current (a.c.) polarography. The effect of surface active substances on the differential capacitance of the electrodes in solutions of indifferent electrolytes was investigated by Proscumin and Frumkin, who used a mercury-pool electrode [42]. They established that surfactants, due to their adsorption on electrodes, caused a marked lowering in the differential potential versus potential curves at potentials corresponding to the vicinity of maximum in electrocapillary curve (Chapter III,3.3). At either end of this region of prevailing surfactant adsorption sharp capacity peaks were observed, and at potentials
further away from these peaks the capacitance was
unaffected by the presence of surfactants (Fig. II-34). These observations were consistent with surfactant adsorption and desorption processes taking place at different potentials. Almost seventeen years after the work by Proskumin and Frumkin, Breyer and Hacobian and
156 Doss and Kalyanasundaram independently introduced the method of investigating these adsorption and desorption processes by measurement of an alternating current [42]. Breyer and Hacobian called their method tensammetry (from the combination of the terms "surface
~3
Potential Fig. II-34. Effect of surfactants on the differential capacitance tension" and "ammetry", i.e. current measurements). The new term was introduced in order to draw a clear distinction between "regular" a.c. polarography, involving electron transfer processes, and polarographic adsorption-desorption, not accompanied by electron transfer. In tensammetry the a.c. current, approximately proportional to capacitance, is measured as a function of the applied d.c. potential, E. To measure the capacitance, one superimposes an a.c. potential, U, of frequency foyer the d.c. potential E. In the corresponding polarogram, the surfactant adsorptive range, corresponding to a decrease in capacitance, lies between the positive and negative tensammetric waves (ET+, and ET_, respectively) characteristic of surfactant desorption (Fig. II-35). The tensammetric curve shown in Fig. II-35 closely resembles the differential capacitance versus potential curve shown in Fig. II-34. The reduction in capacitance as a function of surfactant concentration and the height of the desorption wave are used for the quantitative determination of surfactants. The shape of the desorption wave in some cases makes it possible to distinguish surfactants of different types.
157
desorption " " ~ i
~
E < +.a
adsilpti~
Er-
Potential
ET§
Fig. II-35. Schematic tensammetric curve Potentiometric determination of surfactants is performed using various surfactantselective electrodes. The procedure consists of dipping a surfactant-selective electrode into
the test solution, measuring the electrode potential, and determining the concentration from voltage data via the Nernst equation. The response of the surfactant-selective electrodes varies day by day and is effected by different non-surfactant ions. For these reasons potentiometry is most often used as a technique for the determination of titration end-points rather than for the direct measurement of potential. One must also be aware that surfactant-selective electrodes can be used reliably only below the critical micellization concentration (CMC, see Chapter VI,3) of a surfactant. Electrodes suitable for the potentiometric determination of surfactants are either specially designed liquid or solid membrane electrodes or ion-selective electrodes that in addition to being selective to a particular ion, also quantitatively respond to surfactants. For example, a nitrate ion-selective electrode responds to anionic surfactants, a calcium ionselective electrode is sensitive to quaternary ammonium salts, and a barium ion-selective electrode can be used for assaying polyethoxylates [43 ]. In some cases it is possible for one to perform potentiometric determination of a counter-ion, e.g. one can titrate alkylpyridinium chloride or bromide salts with silver nitrate solution using silver wire as an indicator electrode [38]. Radiochemical methods can be effectively used to measure low concentrations of surfactants. These methods can be especially advantageous in determining adsorption
158 isotherms, particularly under conditions when the surface area of the adsorbents, and consequently the amount of adsorbed material, are not very high. The concentration of radioactive surfactant is determined via the registration of radioactive emission, e.g. with Geiger or scintillation counters. Various surfactants labeled with different radioisotopes are commercially available. Despite this high sensitivity and accuracy, the use of radiochemical methods is limited because of the requirements of special equipment and safety concems in handling radioactive materials. To conclude our discussion of analytical methods, we must briefly address the use of
chromatographic methods. Gas chromatography (GC), gas-liquid (GLC), high
performance liquid chromatography (HPLC), thin-layer chromatography (TLC) and supercritical fluid chromatography (SFC) can all be used for the analysis of surfactants. Among the chromatographic techniques HPLC is probably the most practical one. All chromatographic techniques involve two major stages: s e p a r a t i o n and d e t e c t i o n . Depending on the type of chromatography and type of surfactant to be analyzed, an appropriate combination of mobile phase and stationary phase is chosen in order to achieve successful separation. Detection is the stage at which actual determination of the surfactant concentration takes place. The detection will normally involve one or more of the analytical methods that we have already described. Typical detectors used in liquid chromatography are UV, refractive index, conductivity and fluorescence detectors; gas chromatography utilizes flame ionization (FID) and mass spectroscopic (MS) detectors. The surfactant concentration can be obtained by assessing peak heights or peak areas in chromatograms and by using calibration curves. Thus, quantitative analysis by chromatography, similarly to many other techniques, requires standards. HPLC is the preferred technique for determining ionic surfactants in mixtures. Anionics and cationics are usually separated by reversed-phase chromatography, in which a salt containing the organic ion is added to the mobile phase (the so-called paired ion liquid chromatography). The organic ion (either a cation or an anion, depending on the type of surfactant) forms an ion pair with the surfactant ion. The ion pairs formed are usually retained by the non-polar stationary phase longer than the surfactants themselves. In cases when the separated products are not readily analyzable by the detector, a postcolumn chemical
159 treatment is applied in order to obtain compounds more suitable for analysis. For nonionics both normal phase and reversed-phase chromatography can be used. The separation is based on the significantly differing polarities of hydrocarbon and polyethelene oxide chains, i.e., separation can be done either on the basis of the alkyl chain length or on the basis of the ethylene oxide chain length. HPLC analysis of amphoteric surfactants is difficult due to their low detectability and is hardly ever performed. The analysis of surfactants by GC and GLC is difficult because surfactants are not volatile. To overcome the problem of volatility, the surfactants can be either derivatized to yield more volatile products,
or pyrolyzed to hydrocarbons.
In the case of
alkyl(aryl)sulfonates desulfonation is often used [31 ]. The resulting chromatograms usually contain several peaks and may be difficult to use for quantification. Thin layer chromatography is a rapid and inexpensive method that is often used to separate surfactants into classes, i.e. to separate ionic surfactants from nonionic ones. Typical stationary phases used in TLC are usually unmodified or modified (e.g. hydrophobized) silica or alumina. Quantitative analysis using TLC is based on one's ability to isolate the spot on the TLC plate and determine the concentration of active material in it. For example, the surfactant can be extracted into an appropriate solvent, and then analyzed by a suitable technique. Alternatively, one can develop the chromatogram with the suitable reagent to produce colored compound and then analyze the colored spot with optical densitometer. Supercritical fluid chromatography (SFC) is a rather new chromatographic technique that utilizes liquid CO2 as a mobile phase. The separation mechanism in SFC thus resembles that in HPLC. A flame ionization detector that has linear response over broad range of concentrations is used in SFC. Because of the peculiarities associated with the supercritical state, such as high diffusion coefficient and high solvent strength ofsupercritical phases, SFC gives higher resolution than HPLC and does not require the volatility of analyzed compounds that is essential for analysis by GC. SFC is applied mainly to nonionic compounds, because ionic surfactants are generally insoluble in liquid CO2.
160 References Q
o
~
0
5.
11,
0
0
Q
10. 11.
12.
13. 14. 15. 16.
Rusanov, A.I., Phase Equilibria and Surface Phenomena, Khimia, Moscow, 1976 Guggenheim, E.A., Thermodynamics, North-Holland Publishing Company, Amsterdam, 1959 Gibbs, J.W., "The Collected Works of J.W.Gibbs",vol.1, Thermodynamics, Longmans, Green, New York, 1931 Rusanov, A.I., Kolloidn. Zh., 49 (1987) 688 Rehbinder, P.A., "Selected Works", vol.1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) McBain, J.W., Colloid Science, D.C. Heath and Company, Boston, 1950 Adamson, A.E., Gast, A.P., Physical Chemistry of Surfaces, 6th ed., Wiley, New York, 1997 Eisenberg, D., and Kauzmann, W., The Structure and Properties of Water, Oxford University Press, New York, 1969 Tanford, C., The Hydrophobic Effect, 2nded., John Wiley and Sons, New York, 1980 Ovchinnikov, Yu.A., Bioorganicheskaya Khimiya(Bioorganic Chemistry), Prosvesheniye, Moscow, 1987 (in Russian) Dukhin, S.S., Kretzschmar, G., and Miller, R., Dynamics of Adsorption at Liquid Interfaces, in "Studies in Surface Science", vol. 1, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1995 Kralchevsky, P.A., Nagayama, K., Particles at Fluid interfaces and Membranes, in "Studies in Surface Science", vol. 10, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 2001 Izmailova, V.N., Yampolskaya, G.P., in "Studies in Surface Science", vol. 7, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Rehbinder, P.I., Izmailova, V.N., Alexeeva, I.G., and Schukin, E.D., Dokl. Akad. Nauk SSSR, 206 (1972) 1150 Lu, J.R., Thomas, R.K., Penfold, J., Adv. Coll. Interface Sci., 84(2000) 143 Dynarowicz- L~Itka,P., Dhanabalan, A., Oliveira, Jr., O.N., Adv. Coll. Interface Sci., 91 (2001) 221
161 17.
18.
19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
Lekner, J., Theory of Reflection of Electromagnetic and Particle Waves, in "Developments in Electromagnetic Theory and Applications", J. Heading (Editor), Martinus NijhoffPubl., Dordrecht, 1987 Lee, E.M., Thomas, R.K., Penfold, J., Ward, R.C., J. Phys. Chem., 93 (1989) 3 81 Simister, E.A., Lee, E.M., Thomas, R.K., Penfold, J., J. Phys. Chem., 96 (1992) 1373 Lu, J.R., Lee, E.M., Thomas, R.K., Acta Cryst., A52 (1996) 11 Muramatsu, M., in"Surface and Colloid Science", vol. 6, E. Matijevid (Editor), Wiley, New York, 1973 A.A. Abramson and E.D. Schukin (Editors), Surface Phenomena and Surface Active Substances, Khimiya, Leningrad, 1984 (in Russian) Porler, M.A., Handbook of Surfactants, Blackie and Son, Glasgow, 1991 Flick, E.W., Industrial Surfactants, Noyes Publ., Park Ridge, New Jersey, 1988 Ash, M., Ash, I., Handbook of Industrial Surfactants, vols. 1-2, 3~ ed., Ashgate, 2000 Swisher, R.D., Surfactant Biodegradation, 2n~ ed., in "Surfactant Science Series", vol. 18, Dekker, New York, 1987 Stacke, H.W., Anionic Surfactants, Dekker, New York, 1995 "Surfactants and Colloids in the Environment", M.J. Schwuger, F.H. Haegel (Editors), Springer-Verlag, New York, 1994 "Surfactant Science Series", vol. 37, D.N. Rubingh and P.M. Holland (Editors), Dekker, New York, 1991 MacRitchie, F., in "Studies in Surface Science", vol. 7, D. M6bius and R. Miller (Editors), Elsevier, Amsterdam, 1998 Rosen, M.J. and Goldsmith, H.A., Systematic Analysis of SurfaceActive Agents, Wiley- Interscience, New York, 1972 Schmitt, T.M., Analysis of Surfactants, in"Surfactant Science Series", vol. 40, Dekker, New York, 1992 Cross, J.T., in "Surfactant Science Series", vol. 4, E. Jungermann (Editor), Dekker, New York 1970 Gabriel, D.M., and Mulley, V.J., in "Surfactant Science Series", vol. 8, J. Cross (Editor), Dekker, New York, 1977 Montana, A.J., in "Surfactant Science Series", vol.19, J. Cross (Editor), Dekker, New York, 1987
162 36. 37. 38. 39.
40. 41. 42. 43.
Reid, V.W., Longman, G.F., and Heinerth, E., Tenside, 4 (1967) 292 O'Connel, A.W., Anal. Chem., 58 (1986) 669-670 Zelenev, A.S., Particle Adhesion in the Presence of Ionic Surfactants, Ph.D. dissertation, Clarkson University, Potsdam, NY, 1997 Gillam, A.E., Stern, E.S., An Introduction to Electronic Absorption Spectroscopy in Organic Chemistry, Edward Arnold Publ. LTD, London, 1958 Waters, J., Kupfer, W., Anal. Chim. Acta, 85 (1976) 241 Sabo, M., Gross, J., Rosenberg, I.E., J. Soc. Cosmet. Chem., 35 (1984) 207 Breyer, B., and Bauer, H.H., Alternating Current Polarography and Tensammetry, Interscience, New York, 1963 Moody, G.L., and Thomas, J.D.R. in "Surfactant Science Series", vol. 19, J. Cross (Editor), Dekker, New York, 1987
List of Symbols
Roman symbols A A as av, by b bj c cc c(s~ E ET F G h k ka kd M, Mr m
adsorption activity absorbance coefficient in Frumkin equation (II.27) coefficients in van der Waals equation constant in Szyszkowski equation (II.18) scattering length densities of atomic species j concentration concentration of surfactant corresponding to the surface condensation average concentration within the surface layer d.c. potential (in tensammetry) potential of tensammetric wave force surface activity Plank's constant Boltzmann constant adsorption rate constant desorption rate constant molecular weight mass
163 N NA
number of moles Avogadro's number g/ refractive index nj number densities of atomic species j vapor pressure P universal gas constant R R reflectance surface area S Sm area per mole lowest possible area per molecule in the dense adsorption layer S1 area per molecule in the adsorption layer SM T absolute temperature a.c. potential (in tensammetry) U V volume Vm molar volume volume per molecule V. l) a rate of adsorption rate of desorption l) d x, y, z Cartesian coordinates
Greek symbols (l
F 5 0
0. 0Br K
~ max
~t ~t(s) Pd V
7t"
activity coefficient constant in the Freundlich adsorption isotherm adsorption thickness of the adsorption layer grazing angle for the beam of neutrons fraction of sites occupied Brewster angle momentum transfer in neutron reflection wavelength of radiation wavelength corresponding to the maximum absorbance of radiation chemical potential chemical potential in the surface layer dipole moment frequency of radiation 3.14159... two-dimensional pressure
164
P P~ "[s~ I'sc
8 80
q~
q)CH2
condensation pressure of two-dimensional vapor scattering length density surface charge density surface tension load tilt angle of surfactant molecule work required to form an interface, given by eq. (II.4) dielectric constant molar absorptivity (extinction coefficient) at wavelength ~, electrical constant shear rate fraction of hydrocarbon chains in the upper portion of adsorption layer surface potential energy of interaction between surfactant polar group and water molecules work of transfer of o n e C H 2 group
165 III. INTERFACES BETWEEN CONDENSED PHASES. W E T T I N G
The laws governing the interfacial phenomena between condensed phases and their vapor (or air) in single- and two-component systems, described in previous chapters, are largely applicable to the interfaces between two condensed phases, i.e., between two liquids, two solids, or between a solid and a liquid. At the same time, these interfaces have some important peculiarities, primarily related to the partial compensation of the intermolecular interactions. The degree of saturation of the surface forces is determined by the similarity in the molecular nature of the phases in contact. When adsorption of
surfactants takes place at such interfaces, it may
substantially enhance the decrease in the interfacial energy. The latter is of great importance, since surfactants play a major role in the formation and degradation of disperse systems (see Chapters IV, VI-VIII). The intensity of intermolecular interactions at the interfaces between condensed phases is one of the critical factors determining the conditions for wetting and spreading. A large number of important technological processes, such as mineral processing (flotational enrichment and separation), are based on these phenomena. The ability to alter interfacial properties by surfactant addition allows one to gain fine control over these processes.
166 III.1. The Interfaces Between Condensed Phases in Two-component Systems
When we described the adsorption phenomena at interfaces between a two-component liquid and its vapor (or air), major attention was given to how the liquid phase composition affected the properties of its surface and the structure of adsorption layers. We did not take into account changes in the vapor pressure (i.e. the total pressure in the system) that take place when the composition of a two-component solution is changed. This
simplified
approach to the solution - vapor interface is justified, if change in the vapor pressure is small and does not affect interfacial properties in any noticeable way. The presence of air or other non-adsorbing gas does not influence interfacial properties either. Under these conditions one can compare properties of different solutions at the same outer pressure by viewing the ternary solute-solvent-air system as a binary one. Binary systems containing two condensed phases (two liquids, two solids, solid and a liquid) differ from the case described above. In such systems under equilibrium conditions at constant temperature and pressure, the compositions of neighboring mutually saturated phases are strictly defined according to the phase diagrams of the corresponding systems. The pressure effect on the equilibrium in systems containing condensed phases becomes noticeable only at very high pressures, and hence will not be considered. Temperature is thus the main variable parameter for the investigation of properties of interfaces in binary systems consisting of condensed phases. This makes such interfaces similar to those in liquid-vapor (or solid-vapor)
167 one-component systems (see Chapter 1,1). Let us now discuss the nature of temperature dependence of the surface tension at the interfaces between condensed phases, and the relationship between the surface tension and the type of intermolecular interactions at such interfaces. The derivative of the surface tension with respect to temperature at the interface between condensed phases in binary systems can be either positive, or negative, or even change its sign when the temperature changes, which makes it different from the vapor-liquid interface in a one-component system. Within a certain approximation one may assume that in binary systems, as in single-component ones, the value r I = -dc~/dT is the excess of entropy within the discontinuity surface. Consequently, for the interface between condensed phases, the excess of entropy can not only be positive (as it was with singlecomponent systems), but also negative. This situation is especially typical for the interface between two mutually saturated liquid solutions. The liquid-vapor system is characterized by only one critical temperature, while in binary liquid-liquid systems both critical temperatures (the higher and the lower) may be present. The higher critical temperature is the upper limiting temperature at which phase separation occurs, and above which two components are fully miscible. Below the lower critical temperature components mix in all proportions, while above it they form two phases. Consequently, several types of systems can be distinguished: systems with a higher critical temperature (such as water- phenol), those with a lower critical temperatures (such as water - ethylamine), those having two critical temperatures and a closed region in which two-phase systems exist (e.g. water -
nicotine). Making the compositions of the phases in contact (expressed, e.g.
168 in mole fractions of one component) infinitely similar to each other in the vicinity of a critical point results in a drop in the surface tension to very small values for both liquid-vapor and liquid-liquid systems. (3"
Lu
x
G"
7"
o-/
T 7',u - - - -
LL x=O
x=l
x=O
CY
x
T
L~-t x=l
x=O
x=l
a b c Fig. III-1. The interfacial tension, o, as a function of the composition,x, of phases in contact and temperature, T, in a binary two-phase system with the upper critical temperature (a), lower critical temperature (b), critical temperature of mixing and a closed region where the separation into two phases occurs (c) In systems with the upper critical temperature, TCo the surface tension decreases with increasing temperature, and consequently the excess of entropy within the surface layer is positive (Fig. III-1, a). In systems with the lower critical temperature, T L (Fig. III-1 b) the increase in interfacial tension is observed above the point at which system separates into two phases; the value of q is hence negative. The latter may serve as evidence for the existence of strong coorientation between molecules within the interfacial layer, which is due to the presence of directed chemical bonding, such as hydrogen bonding. In systems having a closed region of phase separation (Fig. III-1, c) the temperature as a function of surface tension passes through a maximum; the value of o approaches zero in the vicinity of both the lower and higher critical points. In such a case the excess of entropy within the interfacial layer is
169 negative at low temperatures (below the maximum in ~) corresponding to a strong mutual orientation between molecules, and is positive at higher temperatures, which can be related to the destruction of directed bonding at the elevated temperatures (for example, the dehydration of molecules in systems containing water). Similar strong orientation of molecules may occur at the solid-liquid interface as well, especially if the mutual solubility of the components is limited, i.e., both phases consist essentially of one component. According to the data presented by B.V. Derjaguin and N.B. Churaev, in some systems (for example, for water on quartz surface) the structured layers may have a thickness corresponding to several intermolecular distances. The formation of layers with the structure different from that present in the bulk of a phase is of importance for the stability of the disperse systems (see Chapter VII). Similarly to the interfaces between two liquids, the interfacial energy at the solid-liquid interfaces could, in principle, decrease with increasing temperature; however such studies were not carried out due to difficulties in obtaining ~ at such interfaces. The interface between two solid phases is built similarly to the grain boundaries in one-component solid systems. The main difference is in that the grain
boundary
in
polycrystalline
material
must
necessarily
be
thermodynamically unequilibrated, while the interface between two different solid phases may be in thermodynamic equilibrium. The latter state, however, is rarely achieved (except in some geological processes) due to kinetic restrictions related to the low rate of diffusional processes in solid systems. In order to analyze the relationship between the interfacial energy and
170 the type of intermolecular interactions, let us introduce the notion of the work
(or energy) of adhesion, Wa. This value, analogous to the work of cohesion (see Chapter 1,2), is defined as the work required for the isothermic separation of two condensed phases along the interface of a unit area. As a result of such separation, two interfaces with energies of (Yl and u 2 are formed between the individual solid phases and gas. At the same time the interface between two solid phases having the energy c~2 vanishes (Fig. III-2), and work of adhesion is thus W a -
t~ 1 at- ~ 2 -
(y 12
9
(III. 1)
According to Dupr6, the work of adhesion, Wa, is a characteristic of the similarity between neighboring phases, i.e., it shows the degree of saturation in the uncompensated surface forces upon contact. The interfacial energy, o~2, on the contrary describes the intensity of the remaining uncompensated interactions at the interface between neighboring phases.
/
Gas
/ Fig. III-2. The definition of the energy of adhesion Wa Let us start by discussing the extreme case when the components A and B (A makes phase 1, and B makes phase 2) are essentially mutually
171 insoluble (Fig. III-3, a). Let us assume that both phases are similar in their structure, i.e. consist of atoms of equal sizes and have equal coordination numbers, ZA = ZB = Z, and number of bonds in a unit cross-section,
Zsn s (see
Chapter I, 2). Within the approximation in which only interactions between the closest neighbors are considered, the work of adhesion, Wa.is related to the interaction energy between unlike atoms (molecules) by an expression similar to eq. (I.6): -
-
-
9
In the above expression UAUis considered to be negative. The interfacial energy c~2 can then be defined as the total of the surface energies of the individual phases
minus the work of adhesion. In the case
considered, the simplest case, with the use of expression (I. 11) for the work of cohesion of both phases, the interfacial energy is 1 G12 --(3" 1 + 0" 2 -- W a - =- Z s n s ( - UAA - UBB
2
+ 2 UAB)
-
Zs Z
nsUo,
(III.2)
where UAAand UBBare the interaction energies between like atoms (molecules). The value u0,defined as 1 Uo -
is referred to as the
Z[UAB
energy
-
a-(UAB
Z
of mixing
+
UBB)],
of the components. This value
determines the mutual solubility of the components, and an estimate for it can be obtained by analyzing the shape of a phase diagram for a binary system. When mutual solubility of components becomes noticeable, the excess
172
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1
1
l
l
l
l
2 ,
l
l
l
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A
A
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BABBBBBBBB BBBABBBBBA ABBBABBBAB
m
1
AAABAAABAA BAAAAABAAA AAABAAABAA
a
b
Fig. III-3. The compositions of phases and of interfacial layers when components A and B forming phases 1 and 2 are completely mutually insoluble (a) and noticeably soluble (b) interracial energy, ~, decreases. The decrease in G occurs for the following reasons: first, within the interface some bonds of the A-B type are replaced by bonds of A-A and B-B types; second, the number of A-B bonds in the bulk of both phases increases. The free energy density decreases within the interfacial layer, but may increase in the bulk. As a result the excess of energy density, i.e. the surface energy, G, decreases. The approximate treatment given above was developed for the interface between two solid phases, however, the approximation is also valid for interfaces between two liquids or a solid and a liquid. As emphasized earlier, similarity in the composition of contacting phases in the vicinity of the critical temperature leads to a decrease in the interfacial tension to very small values, at which the spontaneous formation of thermodynamically stable
liophilic colloidal systems is possible (see
Chapter VI). As in one-component systems, the value of cyin binary systems can be subdivided into a "non-specific" dispersion component, o d ,and a nondispersion component, (y",which is determined by the types ofintermolecular interactions in the neighboring phases. The dispersion component of the
173 surface energy of the interface between condensed phases can be described by an equation similar to eq. (I.10). In this expression, like in eq.(III.2), the interactions between molecules of different types (i.e., A with A, B with B, and A with B) are summed and described by the corresponding Hamaker
constants AA, A B, and AAB," AA +
AB - 2 AAB 24rtb 2
A* 24rtb 2 '
where b is the effective intermolecular distance. The quantity A*, equal to A* - A A + A B - 2 AAB ,
is referred to as the complex Hamaker constant. The Hamaker constant AAB, which describes interactions between unlike molecules, is proportional (~) to the polarizability, am, and the concentration of molecules of types A and B. If the ionization potentials of molecules of A and B are close, the following frequently used approximate relationship is valid --~ 4AAAB
AAB ~ a A ~ B n A n B
,
i.e., the Hamaker constant describing interactions between unlike molecules is close to the geometric mean of the Hamaker constants of like molecules. Thus, o-d2 ~
--
,
and the total free energy of the interface between condensed phases is given by the Girifalco-Good-Fowkes equation:
174
O"12 ~ n
where o~2
--
+ (5"n12
,
is the component of the surface energy stipulated by the
uncompensated in the surface layer non-dispersion interactions between phases 1 and 2. The relationship
between the dispersion and non-dispersion
components of the surface energy depends on the nature of the phases in n
contact. If one phase is polar (~>0), and the other one in non-polar (~=0, and ~2 ~o~), then one can write that If both phases are non-polar, the non-dispersion component ~n is close to zero. The estimates for the dispersion and non-dispersion components can be 13"12 ~ ( ~
O'd
--~2dl
+O'nl
"
(III.4)
obtained from the investigation of the wetting of solid surfaces by liquids (see Chapter III,4). At the interfaces between two liquids an empirical rule, Antonow's
rule, is often valid. According to this rule, the interfacial energy (Y~2equals the difference between the surface tensions of more polar (cy~)and less polar (c~2) liquids, namely (5" 12 -- (y 1 -- (5"2 "
(III.5)
It is important to realize that for two mutually soluble liquids the values of o~ and c~2 correspond to the surface tensions of their saturated solutions. The latter is especially important for polar liquids, as the surface tension can be significantly lowered by the dissolution of a less polar (and, therefore, surface
175 active) component. Table III. 1 shows that Antonow' s rule is valid with a high degree of accuracy for interfaces between various organic liquids and their aqueous solutions. A comparison ofAntonow' s rule (III.5) with the expression for the work of adhesion yields Wa - (3"1 + 0"2 - ~
- 2cY2 - Wc(2) ,
i.e., the work of adhesion in the described case equals the work of cohesion of TABLE III.1. The values of the interfacial tension G12 of mutually saturated solutions, as determined experimentally and estimated from Antonow's rule [1 ].
The surface tension of the less polar phase %, mN/m
The surface tension of the aqueous phase ~ , mN/m
benzene
28.8
ether
Liquids
Interfacial tension, (~12,mN/m experimental
evaluated from Antonow's rule
63.2
34.4
34.4
17.5
28.1
10.6
10.6
aniline
42.2
46.4
4.8
4.2
CC14
26.7
70.2
43.8
43.5
nitrob enzene
43.2
67.9
24.7
24.7
amyl alcohol
21.5
26.3
4.8
4.8
a less polar liquid. One can say that the formation of a new interface, when one breaks the contact between liquids with substantially different polarities, occurs along the less polar phase, in which interactions are weaker as compared to those in a more polar phase and at the interface. During this process, the adsorption layer with a composition characteristic of the less polar phase stays on the surface of the more polar phase (Fig. III, 4).
176
2
/
BBBBBBABBB BABBBBBBBB BBBBBABBBB Wa ~ Wc(2) BBBABBBBBB AAAAAABAAA ABAAAAAAAA
Fig. III-4. The interfacial composition of condensed phases when Antonow's rule is valid 111.2. Adsorption at Interfaces Between Condensed Phases
Let us briefly address the basic laws governing adsorption phenomena at interfaces separating two condensed phases when the third component, surface active with respect to the said interface, is introduced into the system. According to the polarity equalization rule, originally formulated by Rehbinder [2], the surface activity of the introduced component is determined by its ability to compensate for the striking difference in polarities of two unlike substances with low mutual solubility. Such smoothing of the threshold between
polarities within the
interfacial discontinuity surface is possible when the polarity of the introduced third component is intermediate with respect to those of two other components, that make up the phases in contact. The smoothing is especially effective, when it is achieved via the adsorption of diphilic molecules containing segments with different polarities. These molecules are the organic surfactants that can smoothen the difference in polarity between water and any solid or liquid hydrocarbon phase ( further referred to as "oil"). When the adsorption takes place, the polar heads of the surfactant molecules are oriented towards water, while the hydrocarbon chains face the oil phase (Fig. III-5). As
177 a result, the surfactants form an intermediate layer which to a great extent, or even completely (at high values of adsorption), levels off the difference in polarities between two phases. Using this typical example of the adsorption of organic surfactants at the oil-water interface we will discuss the main laws governing adsorption phenomena in ternary systems.
__ __ Water ~
Fig. III-5. The orientation of adsorbed surfactant molecules at the water/non-polar phase ("oil") interface
The solubilities of water in hydrocarbons and of hydrocarbons in water are negligible. The surfactants can be classified as water soluble, oil soluble, and those soluble in both phases, depending on their ability to dissolve in either phase, which is determined by the structure of surfactant molecules. Water soluble surfactants usually contain either a charged polar group or a rather large polyethyleneoxide group and have hydrocarbon chains of moderate length, not exceeding 16 - 18 carbon atoms. These surfactants, when dissolved in water, often form micelles (see Chapter VI for more details). Oil-soluble surfactants are usually the substances that are insoluble
178 in water but often form adsorption layers at the air-water interface, which may be studied with the Langmuir balance. These surfactants contain one or several long hydrocarbon chains and almost always contain a weakly dissociating or nonionic polar group. Surfactants that are substantially soluble in both oil and water are nonionic substances with a relatively short hydrocarbon chain, e.g. lower members of homologous series of acids and alcohols. When equilibrium is established, these surfactants are distributed between the aqueous and oil phases, and at low concentrations Henry's law is obeyed, i.e 9 Cw/Coi I - K ,
where c w and Co~~ are the surfactant concentrations in the aqueous and oil phases, respectively, and K is the partition coefficient. When water soluble surfactants adsorb at the interface between a liquid hydrocarbon and water, the trends in adsorption are very similar to those established for the air- solution interface (see Chapter II). The Traube rule remains valid, and the dependence of the surface tension on concentration can be described by Szyszkowski's equation (II.18). Moreover, at identical surfactant concentrations, the absolute values by which the surface tension is lowered at water - air and water - hydrocarbon interfaces are not that different. The surface tension isotherms for these interfaces are parallel to each other (Fig. III-6). That is due to the fact that the work of adsorption per CH 2 group, given by eq. (II. 14), is determined mostly by the change in the standard part of the chemical potential of the solution bulk, go. Similar to the air-water interface, the energy of surfactant adsorption from an aqueous solution at an
179 mN CI~
In
"--72
~50
c Fig. III-6. Surface tension of surfactant aqueous solutions at solution-air (1) and solutionheptane (2) interfaces
oil-water interface is governed by hydrophobic interactions between hydrocarbon chains in the bulk, i.e., has an e n t r o p i c n a t u r e . The situation is different when oil-soluble surfactants dissolved in a liquid hydrocarbon adsorb at the same interface: the extension of the hydrocarbon chain length results in only a small decrease in the surface activity. This is related to a small increase in the solubility of surfactants in oil upon extending the hydrocarbon chain length. The energy of surfactant adsorption from the oil phase at the water - oil interface is controlled by the hy d r at i o n o f t h e p o 1ar g r o up s, which takes place when surfactants move to the interface from the oil bulk. Everything we have just said about the adsorption of water- and oilsoluble surfactants is also true (at least at low concentrations) for the surfactants soluble simultaneously in both aqueous and hydrocarbon media. In this case an equilibrium is established between surfactant solutions in the aqueous and oil phases and the adsorption layer formed at the interface. At
180 low concentrations a simplified Gibbs equation can be applied to both phases, yielding Coi 1
d~
c w d~
F _
RT dcoi~
RT dc w
Therefore, Cw
Co,,- Gw
- CwX, Coil
where Gw and Goi~are the surface activities corresponding to the adsorption from the water and oil solutions, respectively. The partition coefficient, K, is approximately proportional to the ratio of the surfactant solubilities in the water and oil phases. This coefficient (similar to the surfactant solubility in water) decreases by a factor of 3 to 3.5 when the surfactant hydrocarbon chain by is lengthened one CH 2 group. At the same time the surfactant solubility in oil does not change significantly with the extension of hydrocarbon chain length. While the surface activity of surfactants in aqueous solutions is increased by the same factor on transition
to the next member in the
homologous series, the surface activity during adsorption from oil is not significantly influenced by differences in the chain length [3]. Interesting features are observed in two-phase systems containing surfactants soluble in both water and oil, especially when they are present at high concentrations. The increased content of a component with intermediate polarity in both phases results in smaller differences in polarity of contacting phases and leads to a very significant lowering of surface tension on top of that caused by the adsorption. The surface tension may reach extremely low
181 values. At the same time, mutual solubility between neighboring oil and water phases sharply increases, and their compositions become more and more similar, up to total miscibility (see Chapter VI,2). The surface tension at the water-hydrocarbon interface may also drop to very low values when one introduces micelle-forming surfactants and surfactant mixtures, especially those that are oil- and water-soluble. The possibility to reach very low values of surface tension is a significant feature of liquid-liquid interfaces, which accounts for a substantial difference between such interfaces and
gas-liquid and gas-solid ones, at which, even after
reaching limiting values of adsorption, the surface tension remains high, i.e. the system remains lyophobic (See Chapter IV). Let us now proceed with the discussion of the general trends of surfactant adsorption at solid surfaces. The most common method used to monitor adsorption phenomena in such systems is to study the concentration dependence of surfactant adsorption. In these studies solid surfaces with high specific surface areas, such as powders and highly porous adsorbents, are used. If the surface area of adsorbent, S~, is not known, then the total amount of substance accumulated per unit weight of adsorbent, F*, is determined. The latter can be established from the decrease in the concentration of adsorbing substance, Ac, in a solution of known volume, V, after adsorption equilibrium has been established, i.e." AcV F
m
where m is the weight of adsorbent. If the adsorbent surface area, S~, is known (usually from gas adsorption measurements), the surfactant adsorption at the
182 surface of the adsorbent can be estimated as F -
F*/S~. The investigation of the
concentration dependence of adsorption, F, or of F* allows one to obtain information regarding the structure of surface and adsorption layers. According to the polarity equalization rule, substances with polarity intermediate to those of phases in contact have the highest tendency to adsorb. Consequently, at interfaces between aqueous solutions of organic surfactants and non-polar solids (such as wax, charcoal, and carbon black, including activated carbon with a high surface area) the adsorption layers, in which the hydrocarbon chains are oriented towards the solid surface and the polar groups are facing the aqueous solution phase (Fig. III-7, a), are formed. The formation of such adsorption layers is one of the main factors responsible for the surfactant detergency action. (See Chapter VIII, 6).
.__
(
.
.
.
.
. m
m
Oil
a
b
Fig. III-7. The orientation of surfactant molecules adsorbed at different interfaces: a - nonpolar solid/surfactant aqueous solution; b - polar solid/surfactant solution in non-polar liquid (oil phase) [4] Conversely, when polar solids or powders (oxides, carbonates, silicates, alumosilicates, e.g. chalk, clays, etc.) are exposed to an oil phase containing oil-soluble surfactants, the adsorption layers in which polar heads are facing the surface and the hydrocarbon tails are floating in oil are formed (Fig. III-7, b). This process is of great importance for the incorporation of
183 polar fillers and pigments into the oil or low polarity polymer phase. At sufficient surfactant concentrations the dense adsorption layers, which can radically change interfacial properties are formed in both cases (see Chapter III,2).
An important case addressed in numerous studies is the adsorption of anionic and cationic surfactants from aqueous solutions on polar surfaces such as metal oxides [5,6]. The shapes of experimental adsorption isotherms, which represent a relationship between the adsorption, F, and the equilibrium surfactant concentration, c, have been thoroughly investigated and some common features noted. A typical adsorption isotherm plotted on a loglog scale can be subdivided into four regions ( Fig. III-8, a), the interpretation of which and
CMC
3
1
log c
a
b
Fig. Ill-8. A model S-shaped four-region isotherm for the adsorption of ionic surfactant on an oppositely charged surface (a); The structure of the adsorbed surfactant layers corresponding to different regions of the adsorption isotherm (b)
184 considerations regarding the structure of the adsorption layers within them have been offered by a number of authors [7-9], as schematically illustrated in Fig. III-8, b. In region 1 the surfactant adsorption takes place through ion exchange with ions in the electric double layer (see Chapter III,3) and obeys Henry's law, and the interactions between surfactant ions and the adsorbate surface are purely electrostatic. The steep increase in F in region 2 is due to the additional interaction between the hydrocarbon chains of surfactants, resulting in local monolayer surfactant associated structures, referred to as hemimicelles (Fig. III-8, b). The original charge of the surface is neutralized by the adsorbing
surfactant molecules, i.e. the isoelectric point (i.e.p.) is reached. Further uptake causes a charge reversal of the adsorbent, which decreases the isotherm slope (region 3, Fig. III-8, a) due to electrostatic repulsion between the adsorbing surfactant ions and the adsorbent surface bearing the same sign of charge. In this region local bilayer structures called admicelles are formed (Fig. III-8, b). The electrostatic repulsion on one hand, and van der Waals attraction between surfactant hydrocarbon chains on the other, are the main factors responsible for the formation of such structures. The bilayer is nearly complete around the so-called critical micellization concentration (CMC, see Chapter VI,3) at the end of region 3. As will be discussed in Chapter VI, above this concentration surfactant molecules are combined into associates, referred to as micelles. Above the CMC the plateau occurs (region 4), because surfactant ions are simultaneously at equilibrium with both the micelles and the adsorbent, and their concentration in solution remains constant. Some authors observed a maximum (often followed by a minimum) in the plateau region of the isotherm [8]. The possible explanations for such observations include (1) the exclusion ofmicelles from the near-surface region due to electrostatic repulsion between the adsorbent particle and the micelles and between the former and the individual surfactant ions; and (2) the repulsion between various areas on the surface of adsorbent particles upon excessive adsorption of surfactants. Since the charge of many polar surfaces (especially of metal (hydrous) oxides) is determined by the concentration of H30 +and OH- ions, the surfactant adsorption is strongly influenced by the solution pH. Changes in the pH may drastically affect the value of F, the concentration ranges corresponding to different regions on the isotherm (Fig. III-8), and the shape of the isotherm itself.
185 Theoretical modelling ofsurfactant adsorption uses as a starting point the theoretical isotherms derived from statistical and kinetic data for the gas-solid interface. The most common model is the one based on the Langmuir adsorption isotherm (see Chapter II,2): 0
~-Bc~ 1-0
where the coverage, 0, is defined as the ratio of the adsorption, F, to the maximum adsorption corresponding to the substance adsorption in a single layer, Fro,• ; B is the adsorption coefficient. For adsorption from extremely dilute solutions 1-0 ~ 1, and the above equation reduces to Henry's adsorption isotherm, 0 = Bc. Koopal et al [9] accounted for the effect of the architecture and flexibility of surfactant molecules, as well as for their interactions with solvent molecules. His result turned out to be similar to the expression obtained previously in a monolayer Frumkin-Fowler-Guggenheim (FFG) adsorption model [9], and in its simplified form can be written as
0
~ = B'cexp[2l(~12 -0.5)O ] 1-0 where ~12 is the average Flory-Huggins interaction parameter [10], describing the solventsurfactant hydrocarbon chain segment interactions, and l is the hydrocarbon chain length of surfactant molecule. Both the adsorption coefficient, B ~ and the ~12 parameter are functions of l. The parameter B' also (in an indirect form) accounts for the loss in conformational entropy of surfactant molecules due to adsorption. It is also possible to account for Coulombic interactions, which results in following expression:
1-0
= B'c expl2/(~; 12 --
0.5)0
zFgta 1 RT
where ~g, is the potential at the center of charge of the adsorbed surfactant ion of valency z, i.e. the potential at the head-group region; F, R and T have their usual meaning of the Faraday constant, universal gas constant and absolute temperature, respectively. The value of ~a depends on the charge due to the adsorbed surfactant ion itself, the surface charge of the adsorbent and the background electrolyte concentration. In many cases ~a is approximated
186 by the electrokinetic ~ potential (see Chapter V, 3 and 4) [ 11]. The above expressions yield no information on the structure of the surfactant adsorption layers at different regions of the isotherm, and thus assumptions regarding the latter have to be made. These equations can only be used for qualitative explanation of the experimental results, rather than for theoretical predictions. The statistical self-consistent field lattice theory for adsorption and association (SCFA), which predicts the four-region isotherm, shown in Fig.III-8, a was developed by Scheutjens et al. [12-14]. Essentially, the theory considers the equilibrium distribution of segments perpendicular to the homogeneous surface by minimizing the free energy of the systems. This theory allows for gradual changes in the structure of the adsorption layer as a function of surfactant equilibrium concentration, and predicts the presence ofhemimicelles and admicelles, making it different from all previous models. A comparison between the results of the SCFA calculations and experimental data [9] indicated, however, that the SCFA model somewhat overestimates adsorption and is in only semiquantitative agreement with experiments. The actual mathematical treatment of the SCFA theory is rather cumbersome and will not be given here.
One of the essential features of the solid-liquid interface is that the adsorbing substance may not only be bound to the surface by relatively weak physical forces, but also may form true chemical bonding with molecules or ions located at the surface of the solid phase. This phenomenon, referred to as the
chemisorption,may seem to invalidate the polarity equalization rule" at the
interface between a polar crystal (e.g. silicate or sulfide) and a polar medium (water) the adsorption due to chemical bond formation may occur in such a way that the hydrocarbon chains are facing the water phase (Fig. III-9, a). At sufficiently high concentrations of chemisorbing surfactant, when the entire solid surface is covered with a monolayer, the formation of a second, oppositely oriented, surfactant layer starts, i.e., "regular" surfactant adsorption
187 --------VVater- -
-
-
~",~\\\\\\\\\\\\\\\\'<
a
b
Fig. III-9. The orientation of chemisorbing surfactant molecules at :he interface between a solid phase and an aqueous solution at (a) - low, and (b) - high bulk surfactant concentrations
on the non-polar surface occurs (Fig. III-9, b). The ability of chemisorbing surfactants to form adsorption layers at polar surface is used in the enrichment of ores by flotation. In such layers hydrocarbon chains are oriented towards the aqueous phase.
In studies of surfactant adsorption at the solid-aqueous solution interface one uncovers interesting features related to the shape of adsorption isotherms in the region corresponding to high surfactant concentrations [15]. Let us discuss the shape of the adsorption isotherm by describing the adsorption in the whole range of surfactant concentrations (from x = 0 to
x = 1)
from a solution containing infinitely miscible
components. To do so, let's compare the surfactant concentration at the surface, x (S~,and in the bulk, x. In a system that does not separate into individual phases, an increase in bulk concentration, x, also corresponds to an increase in the surface concentration, x (S). Depending on the nature of the surfactant and solid surface, one may observe two types of x (s) =f(x) dependencies (Fig. III-10). For a high surface activity of adsorbing component at low solution concentrations, x, one observes a steep rise in x (~ until surface saturation is reached (x(s~ =1), as shown by curve 1 in Fig. III-10. At low surface activity of the adsorbing substance, x (s~=J(x) may be an S-shaped (curve 2). The intersection point "A" corresponds to the identical compositions of the surface layer and the bulk solution, i.e., a kind of "surface azeotrope" is formed. Different types ofx ~) =f(x) functions correspond to adsorption isotherms, F(x), of
188 different shapes. The adsorption can be expressed through the surface layer concentration excess and the surface layer thickness 5 as given by eq. (II.2): (s)
r-
x /5 '
Vm(S Vm
(III.6)
where Vm (s) and Vmare the molar volumes of surfactant in the surface layer, and in the bulk, respectively. In the case of strong surface activity (x(s) =fix) dependence of the first type) the surface concentration exceeds that of the bulk (i.e., x (~) >x) in the entire range of bulk surfactant concentrations (up to x = 1). In this case the adsorption isotherm, F(x), contains a maximum (curve 3). In the range of high surface concentrations (x(~ -~ 1) the adsorption, F, decreases essentially linearly with an increase in solution concentration. The extrapolation of this linear portion of the adsorption isotherm, F(x), to zero bulk concentration (x = 0), yields the limiting value of adsorption
Fma x =
~lV(ms), in
agreement with eq.(III.6). In Fig.
(III. 10) this value corresponds to F/Fmax = 1 at x -' 0.
1
1
P/P.,.~
/
0 P/P.,a~ < 0
Fig. III-10. The surface concentration, x (s), and relative adsorption, F/Fmax, as a function of the bulk concentration, x, for the adsorption at the solid/surfactant solution interface at strong (curves 1 and 3), and weak (curves 2 and 4 ) surface activity of the adsorbing component. The dashed lines are characteristic of the situation when there is no surface activity
189 The adsorption isotherm corresponding to the second type of x (s) =J(x) dependence (the Sshaped curve) contains a maximum and a minimum in the region of negative F values (curve 4). The "surface azeotrope" point occurs in this case at zero adsorption. It is noteworthy that in such systems both components are surface active at low concentrations, and surface inactive at high concentrations. The described shape of the adsorption isotherm is observed in the case of solutions that are not too far from the point at which phase separation occurs.
Another peculiarity of adsorption at solid interfaces that deserves special attention is the role of the mosaic structure (non-uniformity) of the surfaces, and primarily the role that various structural defects play in the adsorption and chemisorption phenomena.
The latter is of particular
importance in the chemisorption of inorganic ions at polar solid interfaces. As will be shown further, these ions are responsible for charging the surface. The adsorption from solutions at interfaces between two liquids and at solid-liquid interfaces is of importance in various applications. Many aspects of use of adsorption layers for monitoring the properties of disperse systems will be addressed in the subsequent chapters; here we will just briefly list some typical examples of practical applications that involve adsorption from solutions on solid adsorbents. The adsorption from solutions on finely dispersed powders and porous adsorbents is used for the removal of dissolved toxic components, as well as for concentrating and entrapping valuable substances from dilute solutions. In agreement with the polarity equalization rule, surface active substances dissolved in aqueous medium can be removed by adsorption on non-polar adsorbents (such as activated carbon), or on adsorbents that are capable of chemisorbing the surfactant polar heads. In order to increase the effectiveness
190 of the purification ofwastewater from dissolved surfactants, finely dispersed systems, obtained by precipitation from supersaturated solutions (see Chapter IV, 5), are often used. As similar method also allows one to recover dissolved electrolytes. To purify a non-polar medium from oil-soluble surfactants, e.g. for the improvement of the dielectric properties of transformer oils, one utilizes polar adsorbents, such as clays and zeolites. Adsorption from liquids onto solid adsorbents is widely utilized in liquid chromatography (HPLC, see Chapter II,4), described in detail in physical chemistry and instrumental methods textbooks, e.g. in [16]. The separation by HPLC is based on adsorption-desorption kinetics, i.e. on how long various dissolved compounds remain present in the adsorbed state. The average time, ta, during which molecules are present in an adsorption layer is determined by the energy of adsorption, P0 - P0(s), of a substance"
This energy, as shown in Chapter II, can be evaluated from the ratio between the concentrations of a component at the surface and in the bulk: C
(s)
P0 - ~t(0s) - R T l n ~ Thus, c (s) t fl ~ ~ . r
The multiple repetition of adsorption and desorption acts as the solution under pressure flows through a layer of adsorbent causes a delay in
191 the retention time of more surface-active components, which allows for their determination, or separation from other substances that are less surface active. HPLC is widely used in the fractionation of aminoacids, nucleic acids, proteins, and other biopolymers, in the separation of various enzymes and drugs (e.g., penicillin, tetracycline, various alkaloids, et al). Adsorption from solutions is often used as a relatively simple method for the determination of the specific surface area of adsorbents [15]. The decrease in the surfactant concentration, Ac, is determined after a solution of known volume, V, has been equilibrated over an adsorbent of known weight, m, and the adsorption isotherm, F*(c), is established (note that the adsorption in this case is expressed in moles per gram). The limiting adsorption, F'm,x , can be obtained from the Langmuir equation, e.g. as c/F* =
0t/F*ma x +
c/F*max
(see Chapter II, 2). Using independently acquired data for the area occupied by one surfactant molecule, s~, on a similar solid substrate, one can obtain the specific surface area (in m2/g) 9 S 1 - Fmax s1 N a . Surfactant adsorption at interfaces between condensed phases is used to control wetting, which will be described in some detail later in this chapter. Let us now briefly touch on the adsorption phenomena taking place at the grain boundaries ofpolycrystalline solids. Grain boundaries are similar to other interfaces in the sense that the adsorption at these boundaries is also described by the Gibbs equation. In the case of grain boundaries surface active substances are those that lower the free energy of the grain boundary. These substances may spontaneously concentrate at the grain boundary (this process
192 is sometimes referred to as the segregation of admixtures at grain boundaries). There are, however, some peculiarities that are specifically related to the adsorption phenomena at grain boundaries. First, the adsorption processes that take place in solids are usually non-equilibrium: due to a low rate of diffusion, the equilibrium between the bulk volume and the grain boundary is established extremely slowly. Second, in polycrystalline substances the spectrum of the adsorption energies of admixtures is very broad, which is related to both the difference in energies of grains with different orientations (see Chapter I, 2), and the inhomogeneity of the grain boundaries themselves, i.e., the presence of "defective" sites with non-uniform energy density and even voids of different sizes. The adsorption of admixtures on grain boundaries is a highly selective process. One of the reasons for this high selectivity is the preferential adsorption of components, whose atoms are close in size to typical defects at grain boundaries. Admixtures adsorbed at grain boundaries in polycrystalline substances can cause significant changes in the properties (primarily, the mechanical ones) of polycrystalline materials. For example, steel becomes brittle owing to the adsorption of sulfur and phosphorous impurities at grain boundaries. It is thus important to remove impurities that can cause such serious defects. Alternatively, the negative effect of such impurities can be minimized by introducing more surface active but harmless substances that may be capable of displacing the harmful impurities from grain boundaries.
193 III.3. Adsorption of Ions. The Electrical Double Layer (EDL)
Many properties of disperse systems are related to the distribution of charges in the vicinity of the interface due to the adsorption of electrolytes. The adsorption of molecules is driven by the van der Waals attraction, while the driving force for the adsorption of electrolytes is the longer-range electrostatic (Coulomb) interaction. Because of this, the adsorption layers in the latter case are less compact than in the case of molecular adsorption (i.e., they are somewhat extended into the bulk of the solution), and the discontinuity surface acquires noticeable, and sometimes even macroscopic thickness. This diffuse nature of the ionized adsorption layer is responsible for such
important features of disperse systems as the appearance of
electrokinetic phenomena (see Chapter V) and colloid stability (Chapters VII, VIII). Another peculiar feature of the adsorption phenomena in electrolyte solutions is the competitive nature of the adsorption" in addition to the solvent there are at least two types of ions (even three or four, if one considers the dissociation of the solvent) present in the system. Competition between these ions predetermines the structure of the discontinuity surface in such systems i.e. the formation of spatial charge distribution, which is referred to as the
electrical double layer (EDL). The structure and theory ofthe electrical double layer is described in detail in textbooks on electrochemistry. Below we will primarily focus on those features of the EDL, which are important in colloid science.
194 III.3.1. Basic Theoretical Concepts of the Structure of Electrical Double Layer
In agreement with the simplest model designed by Helmholtz, the spatial distribution of charges in the vicinity of an interface can be viewed as an electrical (ionic) double layer, modeled as two plates of a capacitor separated by the dispersion medium of thickness 6. One of these plates is formed by the potential determining ions (PDI), firmly fixed at the surface, while the other one is formed by counter-ions, "floating" in the dispersion medium. The ions with charges of the same sign as the surface, the so-called
co-ions, are displaced into the solution bulk. This distribution of charges gives rise to a potential difference, Aq~, between the phases in contact, and in the present model is represented by a linear drop of potential between capacitor plates (Fig. III-11). The charge density at the plates of a capacitor with a potential difference of Aq~ and dielectric constant s is given by ss0AqW6. q~
(9
\ 0
e
g
x
Fig. III-11. The potential difference between two neighboringphases
195 Modern theory describing the structure of the EDL was developed by G. Gouy, D. Chapman. O. Stern, A. Frumkin, D. Graham and others, and is based on the analysis of the electrostatic interactions between ions in the double layer and comparison of these interactions with the intermolecular interactions and thermal motion of ions [ 11,17]. The equilibrium in a system where phases in contact bear different electric potentials, q~, is determined by the condition of the equal electrochemical potentials of ions. The electrochemical potentials of ions of
i-th kind, ~i,, are related to their chemical potentials, bt~ ,via the following relationship: g i -- ~ i
-~ Zi
eq~Na
9
In the above expression, z~e is the charge of the ion (with the sign taken into account); e is the elementary charge; N A is Avogadro's number; the eNA product is Faraday's number. At low concentrations of ions (number of ions per m3), nl, the equilibrium conditions where the electrochemical potential is constant within the entire system can be written as ~, - g,0 + RTlnn~ + z~eq~N A - const
(III.7)
The above expression includes all three major factors describing the behavior of the ions in the system: their molecular interaction with the medium, g,0, their contribution to the thermal motion, RT Inn,, and their interaction with the electric field, z~eq~NA. Equation (III.7) should remain valid for all ions present in the system. It may
196 occur sometimes that particular ions are not present in either or both phases in contact. In the latter case some ions are present at the interface only, i.e., surface dissociation (typical for silicates and alumosililicates) takes place. It is also possible for polarization of the surface to occur in some instances. This can happen when, due to kinetic restrictions, equilibrium is not reached for one of the ions and the potential difference between phases in contact can be altered without changing the composition of the phases by applying external voltage. If surface polarization does not take place, change in the potential difference between phases in contact is always related to change in the composition; differentiation of eq. (III.7) readily yields the Nernst equation: kT -dq~ - ~ d Inn i zie
.
It is noteworthy that the potential difference between phases in contact, q0, can not be experimentally measured, but the change in this quantity can be relatively easily determined. In the vicinity of the interface the values of g~0change for different ions from values corresponding to the bulk of one phase to those corresponding to the bulk of another phase. This results in ion distributions in which the q0(x) function becomes a lot more complicated than the one described by the Helmholtz model. One usually identifies the potential of the surface of the
solid phase with respect to the dispersion medium, %, which is not an experimentally assessable quantity. If in a solid there are no excessively accumulated charges of one sign present near the surface, the potential % represents the potential difference between phases in contact. Similar to the
197 bulk, the behavior of ions in the vicinity of an interface is determined by three factors that in turn determine the double layer structure, namely the molecular forces (i.e., the specific adsorption interactions at the interface), the electrostatic interactions between ions themselves and between ions and the charged interface, and the thermal motion of ions. Due to the short range of the adsorption interactions, it is possible for one to subdivide the EDL into two main parts (Fig. III-12)" a dense part, that is closer to the surface
(the Stern-Helmholtz layer), within which the
adsorption forces are of importance, and a diffuse part (the Gouy - Chapman layer), which is further away from the surface, and within which the adsorption forces are negligible. The major task in EDL theory can be defined as the problem of finding the quantitative distribution of the concentrations of all ions, n,, present in the system and that of the potential at any point in the solution, qo, as a function of the distance from the surface, x (if confined to a single dimension). qo0
(- I () J
~-) i',, | E) + \\\
E3 \
p
|
\
", \ (3 \
I
I
0
d
X
Fig. III-12. A schematic representation of the structure of the EDL
198 The electrical double layer can be established through various means, which include: 1) the transfer of a particular ion from one phase to the other when an electrochemical equilibrium is being established. For instance, a silver plate immersed into an AgNO 3 solution with a rather small Ag + concentration becomes negatively charged due to the transfer of some of the Ag § ions from the metal surface into the solution; 2) the ionization of molecules present at the solid phase (e.g. the surface dissociation of silicates in contact with aqueous solutions); 3) "completion"ofthe solid surface with ions present in the dispersion medium. A classic example of this is the formation of a layer of potential determining ions on the surface of an AgI crystal; such ions are I ions in the case of contact with a KI solution, or Ag + ions in the case of contact with an AgNO 3 solution); the driving force for such completion of the surface is the
specific chemical interactions of ions in the crystal lattice; 4) the polarization of a surface by an external electric field. An example of this is the charging of a mercury surface in electrolyte solutions. According to the Stern-Graham model, the dense part of the EDL, adjacent to the solid surface charged with the potential determining ions, may, in turn, consist of inner and outer layers. The inner part, located in the direct vicinity of the charged surface, is formed by the specifically adsorbed ions, which are completely, or partially dehydrated (the so-called inner Helmholtz plane). The outer part, referred to as the outer Helmholtz plane, consists of hydrated ions that do not reveal such strong specific adsorption. The ions, specifically adsorbed within the inner part of Stern- Helmholtz plane, may
199 bear a charge not only of the opposite but also of the same sign as the potential determining ions. The latter depends on the ratio of the energy of electrostatic interaction between the ions and the charged surface, z,eq)a~ (gd~ is the potential at the border of inner part of the Stern-Helmholtz plane), and the energy of the specific molecular interaction between the ions and the surface, tI)i. W h e n discussing the role of the EDL in colloidal phenomena, in most cases it is sufficient to treat the Stern-Helmholtz layer as a single layer of thickness d (see Fig. III- 12). The subdivision of the dense part of the EDL into two sublayers might be of significance in the analysis of electrochemical phenomena.
According to Stem, the charge in the dense part of the EDL can be determined from Langmuir's theory ofmonomolecular adsorption, discussed in Chapter II. The work required to transfer the i-th ion from the solution bulk to the dense part of the EDL, W~(x = d) should contain the term -(I) = -(~t0 - ~t~s) )/NA, describing the purely adsorption interaction of the ion with the surface (see Chapter II,2), and the term due to work against electrostatic forces of the interaction of the ion with a charged surface, z~e%,where % is the dense layer potential. Thus, the term (!% - la~s) )/NA in eq. (II.21) should be replaced with ~ -
zjetgd,and the
adsorption of the i-th ion as a function of its bulk concentration, n~0,can be expressed as F i - Fmax 1 +
F ~i -z~eq)d max exp 2dn~o kT
where 2d is the adsorption layer thickness;
Fmax is the
limiting adsorption, determined by the
number of adsorption sites per unit interfacial area. Due to strong mutual repulsion between the ions, high values of adsorption, comparable to ['max, are usually not reached, and one can use the approximate relationship, corresponding to the initial linear region of the Langmuir
200 isotherm, namely
F i ,~ 2 d n i o
exp
tJs i -- z i e q) d .)
kT
"
The total number of charges per unit area (surface charge density) in the StemHelmholtz layer, pc, is then given by
zieg) d t " Pcl - 2 Fizie- 2 d e 2 zinioexp ( (~)i -k--T -
i
-
i
Consequently, even in the absence of specific adsorption (~i = 0), the charge of the dense layer is not zero" in this case the dense layer is formed by counter-ions held in the vicinity of the surface solely by electrostatic forces.
Using the Gauss theorem for the surface charge density, Ps, in a plane (x - 0), and assuming linear change in the potential between x=0 and
x=d, it
is possible to write (q~o -q~d) x=O
-
~;d~:O
d
'
where e d is the dielectric constant in the Stem-Helmholtz layer, the value of which can be significantly smaller than that of the dispersion medium. The quantity equal to Cd = p,/(%
- q~d)= ed eo/d is the integral capacitance
of a flat
capacitor formed by the charged surface and the ions of a dense layer. The condition of electroneutrality for the EDL as a whole, requires that Ps + Pd + P5 - 0, where P~ is the charge per unit interfacial area in the diffuse part of the EDL.
201 The determination of the adsorption potentials of different ions in disperse systems, q)i, is a task of substantial difficulty that one is not always able to carry out. The value of q~dcan not be measured either. These obstacles limit the use of the Stern-Graham theory as a source of quantitative information. However, at the same time this theory allows one to explain the phenomena of surface charge reversal that take place when electrolytes are introduced into the system. These phenomena are observed during the measurements of the electrokinetic potential, r (see Chapter V, 3 and 4). Let us now examine how the potential q0 changes within the d iffu s e part of the EDL, assuming that q~=0 in the bulk of the dispersion medium. The theory describing this part of the EDL was developed by Gouy and Chapman, who compared the energy of the electrostatic interaction of the ions with the energy of their thermal motion, assuming that the concentration of ions in the EDL was consistent with the Boltzmann distribution"
n~-n;0 exp(- ~ )
.
Gouy and Chapman thus utilized an approximation of the ideal solution of ions. They also assumed that the quantity W, is of purely electrostatic nature, and is equal to the work required to move the charge of zle from the depth of the solution, located at an infinite distance far away from the surface, to a given reference point (x,y,z): W~ - z,e(p ( x , y , z ) .
Consequently, the concentration of the i-th ion can be determined from the
202 expression n i - nio
E
exp -
zieq~
(x,y,z)
(III.8)
.
kT
The counter-ion concentration is higher within the diffuse double layer (z, and qo are of opposite signs), and decreases to the bulk value as one moves further away from the surface (Fig. III-13). Oppositely, the concentration of co-ions (bearing the same charge as the surface) in the diffuse layer is lower and increases as one moves away from the surface. This excess of counter-ions and lack of co-ions within the diffuse double layer gives rise to an excessive volume charge density,
9v,
which can be determined by summation of the
concentrations given by eq. (III.8) over all types of ions present in the dispersion medium, i.e.,
9v-
~. n ? p . l
tt+ + t't
/,t +
x
Fig. III-13. The change in the concentration of co-ions, n§ counter-ions, n-, and the total ion concentration, n = n++n -, in the diffuse part of the EDL The relationship between volume charge density, Pv, and the distribution of potential is described by the Poisson equation" ~0div(e grad q~
-
~;~;0V2q ) - - 9 v
- - ~
nizie, i
(III.9)
203 where
V 2 is
the Laplace operator; e is the dielectric permittivity that is assumed
to be constant. Substitution of eq. (III.8) into eq. (III.9) yields the
Poisson-
Boltzmann equation, which is the basic theoretical relationship describing the diffuse part of the EDL [ 18] :
i Z z en oexp( ) ~0
i
For the double layer at the flat interface (one-dimensional model) the PoissonBoltzmann equation can be written as
dx 2 = - ~ e f , o
i zienio exp
-
kT
1
"
For the usual case of a symmetric electrolyte (z+ = -z_ = z; no(+)- no(.) = no):
d2q~ = dx2
{ E j E 1}
zen-------~~exp - ze q~(x) _ exp q~(x_______~) ze ~;~;0
kT
kT
"
Introducing hyperbolic functions ~ one can write the Poisson-Boltzmann equation in a simpler form, namely
d2q) = - 9v(x) 2 z e~n ~ sinh [zeq)(x) 1 dx 2 ~ o ~ o kT " The appropriate boundary conditions for this
(III- 10)
second-order non-linear
differential equation can be stated as follows:
1The basic hyperbolic functions are: sinh(y) = (eS-eY)/2; cosh(y) - (eS+e-Y)/2; tanh(y) = sinh(y)/cosh(y) = (ey - eY)/(ey + e y) = (e 2y- 1)/(e 2y+ 1). Also, when y<~l sinh(y)~tanh(y)~y, and cosh(y)~ 1+y2/2. When y~ 1, tanh(y)--1
204 1) At the border between the diffuse part of the EDL and the dense Stem-Helmholtz layer 1
x=d
=- ~(Ps
EEO
1
+Pal) - ~ P s ,
(III.11)
~;~;0
where P~ is the charge per unit area of the diffuse layer, given by O0
P~ - Ipv(x)dx ; d
2) In the bulk of the solution x ~ oo,9 q~--~ O,9 dq~ - - > 0 . dx In order to carry out first integration of eq.(III. 10), let us multiply both sides by (dqo/dx) dx. Then, taking into account that 1
dxdx 2
2dx
'
and I sinh(y) dy - cosh(y) + const, we obtain
2
2zeno f sinh[Ze(p(X) ldg~~o kT
2k0r0 Ize x l } ~cosh
kT
+ const .
Using the boundary conditions for the bulk, i.e., q0=0 and dqo/dx=0 at x -~oo,
205 we find that the integration constant is const. = -cosh(0) = -1. Then, considering that cosh(y) - 1 - 2 sinh 2(y / 2), we obtain
dq~dx- - I
8kTn~
sinhlZeg~(x)l'2kT
In the above equation the minus sign is set in agreement with Fig. (III-13), which indicates that the signs of
cp(x)
and dqo/dx should be opposite to each
other. It is also noteworthy that the higher the electrolyte concentration, no, the faster the decay of q~ with increasing distance. The substitution of eq. (III-11) into the first integral of the PoissonBoltzmann equation (III. 10) yields the expression for the charge per unit area of the diffuse layer, namely P8 = ~ 0
x=d
-
2kT
1
,
(III. 12)
where the minus sign emphasizes that at positive potential at a distance where x = d counter-ions in the diffuse layer bear a negative charge. Taking into account that
f d(y/2)=lntanh(4sinh(y/2)
/ + const,
we can carry out the second integration of the Poisson-Boltzmann equation (III. 10). Using the boundary condition stating that at x = d qo - q~d, we obtain
206
tanh(Zeq)d~ 1
In L
8k---T-n~ ze ~o 2kT
4k_______T_- _ ]
(x-d)--
I
2z2e2no ( x d) ~0kT '
\ 4kT / which upon rearrangement yields
tanh[Ze'xltanh/ze%
(III.13)
--1=~5-I eeokT
(III. 14)
exp [-K(x - d)],
where
K
222e2no
The quantity 1/re,referred to as the effective ion atmospherethickness, was introduced in the Debye-Htickel theory of strong electrolytes, which was developed later than the Gouy-Chapman theory. If the difference between % and q9d is ignored, equations (III.9) and (III.8) can be written in simplified form:
tanh[Zeg(x) 1 4kT
tanh(zeq~~ exp (-rcx) 4kT
(III.15)
and
ze(Po]
Os - - 4 8~0kTn0 sinh 2kT ]"
(1II.16)
Such an approximation is valid only in the case of the weak adsorption activity of ions and in very dilute solutions, where due to a large diffuse
207 double layer thickness, 8, the difference between the potential q0d at a distance d ~ 8 from the surface and the potential % is very small. The approximation of q~e with q~0 does not cause any significant confusion, since it is always possible for one to go back to the original eqs. (III.9) and (III. 12). In the analysis of colloidal phenomena the behavior of the q0(x) function at large distances from the surface, where qo is small compared to
4kT/ze, is of particular importance. Since zeq)(x)/4kT~ 1 eq.(III. 15) can be written as q~(x) z 4kT tanh
for small y,
zeq)o) 4kT exp ( -
tanh(y)~y, at
~: x ) .
(III.17)
ze
In the case of w e a k l y c h a r g e d s u r f a c e s , when also % is small compared to
4kT/ze, eq.(III. 17) can be simplified further: q~(x) - q~o exp (-v~x),
(III.18)
meaning that the potential in the diffuse layer is proportional to the surface potential, and exponentially decays with distance from the surface. At low potentials one can replace the hyperbolic sine in the eq. (III. 16) by its argument (i.e., for
zeq~o/2kT) as well, which allows one to write the
expression for the charge density of diffuse layer as ,
= -
2kT
--880
~ ssokT
q~0 - -
ss 0 5 ,
which describes the charge density of a flat capacitor with capacitance per unit area, C~. = se0/8 = ss0K, as a function of potential. The quantity 8=lhc,
208 corresponding to the distance between plates of such a capacitor, is referred to as the effective thickness of the diffuse part of the EDL. The quantity ~:, the inverse to the ionic atmosphere thickness, 6, characterizes the steepness of potential decay as one gets further away from the surface; see eqs.(III.17) and (III.18). The higher the electrolyte concentration in the system (the higher to), the steeper the potential decay with increasing distance from the interface (Fig. III-14). For
1:1 electrolyte
calculations based on eq. (III.14) yield the ionic atmosphere thickness, 3 x 10 -~~ x c-~/2 m (provided that the concentration, c - n0[NA, is expressed in mol dm-3). Thus, in a 1 mol dm 3 solution 6 ~ 0.3 nm; in a 10 .2 mol d m -3 solution 6 - 3 nm, and in solution with c - 1 x 10 .7 mol dm -3 6 = 1 gm.
(Do
c 1 < c 2 < C3
"..N.., '\
1/K3=83
1/K:2=~i2
1/Kl=S1 X
Fig. III-14. The effect of electrolyte concentration on the decay of potential, qo, in the EDL
Another extreme case of interest is that of a s t r o n g l y c h a r g e d i n t e r f a c e , for which % ~ 4kT/ze, and tanh (zeq)o/kT) ~ 1. Consequently, one can write
209 4kT q~(x) ~ ~ e x p
(-Kx).
(III. 19)
ze
A comparison of eqs. (III. 18) and (III. 19) indicates that the potential always experiences exponential decay with distance at long distances from the surface. For a weakly charged surface q0(x) o~%, while for a strongly charged one the magnitude of the surface potential, %, has no effect on the potential distribution in the part of the diffuse layer that is sufficiently distant from the surface. This can be explained by the strong interaction between counter-ions and a strongly charged surface; the counter-ions located iv. the vicinity of such a surface s c r e e n the charge of the latter. Thus, the potential distribution at a distance far from the strongly charged "wall" (surface) is determined by the ionic atmosphere thickness, 8, and the value of qor = 4 k T / ze, which characterizes the ability of thermal motion to counteract
electrostatic
attraction between ions and a charged surface (determined by the charge of the ions, ze). This rather clear and simple approximation, often used in colloid science, makes it possible for one not to take into account the details of structure of the dense part of the EDL when describing the diffuse layer of counter-ions. For a 1"1 electrolyte at room temperature q0r ~ 100 mV. Figure III-15 shows the change in q~(x) due to a gradual change in q00. Changes in the trends of decrease in potential with increasing distance are best observed if log- lin coordinates (Fig. III-15, b) are used. At low surface potentials the In q~=f(x) graphs are represented by straight parallel lines 1 and 2 with slopes equal to -~c. In the case of high % curves 4 and 5 are steep at short distances from the surface, while at longer distances they approach a
210 common straight line, parallel to lines obtained at low %. The intercept of this "asymptotic" line yields the value of In q~r. hag)
q~
!
4kT m
N
5
q~r
--
,
ze
in
4kT
1
!
I
\ 3 2
l/~c
2/~: a
3/~:
x b
Fig. III-15. The potential, % (a) and its In qo(b) as a function of distance from the surface for different values of the surface potential, % Fig. III- 16 shows the potential at some particular distance, x > 6 = 1/~:, as a function of the surface potential, %. The asymptotic lines characteristic of the initial and final portions of the curve correspond to simplified eqs. (III.18) and (III.19), and describe reasonably well the properties of distant regions in the diffuse layer (the intersection point of these asymptotic lines corresponds to % - q~r ; at this point the true value of the potential is lower than the approximate value by about 20% ). Thus, at large distances from the surface the potential at a given fixed point x is proportional to the surface potential, q00,at low q00, and is independent of it (of the q~din the general case) at high q~0.
211
q~ tprexp(-~x) 9
/ 0
~ - q0~
/'i I ............ hot
%
Fig. III- 16. Potential qo as a function of surface potential % at some particular distance x > > 1/~:
As we have seen, the structure of the diffuse part of the EDL is determined by the ratio of the potential energy of the electrostatic attraction between counter-ions and the charged surface to the thermal energy of ions. This ratio is given by a dimensionless function, ze%/4kT (or zeq~d/4kT). When the potential energy of interaction between ions and the charged interface is small (zeq)o/4kT < 1), the potential decays exponentially with increasing distance, and its value at any reference point in the diffuse layer is proportional to the surface potential, %. Conversely, if the potential energy of attraction between the ions and the interface exceeds the kinetic energy of their Brownian motion (ze% / 4kT> 1), the surface charge is majorly compensated in the direct vicinity of the charged surface, i.e., the counter-ions present at short distances from the surface effectively screen its cbrge. It is important for one to remember that if the surface potential is high, at short separation distances eq. (III.17) should be replaced by the more accurate eq. (III.18), which takes into account the structure of the dense portion of counter-ion layer, as well as the individual size of the counter-ions. It can be verified that the asymptotic eq. (III. 15) can be readily obtained by
212 extending the integration limit of the Poisson-Boltzmann equation to the interface, i.e. to x - 0. This means that the centers of the ions can be located directly at the interface. The latter has no significant effect on the distribution of potential at large distances from the surface, especially in the situations
ze%~/ 4kT >
when the adsorption layer potential, q~d, is sufficiently high and
> 1. In some cases these distant diffuse layers of counter-ions are the ones that determine colloid stability (see Chapters VII, VIII). The presence of a diffuse layer with elevated concentration of counterions and lowered concentration of co-ions in the vicinity of an interface gives rise to many electric and filtration phenomena taking place in disperse systems. It is of significance that the diffuse layer has an increased total concentration of current carriers (Fig. III-13). For the simplest case of a symmetric electrolyte in agreement with eq. (III.8) one can write that
n
-4-
+n
- no
exp
-
+ exp
kT
=
kT
(III.20)
=2n~176 "kT1 The above treatment is valid for flat interfacial double layer. In disperse systems the EDL can be treated as flat when the size of dispersed particles is substantially larger than the thickness of ion atmosphere. If this is not true, one has to write the Poisson-Boltzmann equation in its complete form, namely: d2____~_~+ d2q)4 d2q) _ dx2 dY 2 dz2 _
~
~
m
.
9v_2zenosinhlZe~p(x)1 ~
~:gO
m
~
~3~;0
,
kT
This equation cannot be solved in quadratures even for the simplest model of
213 spherical or cylindrical particles. The results of the numerical integration of this equation are available for different geometries and cover a broad range of surface potentials and values of ion atmosphere thickness. P.Debye and E. Htickel offered an approximate solution of the above equation for the system consisting of weakly charged spherical particles of radius r, when zeq)o / k T < 1, and sinh (ze% / kT) ~zeq~o / kT. The Poisson-Boltzmann equation, written in spherical coordinates, in this case appears as:
e(R)
R 2 dR
where R is the distance from the center of a particle. The solution of this equation reads r
q~ (R) - q~o ~ - e x p [ - K (R - r ) ] .
This result reflects both the common decay of potential as the distance from the center of a charged sphere (r/R term) increases, and a more rapid exponential decay due to the presence of the diffuse layer (exponential term). Consequently, the potential of a sphere surrounded by diffuse layer decays with distance faster than the potential near the particle in a dielectric medium, or the potential in the vicinity of a flat interface with a diffuse layer. One can say that the distant regions with low potentials are mostly "proliferated" around the charged particle, while those with high potentials occupy a small volume in direct vicinity of the particle surface. For potentials at large distances from the surface of strongly charged particles one can use an expression similar to eq. (III. 19)"
q~(R) -
4kT r ze
exp [ - ~ : ( R - r ) ] .
R
In later sections (in particular in those devoted to colloid stability) we will limit our discussion by considering the flat double layers only.
214 III.3.2. Ion Exchange
Changes in the electrolyte composition of the dispersion medium electrolyte cause some particular changes in the structure of the electrical double layer (EDL), and are followed by ion exchange, during which some of the newly introduced ions enter the double layer, while some of the ions previously located in the EDL return to the solution bulk [19-20]. The nature of the changes to the EDL is determined by the ability of the introduced coions and counter-ions to enter the solid phase, their tendency to become specifically adsorbed at the interface, and the ratio of their charge to that of the ions forming the EDL (the latter is mostly related to counter-ions). One can identify two extreme cases: the indifferent electrolytes, which do not affect the surface potential, r
and
non-indifferent
electrolytes, which are capable of
changing %. The electrolytes of the latter type usually contain ions that are able to enter the crystalline lattice of solids, for instance through isomorphic substitution with ions forming the lattice. The ion exchange process in solutions of indifferent electrolytes can be described in the most general way by the Nikolsky equation, which for rather concentrated (non-ideal) solutions can be written as
c~/z~ 1/z2 C2
a ll/zl -
-
k12
1/z 2 " a2
In this equation a, c, and z are the activities, concentrations, and charges of ions of type 1 and 2, respectively. The ion exchange constant, kl2, is related to the corresponding adsorption potentials, ~ and q)2, by
215
k12-exp((I)l - cI)2./ " kT Depending on the nature of the introduced electrolyte, the ion exchange can affect different regions of the EDL: the diffuse and adsorption regions, and even the layer of potential-determining ions (in which case it is, however, more appropriate for one to talk about the build-up of the crystal lattice of the solid phase with the constituent ions of introduced electrolyte). The diffuse layers of counter-ions are the ones that undergo exchange most easily. Disperse systems consisting of positively charged particles or macromolecules that are surrounded by diffuse layers consisting of exchangeable anions, are referred to as anionites, while systems consisting of negatively charged particles (macromolecules) that are capable of exchanging cations, are referred to as cationites [21 ]. In finely dispersed systems changes in the ionic content of layers containing potential determining ions or counterions may cause a significant change in the composition of the colloidal particles. For example, a particle with a diameter d ~ 10 nm contains (d/d) 3 (30) 3 ~ 3 x 104 ions (assuming that the average ionic diameter d; = 0.3 nm). Out of that many ions
4%dZ/%di 2 ~
4x 103 ions (more than 10%) are located at
the surface, i.e., changes in the ion content of the surface layer may affect a significant portion of the matter making up the particle. The ability of disperse systems to participate in ion exchange is characterized by the exchange capacity, equal to the number of gramequivalents of ions taken up by one kilogram of a substance. Since the ion exchange ability is strongly dependent on the pH, concentrations and
216 composition of the medium, the exchange capacity is usually determined under certain standard conditions, i.e., one uses the conditional ion exchange capacity. For instance, in soil science the exchange capacity is usually determined at pH 6.5, using Ba 2+ as exchangeable ions at an electrolyte concentration of 0.1 N (usually BaC12.which is normally not present in soil). Ion exchange processes play an important role in nature and technology. For example, clay m i n e r a l s reveal a strong ion exchange ability. These minerals are alumosilicates with a lamellar structure (the interlayer distance is -~
nm). The potential determining ions in these
materials are silicic acid surface groups, and the cations play the role of exchangeable counter-ions. Depending on medium composition, the counterions may be Na +(Na-clays), Ca 2+or others. Ion exchange in clays plays an important role in the formation of the so-called s e c o n d a r y ore depo sits: hydrothermal waters containing ions of heavy metals enter the strata rich in clay minerals, where they undergo ion exchange, leaving the heavy metal ions behind, and washing the light ones out. The influence of the adsorption activity of ions on their geochemical fate can be clearly followed by looking at the distribution of potassium and sodium in nature. These elements have approximately equal abundance in the Earth's crust (2.4 and 2.35%, respectively), while sea water contains mostly sodium (there are about 10.8 g of sodium and only 0.4 g of potassium in 1 kg of sea water). Ion exchange taking place in clay deposits at the sea bottom is the reason for such an enrichment: sodium originally present in clays becomes nearly completely displaced by potassium. The ability of soils to actively participate in ion exchange determines
217 their functioning and fertility [22]. Soils are complex disperse systems containing finely dispersed insoluble polysilicic acids and clays and mineral organic substances formed due to the decomposition of organic matter (the so-called Gedroiz soil absorption complex) [22]. Soil composition, productivity and ability to participate in ion exchange are to a large extent dependent on the climate. The weathering of rocks leads to the formation of various clay minerals with ion exchange capacities up to 1 mol kg -1. In regions with high humidity and low content of organic matter (insufficient heat), the erosion of basic oxides (those of alkali and earth alkali elements), and humic acids, as well as the peptization oftrivalent metals (due to weak binding by organic substances) take place. These phenomena lead to soils impoverished in organic substances and valuable ions and containing increased amounts ofpolysilicic acids. Such coils are, consequently, enriched with clays in which metal cations are replaced with hydrogen ions. All of these factors cause soils (especially podsols) to be acidic and have poor productivity. The exchange capacity of podsols falls within a range of 0.05 - 0.2 mol kg -~. Chernozems are formed in regions with a moderate amount of precipitation and sufficient amount of heat. These soils contain a significant amount of organic matter, most of which is present in the form of poorly soluble humates of divalent metals (as calcium and magnesium salts ofhumic acids). Colloidal particles of humates may undergo heterocoagulation with alumosilicates and silicates (see Chapter VIII), forming finely dispersed highly porous structures with exchange capacities reaching 0.6 - 0.8 mol kg -1. These structures are rich in valuable cations and various nutritious substances. They
218 are able to entrap water due to capillary forces and at the same time are airpermeable. Air permeability is of extreme importance for the life of various microorganisms that improve the structure and productivity of soil. Peat soils, having a content of organic matter comparable to that in chernozems, are usually formed in regions with high humidity, which results in the erosion of cations and their replacement with hydrogen ions. Peat soils are thus acidic. The acidic nature of these soils makes difficult the development of plants that would be able to release the hydrogen ions during growth. Binding of these hydrogen ions released by plants (primarily due to ion exchange) is one of the main functions of a productive soil. The use of peat as an organic fertilizer in acidic soils is practical only if simultaneous exchange of protons with other more valuable ions takes place. The latter is achieved by the addition of either calcium carbonate, which causes the replacement of H § ions with Ca 2- ions, or ammonia aqueous solution, which at the same time plays the role of a valuable fertilizer. Ion exchange processes have enormous importance in various technological applications. Softening and de-ionization of water are the two characteristic examples of processes based on ion exchange [19,23]. Water
softening, or the exchange of Ca 2+ ions with Na § can be carried out using highly porous zeolite-type alumosilicate minerals of the general formula A1203.mSiO2.nH20 [24]. In these materials part of the H § ions can be replaced by metal ions. Both natural and synthetic (permutite) minerals are used. Schematically representing a single exchanging group of the Na form of permutite, Na20"A1203"3SiO2"2H20, as NaP, one can write the ion exchange reaction as
219 2NAP+ Ca 2§ Ca(P)2 + 2Na § . Subsequent treatment of the calcium form of permutate with concentrated NaC1 solution results in the regeneration of its sodium form. Another important practical application of ion exchange is the complete removal of ions from water, widely utilized for the preparation of deionized water, and conversion of sea water to fresh water (i.e. desalination). Highly effective ion exchange resins with exchange capacities reaching 10 mol kg -~ are used for ion removal. Ion exchange resins consist of crosslinked polyelectrolytes that form a three-dimensional network [21 ]. Such structure provides ion exchange granules and membranes with high mechanical strength. In aqueous media the resins swell, allowing all ionic groups within the granules to be available for exchange with the dissolved ions. C a t i o n i c resins usually contain sulfonic groups,-SO3, carboxylic _
groups,-COO , or phenolic groups,
C6H40 ,
the exchange capacity of which
increases with increasing pH. The interaction of the resin H-form with an electrolyte solution results in the exchange of electrolyte cations with H§ ions until a certain pH, determined by the strength of the ionic group, is reached. Cationic resins can be regenerated (i.e., converted back into the H-form) by treatment with acid. Ani o ni c resins contain various aminogroups (-NH3; =NH2; =NH) or quaternary substituted ammonium. The exchange capacity of anionic resins increases as the pH is lowered. These resins allow one to remove dissolved _
anions by ion exchange with OH ions. Anionic resins can be regenerated by treatment with alkalis. Apparently, de-ionized water is produced by the
220 sequential ion exchange of water on cationic and anionic resins. In some cases amphoteric ion exchangers (e.g. activated carbon) are used. According to Frumkin, when activated carbon is saturated with hydrogen, it acts as a cationic ion exchanger, but if saturated with oxygen, it turns into an anionic one. The removal of heavy metals from wastewater is another area in which ion exchange resins are used [21]. The ion exchange method allows one to remove such metals as copper, silver, chromium, and radioactive substances. Ion exchange methods of hydrometallurgy in combination with the use of microorganisms capable of converting the heavy metals present in poor ores into soluble compounds constitute a promising direction in the development of mineral and ore processing.
111.3.3. Electrocapillary Phenomena Information regarding the structure of EDL and the nature of some colloidal phenomena resulting from the interactions between ions and the interface can be obtained from the studies of
electrocapillary phenomena,
focusing at how the interfacial charge influences the surface tension. A complete
description
of electrocapillarity
is
given
in
courses
in
electrochemistry. Here we will only discuss the basic laws governing these phenomena that are important for understanding such colloidal phenomena as the adsorption of anionic and cationic surfactants, nucleation (see Chapter IV, 1), and the Rehbinder effect at charged surfaces (see Chapter IX, 4) The repulsion between charges of the same sign in the interfacial double layer should make an increase in the surface area easier, i.e., it should
221 decrease the interfacial tension u. It is well known from electrostatics that the work Wq required to supply a charge q to a spherical surface of radius r at a potential difference of cp=q/4xeeo r is given by 2
Wq
-
q
87~ggor
=2=georq)
2
One may expect that the specific (per unit area) work of charging is exactly the value of the work "already accumulated" by the interface that is needed to ease the increase in the interfacial area. In other words, the specific work of charging is equal to the potential energy lowering: Wq G O - o((p) - 4~r2
q2
ego(p2
32r~2~;~or3
2r
Differentiation of the above equation with respect to q~yields the L i p p m ann e q u a t i o n , which is the main relationship of electrocapillarity: dcy d(p
~oq~ r
q = p~, 4=r 2
(III.21)
where P~,.is the surface charge density. The investigation of the effect made by the applied potential difference on the interfacial tension can be most conveniently carried out on the ideally polarizable surface of liquid metal (most commonly mercury) in aqueous electrolyte solution. It is important that in these experiments one be able to simultaneously measure the potential difference between phases (with respect to some standard electrolyte) and the interfacial tension. The latter is usually
222 done by measuring the highest level reached by mercury, which is retained in the capillary by the surface tension. At the same time one can also determine the double layer charge density from the current carried by the mercury drops of known area. In agreement with the Lippmann equation, in the absence of surfactants the curve showing the surface tension as a function of the potential difference between phases (the electrocapillary curve) contains a maximum at some particular value of q~ (Fig. III-17). This potential, which corresponds to the
,,,rf,ct,,,t / X ,,,rf,ct,,,t
Fig. III-17. The shift in point of zero cha~'geposition due to adsorbed ionic surfactants maximum in the electrocapillary curve (i.e., to ps=0), is referred to as thepoint
of zero charge. The position of the point of zero charge is determined by the adsorption activity of ions present in solution and by the dipole moment of solvent molecules. In the absence of an externally applied potential the prevailing adsorption of Hg 2+ ions occurs at the mercury surface. These ions, present in the solution that is at equilibrium with the mercury, cause the
223 surface to become positively charged. To balance this charge one has to apply a negative potential, q~<0, and thus the point of zero charge occurs at negative potentials. In the cathode region (to the right from the point of zero charge, i.e. in the region of more negative potentials) the charge at the surface is caused by the electrons that come from an external circuit, while in the anode region it results from the presence of the mercuric ions. The addition of inorganic electrolytes results in changes in the shape of the electrocapillary curves. If the electrolyte contains anions that have a strong tendency to adsorb (anionic surfactants), the surface tension in the anodic region decreases. At the same time, there is no adsorption of anionic surfactants in the cathodic region, and the surface tension does not change (Fig. III-17). The maximum in the electrocapillary curve in this case is thus shifted into the cathodic region. The higher the adsorption activity of the anions, the more drastic the decrease in the surface tension in the anodic region, and at more negative potentials adsorption occurs. It is worth pointing out that in this case the adsorption activity of large organic surfactant anions is especially high. The cations are smaller in size than the anions and, as a rule, have less influence on the shape of the electrocapillary curve. Cationic surfactants, however, adsorb strongly and cause a shift of the maximum into the anodic region. Nonionic surfactants are also capable of influencing the shape of the electrocapillary curve: at moderate applied potentials these compounds adsorb in both the anodic and cathodic regions. At high positive and negative potentials the attraction of dipolar water molecules to the surface is so strong
224 that the molecules ofnonionic surfactants are "squeezed out" into the solution away from the surface. For this reason nonionic surfactants lower the surface tension only in the middle part of the electrocapillary curve. Since the molecules of nonionic surfactants have a dipole moment, the stronger lowering of the surface tension is dependent on the orientation of the dipole moment with respect to the surface, and may occur in either the cathodic (the dipole is oriented towards the surface with its positive end), or the anodic region (Fig. III-18). O
,
,
,
Fig. III-18. Change in the shape of the electrocapillary curve due to the adsorption of nonionic surfactants
In agreement with the Lippmann equation (III.21), the differentiation of the electrocapillary curve, o(q0), with respect to q0yields the surface charge density as a function of the surface potential, the second differentiation yields the value of the differential capacity, which can be compared with the results of the EDL theory. Based on such a comparison one can draw conclusions with respect to the validity of theoretical models and look for ways for further improvements.
225 Investigation of the electrocapillary curves at solid-liquid interface is much more difficult to carry out. It was shown by Rehbinder and W6nstrem, that such curves can be obtained through studies of the effect of surface charge on the mechanical properties of solids (see Chapter IX, 4).
III.4. Wetting and Spreading
Up to now we primarily have been considering the interfaces in twophase systems. The conditions of phase contact in tri-phase systems were only briefly mentioned in relation to the capillary rise and during the description of methods for surface tension measurement
(see Chapter I, 3). Within this
section we will address this issue in greater detail, but restrict ourselves mainly to consideration of systems that consist of mutually insoluble phases and do not contain any substances that might adsorb at interfaces. Let us take a look at a liquid drop on a solid surface. Three types of interfaces can be distinguished in this system: the interface between the solid surface and the surrounding gas (air), the interface between the liquid and the gas (air), and the interface between the solid surface and the liquid. The surface tensions at these interfaces are ~sG, ~LG, and C~si~, respectively. The line along which all three interfaces intersect is referred to as the line of
wetting; a closed line of wetting forms the perimeter of wetting. The angle between the liquid-gas and solid-liquid interfaces, 0, is referred to as the
contact angle [25]. If the surface tensions are viewed as forces acting on a tangent to the corresponding interfaces and applied perpendicular to the unit length of the
226 perimeter of wetting, (Fig. III-19), the force balance is given by the Young equation" O'SG -- O'SL + O'LG c o s O ,
or
COS0 -- (YSG -- (YSL .
(III.22)
(YLG
S
(JSL
(JSG
Fig. III-19. The balance between the surface tension forces acting along the perimeter of wetting Frumkin was the first to point out that while examining wetting one must account for the formation of the adsorption layer or thin film present in equilibrium with the macroscopic liquid phase. Such a layer may be formed either by the transfer of substance from a liquid phase via vapor, or due to the diffusion (migration) of molecules of a liquid along the solid surface. In the latter case Young's equation should be written as COS 0 -- (13"SG -- ~ ) -- ~ SL (~SG
where 7cis the two-dimensional pressure of the equilibrium adsorption layer. Change in the surface tension at the gas-solid interface, CYSG,is of particular importance for the high-energy surfaces, while for the low-energy ones the
227 value o f ~r can be neglected.
The derivation of Young' s equation (III.22) bythe geometric force balance approach is often viewed as being non-rigorous. On a number of occasions doubts were expressed whether asa, ~sL, and ~La can be regarded as real forces acting along the perimeter of wetting. A more rigorous way of obtaining the relationship between the equilibrium contact angle, 0, and the values of specific free interfacial energies is for one to consider the dependence of the system free energy on the shape of a drop of constant volume. Here we will introduce a simpler, but nevertheless rigorous, derivation of Young's equation, based on the conditions of equilibrium of a solid spherical particle on a flat liquid surface in the absence of gravity. Let us assume that the volume of the liquid phase is so large that one can consider the interface to be flat. In the absence of gravity a spherical particle of radius r will be oriented in such a way that the energy of the system is minimized. This means that the change (first variation) in the interfacial free energy when the contact line is shifted by a small distance, 5h, is zero. It follows from geometry that a shift in the position of the contact line by ~Shdownward from its initial position (bold line in Fig. III-20) results in changes in the interfacial areas of the solid-gas (Ssa), solid-liquid (SsL), and liquid-gas (SLB) interfaces equal to
SsG - - SSL - 2 rcqSh / s i n 0 ; SLG -- 2~tr 18h / t a n 0 ,
where rl is the radius of the wetting circle. Consequently, for the equilibrium one has ~)Fs
= (5 SG ~SsG + 13"LS ~SLs + (3"LG SLG /
= 2rtr 1
(YSG 8h(\ sin0
which immediately yields Young's equation.
(YLS sin0
O'LG t --0 tanO
228
G
~h
(YLG
L
0
t
Fig. III-20. The derivation of Young's equation It should be pointed out that in the case of drops of very small size the excessive energy of the perimeter of wetting (the so-called linear tension, a~) may significantly contribute to the energy of wetting. The linear tension can be both positive and negative; at a~>0 the perimeter of wetting tends to further tighten the drop, and the contact angle, 0, should increase as the radius of the wetting perimeter, rl, decreases Oppositely, when ~e<0, the contact angle decreases. The sign and magnitude of the linear tension, a~, are determined by peculiarities of the interaction between the phases in contact in the vicinity of the perimeter of wetting.
Based on the magnitude of the contact angle, 0, the following three cases can be distinguished: 1)wetting of the surface by the liquid ("good wetting") when 0<90 ~ i.e., cos 0 > 0; 2) non-wetting ("poor wetting") when 0 > 0 (cos0<0); 3) spreading ("absolute wetting"), when no equilibrium contact angle establishes, and the drop spreads into a thin film. In agreement with eq. (III.22) the condition where ~sG > CYLs corresponds to wetting; that where (YsG< (YLScorresponds to non-wetting, and the condition where rSsG> CYLS+ rSLGdescribes spreading. The value of Ws = = CysG- ~SLS- CYLG represents the change in the energy of the system when the unit interfacial area becomes covered with a flat layer of liquid, i.e. W~may be
229 viewed as the work of spreading, or as the driving force for the spreading process (this force is applied perpendicular to the unit length of the perimeter of wetting and acts along the solid surface). A comparison of Young's equation with the definition of the work of adhesion, Wa , given by eq. (III. 1), yields cos0 - ~Za -- ~LG O'LG
which allows one to determine the work of adhesion, Wa, at the solid-liquid interface experimentally. Replacing the surface tension of the liquid by its work of cohesion, Wc - 2(YLG,one can write Young's equation as
cos0 -
2Wa - W c
We
.
(III.23)
One can thus establish the following criteria for non-wetting, wetting, and spreading"
Wa < 89 Wc represents non-wetting,
89
< W, < Wc
corresponds to wetting, and Wa > W~ is indicative of spreading of liquid over the solid surface. Ws can then be defined as the difference between the work of adhesion and the work of cohesion, i.e., Ws = Wa - Wc. Since in a vacuum the interaction between condensed phases is always attractive, the work of adhesion is definitely positive (Wa > 0), and, since cos 0 > -1, the contact angle is always less than 180 ~ As a role, contact angles in solid-liquid systems do not exceed 150 ~ In agreement with eq. (III.23), good wetting and spreading are possible when the work of adhesion is large (solid and liquid phases are similar in their
230 molecular nature), and the work of cohesion in the liquid is small (low liquid surface tension). Thus, hydrocarbons and other organic liquids wet nearly all solid surfaces well. Oppositely, liquid metals (those with low chemical reactivity) are characterized by high surface tension (103 mJ/m2), and wet only unoxidized surfaces of solid metals well. Active metals, such as titanium, manganese, and zirconium (so-called deoxidizers) are also capable of wetting the surfaces of some oxides. Since water is a liquid with a rather high work of cohesion (W~ ~ 140 mJ/m2), it wets oxides well and spreads over the surface of some silicates, but does not wet the surfaces of paraffin and fluoroorganic polymers. The work of adhesion (see Chapter 1,1) reflects the degree to which unsaturated molecular interactions between solids and liquids in contact are balanced. The value of cos 0, which is symbatic to the work of adhesion, is also a measure of the degree of similarity between the solid surface and a liquid (liophilicity). Polar surfaces that are wetted by water well
are
hydrophilic, while those poorly wetted (solid hydrocarbons, and particularly fluororinated polymers) are hydrophobic. Since the value of 0 is determined by both the work of adhesion and the work of cohesion, a comparison of the contact angles formed by different liquids at the same solid surface does not allow one to compare the works of adhesion (the degree of similarity in the nature of the liquid and solid) directly. For example, polar surfaces are equally wetted well by both water and hydrocarbons. A comparison of the contact angles in the case of preferential wetting is more representative. Preferential wetting corresponds to the situation where equilibrium is established between the interface separating two immiscible
231 liquids (water (L~) and hydrocarbon (L2), see Fig III-21) and a solid surface. In this case it is convenient to measure the contact angle in the direction of the more polar (with higher surface tension) liquid, which in the present case is water"
cosO -
SL 2 -- (3"SL l
.
(III.24)
(Y L 1L2
Two different liquids may compete with each other, if each of them individually wets a solid surface. In this case the resulting contact angle corresponds to that of a better wetting liquid with nature similar to that of the solid surface. If the surface is better wetted by water than by a hydrocarbon ( 0<900 ), it is referred to as hydrophilic (oleophobic); in the case when the surface is better wetted by non-polar hydrocarbon (0>90~ it is referred to as
hydrophobic (oleophilic). For selective wetting, as opposed to wetting in air, the contact angle, 0, can assume any value between 0 ~ and 180 ~ When 0 = 0 ~ the more polar liquid spreads over the solid surface, forcing the less polar liquid away; when 0 - 1800 the situation is opposite: the non-polar liquid phase completely forces the polar liquid away from the solid surface.
~
b
~
L~
,7,--~/////// ~ / / / / 2 / 2 / / / / / / / , OSL 2
OSL 1
S
Fig. III-21.Balancebetweenthe surfacetensionforcesduringselectivewetting In common practice one makes a judgement whether a particular surface is hydrophobic or hydrophilic based on the measurements of the contact angles formed in air by a drop of water on the surface. The above
232 discussion shows that such an approach is not sufficiently rigorous, and does not allow one to draw conclusions with respect to the oleophilicity or oleophobicity of the surface. In the case of selective wetting, as follows from eq. (III.24), the value of the contact angle is directly related to the surface tension difference at the interface formed upon contact of the solid surface with the two liquids. Hydrophilic materials include such substances as quartz, glass, metal oxides and hydroxides, oxidized minerals, etc. Alternatively, solid hydrocarbons and their fluorinated derivatives, plant leaves, the chitin shells of crabs, animal skin are all examples of objects with a hydrophobic surface. One example of a phenomenon that takes place in disperse systems and is related to selective wetting is the penetration of emulsions through porous filters. If the coarse filter is selectively wetted by emulsion drops, these drops can adhere to the filter surface and become retained. This is sometimes used for the removal of water dispersed in crude petroleum (see Chapter VIII, 3): the oil is filtered through the coarse hydrophilic filter, which retains the water. A selectively wetted filter with a fine porosity may also retain emulsion droplets with sizes significantly exceeding those of pores. These drops are not able to penetrate the filter, as their penetration would require substantial deformation, which in turn would cause a high capillary pressure. Filtration through fine pore hydrophobic filter allows one to remove water from gasoline. A quantitative characteristic of the energetics of wetting and of the nature of a solid surface (its hydrophilicity and hydrophobicity, oleophilicity and oleophobicity), particularly important for finely porous materials and
233 powders, is the
specific heat of wetting, which
is defined as the amount of
energy released upon wetting a unit area of solid surface and is given by the difference between total surface energies of the solid-gas and solid-liquid interfaces [26]. According to Rehbinder, the ratio of the heats of wetting of solid surfaces with water (Hw) and hydrocarbon (Hoi~) represents a criterion for the hydrophilicity of the surface: for hydrophilic surfaces 13 =
Hw/Hoi~>
1, while
for hydrophobic ones 13 < 1. For instance, for activated carbon (hydrophobic surface) 13 = 0.4, for quartz (hydrophilic surface) 13 = 2, for starch (strongly hydrophilic surface) 13 ~ 20. In the case of contact with either water or hydrocarbon, the heat of wetting can be expressed per gram of powder (adsorbent), and hence one does not need to measure the surface area of the powder studied.
Going back to the analysis ofthe influence that the nature of medium has on wetting, let us consider in some detail the case when the interactions at the solid-liquid interface, characterized by the work of adhesion, Wa, are solely of the dispersion type. Such situation occurs during the wetting of low-energy surfaces, such as paraffins, polyethylene, etc. One can assume (see Chapter III,1) that the entire contribution to the surface tension of a nonpolar solid comes from its dispersion component, i.e. ~nso = 0, and %o =GasG 9Then, in agreement with eq. (III.4), one can write the expression for the interfacial energy at the solidliquid interface: 2 n
+ O'LG. e
This equation shows that, since the dispersion components of the surface tension of polar and non-polar phases do not differ significantly (see Chapter I, 2), the surface energy at the interface between a non-polar solid and a polar liquid is mainly determined by the non-
234 dispersion component of the surface tension of the liquid, ~"LG. Substitution of the above expression into Young's equation (III.22) yields
cos0 - - 1+ 2
"SG~d
LG .
(III-25)
CYLG The studies examining the wetting of polymers with low surface energy (Teflon, polyethylene) and metals or glasses coated with saturated surfactant adsorption layers by organic liquids indicated that for liquids belonging to different homologous series the extrapolation of cos 0 =f(C~LG) dependence to cos 0 = 1 yields the same value ~LG"This value of C~LG,corresponding to the zero contact angle was referred by V. Zisman as the critical
surface tension of wetting, %. Since % is independent of the properties of liquids, and is determined only by the nature of a solid surface, Zisman suggested using this quantity as a characteristic of the surface properties of solids. In particular the value of the critical surface tension of wetting of polymers and adsorption films formed by organic compounds is very sensitive to the types of functional groups emerging at the surface, as well as to the packing density of molecules of the solid phase within the interfacial layer. The value of % evaluated from contact angle measurements can also be used in obtaining estimates for the surface energy of low energy solids, GSG.Indeed, in the case of the wetting of a non-polar surface with a n o n - p o l a r liquid, when the surface tensions of both phases are primarily determined by the dispersion components of the interfacial energy, ~sG= --
GdSGand ~LG = ~aLG,the equation (III.25) can be written as
cos0--1
+
2JV 's~ ~LG
Under these conditions the value of % is close to that of the surface energy of the solid, i.e., % = ~,o. The value of CYSGcan be obtained by measuring the contact angle, 0, for several nonpolar liquids and extrapolating linear cos 0 = f ( ~LG~/2) to COS 0 = 1. For instance, for polyethylene ~SG = 31 mJ/m 2. If a non-polar surface for which the surface energy was determined in this way is wetted with a polar liquid of known surface tension (e.g. water), one can obtain the dispersion component ofthe surface tension of the polar liquid, C~dLG,from
235 the contact angle values using eq. (III.25). For water ~dm ~ 20-25 mJ/m 2, and the nondispersion component of the surface energy, GnLG,is close to 50 mJ/m 2. The "unbalanced" value of ~nLGat the interface with liquid hydrocarbon leads to an interfacial tension value, %/o - 50 mJ/m 2
The wetting of solid surfaces by liquids is greatly affected by the state of the s o l i d s u r f a c e , and by its microgeometry in particular, i.e. by the surface roughness. The surface of real solids is never ideally smooth. Figure III-22, a shows a microprofilogram of a region of a zinc surface recorded using a microprofilograph with a diamond needle (magnification: 1000x vertical; 160• horizontal). Figure III-22, b shows a schematic "decoding" of the AB region in this microprofilogram.
a
at
x..
aiz is-/ l
B
,000,t+ 1000
x
b Fig. III-22. Microprofilogram of zinc surface (a), and the schematic decoding of the AB region in this microprofilogram (b) The surface relief can be approximated with a series of microgrooves of depth H and width d; H = (d/2)tan Z, where Z is the angle between idealized flat surface and a side wall of the groove. If the surface is rough, the real surface area, Sreal, is greater than that of the idealized surface, S~dea1. The ratio of the real surface area to the area of its projection onto an idealized flat surface is referred to as the coefficient of roughness, k r [27]:
236
kr
_
Sreal _ d / cos ~; _ 1 Sidea 1 d cos ~;
An increase in the true surface area of the solid results in a corresponding increase in the input from
the solid-liquid and solid-gas
interfaces into the energy of wetting. According to Derjaguin, the expression for the work of adhesion in the case of contact between a liquid and a real solid surface should be written: Wa - k r (O'SG - ~ S L ) + O'LG ,
and the averaged (the "effective") value of the cosine of the contact angle is k r ((Y SG - (y SL ) COS Oef =
O"SG -- (5"SL =
(5"LG
COS 0 =
(5"LG COS ~
COS
The above equation shows that the surface roughness improves the wetting of a solid surface by a liquid (the value of 0el decreases), and makes non-wetting worse (the value of 0el increases). The condition of Z = 0 is sufficient for wetting to turn into spreading. This effect is used in such processes as soldering and gluing: prior to applying glue or solder, the solid surfaces are treated with sand paper, which in addition to removing the impurities makes the surfaces rough. At the same time, surface roughness, especially that caused by the presence of a series of parallel grooves (such as on a mechanically treated surface) enhances the wetting hysteresis phenomena.
Wetting hysteresis (or contact angle hysteresis) is defined as the ability of a liquid to form on a solid surface several stable (or metastable) contact
237 angles, different from the equilibrium contact angle [25,27,28]. For instance, the advancing contact angle, 0a, formed when a drop of liquid is placed onto a solid surface, is significantly larger than the receding contact angle, e r, which originates from the contact of an air bubble with the same surface immersed into a given fluid. Contact angle hysteresis becomes apparent if the solid surface with a drop of liquid on it is tilted: the contact angle in the lower portion of the drop, 0,, is significantly larger than the contact angle in the upper potion, Or, as shown in Fig. III-23. Wetting hysteresis may be due to the surface adsorption of impurities, surface chemical heterogeneity, and other factors. The influence of surface roughness on contact angle hysteresis can be explained as follows. When the drop approaches a groove or a scratch and starts to fill it, the apparent contact angle,
0app, with respect to the idealized flat
solid surface (dashed line in Fig. III-24), increases as compared to the true contact angle, 0. When the number of grooves and scratches present on the solid surface is large, this process results in the average advancing and receding contact angles being different from each other.
III~ ~ Fig. III-23. Shape of a drop placed on a tilted surface
Fig. III-24. A drop of liquid as it moves along a rough surface
238 When the material of solid surface is to a significant extent soluble in the wetting liquid, one also observes the hysteresis phenomena. In this case changes in the profile of the solid surface occur due to a contact with the liquid phase. To understand this, let us recall Fig. III-19, from which one can see that the vertical component of the c~m vector can not be fully balanced by the surface tensions of the two other surfaces. If a liquid is in contact with a solid phase that is insoluble, this vertical component, CYLG,is balanced by the elastic resistance of the solid surface. The situation is entirely different when a drop of liquid (L~) is placed onto the surface of another liquid (L2)" in this case all phases are highly mobile, and the state of equilibrium is described by the vector Neuman equation ~'L1G -t- I~'L2G + I~'L1L2 -- 0 .
Similar conditions exist if a drop of liquid is placed on the surface of a solid which is soluble in this liquid. Due to diffusion through the liquid, the system may come to a true equilibrium described by the Neuman equation: (5"LG + O'SG + (5"SL -- 0 .
Dissolution of the solid surface results in the gradual formation of a groove, and the solid-liquid interface acquires spherical shape (Fig. III-25). In this case one needs two contact angles to describe the equilibrium. These are the angle between the surface of the drop and the continuation of the solid-gas interface, 01, and the angle between the solid-liquid interface and the continuation of the solid-gas interface, 02 (Fig. III-25). The equilibrium in the plane of the solid phase surface is given by
239 SG -- (5" LG COS 01 -k- (3"SG COS 0 2 ,
while for the perpendicular plane one can write" CYLGsin01 - CySOsin0 2 . The contact angles, 0~ and 02, can be measured by freezing the liquid at the section perpendicular to the line of wetting. If the surface tension of liquid, o m, is known, one can simultaneously determine both the surface tension of the solid phase, OsG, and the interfacial tension, OLGby measuring 0~ and 02. This method is referred to as the "neutral drop" method ULG G p
o~
L'~/U/,/,///,/,/,~
E!
"
_
s (JSL
Fig. III-25. A drop of liquid placed on a solid surface that is soluble in that liquid
Similar changes in the profile of the solid surface upon contact with a liquid may also occur at the grain boundaries emerged at the surface. Dissolution of the material of the grain results in the formation of a groove along the grain boundary. For grains of identical composition, oriented symmetrically toward the boundary between them, the equilibrium at the grove vertex is given by q) O'GB C O S - - --
2
2CYsL
where q0 is the so-called two-sided angle (Fig. III-26)
240
\ \ \
Fig. III-26. The equilibrium two-sided angle, % at the groove between two neighboring grains
If the interfacial free energy at a solid-liquid interface is low, while the grain boundary energy is relatively high, the so-called Gibbs-Smith condition, %B >--2CYSL,may be fulfilled. Under this condition the penetration of the liquid phase between grains along their boundaries, resulting in the formation of a thin liquid interlayer (Fig. III-27), is thermodynamically favorable. .o
9
Fig. III-27. Formation of a liquid interlayer between grains in polycrystalline substance when the Gibbs-Smith condition is fulfilled
Such a penetration of the liquid phase along grain boundaries was observed in a number of polycrystalline systems, such as Zn-Ga, Cu-Bi, NaC1-H20. One has reason to assume that a number of geological processes, such as the transport of substances in the Earth's crust and the formation of ore and mineral deposits, are related to this phenomenon. Let us now get back to the issue of the behavior of a 1i q u i d p 1ac e d on the s u r f a c e of a n o t h e r liquid. First ofall, letus emphasize that in the
241 case where Antonow's rule is valid, the work of spreading, Ws, is zero, and hence both angles, 0~ and 02, are zero as well. Consequently, non-zero contact angles may exist only for those liquids which do not follow Antonow's rule [28]. For example, in the CS2 - H 2 0 system the work of spreading is negative" Ws - GL1G- GL2G-
~L1L
2 --
72.5 - 31.5 -48 - -7 mJ/m 2, and thus the
angle (0~ + 02) is non-zero. A special case of non-zero work of spreading may exist in systems that follow Antonow's rule but in which the substance of one of the phases significantly decreases the surface tension of another phase. Such a situation occurs when a drop ofoctyl alcohol is placed on the surface of pure water. Under non-equilibrium conditions, when the adsorption layer has not yet formed, spreading with substantial positive value of W~ may occur. For octyl alcohol such non-equilibrium work of spreading, Ws - 72.75 - 27.5 - 8 = 37.25 mJ/m 2. After equilibrium has been established, i.e., when the phases have become mutually saturated and the adsorption layer of octyl alcohol at the water surface has formed, the work of spreading approaches zero, in agreement with Antonow's rule.
Let us now briefly address the basic rules governing the k i n e t i c s o f w e tt i n g a n d s p r e a d i n g , focusing on the change in the radius ofthe wetted surface, rl, as a function time. If the volume of a drop placed on the solid surface, V, is constant, the increase in its radius results in a decrease in the average thickness of the drop, h
=
V/ltrl 2. As mentioned earlier,
the force per unit of perimeter length, equal to Ws, acts during the spreading of a drop. This force in mostly balanced by the viscous resistance of the spreading liquid phase. The studies by Yu.V. Goryunov, N.V. Pertsov, B.D. Summ, and E.D. Shchukin [29] indicate that this process is in general similar to the decrease in thickness in symmetric flat films (see Chapter VII, 3); the thickness of the fluid layer decreases with time approximately as to consequently the radius of the drop is proportional to
t ~/4.
t -1/z ,
and
Indeed, the r~ ~ t 1/4 dependence in
242 many cases accurately describes the kinetics of spreading during the most lengthy stages. As an example, Fig. III-28 shows the results for the spreading of mercury drops on surfaces of zinc and lead. The data are plotted in log-log coordinates and clearly show an approximate t 0.27dependence. log[Q, (ram)] Hg- Zn
1.0
Hg - Pb
0.5
o i
. 2
,t_ r,-t"
Fig. III-28. The kinetics of spreading of a mercury drop on surfaces of zinc and lead [29] If wetting of a solid surface by liquid is a restricted, it is usually assumed that the surface of a sufficiently small droplet maintains its spherical shape, and that the change in the area of contact between the liquid and a solid surface with time occurs due to the gradual decrease of the advancing contact angle, 0a = 0a (t) to its equilibrium value. The driving force of this process per unit perimeter length is given by
mwet --O'SG --~LG --O'LG cOSOa (t)-- (YLG[COSO0 --cOSOa ( t ) ] ,
where 00 is the equilibrium contact angle. The kinetics of changes in the contact angle, radius, and area of wetted surface are dependent on the viscosity of the liquid, droplet size, and the extent to which the contact angle approaches its equilibrium value; at initial stages it may also be dependent on the inertial forces. The method of molecular dynamics (the numerical experiment of dynamic type) allows one to examine on a molecular level various peculiarities related to wetting and
243 spreading, such as fluctuations (Mandel'shtam waves), a conditionality of the notion of contact angle at every given moment of time, a variety of wetting and spreading mechanisms and their relative probability, an exclusive possibility of establishing down to which nanoscale one can still use macroscopic parameters (including the surface free energy), etc. Figure III-29 illustrates the results of the very first (1976) molecular dynamic two-dimensional ''
t = to
"
~
.::'. ::..-. I oie
ooeeooeloooo0
I
ooo
~ O ~ f'l f'l t"t O f 3 t 3 ~ ~3 Cl f'l t"l s
. . . .
t=
t~ q~e ~ eo
0 o Q~oOo
114l O00oo
llql
e ~
9 _e
9
. . . . . .
9
9
O~
'~ 0 ~
.
.
.
0
t=~
0
o I 9
J
|
eqJO oo'o'o
..
.:.....
Oee ~ iv
J
t=~
oo6oe~
~
oi'w o IQQO
OOOO~o,qloO
a
O
j
"2o'2~gg~ .
O00Q
b
oeJ
9 9 IoOQ
O OoOQ 9 OeQO 9
9e oqDe
oeee
9
...--
c
9
..oo.o"
0 ~.
~ Ot
t,-! t.,,,lt,.,1t,..l t,-i l~ ~ n [~1 t.,~lt...,lcl t31x~zl
9
~..o.:.:o"~I d
Fig. III-29. The molecular dynamic simulation of the behavior of a droplet of liquid placed on a solid support; a - spreading, b - gathering of liquid into a droplet in the case of nonwetting, c - fluctuations of a droplet under the condition of non-wetting, d - effect of a surfactant on wetting process [29] observation of the behavior of an extremely small droplet (consisting of only nineteen molecules) placed on a solid support [29]. This Figure shows the arrangement of molecules at different times, t=0, t~, t2, and t3 with properly selected parameters of van der Waals interaction inside the droplet, the solid support and between the two. Under the condition of perfect wetting (when the work of adhesion equals work of cohesion in a liquid phase), the droplet spreads over the solid surface (Fig. III-29, a). Upon the transition to non-wetting (the work of adhesion equals half the work of cohesion), the liquid molecules gather back into
244 a droplet, which then undergoes fluctuations (Fig. III-29, b and c, respectively). When the surface active component is introduced into a droplet, an improvement in wetting and dispersion is observed (Fig. III-29, d).
III.5. Controlling Wetting and Selective Wetting by Surfactants
The lowering of the interfacial tension due to surfactant adsorption allows one to finely control the wetting of solid surfaces by liquids. Let us now demonstrate how such controlled wetting is done by discussing the most important and typical use of organic surfactants for the hydrophilization and hydrophobization of surfaces. There are two major means by which one can introduce surfactants into the system in order to gain control over the properties of solids. First, direct addition of surfactant into the liquid (or one of the liquids) which is then brought in contact with the solid surface. Second, prior firm immobilization of the surfactant adsorption layer at the solid surface. Such a layer modifies the solid surface. Below we will discuss each of these processes in some detail. When s u r f a c t a n t is i n t r o d u c e d into a liquid phase brought into contact with a solid surface, the adsorption takes place at both the airliquid and the solid-liquid interfaces. A slower process of surfactant migration along the solid surface may occur as well. In agreement with Young's equation, the decrease in surface tension of a liquid due to the adsorption of a surfactant causes the value of (CysG- GSL)/CYLG= COS 0 to increase (Fig. III-30, curves 1, 1' ), thus making wetting better only at the surface that is "philic" with respect to the applied liquid. Consequently, the substances adsorbed at the air-liquid interface act as weak wetting agents, and are most commonly
245 used for the purpose of improving wetting of polar surfaces with water. Based on their mechanism of action, these substances belong to surfactants of the first group in Rehbinder's classification (see Chapter II,3). I~
- ~
1
OLG
/
spreading
1 cosO I
G
~
f
c
Fig. III-30. The isotherms of hydrophilization (curves 1, 1 ", and 2 ) and hydrophobization (curves 3 and 4) of solid surfaces by surfactant solutions
The interfacial properties change more dramatically if surfactants of the three other groups, capable of adsorption at a solid-liquid interface, are introduced. In the case where physical adsorption ofsurfactants, corresponding to the polarity equalization rule, takes place, wetting improves greatly, up to a transition from complete non-wetting to wetting and even spreading (Fig. III30, curve 2). The surfactant concentration at which the transition from nonwetting to wetting (cos 0 = 0) takes place is referred to as the wetting inversion
point, ci. Water can wet the surfaces ofhydrophobic materials in the presence of various surfactants that are able to adsorb at oil-water interface. The hydrophobization of polar surfaces by chemisorbing water soluble surfactants is of outmost practical importance. The use of flotation aids discussed later in this chapter is based on the specific chemisorption of
246 surfactants at the surfaces of the mineral. A significant feature of surfactant chemisorption is its irreversibility. Another aspect of interactions between a chemisorbing surfactant and a solid surface that has to be accounted for is the possibility of so-called "over-oiling", which may occur at high surfactant concentrations (see Chapter III,2). If "over-oiling" takes place, the wetting isotherm contains two inversion points (Fig. III-30, curve 3): as the surfactant concentration increases, the surface first becomes hydrophobic, and then hydrophilic again due to the formation of a surfactant bilayer. To control p r e f e r e n t i a l w e t t i n g , one can introduce surfactants into both aqueous (water soluble surfactants) and oil (oil-soluble surfactants) phases. Depending on the nature of the surfactant, either hydrophilization or hydrophobization of the surface may take place. As shown in Fig. III-30 (curve 4), oil-soluble surfactants are only capable of oleophilization due to their physical adsorption and chemisorption at a polar surface (it is worth reminding here that in the case of selective wetting the contact angle is measured in the more polar phase, i.e., in water). In the case of selective wetting of a hydrophobic surface, oil-soluble surfactants are able to adsorb only at the water-oil interface; their adsorption results in the contact angle increasing. The hydrophobization of the surface may result in ((3"SL2 - (~SL1)/(YL1L2 being less than -1, which corresponds to the case of the spreading of oil along the solid/water interface, and complete displacement of the aqueous phase away from the surface. Similarly, if complete hydrophilization of the solid surface takes place, the aqueous phase totally displaces the oil phase. Contact angle hysteresis is typical in all cases when surfactants are directly introduced into a liquid phase that is brought into contact with a solid
247 surface. Decrease in the contact angle (or increase in the case of chemisorption) occurs gradually, when adsorption occurs on surfaces as they are covered by a liquid phase. The role of adsorption kinetics and the diffusion of surfactants is especially important in controlling capillary impregnation. According to studies by N.N. Churaev, the solution impregnating the capillary quickly loses its dissolved surfactant due to adsorption of the latter on capillary walls, so the rate of impregnation may be limited by the diffusional transport of surfactant from the bulk of the solution to the menisci in the pores. M o d i f i c a t i o n of the solid phase by prior attachment of a surfactant adsorption layer may be carried out in different ways, depending on the nature of the surfactant and its interaction with the solid surface. For instance, one can dissolve surfactant in a liquid with polarity opposite to that of the surface, and surfactant adsorption will take place at the interface, in agreement with the polarity equalization rule. Such a method allows one to hydrophilize non-polar surfaces by treating them with aqueous surfactant solutions, or oppositely, to hydrophobize surfaces by adsorbing surfactants dissolved in a hydrocarbon. A more convenient way of surface hydrophobization is, however, for one to chemisorb appropriate water soluble surfactants, and thus avoid dealing with flammable and toxic hydrocarbon solvents. The use of chemisorbing surfactants is advantageous, since it allows one to achieve firm binding of the adsorption layer to the solid surface. It is, however, important to avoid "over-oiling" the latter. Firm binding of the adsorption layer can also be achieved by utilizing irreversibly adsorbing polymeric surfactants.
248 This use of surfactants as controlling agents for wetting, selective wetting and surface modification has numerous practical applications, some of which are described below. The broad application of surfactants in the textile industry is primarily related to their use as so-called "textile aides", applied at different stages during the processing of natural and synthetic fibers. These stages include "oiling" of the crude yarn (hydrophobization of fibers in order to prevent their damage and decrease their adhesion) and fabric "softening" (the adsorption modification of fabrics). Surfactants are also used to aid fabric dyeing and printing on fabric surface, as well as during special treatment of fabrics, such as in applying antistatic and hydrophobizing "water repelling" coatings. Surface modification is broadly used in controlling the surface properties of fillers in rubber, synthetic polymers and other materials (see Chapter IX). The surfactant adsorption layers that make surfaces hydrophobic are used to prevent caking in hygroscopic powders (fertilizers), as anticorrosive agents and in numerous other processes. Surfactants aid in the application of herbicides to plants when they are present as wetting agents in water-based formulations and emulsions. The leaf surface is hydrophobic, and in order to ensure herbicide adhesion one has to make it hydrophilic, which is achieved by surfactant addition. The use of surfactants allows one to increase the effectiveness of fire extinguishing formulations, which is especially important in fighting peat fires, since the surface of dried peat is hydrophobic, and water does not penetrate it in the absence of a surfactant. Aqueous solutions of wetting agents are also used in
mines to
249 minimize the formation of coal dust. Hydrophilization of surfaces is also of importance in applying light-sensitive layers onto polymer tape in the manufacturing of photographic materials. Controlling selective wetting is of outmost importance in various industrial processes and applications, among which o i 1 r e c o v e ry is probably one of the most important. Crude oil is a multicomponent complex mixture of substances containing natural high molecular weight surfactants. Oil is stored within the Earth' s crust in hydrophilic rocks, together with heavily mineralized water. The surface active substances present in oil due to their adsorption at the rock surface cause oleophilization of the latter, and thus allow oil to selectively wet it. It is noteworthy that hydrophobized regions of rocks are distributed between hydrophilic regions in contact with water. After the oil well has been drilled it is usually necessary to ease the access of the oil to it, i.e. to "open the collector". The latter can be achieved by pumping solutions of chemisorbing surfactants into the well. Such solutions convert hydrophilic regions of rocks into the hydrophobic ones, and thus make oil transport through the cracks and capillaries of the crust to the bottom of the borehole easier. During further use of the oil well it is important for one to be able to extract from the bed as much oil as possible. Typically, no more than 50-70% of the oil can be recovered, even from the best performing wells, and in most cases the recovered amount is 30-40% or less. To enhance the degree of oil recovery, special complex formulations referred to as surfactant micellar solutions (see Chapter VI,4 ) are introduced into the satellite wells drilled around the operating hole. These additives improve selective wetting by water, and thus force the oil to move towards the
250 operating hole, from which it is then recovered. Detergency, i.e. the process of removing non-polar contaminants from polar surfaces (see Chapter VIII, 6 ), is based on controlling selective wetting. Another example of the utilization of selective wetting monitoring is in offset printing. This process is based on the difference in the wetting of the printing elements and the gaps between them by the ink. The printing elements are hydrophobized when brought in contact with the image-carrying matrix, while the gaps between the printing elements remain hydrophilic. Hydrophobic ink selectively wets the printing elements without adhering to the blank area between them, and the image formed thereby is subsequently transferred to the paper. The ability of surfactants to make surfaces hydrophobic through chemisorption is utilized in the manufacturing of pigments for oil-based paints. The synthesis of finely dispersed pigments is carried out in aqueous solution, to which chemisorbing surfactants are subsequently added. These surfactants make the pigment surface hydrophobic, so these pigments can be transferred into the oil phase.
111.6. Flotation
The flotational separation and enrichment of minerals is one of the most broadly used technological applications that utilizes the control of wetting. Flotation is usually classified as foam (froth), oil, or film flotation, and is based on the difference in wetting of the valuable (flotated) mineral that is to be extracted or concentrated, and the gangue (a barren rock). In froth and
251 film flotation the valuable mineral, poorly wetted by water, accumulates at the air-water interface, while the well wetted gangue is transferred into the water phase. In froth flotation crushed ore is intensively mixed with water through which air is bubbled. Rather small particles of the valuable mineral are carried by the air bubbles to the water surface, where they accumulate as froth. The froth is then mechanically removed for further processing. In the case of film flotation the particles of crushed ore are poured out onto a surface of flowing water. The particles of flotated mineral remain on the surface, while those of the gangue sink. The process of oil flotation is used less frequently than film and froth flotation. In this process oil drops are used as carriers for the particles offlotated mineral, i.e., in this method one forms an emulsion instead of froth.
Let us now describe the principles of flotation in detail using froth flotation as an example. Let us assume that a particle of radius r is placed at the air-water interface (Fig. III31). The interface can be either flat or that of a rather large air bubble. In the absence of gravity equilibrium is reached when the angle between the flat water surface and that of the particle equals the contact angle, 0. The distance between the flat water surface and the plane I
//
H
Fig. III-31. A hydrophobic spherical particle placed at the surface of water in the absence (/) and in the presence (I/) of gravity
252 of horizontal particle diameter is given by H = r cos0. If the particle is wetted by the water (0<90~ H is positive, and more than half of the particle is immersed in the water. If water does not wet the particle (0>90~ H is negative, and the particle is immersed in the water only to an insignificant extent. The radius of the wetting circle formed by the edge of the meniscus is equal to r L = r sin 0. If the particle density is greater than that of water, gravity forces the particle to immerse into the water, and the water surface sags, forming an angle ~ with the horizontal plane. This causes H and rL to change. Assuming that the contact angle remains constant (no contact angle hysteresis), the new values,/4* and r* (position II), are given as H*
-
r c o s (0 - Z) ,, rL* - r sin (13 - ;~)
,
Since both expressions contain the angle ~, it is convenient for one to use it further as a variable, replacing/4*. The flotation force, F, is given by F - 2rtr L cyI~G sin Z - 2rtr sin (0 - Z) sin Z cy L~ 9 The maximum value of the flotation force can be found by taking a derivative of above expression and setting it equal to zero: dF
= 2rtPCyLG [ s i n ( 0 -
Z)cosz-
sinzcos(0-
Z)]-
dz = 2rtrcyLG s i n ( 0 - 2 Z ) - 0 . This relationship apparently holds when X= 0/2, and thus the maximum particle weight force, Fmax, with the buoyancy force accounted for is given by
Fma x - 27~rcrLC; sin 2 (0 / 2 ) . The above equation shows that flotation is possible at any non-zero contact angle, but the smaller the angle, the smaller this force is. Flotation is a complex physico-chemical process, in which other factors, besides wetting, may play a significant role. These factors include hysteresis phenomena and
253 interactions between particles and bubbles as they approach each other at close distances upon the collapse of the dispersion medium films that separate them. For flotation to be possible, the flotation force must exceed the combined gravity and buoyancy force of the particles. This is achieved by milling minerals down to the right particle size. The particles of flotated mineral and gangue that are to be separated differ in contact angles at the solid-liquid interface, but are usually of similar sizes. The gangue particles undergo flotation and interfere with the separation process despite the small values of contact angle, 0, if their size is significantly smaller than that of flotated mineral particles. The effective enrichment of ores and minerals by flotation, including the possibility of separating materials with similar chemical properties, is achieved by employing a variety of different surfactants which selectively hydrophobize the surface of minerals that undergo flotation, and hydrophilize the surface of those that do not (or the other way around). Due to the relatively small surface area of the minerals treated, the consumption of surfactantbased flotation aids can be as low as 100 grams per tonne, which allows one to use rather complex and expensive surfactants in fine tuning the surface properties of the minerals separated. Commercial flotation aids can be subdivided into the following groups, based on the mechanism of their action [30] : 1. C o 11e c t o r s are surfactants that typically chemisorb at the surface of the flotated materials and make the latter hydrophobic. A clear chemical specificity ofchemisorption allows one to carry out selective flotation, which is especially effective in stage separation of multicomponent minerals and
254 ores. Anionic surfactants are used as collectors for the flotational separation of minerals that are basic in nature, while cationic ones are used primarily for the separation of silicates. In the separation and enrichment of polymetallic sulfide ores the anionic surfactants containing thiol, -SH, thion, =S, and a combination of these two functional groups serve as effective collectors.
X a n t h a t e s , ~ o c ~~s__js] Me
aeroflots),
F(RO)2~ s P~
L
S
+
, and t h i o p h o s p h a t e s (the so-called
- Me+ ,capable of forming slightly soluble
compounds with di- and polyvalent metals belong to the class of collectors. As an example of the fine use of collectors, one can point out at the separation of minerals with very close properties, such as sylvite (KC1) and halite (NaC1) by flotation in saturated solutions using long chain amines or other surfactants as collectors. 2. A c t i v a t o r s enhance the selectivity action of collectors. These (often surface inactive) substances are primarily the electrolytes, which modify the particle surface, e.g. by ion exchange, and make the adsorption (chemisorption) of collectors easier. One example of an activator is sodium sulfide used to aid the flotation of oxidized non-ferrous minerals. Substances that control the solution pH, and thus create the desired ionization state of the surface of mineral particles also belong to the class of activators. 3. Depressants are surfactants or inorganic electrolytes that improve wetting by water of minerals whose flotation is undesirable. Typical examples
255 of depressants include starch, dextrin, carboxymethylcellulose, and other wetting agents. A solution of sodium silicate ("liquid glass") is often used to prevent flotation of barren silicate rock. 4. Weak foaming (frothing) agents ensure the formation of a moderately stable froth that is capable of holding mineral particles in it (sometimes the collectors themselves can act as such agents). In order to make further processing easier, the froth layer should contain as little water as possible. Nowadays most ores undergo treatment by flotation. As rich ore deposits become exhausted, the role of flotational enrichment becomes more and more significant. Flotation is used even with such relatively cheap mineral fossils as coal and sulfur.
References 0
~
~
0
0
Q
Shchukin, E.D., Pertsov, A.V., Amelina, E.A., Colloid Chemistry, 2 nd ed, Vysshaya Shkola, Moscow, 1992 (in Russian) Rehbinder, P.A., "Selected Works", vol.1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Abramzon, A.A., Surface Active Substances: Properties and Application, Khimiya Publ., Leningrad, 1981 (in Russian) J6nsson, B., Lindman, B., Holmberg, K., and Kronberg, B., Surfactants and Polymers in Aqueous Solution, Wiley, Chichester, 1998 Hough, D.B., and Rendall, M.H., in "Adsorption from Solutions at the Solid / Liquid Interface", C.D. Parfitt and C.H. Rochester (Editors), Academic Press, London, 1983 Chander, S., Fuerstenau, D.W., and Stigter, D., in "Adsorption from Solutions", R.H. Ottewil, C.H. Rochester, and A.L. Smith (Editors), Academic Press, London, 1983
256 o
111
,
10. 11. 12. 13. 14. 15. 16.
17. 18.
19. 20. 21. 22. 23. 24. 25.
26.
Rosen, M.J., Surfactants and Interfacial Phenomena, 2nd ed., Wiley, New York, 1989 Dobifig, B., in "Surfactant Science Series", vol. 47, B. Dobifig (Editor), Dekker, New York, 1993 Koopal, L.K., in "Surfactant Science Series", vol. 47, B. Dobifi~ (Editor), Dekker, New York, 1993 Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953 Hunter, R.J., Zeta potential in Colloid Science" Principles and Applications, Academic Press, London, 1981 Scheutjens, J.M.H.M., and Fleer, G.J., J. Phys. Chem. 83 (1979) 1619 Scheutjens, J.M.H.M., and Fleer, G.J., J. Phys. Chem. 84 (1980)178 Van Lent, B., and Scheutjens, J.M.H.M., Macromolecules 22 (1989)1931 Kipling, J.J., Adsorption from Solutions of Non-Electrolytes, Academic Press, New York, 1965 Jeffery, G.H., Bassett, J., Mendham, J., Denney, R.C., Vogel's Txtbook of Quantitative Chemical Analysis, 5th ed., Longman Group, Essex, UK, 1989 Usui, S., in "Surfactant Science Series", vol.15, A. Kitahara and A.Watanabe (Editors), Dekker, New York, 1984 Lyklema, J., Fundamentals of Interface and Colloid Science, vol.2, Academic Press, London, 1995 Helfferich, F., Ion Exchange, McGraw Hill, New York, 1962 Harland, C.E., Ion Exchange Theory and Practice, 2n~ ed, RSC, Cambridge, 1994 Dorfner, K., Ion Exchangers, Properties and Applications, Ann Arbor Science Publ., Ann Arbor, 1972 Kelley, W.P., Cation Exchange in Soils, Reinhold, New York, 1948 The NALCO Water Handbook, 2no ed., F.N.Kemmer (Editor), McGraw Hill, New York, 1988 Amphlett, C.B., Inorganic Ion Exchangers, Elsevier, Amsterdam, 1964 Miller, C.A., Neogi, P., Interfacial Phenomena: Equilibrium and Dynamic Effects, in "Surfactant Science Series", vol. 17, Dekker, New York, 1985 Neumann, A.W., in "Wetting, Spreading and Adhesion", J.F. Padday (Editor), Academic Press, London, 1978
257 27. 28. 29. 30.
Good, R.J., in "Surface and Colloid Science", vol. 11, R.J. Good and R.R. Stromberg (Editors), Plenum Press, New York, 1979 Johnson, Jr., R.E., Dettre, R.H., in "Surface and Colloid Science", vol.2, E.Matijevid (Editor), Wiley-Interscience, New York, 1969 Summ, B.D., Yushchenko, V.S., Shchukin, E.D., Colloids Surf., 27 (1987) 43 Hornsby, D., and Leja, J., in "Surface and Colloid Science", vol. 12, E.Matijevid (Editor), Plenum Press, New York, 1982
List of Symbols
Roman symbols A A*
Hamaker constant complex Hamaker constant a activity of ions B,B" adsorption coefficient integral capacitance of Stem-Helmholtz layer capacitance per unit area C concentration C(s) average concentration within the surface layer surfactant concentration in oil phase Coil surfactant concentration in aqueous phase Cw d distance, also particle diameter r elementary charge F flotation force G surface activity specific heat of wetting by oil Ho. specific heat of wetting by water /-/w h small distance K partition coefficient k Boltzmann constant ion exchange coefficient kl2 coefficient of roughness kr m mass Avogadro's number NA /7 number of molecules (atoms) per unit volume
258 number of molecules (atoms) per unit area charge universal gas constant surface area specific surface area ideal surface area ideal real surface area real area occupied by one surfactant molecule Sl absolute temperature T L lower critical temperature re upper critical temperature :rc average time ta /// interaction energy energy of mixing H0 volume V Vm molar volume in the bulk Vm(s) molar volume in the surface layer V/a work of adhesion work of cohesion work of ion transfer work of charging Ws work of spreading mole fraction x, y, z Cartesian coordinates Z coordination number J~s
q R S S1
Greek symbols ~m
]3 F F* 5 q 0 0 0a
polarizability Hw / Hoil ratio adsorption adsorption per unit weight small distance electrokinetic potential excess of entropy contact angle surface coverage advancing contact angle
259 0app
0~ K
g g 71;
Ps Pv
(~LG GSG (~SL
O Z g ~o
Aq~ 8~
apparent contact angle receding contact angle inverse thickness of electrical double layer (Debye-Hackel parameter) chemical potential electrochemical potential 3.14159... two-dimensional pressure surface charge density volume charge density specific surface free energy, surface tension dispersion component of specific surface free energy non-dispersion component of specific surface free energy specific surface energy at liquid/gas interface specific surface energy at solid/gas interface specific surface energy at solid/liquid interface adsorption potential the angle at the side wall of the groove Flory-Huggins interaction parameter dielectric constant electrical constant electric potential potential difference energy of the perimeter of wetting
260 IV. THE F O R M A T I O N OF DISPERSE SYSTEMS
Disperse systems occupy an
intermediate position between
macroscopic heterogeneous systems and molecular solutions (homogeneous systems), and thus can be prepared via two main paths: by d i s p e r s i n g t h e m a c r o s c o p i c phases (dispersion path), or by c o n d e n s a t i o n , either in true s o l u t i o n s or in h o m o g e n e o u s single c o m p o n e n t s y s t e m s (condensation path). In most cases the formation of disperse systems requires work. This work can either be introduced from the outside, e.g. in the form of mechanical work, or be generated within the system through various processes, including chemical reactions. The resulting disperse systems are thermodynamically in disequilibrium, and their prolonged existence is impossible without additional stabilization. In the absence of the latter the system is unstable and a particular particle size or particle size distribution can not be maintained, i.e.,
the
particles spontaneously coarsen and finally the disperse system collapses. As a result, the system may even become separated into macroscopic phases. These systems in thermodynamic disequilibrium are referred to as lyophobic. A number of disperse systems may form as a result of the spontaneous (i.e. without applying any external work) dispersion of a macroscopic phase. These systems, referred to as lyophilic, are thermodynamically in equilibrium and do not require any additional stabilization. This chapter is primarily devoted to the formation of lyophobic disperse systems. It is normally assumed that these systems have been stabilized in some way. Throughout this chapter, along with the discussion of
261 the basics of the thermodynamics of disperse systems, we will devote most of our attention to the condensational formation of lyophobic colloids as a result of nucleation taking place in metastable systems. We will leave the discussion of the basic laws governing the mechanical dispersion processes to Chapter IX, which covers the principles of physico-chemical mechanics.
IV.1. Thermodynamics of Disperse Systems" the Basics
Let us discuss the formation of a disperse system through the dispersion of a macroscopic phase. As an example, let us look at the separation of spherical particle of radius r from a macroscopic phase, A, in thermodynamic state I (Fig. IV-l). This process corresponds to a transition from thermodynamic state I to thermodynamic state II. II _
i
.
i
i|l
%h - % +
/
~x
A ~-1V
III
Wd = 4 x r - c r
i
ll|
|
B
~
~1~x 4.~r 3
web =
3v.,
(~v
- ~
)
Fig. IV-1. A diagram schematically showing the thermodynamics of the formation of a single particle of the dispersed phase
The formation of an individual particle at constant temperature and pressure requires additional work equal to a change in the Gibbs thermodynamic potential of the system, A ~i~ii. Since the state of matter and
262 the chemical composition of a macrophase A do not undergo changes in the dispersion process, the work required to form a single particle (the work of dispersion) is determined solely by the energy required to form a new surface with an area S=4~r 2. Using the definition of surface tension, c~, this work, Wd, can be expressed as (IV. 1)
Wd - /~ ~ - ~ II - 4 7 z r 2 CY.
For non-spherical particles one can write this equation in a more generalized way, i.e." (IV.2)
W d - A ~:~_>Ii - o t d 2 c y ,
where d is the linear dimension of particles (diameter, edge length of a cube, etc); a is the particle shape coefficient (for spherical particles a = ~). In agreement with eq. (I. 13), the chemical potential of the substance in the particle (state II), g~, is greater than the chemical potential in the initial stable phase (phase I), ~tv, by the amount
d(cyS)
cyd(4rtr 2 )
2(yVm
where N is the number of moles and Vm is the molar volume of the substance forming the spherical particle. The same relationship can be obtained by multiplying the capillary pressure, Ap=2~/r, by the molar volume, Vm, assuming that the substance is incompressible. Let us now address the basic thermodynamics of the formation of
263 disperse systems under conditions when the latter is followed by a change in the aggregate state or chemical composition of the dispersed phase in comparison with initial macrophase, i.e., the process is accompanied by a phase transition, such as the formation of drops of liquid inside a vapor or another liquid, the formation of crystals in a melt or in solution, etc. For simplicity let us examine the formation of a single-component particle from some macroscopic (not necessarily single-component) phase B (state III) with the same pressure. The chemical potential of the substance in phase B, gx, is different from the chemical potential of the substance in the particle, [.tr. This potential change is consistent with the transition of the system from state III to state II. Since macroscopic phases A and B, corresponding to states I and III, respectively, differ either in their aggregate state or in chemical composition, the chemical potentials of the substance in these phases are not the same, i.e., g~ ~ g v ~ ~b. On the one hand, the work of formation of a particle in a process related to changes in the phase state is given by the change in potential, ~', upon transition from state III to state II, Wph= A ~ . i *.. On the other hand, one can also represent this quantity as the summation of changes in the potential, ~ , during the transition from state III to state I, followed by the transition from state I to state II, i.e.,
A~III_,I I "- A~III__,I-1-A,~I_.I
I.
Here
A ~iII ~i corresponds to the "chemical work" that is needed in order to transfer N moles of substance from phase B (state III) into stable macroscopic phase A (state I),
Wch" 47rr 3 ~ c h -- / ~ I I ~
I =
3G
264 The quantity A ~ -H = Wd is defined by eq. (IV. 1). In general the expression for the work of formation of a particle during the phase transition can be written as
Wph -
W d -t- Wch - A ~ I I ~
II =
47tr 3 4rtr2cY + ~ ( ~ t v
3Vm
- ~tx)
(IV.3)
Consequently, the work of particle formation from a macroscopic phase in a different aggregate state, Wph,differs from work of formation of particles from a macrophase in the same state of aggregation and composition, Wd, by the amount of the "chemical" work, Wch,which is a function of the difference in substance chemical potentials in macroscopic phases A and B, and is related to the energy of phase transition. Of special importance, especially in the description of the nucleation, is the case of particle formation under a condition of equilibrium between states Ill and II. The equilibrium between states III and II implies that the corresponding chemical potentials are equal, namely lt-tx = g r
"
In this case the eq.(IV.3) for the work of particle formation can be written as 4rtr 3 3Vm
Wph - 4rtr2cy + ~ ( l a v
- gr)-
4rtr2cr
4rtr3 2or 1 orS. (IV.4) =3 r 3
This equation was originally derived by Gibbs. Thus, at the equilibrium between the particle and mother liquor, the work of particle formation is 1/3 of the work of formation ofthe particle surface:
Wph--1/3Wd .The 2/3
Wd are
compensated for by the "chemical" work related to the gain in energy due to
265 phase transition. Taking this into account one can write the general expression for
mph(eq. (IV.3)) also
as 1
4z~r 3
Wph - - ~ (yS - + - ~ ( ~ t
3Vm
r - [.tx);
(IV.N)
When gx = g~, the above equation becomes identical to eq. (IV.4), while when g~ = g v it turns into eq. (IV.I). Above we have presented the thermodynamic equations that describe a "single act" in the formation of disperse systems: the s e p a r a t i o n of a s i n g l e p a r t i c l e . In order to make a transition from the description of the thermodynamics ofthe formation of a single particle to the analysis of changes in the free energy upon the formation of an e nt i r e d i s p e r s e s y s t e m one has to account for the number of particles present in the system. To accomplish this, we need to introduce a factor, corresponding to the number of particles formed in a given volume, N~, into the expression for the work of formation. A more thorough analysis reveals that one also has to account for the participation of particles in thermal motion, i.e., for the entropy of the colloidal system. The participation of colloidal particles in thermal motion (the entropic factor) was taken into consideration, mostly indirectly, in earlier studies dealing with the molecular-kinetic properties of disperse systems. Volmer was the first to realize the importance of the role that the thermal motion of colloidal particles played in controlling the formation and stabilization of disperse systems. However, the attempt to compare the work of surface formation and the entropic factor directly, undertaken by March, was not successful, since it was applied only to systems with high interfacial energy.
266 Volmer's concepts were generalized in works by Rehbinder and Shchukin ~ [1,2], who gave a general and consistent description of the role of the entropic factor in the stabilization of disperse systems. These works established guidelines for further studies in the area. Following the logic used by Rehbinder and Shchukin, let us make an estimation of the increase in entropy during the formation of a disperse system upon the dispersion of a stable macroscopic phase. Let us treat the disperse system containing ~/'~ particles (or N~ - ~4r~/N A moles of particles) in N2 moles of solvent as an ideal or regular solution 2 . The entropy increase due to the formation of a disperse system can then be expressed as the increase in entropy related to mixing"
A d ' - R N 1In + k ~1 In
N1 + N 2 N1
+ N2
In
N1 + N21 N2
+
N1 + N 2 N1 + N2.1 N1 + N 2 N A In N2 .
In a real colloidal system the number of dispersed particles is much smaller than the number of solvent molecules, i.e., N1/N2 <<1. For a system in which the concentration of dispersed particles (with a radius of, say, ~10 -8 m) is ~0.1% by volume, the N1/N2 ratio is ~ 10 -8. This allows one to write an approximate expression for the formation of the colloidal system"
~The symbols used in the original paper were changed so to be in line with the text
267
k l(lnN2 + N1 or (IV.6) where -
+
N1
~ln
(IV.7)
~15to30.
N1
If the composition and aggregate state remain unchanged throughout the dispersion process, the total change in free energy upon the formation of a system with ~4/'~ identical particles can be written as
A , ~ - - J4~147~r2o- TAS,
(iv.s)
A J - ~ (4~:r2c~ - 13kT)M~l.
(IV.9)
or approximately as
When the phase composition is altered, the change in free energy is given by [3, 4]" -
la i4-3~ r 2o r - 13kT+ ~43V(r3 m
1
r - g~) J ~ l ,
where gr and g~ are the chemical potentials of substance in the particle and in initial macroscopic phase, respectively. If the gain in the entropy compensates by an increase in the free energy
268 related to the development of a new surface, so that A 5C<0, the formation of a colloidal system becomes thermodynamically favorable, and may occur spontaneously. At constant phase composition the dispersion process is spontaneous if 4=r 2o < ]3kT. According to Rehbinder and Shchukin the above condition has a simple physical meaning: if the particles are of colloidal dimensions, and the interfacial tension is low, the spontaneous separation of particles from a macroscopic phase becomes possible, since the work required to form a new surface is compensated by the gain in energy upon the involvement of the newly formed particles in thermal motion. There is a critical value of the specific surface free energy, %, below which the spontaneous dispersion process becomes possible. This critical value is given by
CYcr
13kT 13 kT kT t 4rtr 2 cz d 2 - 13 d 2 '
(IV.10)
where a is the particle shape coefficient, see eq. (IV.2); d is the particle linear dimension, and 13'=13/a. The value of % is thus a function of the size and concentration of dispersed particles, and at room temperature for particles with a radius o f ~ l 0 -8 m it is on the order of tenths and hundredths of mJ/m -2. The disperse system is thermodynamically stable if it forms spontaneously from the macroscopic phase at o < % and does not show any tendency towards further fractionation into individual molecules. Rehbinder suggested calling such systems lyophilic colloidal systems.
In contrast,
269
lyophobic disperse systems, in which the interracial energy exceeds (usually by several orders of magnitude) the critical value, ~cr, are thermodynamically unstable with respect to separation into individual macroscopic phases, and can not be formed by spontaneous dispersion. Along with typical lyophobic and lyophilic systems, there are also various systems of intermediate nature. Depending on the degree of similarity between the dispersed and continuous phases, and on the size and concentration of the dispersed particles, thermal motion may play different roles in such systems. The equilibrium number of particles of radius r per unit volume of a lyophilic colloidal system, n~ (r), is given by the expression n l ( r ) _ exp(4~r2cY-)
(IV. 11)
and may be viewed as the "colloidal solubility" of the dispersed phase as individual particles of radius r. The total colloidal solubility can be determined by carrying out the summation in eq. (IV. 11) over the particles of various sizes. Since the colloidal solubility is an exponential function of the surface tension, its value may vary within a broad range. The concentration of particles that corresponds to a saturated colloidal solution may be quite high at small values of c~Z~cr, while it is negligible at regular values of interfacial energy, ~)~cr (for a given particle size). Based on this criterion, one can draw a borderline between lyophilic and lyophobic systems. Consequently, this approach reflects in the most general way the specifics of the colloidal dispersed state, and introduces into the thermodynamic description of disperse systems two terms different in nature
270 but c o m p a r a b l e in order o f magnitude. These two terms are the w o r k o f dispersion and the free energy decrease due to the entropy gain resulting from the i n v o l v e m e n t o f the newly formed particles in Brownian motion.
The use of particle concentration as an independent variable makes the description of the thermodynamic properties of disperse systems more like that of true (molecular) solutions, i.e. of micro-heterogeneous and homogeneous systems. Owing to the intermediate "position" of disperse systems between typical heterogeneous systems (including macroscopic phases) and homogeneous solutions, the peculiarities associated with the dispersed state of the substance become more important as the particle size decreases. At the same time, the properties of disperse systems that resemble those of macroscopic phases become less important. The presence of a well-established interface with a particular value of surface energy attributed to it is characteristic of coarsely disperse systems. The number of molecules making up the particles in such systems is rather large, and thus one can talk of the statistical (averaged) properties of particles. At the same time, in such coarsely disperse systems one can already observe characteristic differences between the properties of individual particles and those of macroscopic phases: the chemical potential of the substance that formed the disperse phase becomes dependent on the particle size (see Chapter I, 3). As the degree of dispersion in the systems increases, i.e., when the particle size becomes comparable to the surface layer thickness, the notion of the surface energy of the system (and, therefore, ofthe meaning of~) becomes more conditional. A higher involvement of dispersed particles in thermal motion, i.e. the statistical nature of an ensemble of a large number of particles, becomes a characteristic property of such finely disperse (colloidal) systems. According to Hill [5], this transition from the statistical properties of molecules forming a single particle to those of an ensemble of particles is especially typical for the colloidal state. Such a combination of features characteristic of two-phase and one-phase systems allows one to utilize different approaches in investigating colloidal systems. On the one hand, one can treat dispersions as two-phase systems with some peculiar properties, taking into
271 account the dependence of the chemical potential of the particle substance on the degree of dispersion and the entropy of the mixing of particles with molecules in the continuous phase. On the other hand, a finely disperse system can be conditionally regarded as a one-phase colloidal solution containing large "molecules", i.e. the particles. In the latter case the surface energy of one mole of particles (6.02x 1023 particles) can be viewed as the energy of particle "dissolution". Such a junction of concepts (e.g. surface energy - heat of dissolution; disperse system - true solution, etc) in the description of the transition from macroscopic phases to coarsely and finely dispersed systems and further to true solutions clearly shows how the accumulation of q u a n t i t a t i v e changes in the system results in the appearance of its q u a l i t a t i v e l y novel states and consequently, of concepts describing such states. As outlined by Hill, ambiguity in the definition of the chemical potential of the substance that makes up the disperse phase is an important peculiarity of the disperse state. Indeed, let us consider a system of large volume containing "one mole" of particles of radius r. The excess in the chemical potential of the dispersed substance, regarded as the work of reversible isothermal transition (taken with the appropriate sign) of a mole of substance from the system into a macroscopic phase of the same composition and aggregate state, can be defined in two principally different ways. On the one hand, one can remove one molecule from each particle, thus keeping the number of particles constant but altering their size. On the other hand, it is possible to remove from the system A j g '= Vm / 4/3rW3 of particles, totally containing one mole of substance, without changing the size of the remaining particles. In the first approach, defining the change in the chemical potential, Ag,., one does not take into account the change in entropy due to the formation of the disperse system. This approach is identical to the one described in Chapter 1,3. In contrast, the second approach accounts in a substantial way for the contribution of entropy factors to Ag',.. The differentiation of eq. (IV.8), assuming ~ = const with respect to the number of moles, N1=4/3 ~r 3~ / V m ' , yields Ag r = (2cy/r)V m. Taking the derivative of the same expression with respect to the number of moles, N,, and assuming that r = const., one obtains
A~t 4=r2 3Vm ( 4=r 2 - k T l nN2NA) ~ .
(IV.12)
272
If the system contains particles of different sizes, in the above equation ~ = .~l(r), and the product N 2NA - ~ 0 yields the total number ofkinetically independent units in the system. The latter include the solvent molecules, dispersed particles of all sizes and some dissolved molecules of the dispersed phase. These two approaches for the determination of the excess chemical potential of the substance of the dispersed phase, A~t,. and A~tr', are used in the analysis of different aspects related to the equilibrium state of disperse systems. The first approach was utilized in Chapter I, 3 in the derivation of the Kelvin equation, when we examined the equilibrium between the dispersed particle and the continuous phase. The second approach accounts for the involvement of particles in thermal motion and therefore envisions both generation and disappearance of a particle as a whole, and thus allows one to describe the equilibrium between particles of different sizes. The equilibrium particle size distribution corresponds to a condition of constant chemical potential for particles of different sizes (including those of molecular dimensions), i.e.
Ag r' =
const. The expression for the equilibrium number of
particles of a given radius 2, r, can be obtained from eq. (IV.12) as
4 rrr2 ~ - - -
4
rtr
3 ~x - btv
3 ~4~1(r) = ~ 0 e x p
-
~0
exp
kT
In terms of a unit volume of the disperse system the above expression can be written as
4 7cr 2 ~ _ ~4 7~r 3 ~x - g v 3 n 1( r ) = n 0 e x p
- n o exp
-
,
( I V . 13)
kT
2 In order to determine the total equilibrium number of particles of different radii, one has to carry out the summation over all radii ranging from the size of a molecule to ~o. It is also necessary to take into account that the difference between the volumes of individual particles is not less then the volume of a single molecule
273 where nl and no are the number of dispersed particles and the number of kinetically independent units per unit volume of the system, respectively. Consequently, the equilibrium particle size distribution and particle content are described by the Boltzmann distribution of the free energy values. The value gx - gv characterizes the degree of "unsaturation" of the system with the dispersed phase, or the degree of system's metastability. When gx = g v the system is saturated, i.e. it is in equilibrium with a macroscopic phase; when g, < gv it is not saturated, and when g, > gv the system is supersaturated with respect to the macroscopic phase.
IV.2. Thermodynamic Principles of the Formation of New Phase Nuclei
The formation of a disperse system as a result of the generation and successive growth of primary particles (nuclei) of a new stable phase may take place in any metastable system. The metastability, arising as conditions shift away from "normal" equilibrium conditions, may arise from deviations in the chemical composition of the phases (supersaturation) as well as from physicochemical action (changes in the temperature and pressure).
IV.2.1. General Principles of Homogeneous Nucleation According to Gibbs and Volmer
The thermodynamics of the formation of disperse systems shows that the formation of particles whose aggregate state or chemical composition differs from that of the mother medium requires work, the value of which, for each spherical particle of radius r, is given by eq.(IV.5):
Wph - W(F) = 4z~r2c~
4 zcr3 3 vn
274 where the superscript "n" stands for the n e w phase. The above relationship shows that the work of formation of nuclei of a new phase depends on the extent to which the mother phase is metastable, i.e. it depends on the excess of the chemical potential of the substance in the initial phase, btu,over the chemical potential of the same substance in stable m a c r o s c o p i c phase, btv" bt~ - btv = -Ap. The value of -Abt>0 describes the extent by which the mother phase deviates from the stable state, i.e. the degree of penetration of the system into the metastable region. When applied to particular systems, the latter can be expressed through the corresponding thermodynamic parameters (e.g., temperature, pressure). In eq. (IV.3) the first term is positive and increases proportionally to r 2 with increasing r. The second term may be negative (in the case of supersaturation -Abt>0 ), with absolute value increasing proportionally to
r 3.
Thus, in the case of supersaturation, the W(r) curve must pass through a maximum. This maximum is characteristic to some critical particle size, rcr, corresponding to the critical nucleus of a new phase. The critical nucleus of size Gr exists in equilibrium (unstable equilibrium) with the mother medium, i.e. the bt~ - bt~ condition (bt~ is the chemical potential of the substance in the nucleus) stays valid. For such a nucleus, one can write in agreement with eq. (I. 13) that A l a - la V - ~ x - gv
-lCtr
-
2r -~V~. rCF
The position of a maximum on the W(r) curve, corresponding to the work required for the formation of a critical nucleus, Wc~,can be obtained by setting the derivative of the work with respect to the radius equal to zero, i.e
275 dW(r)/dr - 0
(the second derivative is negative:
d2W(r) / dr2
size of
the critical nucleus is thus given by
rcr =
2cyVn iAbtI .
(IV. 14)
The substitution of eq. (IV. 14) into eq. (IV.3) yields an expression similar to eq (IV. 4): mcr -
4~rZcy
4=rc3r 2cyVn 3 Vn rcr
4 2 1 3 71;Pcr(5" - o ' S cr ,
where Scr is the surface area of the critical nucleus. One can also exclude the surface tension from the expression for the work of formation of the critical nucleus, i.e 4 rtr2cy - 4 rtr 2 rcr ( - a ~n ) Wcr - --33 cr 2Vm
where
1
Vcr
2 (-a.)
vd
(IV.1 5)
Vcr is the volume of the critical nucleus (a similar relationship,
expressed in terms of pressure rather than chemical potentials, was derived by Gibbs). The above expression will be utilized later, during the discussion of heterogeneous nucleation. Finally, we can express the work of the critical nuclei formation using only the parameters associated with macroscopic phases, i.e." 16~cy3 ( v n ) 2 W =
3
2
"
(IV. 16)
276 In the absence of supersaturation (g~ = g v) the parabolic,
W(r)=4rtr2o (Fig.
IV-2);
W(r) dependence
is
re~ --, oo, and also Wcr--' oo. Upon
penetration into the metastable region ( ~ > ~tv), the maximum appears in the
W(r) curve,
i.e. Wcr and rcr both have finite values that decrease as the
supersaturation, lAst I, increases. The work of the critical nucleus formation, W,, can be thus viewed as the height of the energy barrier that one needs to overcome in order to make further spontaneous growth of nuclei possible.
WII
~-4
JT~/cr= 167t(~3 (Vmn)2
3 (Al~) 2
r
-Ala Fig. IV-2. The influence of the extent of metastability, -Alx, on the profile of the energy barrier, radius re,, and the work of the critical nucleus formation, We,
Particles with radii r < rcr are unstable and disappear, as the work,required for the formation of their surface, grows with increasing radius, r, faster than does the work associated with the change in the thermodynamic Gibbs potential, corresponding to the phase transition.
The growth of particles to sizes
exceeding rcr results in a decrease in the system's energy" the gain in energy achieved when new portions of substance undergo phase transition compensates the work associated with the particle surface area increase. The
277 larger the particle size, the greater the extent of this compensation. Such particles are thus unstable and continue to grow spontaneously; the chemical potential of the substance in the particle, gr, decreases, tending to approach the chemical potential of the substance in the macroscopic phase. In agreement with eq. (IV. 15) the work of critical nucleus formation is inversely proportional to the second power of the supersaturation, and thus a noticeable supersaturation is required for a new phase to spontaneously form in homogeneous system. The frequently observed new phase formation that occurs at low supersaturation (and even in the absence of the latter) is caused by the presence of foreign inclusions, that cause the process to follow the heterogeneous path.
The derivation of the main equations describing the thermodynamics of nucleation in homogeneous systems can be approached from a somewhat different viewpoint, following the logical scheme introduced by V.P. Skripov [6]. Let us still assume that if a very small nucleus of a new phase is formed inside the large bulk of an initial phase, the state of the substance in the mother phase remains essentially unchanged, and all changes in the system's energy are due to changes in the state of the substance within the nucleus itself. In general, for nuclei of arbitrary size (including the non-equilibrium ones) the change in the system's free energy occurs due to 1) the appearance of an interface between new and old phases with the corresponding surface energy; 2) the mechanical work related to the change in pressure inside the nucleus, as compared to that in the surrounding medium, due to the action of capillary forces, and 3) the change in the chemical potential of the substance upon its transition from an old phase into the new one that is thermodynamically more stable. Consequently, the work of formation of a spherical nucleus of radius r in a onecomponent system is equal to
278
W(r)
3 - 4 r t r 2 (Y + -4 71;r3 [ p _ p n ( r ) ] ~ 4 rtr [~t n ( r ) - ltI], 3 3v n
where p and pn are the pressure; g and gn are the chemical potential of substance in the initial and new phase, respectively. The chemical potentials, g and lan, are functions of the pressures, p and p", and g(p) and lan(p n) are determined by the corresponding state equations for these two phases. To determine the work of critical nucleus formation, Wet, one can use the mechanical equilibrium conditions:
pn(rcr)- p - Ap -
2cy ~cr
and those of chemical equilibrium: ~ n ( r c r ) -- ~.l -- 0 . Then one can write the expression for Wc~as
Wcr
-
4rt'~2Cy'cr
-
-
4
= r
3 2or cr rcr
=
-
4 rt"2Cr'cr 3
-
-
1 3
c y S
cr ,
which is identical to eq. (IV.4). Alternatively, Wet can be written as 16rtcy 3
WCF
3 [ p n (rcr)_ p]2 '
where the quantity [pn(rcr)-p] describes the penetration
into metastable region
(supersaturation), expressed, in contrast to eq. (IV. 15), in terms of pressure. The initial state is compared with the new state ofthe substance in the particle, and not with that of a stab le phase.
The t h e r m o d y n a m i c relationships (IV.14) and (IV.16) establish the functional d e p e n d e n c e o f
Wcr and rcr on the extent o f metastability in the
279 initial mother phase. It is, however, not possible to obtain numerical estimates for these quantities, based on the values of parameters known for the initial phase, since these expressions also include still unknown parameters associated with the nuclei of the newly forming phase. In order to express Wc~ and rc~ solely through the parameters of the m a c r o s c o p i c phases, one must use the corresponding equations of state. It is evident that for different types of phase transitions one should use different equations of state, depending on the structure of the new and old phases. Let us now discuss the principles of nucleation thermodynamics in relation to a number of particular examples illustrating the formation of a new phase within a metastable initial phase.
IV.2.2. Condensation of the Supersaturated Vapor In this case one can use pressure, p, as a parameter characterizing the initial metastable system. The degree of penetration into the metastable phase, -A~t, should, consequently, be expressed via the deviation in the supersaturated vapor pressure,p", from the equilibrium saturated vapor pressure existing over a flat interface, Po. Using eq. (I. 14), one can write tt
-A~t~RTln
p P0
.
The work of critical nucleus formation is then given by [7] 16rto3 (Vm) 2 ~Vcr --
3(RTln) 2
P0
280 where P"/Po - a is the vapor supersaturation, Vm' is the molar volume of liquid.
IV.2.3. Crystallization (Condensation) from Solution Analogously to the previous case, one can discuss the process of the separation of the solid or liquid phase (of molar volume Vm) out of a solution with supersaturation c~=c/co, where c and c o are the concentrations of the supersaturated and saturated solutions, respectively. In the case of the ideal solution, the expression for the work of critical nucleus formation can be written as 16no.3 Vm 2 ~Vcr =
3( Tln
9
If the solution is non-ideal, the above expression will also contain activity coefficients.
IV.2.4. Boiling and Cavitation During boiling and cavitation the nuclei of a new gaseous phase form inside a metastable liquid phase. If boiling takes place in an open vessel, the liquid evaporates into an unbound volume (into the atmosphere), and the vapor pressure over the flat liquid surface does not rise, i.e. boiling takes place at the atmospheric pressure, Patm"(In a closed vessel, in which most of the available volume is occupied by liquid, boiling is hardly possible, since the system would approach equilibrium earlier than the required supersaturation could be reached). Consequently, at boiling the pressure in the critical nucleus of radius rc~, p"(rc~), exceeds the atmospheric pressure by 2o/r~r. During cavitation the formation of so-called cavitational bubbles occurs under conditions corresponding to the stretching of the fluid when
281 the pressure in the latter is negative, p'<0. For example, the formation and subsequent collapse of cavitational bubbles may take place on screw propellers, causing the accelerated wear of their surface. The vapor pressure inside a cavitational bubble, p"(rcr), is only slightly lower than the pressure of saturated vapor equilibrium with the flat surface, P0, while the negative pressure in the liquid may be quite high: Ip'l)~po (Fig. IV-3). As a rule, boiling and cavitation are related to heterogeneous nucleation, and thus the described case of homogeneous formation of vapor bubbles inside a liquid phase is only related to very specific conditions under which no wall effects are present. P
P0
P;; V"-Vm Vo tr
p'l Fig. IV-3. To the analysis of the equilibrium of a vapor bubble in a liquid The formation of a critical nucleus during the boiling of a stretched liquid occurs when the chemical potential in the vapor nucleus and in the liquid are equal, i.e. ~t"[p"(rcr)]= = g'(p'), in agreement with Sckripov's scheme 3. Taking into account the corresponding equations of state for the liquid and gas one obtains
3 The scheme shown in Fig. IV-1 implies that the new phase, A, can exist at a pressure equal to that in the old phase, B. This scheme describes (at least approximately) any phase transitions that may take place, except cavitation. In the case of cavitation the vapor bubbles, in which the pressure is positive, are formed in the stretched liquid under negative pressure. In order to include cavitation (and a similar process of continuity rupture taking place in the solid phase) one has to adopt a more complex scheme
282 RTln p'' (rcr) = Vm ( P ' Po
P0).
This corresponds to the equal areas of rectangle 1 and the shaded area 2 in Fig. IV-3. If the difference between p"(rcr ) and Po is small, one can write that
RTln P"(rcr)~Po
RTIP"(rcr)-ll'~Vm'[P"(rcr)-po P0],
where Vm" is the molar volume of vapor. Consequently,
v - ( p ' - p o ) ~ v-' [p" (rc~) - po ]
and ptt(rcr ) _ p t ~ ( P O -
JOr) 1 -
Vrnt ,
Substituting the above relationship into eq. (IV. 16), and taking into account that V m, (( Vm#,
one obtains the equation for the work of critical vapor nucleus formation inside stretched liquids, namely
16n~ 3
Wcr m
3@0 - p')2
9
IV.2.5. Crystallization from Melt In the case of condensation from melt both the newly generated (new) and initial (old) phases are incompressible. Hence, a moderate increase in pressure is not related to the performance of any significant work, and is not an effective way of penetrating into the metastable region. The desired effect may be achieved by changing the temperature, T. The diagram of phase equilibrium (Fig. IV-4) indeed shows that a significant change in the
283 equilibrium pressure of the solid and liquid phases (and, consequently, the ability to reach high supersaturation, -Ag (Fig. IV-5)) corresponds to a relatively small change in the temperature of the melt. Supersaturation in the initial phase is thus usually reached by supercooling of the melt. Applying the Gibbs-Helmholtz equation to the process of melt solidification one can write A ~ ( T ) - - ~0~ + T ~ A ~ ( T ) ,
(IV.17)
c3T
or
c3T P
zXp
T
1_
T2 "
2
~s
L
AT Fig. IV-4. The reciprocal relationship between changes in temperature, AT, and pressure, Ap, along the curve of equilibrium during crystallization (melting)
\
II
I",<
!
1
I
T Tmp T Fig. IV-5. Chemical potential of solid, g, and liquid, g', phases as a function of temperature, T
In the above equations Ag(T) = g(T) -g'(T) < 0 represents the difference between the chemical potentials of the solid, g, and liquid, g', phases, i.e., the change in chemical potential that occurs due to solidification of a metastable (supercooled) liquid, and which characterizes supersaturation in the system; ~'is the specific (per mole of substance) heat of melting, which is assumed to be constant. The integration of eq. (IV.17) over the range
284 between the melting point, Tmp,which corresponds to Ag=0, and temperature Tyields A
t(T) _
fl
r
~"
1 t
_
rmp
AT
rmp r
or AT rmp
where A T - Tmp- T > 0. Substitution of the above relationship into eq. (IV. 15) readily yields the work of critical nucleus formation in the melt, namely
~cr
--
16 3 ( Vm Tmp./2 --71;(3' 3
~AT
where Vm is the molar volume of the solid phase.
IV.2.6. Heterogeneous Formation of a New Phase
H o m o g e n e o u s nucleation takes place only when the system does not contain any surfaces on which the formation and growth of nuclei can occur at a sufficient rate. If such surfaces are present (e.g. vessel walls, or especially the surfaces of foreign inclusions), h e t e r o g e n e o u s nucleation may become much more probable, depending on the nature of the surfaces. If seeds of the substance forming a new phase or those of a substance with similar structure and properties are introduced into the system, the new phase starts to form on the surface of such seeds.
285 In order to investigate heterogeneous nucleation at the existing interface one needs to analyze the conditions of equilibrium of such a nucleus with the medium. In the simplest case of non-crystalline nuclei (vapor or liquid), their shape is defined by the contact angle, 0 (Fig. IV-6). In agreement with the Young equation
COS0 --
O"13 -- 13"23 O12
where 0~3, o23, and 0~2 are the specific surface energies attributed to the interfaces between corresponding phases (1 - initial phase; 2 - nuclei of a new phase ;3 - foreign inclusion).
2
j/////~
/,~///////h ~zs 3
(i13 rl
3
a
b
7///'///////////////////I
3
c
Fig. IV-6. The shape of nuclei when there is condition of a complete non-wetting (a), poor wetting (b), and good wetting of the new phase substrate (c)
Regardless of the aggregate state of the phases, the contact angle, 0, is measured in this case inside a new phase. One must realize that depending on the properties of the new and initial phases, the value of angle 0 may vary between 0 and 180 ~ as in to the case of preferential wetting. The shape of a nucleus will, therefore, be determined by whether the initial or the new phase better wets the surface of an inclusion. The case of complete non-wetting of a surface by the new phase (i.e., complete wetting by the initial phase)
286 corresponding to 0=180 ~ is represented by Fig. IV-6, a. This situation is possible when vapor bubbles form in a liquid that completely wets the solid surface. The case of preferential wetting by the initial phase, when 90 0 < 0 < < 180 ~ is given in Fig. IV-6, b, while better wetting by a newly formed phase, 0<90 ~ is shown in Fig. IV-6, c. This last situation may occur in the case of the condensation of the vapor of a poorly wetting liquid, or in the case of the boiling of a non-wetting liquid. The nucleus height, H, and the radius of the contact line of all three phases, r~, are related to the nucleus radius, r, and contact angle, 0 (Fig. IV-6, b), as H = r ( 1 - cos0) ; r 1 = r sin0. Since the volume of a spherical segment is given by
V-
1 rcH2 (3r 3
H)
for the volume of a heterogeneous nucleus one can write V ( r ) - - T1 r r 3 (1 - cosO) 2 (2 +cosO) - - -4~ r 3 f ( O ) .
3
3
In the above expression the quantity
1 f ( 0 ) - -7 (1 4
cos0)
2
(2 + cos0)
is the ratio of the volumes of truncated and fully spherical nuclei with the same surface curvature radii; J(0) decreases from 1 to 0, when 0 is varied between 180 and 0 ~ It is quite obvious that the surface curvature radii of the critical nuclei
287 are the same, in both homogeneous or heterogeneous nucleation. Indeed, equilibrium conditions for the portions of a nucleus distant from the region of contact with the solid surface are independent of whether such a solid is present or absent. One may expect (a more detailed derivation will be given below) that eq.(IV. 15) remains valid for a heterogeneous nucleus, i.e. the work of a critical nucleus formation is proportional to the nucleus volume. Then the work of heterogeneous formation of critical nucleus, Wcrhet, equals
the work of
homogeneous formation of a critical nucleus, Wc~h~ multiplied by the ratio of nuclei volumes, i.e. by the value of J(0), namely W c het r -
f(O)"cr
W hom
9
Since f(0) may vary from 1 to 0, depending on the angle, 0, the work of heterogeneous formation decreases from its maximum value, corresponding to the work of homogeneous formation, Wcrh~ at 0=1800 (complete nonwetting of the surface by a new phase) to zero at 0=0 ~ (perfect wetting). Consequently, in the case of good wetting of the surface with a new phase, the formation of this new phase may take place, even at a very low supersaturation, at which the new phase formation in a homogeneous system is impossible. 4 If nucleation takes place at a rough surface, the work of critical nucleus formation may be lowered even further, due to the additional decrease in
4 The conditions of heterogeneous nucleation, as well as the work of formation and the curvature radius of the critical nucleus, may be significantly influenced by the linear tension of the perimeter of wetting, • if • the formation of heterogeneous nuclei becomes easier
288 critical nucleus volume, taking place in cavities (Fig. IV-7). The latter is the reason for the easier boiling of liquids in the presence of capillaries or porous materials ("boiling chips").
"/////~///////A~///////. Fig. IV-7. Nucleation at different regions of a rough surface Thus, the presence of solid surfaces (especially rough ones), selectively wetted by the new phase in the presence of the initial phase, considerably favors the generation of the new phase due to lowering of the work of critical nuclei formation. The better the wetting, the less work is required. For this reason one can observe purely homogeneous nucleation only in the absence of foreign inclusions and under conditions of perfect wetting of vessel walls by the initial phase.
Let us now examine heterogeneous nucleation in more detail. In agreement with the Young equation (III.7), the increase in the surface free energy during heterogeneous nucleation, is given by A,-~ss - S12~
+ $23 (u23 - (3 13 ) = (3 12 (S12 - $ 2 3 c o s 0 ) ,
where $12 and $23 are the interfacial areas of the nucleus-medium and nucleus -inclusion interfaces, respectively. The surface area of a spherical segment, S~2,is given by S12
-
2rcrH-
rc ( H
2 + rl2 ) ,
while the area of contact between the nuclei and the inclusion, $23equals t o account the expressions for H and rl one further obtains
grl 2 .
Taking into
289
A ' ~ s - o" 12 (S12 - $23 cos0) - 71;012 ( H2 + ?.2 _/.2 cos0) = gr2cYl2 [(1- cosO) 2 + s i n 2 0 ( 1 - c o s O ) ] - gr2cYl2 [(1- cosO)+ + ( 1 - c o s 2 0 ) ] ( 1 - cosO) - rl:r2cYl2 ( 1 - cosO) 2 ( 2 + c o s O ) - 4 r t r 2 (Yl2f(O) 9 Thus, J(0) describes both the ratio of the volumes of the heterogeneously and homogeneously formed nuclei of equal radii, V het and V horn,respectively, and the ratio of the free surface energies of their formation, ~S het and ~gr sh~
Consequently,the work of heterogeneous
formation of a nucleus of any size (notjust of the critical one) can be obtained by multiplying the work of homogeneous formation of a nucleus with the same radius, Wh~ by f(0), which is independent of radius, namely
vhet W het ( r ) - A ,_~shet -t- A ~ ~
Vm
=f(0)
A,_~sh~ + A ~
vhom gm
m I -f(O)Wh~
IV.3. Kinetics of Nucleation in a Metastable System
The thermodynamics of new phase nuclei formation in a metastable macroscopic system indicates that an energy barrier opposing the formation of nuclei exists for different phase transitions and under different conditions of nucleation (i.e., heterogeneous or homogeneous). Under these conditions the generation of nuclei can be viewed as a fluctuational process as a result of which the system overcomes the energy barrier. By analogy with other similar processes one may expect the frequency of nuclei formation, J , to be an exponential function of the energy barrier height, i.e. of the work of critical nucleus formation, W~ [7,8]:
290
J - J0 exp(- -~--~) .
(IV.18)
The value of Wc~decreases with deeper penetration into the metastable region, and upon the introduction of surfaces selectively wetted by the new phase. Conversely, the pre-exponential factor, J0, is independent (or weakly dependent) of the extent of penetration into the metastable region, and is determined by the mechanism involved in the overcoming of the potential barrier by a new phase nuclei. Following the approach introduced by Ya. Frenkel, let us apply the concepts of particle size distribution to the analysis of the kinetics of the formation of new phase nuclei during phase transitions. Equation (IV. 13), describing the particle size distribution, yields the particle concentration distribution curve with respect to particle radii, n(r) when applied to metastable system ( Ala = gv -gx<0; also assuming that the interfacial tension is independent of particle size distribution). The particle radii distribution curve contains a minimum (Fig. IV-8). The shape of the curve indicates that the formation of large particles only occurs in a
.(r) A
Ap<0
n(rcr)
"~
2
/ b
rcr
r
Fig. IV-8. Equilibrium particle size distribution in a metastable system for which A~ = ~v~x<0, according to eq. (IV. 13) - curve I; the steady-state distribution corresponding to the formation of new phase nuclei - curve 2
291 state of thermodynamic equilibrium. One has to realize that the initial state of system (homogeneous molecular solution) is represented by point A in the left portion of the diagram (Fig. IV-8). The equilibrium distribution curve should gradually proceed from the smallest particles to those of larger size, and particles exceeding the size corresponding to the minimum in the n(r) curve should become larger. Within the first approximation one can assume that the quasi-equilibrium distribution, described by the left branch of the curve only (i.e. to the left of the minimum), will be established fairly quickly. Then n(rcr ) gives the concentration of critical nuclei; in order to obtain the frequency of generation of nuclei larger than the of critical ones ("supercritical nuclei"), one has to normalize this concentration over some characteristic time during which critical nuclei exist, tcr. This time, tc, can be estimated as the average time needed to add another molecule to the critical nucleus, which transferres that nucleus into the "supercritical state". Using eq. (IV. 13) to determine the concentration of particles with radius equal to rc~ in the equilibrium system, n(rcr), one can write an estimation for the frequency with which nuclei overgrow the critical radius, rc~ (i.e. the frequency of overcoming the energy barrier), namely
4 Xrc2rcy -~ j=
=
tcr
nOexp(_Wcrl_.nOexp tc~ kT] tc~
3 Vm Ag kT
'
where -Ag = g , - gv>0.
Consequently, one can regard the pre-exponential factor in eq. (IV. 18), J0, as the quantity determined by the ratio of the number of molecules per unit volume in the metastable phase, n 0, to the critical nucleus life time, tcr. Such treatment emphasizes that the formation of a nucleus is not the result of the simultaneous collision of a large number of molecules (the
292 probability of which would be very low), but rather a gradual growth of particles up to the critical size. According to Ya.B. Zeldovich this process can be viewed as "the diffusion of particles in the space of sizes". Among the large number of particles that are generated accidentally and that get dissolved prior to reaching critical size, there are a few ("the most stubborn") that, due to prolonged fluctuations, turn into critical nuclei and further into particles of a new phase. A more thorough consideration must account for the fact that the rapid growth ofsupercritical nuclei causes changes in the shape of distribution curve (dashed line in Fig. IV-8). At the same time one should also take into account the time needed for the particle size distribution curve to acquire a shape close to that at equilibrium. Assuming that the pre-exponential factor in eq. (IV. 18) is determined by the ratio of the number of molecules per unit volume of metastable phase, n 0, to the life time of critical nuclei, tc~, let us examine how the nature of the phase transition influences the nucleation frequency. S -1, where Scr is the surface area The quantity tc~ can be written as (c~q)
of the critical nucleus, and q is the frequency of addition of molecules to the unit area of a newly formed phase. In the case of the c o n d e n s a t i o n of a v a p o r with pressurep", the frequency of the attachment of molecules to a unit of surface area of the critical nucleus may be considered to be equal to the collision frequency of molecules with the surface. The latter, in agreement with molecular-kinetic theory, is given by Pf
q
m
P x/2 rcmkT '
293 where m is the mass of a molecule. Consequently, in this case the estimate for the frequency of nuclei formation can be written as J ~
noScrP" e x p ( ,/2 mkr \
Wcr
kT)"
For particles with a radius o f ~ l 0 nm, at a pressure corresponding to that of water vapor at room temperature, ~20 mm Hg or 3 • 103 N m -2, and for a mass of water molecules of~3 x 10 .23 g, the above estimation yields tc~~ 10 -~~ s; for n0~6• 10 23 molecules per
m 3
the value of J0 is ~ 1034 m
-3 s -1.
It is worth
mentioning that such an estimation results in a pre-exponential factor somewhat higher than that obtained from a more accurate theoretical treatment. If phase transitions occur in c o n d e n s e d p h a s e s , a nucleus is in contact with a large number of molecules in the initial metastable phase. In this case the lifetime of the critical nucleus is determined by the number of molecules present on its surface, Sc~/b 2 (b is the intermolecular distance), their oscillation frequency, v M, and the height of the potential energy barrier, U, that a molecule has to overcome in order to become attached to the nucleus surface. The frequency of thermal oscillations of molecules can be obtained from the expression v M = k T / h ,
where h is Planck's constant. The latter
allows one to write the pre-exponential factor, J0, as
J~176 which yields J0 ~ 1045 m
-3 s 1 a t
h exp - k T
U = 0.
'
294 The energy barrier, U, that molecules have to overcome in order to become attached to a nucleus surface is of importance in the solidification of silicate melts and organic liquids, especially of polymeric substances. In this case U signifies the activation energy in the process of the diffusion of a molecule (or its segments) from the bulk of the liquid phase to the surface of a nucleus. A drastic decrease in the diffusion rate in such liquids, related to the increase in viscosity as the temperature is lowered, causes a maximum to appear in the curve representing nuclei formation frequency as a function of temperature. The position of this maximum corresponds to some supercooling, AT*, as shown in Fig. IV-9.
I
0
AT*
AT
Fig. IV-9. The frequency of formation of a new phase nuclei, J, as a function of the supercooling, AT
Upon the rapid cooling of these liquids to temperatures significantly lower than Tmp- AT~, the rate of formation of crystalline nuclei becomes very low, while the viscosity becomes very high, and the liquid undergoes a transition into an amorphous glassy state. The rates of nucleation and nuclei growth in glasses are so low that a metastable amorphous state can exist for extended periods of time. For instance, common silicate glasses reveal a significant degree of crystallization (devitrification) only after storage for
295 hundreds of years. There are, however, methods used to artificially catalyze fine particle crystallization of silicate glasses. These methods, aimed at partial fine crystallization of silicate glasses, are based on introducing seeding additives (such as, e.g. titanium dioxide) and the appropriate thermal treatment. The resulting materials, referred to as sitalls, while maintaining high strength at normal temperatures, show significantly improved strength to high temperatures. These materials are not as brittle as regular glasses, and their strength is less influenced by surface defects and aggressive media. There are also methods for the "amorphization" of some metallic alloys. Such "amorphous metals", formed by rapid cooling, have a number of unusual valuable properties.
IV.4. The Growth Rate of Particles of a New Phase
The chemical potential of the substance forming a nucleus of size greater than critical (after the potential energy barrier has been overcome) is lower than the chemical potential of the substance in the initial phase. This difference in chemical potential is the driving force for the growth of nuclei, i.e. for the growth of a colloidal particle of the new phase. As the particle size increases, the chemical potential of the substance making up the particle continues to decrease and the force promoting the growth, given by the difference in chemical potential between the surrounding medium and the particle, increases. In systems where the concentration of substance in the new phase is much greater than in the initial phase (e.g., in the case of condensation from
296 vapor or solution), particle growth results in a decrease in the amount of condensing substance in the immediate vicinity of the forming particle. Here particle growth is limited by the rate ofdiffusional transport of substance from the bulk of the initial phase to the particle surface. Hence, the growth rate in this case is primarily determined by the rate of diffusion of the substance, as well as by the rate of attachment of molecules to the particle surface. If substance concentration in the initial phase is close to that in a newly formed phase (as, in the case of melting or crystallization from melt), or if in the new phase the concentration is much smaller than in the initial one (e.g., during boiling), the particle growth rate is primarily determined by the rate at which the molecules travel across the interface, and by the rate of dissipation (during crystallization) or supply (during melting or boiling) of the phase transition heat. Let us discuss the growth of new phase particles upon crystallization from a solution, a situation frequently encountered in colloid science. Depending on the conditions under which crystallization takes place, it is possible for the crystallization to occur in two limiting regimes: the particle growth may be either diffusionally or kinetically controlled.
In the case of
diffusional control, the growth rate is determined by the diffusional transport of molecules from the solution bulk to the particle surface, while in the case of kinetically controlled particle growth the growth rate is determined by processes taking place at the particle surface. During diffusionally controlled particle growth the rate at which the radius of the spherical particle, r, increases, is related to the total flux of substance to its surface,js (moles s-~ ), as
297
dr dt
=
jsVm 4~r
2 '
(IV. 19)
where Vmis the molar volume of substance forming the particle. This situation is typical when the droplets of new liquid are formed in solution. The flux,is, is positive when directed towards the surface of the particle. The flux of substance towards the particle surface is determined by the gradient in the
js-4=r 2D(d~-~) - 4 = r 2 D ~Ac r
concentration of the dissolved substance in the vicinity of the surface, (dc / dR)r ,
and by the value of the diffusion coefficient, D (see Chapter V):
In the above expression Ac is the difference in substance concentration at the particle surface and in the bulk, Ac = Co [exp(Ag(r)/RT) - 1]; Co =
c(r) is the
solubility of particles of radius r; Ag>0 is the difference in chemical potential between the bulk of initial phase and the solution near the particle surface; 6 is an effective diffusion layer thickness. Let us now address the diffusion of dissolved substance towards the particle surface in more detail. Let us assume that the diffusion is a quasisteady state process, i.e. the distribution of concentration as a function of distance from the surface,
c(R), changes slowly with time, and at any given
moment the total flux of substance across a sphere of arbitrary radius R can be regarded as essentially constant, i.e. Js
-
4~R 2 D ~dc dR
const.
298 The integration of the above expression using the boundary conditions of c Co at R - r, and c = c o + Ac at R -~o~yields
Js - 4~r2 D~Ac.
(IV.20)
A comparison of this expression with eq. (V.12) indicates that diffusion towards a spherical particle is similar to diffusion towards a flat surface through a solution layer of thickness 15- r. Substitution ofeq. (IV.20) into eq. (IV. 19) yields the expression for the rate of diffusionally-controlled particle growth: dr
dt
:
D A c Vm
.
(IV.21)
r
K i n e t i c ally- controlled particle growth is typical in the crystallization process. The addition of molecules (atoms, ions, etc.) to an ideal flat surface may involve additional difficulties, such as those encountered during the formation of the nuclei themselves, namely, the molecules arriving at the surface of the crystallite associate into a new crystalline plane. Such a nucleustype "island" (a two-dimensional nucleus) has an excess of energy attributed to its side surfaces (Fig. IV-10). The energy change that occurs in the system
b! r
Fig. IV-10. Schematic representation of a two-dimensional nucleus
299 during the formation of a two-dimensional nucleus of square shape with a side of length d may be written as W ( d ) - 4 K d + b d 2 Abt
Vm The linear tension of the side face ("the step"), ~, is approximately equal to the product of the specific interfacial energy and the nucleus height (size of a molecule): ~ ~ bey. The work of formation of the critical two-dimensional nucleus is given by: 4K2Vm
/~scr = blAlti [ ~
4(y2bVm
lag ]
9
(IV.22)
Equation (IV.22) can be derived similarly to eq. (IV. 16). If one directly compares eq. (IV. 16) to eq. (IV.22), it becomes evident that the ratio of the works of formation of three- and two-dimensional critical nuclei is determined by a dimensionless quantity, (yVm/ ]A~ ]b, which can also be written as (sb2/[ Ag[(b ~ / Vm) , where the numerator represents the surface energy of a single cell in a crystal lattice, and the denominator is the chemical potential difference per single molecule. At low supersaturations the work of formation of a three-dimensional nucleus is significantly higher than that of a two-dimensional one. Nevertheless the value of the latter is sufficiently high to retard particle growth. Experimental studies of crystallization indicate that in real systems there is no significant kinetic retardation present in the case when crystal growth takes place at low saturations. This is related to those defects that are present in the structure of real crystals, and in particular to the presence of a
300 special type of linear defects, known as s c r e w dislocations [9]. A dislocation of this type transforms the parallel crystal planes into a single helix (Fig. IV11), and the growth of crystal occurs due to a continued building of the helix, rather than due to the formation of new atomic planes. While the helix is continuously being built-up, there is always a step present at the surface, and the attachment of new atoms to that step is preferred over the attachment to a flat surface of crystal. For example, in the case of a primitive cubic lattice (Fig. IV-12) the atom located at the surface of crystal (position A) interacts with a single atom, while the one attached at the step (position B) interacts with two atoms. The attachment of atoms at the kinks is even more favorable: at position C there are three points of interaction. Crystal growth can thus be viewed as the propagation of such kinks due to the diffusion of atoms towards them, either from a solution or along the crystal surface.
c
/
I
Fig. IV-11. Screw dislocation at the surface of a crystal
Fig. IV-12. Schematic representation of possible ways in which new atoms may be added to a crystal surface during the formation of an atomic plane
IV.5. The Formation of Disperse Systems by Condensation
The processes of generation and growth of nuclei form the basis of the condensational formation of disperse systems. Finely dispersed systems can
301 be formed by condensation if on the one hand a sufficiently large number of nuclei of a new more thermodynamically stable phase form, and if, on the other hand, the growth rate of these nuclei lies within a certain (moderate) range. The formation of stable free disperse system also requires that conditions that oppose particle aggregation be maintained. The degree of dispersion in the system being formed is determined by the ratio between the rates of nucleation and growth, and for weakly stabilized systems also by the rate of processes leading to system degradation. One may rather provisionally subdivide the factors responsible for the metastability in the initial system into two categories: chemical (chemical reactions resulting in the formation of highly concentrated solutions of compounds with low solubility, which thus lead to supersaturation), and physico-chemical (changes in the pressure, temperature and composition of phases). There is a variety of chemical methods that one can employ to achieve supersaturation. Essentially any chemical reaction resulting in the formation of insoluble, or volatile (in the case of condensed phases) or nonvolatile (in the case of the reaction between two gases) product may in principle be used for the preparation of disperse system. If there is a need to prepare a disperse system in aqueous medium, one should avoid use of a high concentrations of electrolytes, which could promote coagulation of the system being formed (see Chapters VII and VIII). The processes involving the formation of disperse systems with different degrees of dispersion and concentrations are common in nature and have broad use in various industrial applications. Some typical examples illustrating the preparation of disperse
3O2 systems by chemical reaction are summarized below. Numerous methods used to prepare stable sols of gold and silver are based on the reduction of various salts of these metals with different reducing agents, such as phosphorus (M. Faraday), tannin (W. Ostwald), formaldehyde (R. Zsigmondi), e.g:
2KAuO 4 +
3HCHO + K2CO 3 = 2Au + 3HCOOK + KHCO 3 + H20.
The reactions, leading to the formation of disperse systems by o x i d a t i o n are very common in nature. The reason for this ubiquity can easily be understood" during the raise of volcanic melts and of the gases, fluid phases, groundwaters that separate from them, the mobile phases are transferred from zones of reduction located deep in the crust to zones of oxidation that are located near the surface. A good example that illustrates such a process is the formation of sulfur sols by reaction between the hydrogen sulfide dissolved in hydrothermal waters and oxidizers (sulfur dioxide or oxygen): 2H2S + 02 = 2S + H20 Another typical example is the erosion of ferric silicate present in the erupted minerals. This process leads to the formation of ferric hydroxide hydrosols and silicic acid: Fe2SiO 4 + 8902 +5H20 = 2Fe(OH)3 + Si(OH)4. The formation of hydrosols by the hydrolysis of salts is quite common in nature and has practical importance. The process of salt hydrolysis is used in wastewater treatment. The high specific surface area of colloidal
303 hydroxides formed by hydrolysis allows the
effective absorption of
contaminants, such as surfactant molecules and heavy metal ions. Besides hydrolysis, one can also utilize other exchange reactions in the preparation of disperse systems. It is, however, important to remember that a substantial amount of electrolyte, which is often present in the solution, may result in a loss of colloidal stability. One can sometimes remove excess electrolyte by washing and subsequent peptization of the precipitate. It is advantageous to prepare disperse systems at high supersaturations, which can be reached upon mixing concentrated solutions of reactants. The sols of Prussian Blue, various sulfides, stannic acid and its compound with colloidal gold (Cassian Purple) are all made by this method. The formation of usually coarsely disperse systems during gaseous phase evolution is of importance in the industrial production of various solid foams with valuable mechanical, thermal insulating and sound insulating properties. Examples of such materials include various types of foam concretes (production of these usually involves the evolution of CO2 gas in the reaction between CaCO3 and HC1), foam plastics, and microporous rubber. In nature the degassing of magma leads to the formation of pumice stones and tuffs. Physico-chemical ways of achieving metastability of the initial system are usually related to changes in temperature, pressure (less often), and composition of solvent [10]. The supersaturation (supercooling) of water vapor is the reason for certain meteorological phenomena (cloud formation). The formation of disperse systems upon changes in temperature is the key for the preparation of all polycrystalline materials in metallurgy. Here control of
304 the degree of dispersion is of primary importance in the manufacturing of modern high-strength construction materials. The condensation methods play an especially important role in the preparation of finely disperse systems, which are impossible to make by simple dispersion process. These methods also allow one to control the degree of dispersion (and the degree of polydispersity) in the systems formed. In order to obtain finely dispersed systems, one needs to achieve high supersaturation while limiting the particle growth rate. Usually the finest dispersion is achieved when highly concentrated solution of one reactant is mixed with an extremely dilute solution of another reactant, yielding a product with low solubility. High concentration of the first reactant causes high supersaturation and high nucleation rate, while the low concentration of the second reactant limits further nuclei growth due to retarded diffusion from the dilute solution. An important feature in the formation of dispersions of slightly soluble substances by condensation is the possibility of initially forming particles of a metastable phase of high supersaturation (Ostwald's rule). It was discovered by V.A. Kargin and Z. Ya. Berestneva that the particles so formed may be amorphous. Their further crystallization may lead to disintegration of the particles, probably due to high internal stresses experienced by the system during crystallization. Depending on the nature of the substances, the system in an amorphous state may remain stable for various periods of time, ranging from several minutes (gold sols) to several hours, days, and even years (silicic acid sols). The preparation of the so-called "m o n o d i s p e r s e " colloidal systems
305 consisting of particles of uniform size and shape deserves special attention. Monodisperse systems are excellent objects for studies and modeling of phenomena involved in catalysis, mineral processing, flotation, papermaking, water clarification and other important applications. They are often used as model objects in the investigation of adsorption, interactions between curved surfaces, adhesion, coagulation (see Chapter VII), etc. Monodisperse systems are also responsible for quality and essential properties of many materials. For instance, the quality of recording media is defined by the size, crystallinity and aspect ratio of anisometric magnetic particles; the size of silver halide grains in light sensitive materials is beneficial for the quality of photographic images; the color and purity of inorganic pigments strongly depends on uniformity in particle size; uniform particles are essential for the preparation of high quality ceramic materials. In order to obtain monodisperse systems under conditions of low supersaturations (so that homogeneous nucleation doesn't take place), one may introduce into the system some very small nuclei of a new phase. This "nuclei assisted" method (the Zsigmondi method) is often used to make monodisperse gold and silver sols. Thus, finely dispersed sol of gold nuclei, prepared by the reduction of auric chloride with a solution of phosphorus in ether, is added to the auric chloride solution. Subsequent treatment with a reducing agent, such as formaldehyde, causes growth of the nuclei introduced. The resulting system is monodisperse because of a constant particle growth rate. The nuclei additives act as modifiers of the second kind (in Rehbinder's classification). The nuclei-assisted method of sol preparation is widely used in modern technology, such as in preparation of titanium pigments, sugar
306 processing, etc. Monodispersed colloids can also be obtained by the La Mer method [ 11,12] in which light supersaturation in the system is maintained over a long period of time thanks to a slow chemical reaction. One example of such a process is the reaction between a dilute sodium thiosulfate solution and dilute sulfuric or hydrochloric acid, i.e.
3 Na2S203 + H2SO 4 = 4S + 3Na2SO 4 + H20
The growth of sulfur particles continues for several hours and a fairly monodisperse system with interesting optical properties (see Chapter V) is formed as a result.
La Mer's qualitative interpretation of sulfur particle formation mechanism can be understood from the diagram shown in Fig. IV-13. The concentration of the molecularly dispersed sulfur formed in the above reaction slowly increases until critical supersaturation is reached. At that point nucleation and the formation of solid phase embryos take place. Critical Supersaturation Limit
~
.
Rapid Self- Nucleation Growth by Diffusion
=o iubility / - ' -
'! 'i i I
.
.
I I
.
.
.
.
.
.
.
Time
Fig. IV- 13. Schematic presentation of the mechanism involved in the formation of sulfur sols according to La Mer [11,12]
307 The nuclei formed then grow by diffusion. An important condition needed for the formation of monodispersed particles is single act of nucleation, i.e. that no secondary burst of nuclei occurs. For this conditionto be fulfilled, the newly formedmolecular species of sulfur should be forming at such a rate that their diffusional transport to the already formed solid particles, rather than secondary acts of nucleation take place. Particle growth can be terminated by the addition of KI solution, which reacts with the excess of thiosulfate. However, the excess of potassium iodide may lead to system destabilization due to the removal of the layer of potential determining pentathionic ions, $5062 (see Chapters V and VIII).
Over the decades that have passed since La Mer's work numerous examples of monodispersed particles of various composition, morphologies and properties, as well as methods for their preparation (not limited to condensational formation), were described in the literature. Extensive studies in this area were carried out by E. Matijevid and T. Sugimoto. Examples of monodisperse systems formed by precipitation from homogeneous solutions include dispersions of uniform particles of "simple" composition having different morphologies, such as metal halides, sulfides, phosphates, (hydrous) oxides, etc, various composite particles, including particles of internally mixed composition and coated particles. Both crystalline and amorphous materials can be obtained. Electron micrographs of some characteristic examples of monodispersed colloids are shown in Fig. IV-14. Comprehensive and detailed description of monodisperse systems is given in review articles [ 13-15], book contributions [ 16-19] and references therein. As with the La Mer's monodisperse sulfur sols, the fundamental requirement for the preparation of monodispersed particles is the control of kinetics of the generation of species that serve as precursors for the formation
308
Fig. IV-14. Electron micrographs of some characteristic monodispersed colloidal particles; a - zinc sulfide (ZnS), b - hematite (a-Fe203), c - cadmium carbonate (CdCO3). Courtesy of Professor Egon Matijevid of solid phase. (Not necessarily all ionic species that are present in the medium participate in the formation of solid phase). The methods used to achieve such kinetic control are the deprotonation of hydrated metal cations (so-called
"forced hydrolysis", after E. Matijevid
[18]), controlled release ofprecipitating
anions, and decomposition of metal complexes. Many monodispersed metal (hydrous) oxides can be prepared by means of forced hydrolysis. Uniform particles usually precipitate only under a rather narrow set of conditions that have to be identified experimentally for each particular system. Temperature, pH, ionic strength, nature of the anions in reacting solutions, and the presence of surfactants or polymers, all play important roles in determining the nature and properties of precipitated solids.
Coprecipitation from homogeneous solutions containing more than one kind of reactant may result in the formation ofmonodispersed composite particles. It has been shown that the mole ratios of individual components in the solid phase differ from those in solution, because the rates of precipitation of individual components in the mixture change as the
309 particles grow. This is illustrated in Fig. IV-15, for the case of mixed cadmium/nickel phosphates [20]. I0
.r.,4
Z6 2+
2
~-,0 4"~
[N +]
2
90
a m
180
270
360
t/rain
450
Fig. IV-15. The composition of internally mixed nickel and cadmium phosphate solid particles, presented as [Cd]/[Ni] molar ratio, precipitated from solutions containing three different initial CdZ+/Ni2+ molar ratios, [CdZ+]/[Ni2+] = 2.0, 1.0, and 0.5, as a function of ageing time at 80 ~ (From Ref. [20], with permission) In order to obtain monodispersed colloids via precipitation in homogeneous solutions, one needs to keep the ionic strength low, so that coagulation of the precipitate can be avoided. This imposes a restriction of rather low concentrations of reactants, which can be alleviated with the controlled double-jet precipitation (CDJP) technique, schematically shown in Fig. IV-16. The principle setup consists of a reactor with a constant temperature chamber equipped with a mixer. The mixing chamber is filled with a predetermined amount of solution that may contain various stabilizing, reducing or pH-controlling agents. The reactants are introduced into the mixing chamber at desired flow rates by means of peristaltic pumps. The primary nuclei are generated in the vicinity of the mixer and are then transported into the bulk solution where they grow at the expense of unstable nuclei with sizes smaller than the size of a critical nucleus [21 ]. In CDJP a nucleation zone and a bulk zone ,in which particle growth takes place, coexist in the same solution throughout the entire precipitation process. At early stages of the process the number of stable nuclei increases with increasing supersaturation due to the dissolution of unstable nuclei. These dissolving nuclei provide solute that is consumed by the stable nuclei during their growth. Once the stable nuclei have grown enough to consume the entire amount of solute generated by the unstable nuclei
310 dissolution, the number of stable nuclei ceases to increase and the newly formed nuclei generated in the nucleation zone simply serve as a source of monomer for particle growth. Like in the case of "normal" homogeneous precipitation, the CDJP process involves the stages of accumulation of stable nuclei and particle growth, resulting in the formation of monodispersed particles ranging from several nanometers to several microns. |
Reactant 1 ~
I
' ~ = Reactant 2
I I l l
~
I I
,,
I
,,
>
I
o~
Heating oil , 2>
Nucleation zone
i
Particle growth zone Fig. IV-16. Schematic representation of a controlled double-jet precipitation (CDJP) setup It has been confirmed by different experimental techniques and even observed visually with electron microscopy that a majority of monodispersed colloids consist of much smaller subunits [18]. Also, some spherical colloidal particles obtained by homogeneous precipitation reveal X-ray characteristics of known minerals. Examples of such materials include spherical particles of ZnS and Fe203with X-ray diffraction patterns consistent with sphalerite and hematite, respectively. These findings resulted in the reexamination of La Mer' s single-step nucleation burst - diffusional growth mechanism, and clearly suggested that a g g r e g a t i o n step, not present in La Mer's model, was involved in the formation of monodispersed particles. Recently Privman et al [22] developed a new model that explains the formation of monodispersed particles via the aggregation of primary subunits. This new model allows one to explain the size selection mechanism, i.e. the kinetics of the generation of narrow size distribution, based on the rapid growth of nuclei formed in the supersaturated
311 solution into primary nanoparticles that undergo further aggregation into larger uniform colloids. Among the key assumptions involved in the new model is the absence of any electrostatic repulsion between aggregating precursors, essential for the aggregation to take place. Irreversible capture of precursors by aggregates is another reason for the narrow size distribution of particles in the resulting dispersion. The new model was applied to uniform gold sols yielding numerical results consistent with experimental observation. One important feature of Privman's aggregation model is the fact that it uses interfacial tension at the particle-solution interface as an adjustable parameter, and thus can serve as a means to estimate this often unknown quantity (see Chapter I).
The structure of disperse systems that form during the crystallization of metals and other phase transitions can be effectively monitored by introducing substances that are surface active with respect to the initial and new phases. These surface active substances (modifiers of the first kind, in Rehbinder' s classification) may play a dual role: they can drastically decrease the work of critical nuclei formation, thus increasing the probability of this process, or at the same time, they may retard particle growth by blocking the surface. Both effects cause an increase in the degree of dispersion of systems formed during phase transitions. Finer dispersion is usually a requirement for the preparation of materials with high strength (see Chapter IX).
IV.6. Ultradisperse Systems. Supramolecular Chemistry In recent years the development of novel methods for the synthesis of highly disperse systems and the applications of such systems has drawn a lot of attention both in industry and academia. A new area of science, referred to as the chemistry and technology of ultradisperse systems, also known as
312
nanochemistry has emerged. A considerable effort has been made in the development of computer modeling of the properties of fine clusters, containing just a few atoms or molecules [23,24]. The methods for the generation of such clusters have been developed. These methods allowed one to investigate the structure of these clusters and compare it to the theoretical predictions. One of the methods of synthesis of clusters of uniform size consisting ofjust several atoms is the intrusion of liquid phase (e.g., mercury) under high pressure into zeolites with voids of different volume. High pressure is necessary for overcoming the capillary pressure in order to achieve filling of small voids with a liquid. When the pressure drops, the column of liquid in the thin capillary ruptures, similarly to the column of mercury in the thermometer upon cooling, and monodispersed clusters become trapped in the zeolite voids. Computer modeling and experimental studies of such small clusters both indicated that they form unique crystalline structures, impossible in the case of macroscopic crystals. For example, such structures may contain the axes of symmetry of fifth order. Among the methods used to generate larger nanoparticles (nanometers and tens of nanometers), we should mention nebulization of materials in plasma, which results in spherical amorphous particles with sometimes complex composition (e.g. alumosilicates). Another broadly used method is the generation ofnanoparticles with narrow size distribution in microemulsion systems (see Chapter VI). Another branch of science that emerged over the last decades on the borderline between colloid and organic chemistry is the supramolecular
313
chemistry, for the development of which C.J. Pedersen, D.J. Cram and J-M. Lehn received the Nobel Prize (1987) [25]. This area of science focuses on the studies of the "guest- host"-type compounds, such as the complexes of crownethers or other "molecular containers" with various inorganic ions and organic molecules. At the same time, the objects formed in various self-assembly processes, such as surfactant micelles, vesicles are also classified as supramolecular structures.
IV.7. Dispersion Processes in Nature and Technology
The conversion of solids into a dispersed state is by far one of the most common and heavily used technological processes. At the same time, dispersion is a very important natural phenomenon that to a great degree determines changes in the structure of our planet's surface and the existence of life itself. These processes are responsible for the erosion of the Earth's crust, when existing stresses and moisture can cause the transformation of massive rocks into the finely divided systems that form the basis of Earth's soil layer. In technology the treatment of solid materials by cutting and polishing, the mining and grinding of rocks and minerals prior to flotational enrichment, the construction of mines and tunnels through the rocks and the crushing of such materials as grain, concrete, coal, and paint pigments all constitute examples of dispersion processes. Grinding processes require enormous expenditures of energy, consuming a significant portion of all the electrical energy that is produced.
In the USA the annual cost of such dispersion
314 exceeds 70 billion dollars [26]. The process of emulsification, i.e. the dispersion of one liquid in another, is also of great importance. Grinding of solids is usually carried out in mills of different types [27]. Milling is based on the brittle crushing of materials through collisions of the pieces of the material being refined with milling elements, such as steel or porcelain marbles, or with the walls of vessel in which milling takes place. In order to obtain finely dispersed powder, one may need to carry out milling for many hours and sometimes even days. High-speed grinding can be performed in vibrating mills, in which the drum filled with milling elements and milled material undergoes oscillatory motion with a frequency of several thousand periods per minute. Ground material of high purity can be prepared in stream mills, in which size reduction occurs as a result of collisions between particles moving at high speed. For the preparation of monodispersed particles socalled colloidal mills are used. Grinding in these mills takes place in high velocity gradient fields, such as that in the small gap between a fast rotating cone and a stationary surface; the disperse system is pumped through this gap. Similar devices are also used for the homogenization of emulsions, e.g. of milk. Energy requirements for mechanical dispersion are determined by the mechanical properties of solid phase and are dictated by the desired degree of dispersion. Brittle materials are usually broken down easily while it is rather difficult to crush ductile ones. With respect to the particular degree of dispersion, it is possible to distinguish crushing (coarse dispersion to sizes of several centimeters or millimeters), comminution (to sizes on the order of tens of microns), and fine dispersion. Crushing usually obeys an empirical
315 Kirpichev - Kick rule, according to which the work of crushing, Wc~u~h,is proportional to the volume of crushed material, V [27,28]. Comminution follows the Rittinger rule, i.e. the work ofcomminution,
Wcomm, is proportional
to the surface area of powder formed, AS. In the general case, the work of grinding, according to Rehbinder, is defined by the relationship Wgr - cy*AS + k V . In this expression cy* is the specific work of grinding, which may exceed the surface energy of the solid, c~, by several orders of magnitude. At the same time, for some materials a proportionality between G and ~* is observed. The use of various surface-active media, different in nature and mechanisms of action, plays an important role in the intensification of dispersion processes and in decreasing energy requirements for them. Such media can both facilitate grinding due to an adsorption-related decrease in strength (the Rehbinder effect, see Chapter IX, 4) and, at the same time, prevent aggregation (i.e., adhesion) of the ground particles (see Chapters VIIVIII). For example, in grinding hydrophilic materials that are used in the production of catalysts, ceramics, sorbents and carriers, one usually uses water as a dispersion medium. The use of water, in addition to its adsorption activity towards hydrophilic materials and its strength-controlling abilities, is obviously determined by its low cost and common availability, which are important factors in view of the enormously large industrial scale of these types of production. However, water has a rather high surface tension (~ ~ 73 mJ mZ), which results in the appearance of rather strong capillary attractive forces (Chapter 1,3) that are generated during water removal upon completion
316 of the grinding process. These forces cause particles to aggregate during the drying stage, so one may need an additional stage of disaggregation of the material already ground. The use of surfactant additives allows one to lower the action of capillary attractive forces, prevent aggregation, and carry out grinding under optimal conditions. Investigation of the role that liquid media play in the intensification of the grinding of solid materials is one of the main subjects of physico-chemical mechanics described in Chapter IX. The role of surfactants is equally important in the emulsification and foaming of liquids, which are processes broadly used in various technologies across the chemical, pharmaceutical, and food industries. Finely dispersed sols of metals and alloys in various dispersion media can be obtained by electrospraying. The physico-chemical nature of this process is intermediate between dispersion and condensation. Electrospraying of powders can be most effectively carried out in non-conducting media using high-frequency high-voltage discharge. This method, originally developed by T. Swedberg and T. Bradigg, allows one to obtain various sols, including such exotic ones as the sols of alkali metals in organic solvents. Electrospraying of alloys yields sols with composite dispersed particles.
References
,
3.
Rehbinder, P.A., "Selected Works", vol.1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 Shchukin, E.D., and Rehbinder, P.A., Kolloidn Zh., 20(5) (1958) 645 Rusanov, A.I., Kuni, F.M., Shchukin, E.D., and Rehbinder, P.A., Kolloidn. Zh., 30(4) (1968) 573
317 ~
~
,
,
~
,
10. 11. 12. 13. 14. 15. 16. 17.
18. 19. 20. 21. 22. 23. 24. 25.
Rusanov, A.I., Kuni, F.M., Shchukin, E.D., and Rehbinder, P.A., Kolloidn. Zh., 30(5) (1968) 735 Hill, T. L., Thermodynamics of Small Systems, Bengamin, New York, 1963. Skripov, V.P., The Metastable Fluid, Nauka, Moscow, 1972 (in Russian) Nishioka, K., and Pound, G.M., in "Surface and Colloid Science", vol.8, E. Matijevid (Editor), Wiley, New York, 1976 Heicklen, J., Coloid Formation and Growth: A Chemical Kinetics Approach, Academic Press, New York, 1976 Cottrell, A.H., The Mechanical Properties of Matter, Wiley, New York, 1964 Wedlock, D.J. (Editor), Controlled Particle, Droplet and Bubble Formation, Butterworth, Oxford, 1994 La Mer, V.K., and Dinegar, R., J. Am. Chem. Soc., 72 (1950) 4847 La Mer, V.K., Ind. Eng. Chem., 44 (1952) 4847 Overbeek, J. Th. G., Adv. Colloid Interface Sci., 15 (1982) 251 Sugimoto, T., Adv. Colloid Interface Sci., 28 (1987) 65 Matijevid, E., Langmuir 10(1) (1994) 8 Matij evid, E., in"Controlled Particle, Droplet and Bubble Formation", D.J. Wedlock (Editor), Butterworth-Heinemann, London, 1994 Matijevid, E., in "Fine Particles Science and Technology", E. Pelizzetti (Editor), NATO ASI Series 3, vol. 12, Kluver Acad. Publ., 1996 Matijevid, E., and Sapieszko, R.S., in "Surfactant Science Series", vol. 92, T. Sugimoto (Editor), Marcel Dekker, New York, 2000 Sugimoto, T., Monodispersed Particles, Elsevier, Amsterdam, 2001 Quib6n-Solla, J., and Matijevid, E., Colloids Surf. A, 82 (1994) 237 Sugimoto T., in "Surfactant Science Series", vol. 92, T. Sugimoto (Editor), Marcel Dekker, New York, 2000 Privman, V., Goia, D.V., Park, J., and Matijevid, E., J. Colloid Interface Sci. 213 (1999) 36 Schmid, G., Clusters and Colloids: From Theory to Applications, Wiley-VCH, New York, 1994 Wang, Z.L., Characterization of Nanophase Materials, Wiley-VCH, 2000 Lehn, J-M., Supramolecular Chemistry: Concepts and Perspectives, Weinheim-VCH, New York, 1995
318 26. 27. 28.
Westwood, A.R.C., Ahern, J.S., and Mills, J.J., Colloids Surf., 2 (1981) 1 Lowrison, G.C., Crushing and Grinding, CRC Press, Cleveland, OH, 1974 Lowrison, G.C., in "Dispersions of Powders in Liquids", G.D. Parfitt (Editor), 3~ded., Applied Science Publ., London and New Jersey, 1981
List of Symbols
Roman symbols b r
D d 5T ~sset ~ss~
G H h
Js J k S m NA
N n
P q R R r
intermolecular distance concentration diffusion coefficient linear dimension of particle free energy surface free energy surface free energy of heterogeneous formation of nucleus surface free energy of homogeneous formation of nucleus Gibbs thermodynamic potential height of a nucleus Planck's constant total flux of substance to the surface frequency of nuclei formation Boltzmann constant specific heat of melting mass of a molecule Avogadro's number number of moles number of particles number of particles per unit volume pressure collision frequency of molecules with a surface universal gas constant distance from the surface radius of particle
319 ~"cr
S O"
&r T
rmp t
t. U
Vcr
V het V horn
Vm Vm" mcomm mcr Wc~ush mcrbet W. h~ W~ Wgr Wph mscr
radius of the critical nucleus surface area entropy surface area of the critical nucleus absolute temperature melting point time critical nucleus life time height of potential energy barrier volume of the critical nucleus volume of heterogeneously formed nucleus volume of homogeneously formed nucleus molar volume molar volume of a new phase chemical work work of comminution work of critical nucleus formation work of crushing work of heterogeneous formation of critical nucleus work of homogeneous formation of critical nucleus work of dispersion work of grinding work of particle formation during the phase transition work of critical two-dimensional nucleus formation
Greek symbols
0 K
~t 71; IJ (~cr (l VM
particle shape coefficient ln(N2/N~+ 1) in eq. (IV. 7) small distance contact angle linear tension chemical potential 3.14159... surface tension specific surface free energy specific work of grinding oscillation frequency
320 V. T R A N S F E R P R O C E S S E S IN DISPERSE S Y S T E M S
Structural features of disperse systems, in particular the existence of the electrical double layer (EDL), are responsible for a number of peculiar phenomena related to heat and mass transfer and electric current propagation in such systems. The description of electromagnetic radiation propagation is also included in this chapter. These features are utilized in numerous practical applications and underlie methods used to study disperse systems. These methods include particle size distribution analysis, studies of the surface structure and of near-surface layers, the structure of the EDL, etc. In the most general way the most transfer phenomena can be described by the laws of irreversible thermodynamics, which allow one to carry out a systematic investigation of different fluxes that originate as a result of the action of various generalized forces. The development of theories covering different transfer phenomena in disperse systems is a rapidly growing area of colloid and surface science. Typically this theoretical work utilizes rather complex mathematics and it is by no means complete. Thus, here we shall consider only the most general and commonly accepted approaches that deal with transfer phenomena. At the same time we will do our best to at least mention all types of transfer processes that occur in disperse systems by touching on the present state of theoretical development, and discussing the potential for further studies and applications of such processes.
321
V.1. Concepts of Non-Equilibrium Thermodynamics as Applied to Transfer Processes in Disperse Systems. General Principles of the Theory of Percolations
A systematic analysis of the various processes occurring in disperse systems is based on the study of generalized thermodynamic forces and the fluxes generated by such forces. One may distinguish between direct
processes and crossed (cross-)processes. In direct processes the force and the flux are of the same nature, while in crossed processes they are of different natures. The thermodynamic forces that we will further consider include the chemical potential gradient resulting from the concentration gradient of the dispersed substance or dissolved species, the strength of the external electric field, gravity forces (particle weight), and pressure and temperature gradients. The impact these forces have on disperse systems is strongly dependent on the structure of these systems. One can distinguish between the free disperse systems, in which the particles are freely dispersed in the continuous phase (i.e. in the dispersion medium that is either liquid or gas), and those structured
disperse systems, in which mutually bound particles form a network structure (see Ch. IX) permeated by the continuous phase. The basic fluxes that we will further examine are fluxes of the dispersion medium in structured disperse systems, and those of the dispersed particles in the free disperse systems, as well as electric current and electromagnetic radiation flux. For instance, a gradient in the concentration of dispersed particles results in diffusion, while the action of the gravity field on these particles
322 causes sedimentation (settling). These phenomena, together with thermal (Brownian) motion, viewed at the microscopic level, are classified as
molecular-kinetic phenomena, and are described in this chapter. A pressure gradient existing between the two sides of the barrier formed by structured disperse system with open porosity, (membrane or diaphragm), results in
filtration of the continuous phase. The application of an external electric field to the disperse system causes the flow of electrical current, resulting in
conductivity. All of the above are examples of direct processes. However, along with these direct processes, crossed processes, among which there is a large class of the so-called electrokinetic phenomena, are also frequently encountered in disperse systems. The electrokinetic phenomena include
electrophoresis (motion of dispersed particles in the free-disperse system due to an applied external electric field), electroosmosis (flow of dispersion medium through the membrane due to the applied outer electric field, and the generation of electric currents and potentials during sedimentation and
filtration. These and some other examples of transfer processes involving these forces and fluxes are summarized in Table V. 1. Within the framework set by a linear approximation of irreversible thermodynamics [ 1,2], for two types of thermodynamic forces, F~ and F 2, and two types of fluxes,j1 and j2, one can write that" Jl : C~llF1 + ~12F2
(V.1)
J2 : ~21F1 4- ~22F2 ,
where % are the phenomenological coefficients attributed to the laws that govern particular processes. These coefficients can be either determined
323 experimentally or derived theoretically based on studies of the mechanisms of particular processes. In agreement with the
Onsager reciprocal relations,
based on the principle of process reversibility on a microscopic level, the cross coefficients, a~2 and a2~, are identical. This allows one to limit the analysis to only one of the reciprocal processes (e.g. electroosmosis or the streaming current). One, however, has to remember that the principle of microscopic reversibility can be applied only to processes that take place in identically structured systems.
TABLE V. 1. Thermodynamic forces and fluxes involved in different transfer processes. Thermodynamic force
Flux of matter
Particle concentration gradient
Particle diffusion, Brownian motion, Concentration fluctuations
Gravity
Particle sedimentation (settling)
Sedimentation current and potential
Pressure gradient
Filtration Particle oscillations
Streaming current and potential Colloid vibration potential
Electric potential gradient
Electrophoresis Electroosmosis
Temperature gradient
Thermophoresis Thermoosmosis Thermocapillary phenomena
Solution concentration gradient
Di ffu s iophore s is Particle growth and dissolution Marangoni effect
Flux of electricity
324 Linear approximation relationships of irreversible thermodynamics, eq. (V. 1), also allow one to describe steady-state processes, or equilibrium states, corresponding to zero flux. For a general case one can write
~.
_
O Fj.
(V.2)
t3~ i i
For instance, if ~ is the pressure gradient across the membrane, % and % are the phenomenological coefficients describing the streaming current and the electrical conductivity of the membrane, respectively; F, is then the streaming potential corresponding to zero current, it, across the membrane. For transfer processes involving the motion of dispersed particles taking place in free-disperse systems, one can write the general relationship between the flux,j, and the average velocity of independent particles, a), i.e.:
j - knu,
(V.3)
where the dimension of the coefficient k is determined by the dimension of the flux (if the flux under consideration represents the electrical current, the coefficient k takes into account the effective particle charge), n is the concentration of dispersed particles. If the force
f 1
acts on particles, their
average velocity is given by ~ - F~/B, where the friction coefficient, B, is determined by particle shape. For spherical particles of radius r the friction coefficient is given by the Stokes formula, i.e. B=6xqr, where rl is the dynamic viscosity of the continuous phase. The equation for the velocity can then be written as u-
F , -' . 6xrlr
(V.4)
325 The corresponding flux is given by
F1
j = kn~. 6~rlr
(V.5)
Establishing the values of coefficient, k, and force, F~ constitutes the major task in the theoretical treatment of transfer processes in free-disperse systems. One model example of a
transfer process that takes place in a
structured disperse system is the transfer of the dispersion medium through a single capillary of radius r c that occurs during filtration or in the processes involving various types of osmosis. The proportionality coefficient between the acting force and the resulting flux is referred to as the conductivity (or
permeability) of the capillary, while the reciprocal value is known as the capillary resistance. The extension of this simplest model to real systems with open porosity, in which capillary pores are combined into three-dimensional networks containing merging and branching channels of variable thickness, constitutes a complex problem that is not yet completely solved. To illustrate the level of difficulty involved in solving this problem, it is sufficient to mention that in order to evaluate the flow through parallel capillaries, one has to carry out the summation of the capillary permeabilities, while in the case of capillaries with variable diameters it is necessary to perform the summation of the resistance over individual sections of a single capillary. Needless to say, in describing the flow in a real system one has to combine these two methods of summation. An interesting problem that has been attracting the attention of scientists over the last decades is the analysis of transfer processes that occur
326 in disperse systems when the porosity, H, of the systems increases. The porosity is defined as the ratio of the total volume of the pores to the volume of the disperse system. Let us now turn to some principles of percolation
theory, which addresses this problem [3]. We will base our discussion on the example of the spontaneous formation of liquid phase layers at the grain boundaries of polycrystalline solid substance under the Gibbs-Smith conditions (see Chapter I), i.e., when
(Ygb_>2CYSL,where ~gb is the energy of the
grain boundary, and CysListhe energy of the solid/liquid interface. Since the first quantity strongly depends on the degree of disorientation between grains, one may expect that the Gibbs-Smith condition is valid only for some fraction of the grain boundaries. If this fraction of "active" grain boundaries satisfying the Gibbs-Smith condition, a, is small, the liquid layers will cover only a small fraction of the boundaries. This is illustrated in Fig. V-1 (a), which shows the results of a two-dimensional computer modeling. The greater the fraction of these "active" grain boundaries, the larger the size of the"clusters"formed at the boundaries (Fig.V-l, b-d).
a=0.1
a=0.3
a=0.5
a b c Fig. V. 1. Coverage of"active" grain boundaries with a layer of liquid
a=0.6 d
In agreement with percolation theory, when the fraction of "active" grain boundaries reaches some critical value, acr, an infinite cluster is formed, and the entire polycrystal contained between the surface that is in contact with
327 liquid and the surface on the opposite side is filled with a completed network of permeable grain boundaries. Only after such a network has been formed, can there be macroscopic flux of liquid through the diaphragm. This also relates to the propagation of electric current in the case when the conducting liquid penetrates an insulator, as happens when layers of saline solutions penetrate into ionic crystals. Further increase in the fraction of "active" grain boundaries would result in an increase in the average number of conducting channels per unit area, and in their straightening. In agreement with percolation theory, when a>acr, the flux is proportional to(a-%) ~. The studies by V.Yu. Traskin et al showed that for a NaC1 - water system a ~0.3, and J3~1.4. It is worth pointing out that permeability is a typical critical phenomenon. The power law (in which the power is usually a fraction) characterizing the degree of flux deviation from the critical value is referred to as the "scaling law" (or "scaling"). In our further description of transfer processes we will not address the complicated issues involved in porous medium structures, and hence our discussion will be based on a simple model of transfer through an isolated capillary.
V.2. The Molecular-Kinetic Properties of Disperse Systems
The involvement of dispersed particles in thermal motion, which causes the entropy of the system to increase upon the dispersion of the substance, accounts for similarities in the properties of free disperse systems and molecular solutions. The term "colloidal solutions" was therefore
328 introduced. Meanwhile, the larger, in terms of molecular dimensions, size of particles in colloidal systems results in some peculiarities in the molecularkinetic properties of the such systems, which are discussed below. These peculiarities include first, the sedimentation (settling) of sufficiently large particles in the gravity field; second, a slower, compared to molecular solutions, rate of diffusion, which, as a matter of fact, led Graham to the false conclusion that colloidal systems are incapable of participating in diffusion; third, the possibility of directly observing the thermal motion of colloidal particles, which allowed one to experimentally verify some of the basic principles of molecular-kinetic theory. Thermal motion was observed for the first time by a British botanist R. Brown, and is thus referred to as Brownian motion.
Brown examined the aqueous suspension of pollen particles in the microscope, and found that the particles were in continuous motion. To rule out the possibility that the life cycle of pollen cells was responsible for the observed phenomenon, he conducted similar observation with finely divided matter of both organic and inorganic nature. These experiments confirmed the existence of chaotic motion, which could be observed with highly crumbled matter of any nature. Further in this chapter we will return to the discussion of the theory of Brownian motion, but first let us address the macroscopic laws that govern particle motion in the gravity field (sedimentation) and during diffusion.
329 V.2.1. Sedimentation in Disperse Systems The settling of a spherical particle of radius r in a dispersion medium of viscosity 11and density P0, is driven by the weight force, F~. Correcting for the buoyancy, one obtains: F1 - - -4~ r 3 ( 9 - 9 0 ) g 3
meg,
(V.6)
where 9 is the density of substance forming the disperse phase, g is the acceleration of gravity, and me is the effective buoyancy-corrected particle mass. The above expression in combination with the equation (V.4) yields the particle sedimentation rate" meg
2r2 (9 - 90)g
B
9n
(V 7)
The s e d i m e n t a t i o n f l u x of the dispersed particles,j~, with dimensions m-2s-~, is equal to Js =
megn ,
(V.8)
B where the "-" sign indicates that if the density of particles exceeds that of the dispersion medium, the sedimentation proceeds in the downward direction. The investigation of particle sedimentation (sedimentation analysis) allows one to obtain the particle size distribution function (see Section 7.1 of this chapter).
V.2.2. Diffusion in Colloidal Systems According to Nernst, the force acting on a single dispersed particle
330 may be represented by the negative gradient of the chemical potential per particle. For one-dimensional diffusion along the z-axis only (the vertical axis is chosen for further comparison with sedimentation), one may write
F1- -
dg/dz dlnn ~ = - kT~, N A
(V.9)
dz
where n is number of particles per unit volume. The expression for the diffusionalparticle flux, jd, can then be written as F1 Jd - - n
B
kTdlnn - n ~
B
dz
kT =
dn
6~rlr dz
=-D~
dn dz
(V.IO)
Equation (V. 10) is known as Fick's first law of diffusion. In this equation D is the diffusion coefficient in units ofm2s -1. The diffusional flux,jd, represents the amount of substance that crosses a section of unit area, S, per second in the direction normal to that of diffusion. In the above equation dn/dz is the concentration gradient, which for steady-state diffusion is constant in time at any point within the system. The units ofj and c should be consistent. We will further express the concentration as either c or n, where c is the number of m o l e s of dispersed particles per unit volume (it is assumed that 1 mole contains 6.02x1023 colloidal particles), and n is the particle number concentration, i.e., the number of p a r t i c l e s per unit volume. Consequently, Jd, ~ and Jd,, are expressed in mol m -2 s-1, and m -2 s-1, respectively. According dc to the first definition, Fick's first law can be written as Jd,c - - D ~ and dz' the diffusion coefficient, D, is expressed in the same units as in eq. (V. 10),
331 i.e., m2s-1. The diffusion coefficient is related to the viscosity of the dispersion medium and to the particle size by
D-
kT B
=
kT 6rtrlr
,
(V.11)
which follows from eq. (V. 10). This equation was derived first by Einstein in 1908. It can also be applied to non-spherical particles, if one chooses the proper friction coefficient, B. Studies of diffusion allow one to determine the particle size, or if the latter is known, one can evaluate Avogadro's number ~.
In order to determine experimentally the diffusion coefficient on the basis of Fick's first law, one has to study steady-state diffusion. To do this, one has to maintain constant concentrations, c~ and c2, at both sides of a cylindrical cell with cross-sectional area S and length l, and to measure the flux, i.e. the amount of substance, Am~At, carried through such a cell per unit time. Since the concentration gradient in such an experiment is constant along the entire cell length, l, and is given by
dc dx
c2 - c1
l
F ick's first law can be written as
The possibility of determining Avogadro's number from studies of the diffusion of colloidal particles was of great scientific importance, since it represented a breakthrough in Ostwald's restriction on the direct observation of molecular motion. In contrast, here NA is determined by the observation of the thermal motion of particles whose size significantly exceeds molecular dimensions
332
1 Am =D Jd - S At
Ac
- D c2 - cl
l
1
(V.12)
'
from which it follows that
D
1 Am
l
S At
C 2 -- C 1
In common practice, however, methods based on the studies of non-steady-state
diffusion (when the concentration gradient, dc/dx, changes with time) are typically used to determine the diffusion coefficient. These methods are focused on the evaluation of the concentration distribution, c(x,t), along the length of the diffusion cell. The changes in the distribution that occur during diffusion are then monitored. Typically, a solution of known concentration, Co, is placed in contact with the pure dispersion medium (c - 0) along a plane where x = 0 (Fig. V-2), and the gradual haziness of the initial concentration spike caused by diffusion is monitored as a function of time. r
CO Ax
0
x
x
Fig. V-2. Changes in the concentration distribution that occur as the diffusion process goes on
Non-steady-state, or time-dependent, diffusion is described by a partial differential equation, often referred to as Fick's second law 2, i.e." ~C
c~t
~2C
-
D
(V. 13) (~X 2 "
2 Look further in this section for the derivation of this equation and more detailed analysis of time-dependent diffusion
333 The integration of Fick's second law shows that the solution concentration is a function of
(x/v / 2Dt ), meaning that the motion of a front of solution volume with some concentration c* is described by
x(c*,t)- kx/2Dt,
(V. 14)
where the proportionality constant, k, is a function of the concentration c*. Changes in the concentration profile c(x, t) of dispersed particles within the diffusion cell can be studied by a variety of methods, including colorimetry (for colored substances), nefelometry, those involving the use of radioactive tracers, and others. If the c = c (x, t) dependence is known, one can determine the diffusion coefficient, D, and then obtain the particle size using Einstein's relationship, eq. (V. 11). For example, Herzog employed the diffusion method for the determination of the effective size of sucrose molecules in aqueous solution. The experimental value of the diffusion coefficient that he determined was D - 0.384 cmZ/day. Using eq. (V. 11) and treating the sucrose molecules as spherical particles with a density equal to that of the crystalline sugar (9 = 1.588 g cm-3), one obtains a molecular weight, M - 4/3 ~r 3 pN A = 332 g/mol, which differs only slightly from the true value of 342 g/mol.
V.2.3. Equilibrium Between Sedimentation and Diffusion Equations (V.7) and (V. 11) reflect the increase in the sedimentation rate with increasing particle size, and the increase in the rate o f diffusion with a decrease in particle size. Thus, one may expect that for particles of intermediate size the sedimentation and diffusion fluxes can balance each other, i.e.,j~ + j ~ - 0, which leads to an equilibrium between sedimentation and diffusion:
kT
d lnn dz - - meg
(V.15)
One should immediately emphasize that the above equation does not contain
334 any parameters that depend on particle shape. The integration of eq. (V. 15) yields the equilibrium distribution of particles with respect to the z coordinate"
n ( z ) - n oexp -
megz) kT '
(V.16)
where n o is the particle concentration at the reference level. Equation (V. 16), describing the distribution of particles with respect to height, can be applied both to colloidal systems and to gases. Since for ideal gases the concentration is proportional to pressure, eq. (V. 16) can be written as
p(z)-po exp(-mMgz / kT "
(V 17)
In the above equation mM is the mass of a molecule. Equation (V.17) is referred to as the barometric Laplace equation.
One can use a thermodynamic approach to derive the equation describing the condition of equilibrium between sedimentation and diffusion. The derivation involves the assumption of a constant gravity-chemical potential (i.e. the generalized chemical potential which includes the action of the external gravity field) and the assumption that the laws established for ideal systems can be applied to dilute dispersions, i.e.
la(z) + NA m'gz - const,
~t(z)- ~to + R T l n n ( z ) , where m' =
[(9-9o)/9]m is the effective particle mass. The equation,
then readily obtained:
identical to eq.(V. 17), is
335 NAm' gz + R T In n ( z )
- RT
Inn o ,
NAm'gz
n(z)
ln~no
RT
Equation (V. 17) allows one to determine the height, z, at which the concentration of the
disperse phase is lowered by a certain factor, as
compared to that at the reference level, z - 0. The level at which the concentration decreases by a factor of e may be chosen as a characteristic of the diffuse atmosphere around the dispersed particles, i.e." Z1/e
-- k T
/ m' g -
(V.18)
D / u .
Alternatively one may also chose a level at which the concentration decreases by a factor of 2" Zl/2 -
(ln 2) Z1/e
For molecules of gases, e.g. oxygen, dispersed gold sol ( r ~ 1 nm)
z~/2 ~
suspension ( r~ 0.37~tm), the value
-
0.69 Zl/e
z~/2
.
~ 5 kin; for particles of finely
3.5 m, and for particles in a gamboge
ofzl/2 ~
0.1 ram.
On the basis ofeq. (V. 17), Perrin estimated Avogadro's number from experimental studies of the particle number distribution as a function of height in the gamboge suspensions. His result, 6.7x 1023 mol -~, is quite close to the commonly accepted value. For disperse systems consisting of particles with radii less than 0.1 gm direct observation of the equilibrium between sedimentation and diffusion is difficult or even impossible, as the equilibrium is reached extremely slowly.
336 The characteristic time required to reach equilibrium is the time necessary for the diffusion front to move a distance Z~/e:
t s e d ~"
Z1/e/l).
Since z1/e ~ D/I), one
can write that/sea ~ D/I)2" Substitution of D - kT/6rcrlr and ~) = 2r2g(9 - 90)/911 into this equation indicates that the time needed to reach an equilibrium is inversely proportional to the particle size to the fifth power (i.e., particle mass to the power of 5/3), and to the square of the acceleration of gravity, g, namely"
/sea
kTrl/rS(9 - 90)2g2. Consequently, for particles with radii <
~
0.1gm it would take several years for the equilibrium to be established. Lengthy equilibration periods
increase the probability of unstable
experimental conditions, such as the appearance of convective streams due to random changes in temperature, mechanical disturbances, etc. Owing these factors, finely dispersed colloidal systems can not be studied by direct investigation of their sedimentation or sedimentation-diffusion equilibrium. The sedimentation-diffusion equilibrium can be shifted towards greater sedimentation if the natural gravity field is replaced by centrifugal force fields, in which accelerations are substantially higher than g. Such field can be generated in centrifuges or ultracentrifuges [4]. The centrifugation method, first used by A.Dumansky and further developed in the works by T. Svedberg, allows one to reach accelerations as high as 105 - 106 g, and makes it possible not only to carry out sedimentation, but also to achieve the sedimentational separation
of
macromolecules
with
different
molecular
weights.
Ultracentrifugation used in combination with particle size distribution analysis makes it possible for one to carry out preparatory fractionation in colloidal systems and polymer solutions. Investigation of the equilibrium between sedimentation and diffusion
337 in the ultracentrifuge is one of the methods used to study particle size distribution; since the value of z~/2 is small, an equilibrium can be reached quickly. One usually performs simultaneous investigation of sedimentation and diffusion by analyzing changes in the distribution of particles on the z axis with time. For sedimentation under gravity such simultaneous (within the same experiment) analysis of sedimentation and diffusion can be carried out but with great difficulty and only within a limited particle size range.
V.2.4. Brownian Motion and Fluctuations in the Concentration of Disperse Phase Particles
The thermal motion of the dispersion medium molecules results in collisions of these molecules with the surfaces of dispersed particles and thus causes a displacement in the positions of the latter. For the particles with a size around 1 gm, at an oscillation frequency of ~10 ~2 s-~ there are about 10~9collisions with particle surfaces every second, and on each collision the particle changes the direction and velocity of its motion. Consequently, in a typical experiment one usually observes some averaged trajectories describing sequential particle displacements. The displacement of each particle during some particular time interval, At, can be projected onto an axis x, arbitrarily oriented in space (Fig. V-3). Since each particle experiences a random walk, the corresponding projections of the displacements of the first, second, third, etc. particles, Axe, Ax2, Ax3, ..., have random directions and absolute values. For this reason the average displacement of all particles, Ax - 2
Axt / ,/W, is equal to zero, ~ = 0 , i
338 at sufficiently high ~ , provided that there is no directional fluid flow or particle concentration gradient present in the system. The particles, however, do move and do become displaced from their original positions.
\ /Ax, Fig.V-3. The trajectory of the Brownian motion of particles (random walk)
To characterize the intensity of Brownian motion, one has to carry out the averaging in a way that would result in the addition rather than subtraction of displacements in various directions. That is, one has to average the squares of the displacement projections. Einstein chose the average square displacement of particles, (~c2) ~/2 - ( ~ Ax,2/JV" )~/2, as a suitable characteristic. As the i observation time, At, becomes longer, the particle displacement, (E_.fl)~/2, increases as well. The theory of Brownian motion establishes a relationship between these two quantities. To find this relationship, let us compare the thermal motion of individual particles in the absence of a concentration gradient of the dispersed phase with their collective motion (their drift) in the presence of such a gradient, i.e. during diffusion. Let us assume that within a certain portion of the volume of a column of unit section there is a constant concentration gradient, equal to dn/dx, where n is the number of particles per unit volume. Let us divide this volume
339 using three imaginary planes, A, B, and C, perpendicular to the x axis and separated from each other by a distance ~, which is the average square displacement of particles in each volume over time At (Fig. V-3). These planes isolate two equal volumes, 1 and 2 ,with different particle concentrations in each. If the average particle concentration in volume 1 is n~, and that in volume 2 is //2, then volumes 1 and 2 contain
nl~
and
n2~
particles
respectively. I_..
.~._I~
. . . .
I~..
,
1,--1.,,,I
2/I
,
.__ I
~," 1
F X
A
n~ .
.
.
.
.
B .
.
.
.
.
n2 .
.
.
.
.
C
.
.F-grad
n
Fig.V-4. A schematic representation of diffusion in the derivation of the EinsteinSmoluchowski equation Taking into account the chaotic nature of Brownian motion, let us assume that in each volume one half of the particles has drifted by the distance to the right, while the other half has drifted by the same distance to the left. One may then state that over a time period At a total of 1/2n~ particles will migrate from volume I to volume 2 through plane B, and a total of 1/2n2~ particles will cross the same plane from volume 2 to volume 1. The net particle flux through the middle plane B would then be given by
r
/71 - / 7
2
2At
At the same time the concentration gradient corresponding to the change in
340 concentration at a distance ~ between the midpoints of volumes 1 and 2 equals dn
n2
dx
-
n1
r
By directly comparing these equations with Fick' s first law, one may write that
Jd =
n 1 -
n 2
n2
r
2At
-
n 1
r
which readily yields 2
_
2DAt.
This expression was first derived by A. Einstein and M. Smoluchowski (19051906). The diffusion coefficient can thus be regarded as a bridge between macroscopic, c(x, t) and the microscopic,
~
(At) description of the
diffusion process. Brownian motion theory was verified by many scientists (T. Svedberg, A. Westgren, J.Perrin, L.de Broglie and others), who both observed individual particles and followed the diffusion in disperse systems [5]. The influence of various factors, such as the temperature, dispersion medium viscosity, and particle size on the value of the Brownian displacement, ~, was evaluated. It was shown that the Einstein-Smoluchowski theory describes the experimental data adequately and with high precision. An important contribution of this theory to the development of science is the possibility one now has to determine the value of the Boltzmann constant, k, by studying the motion of individual colloidal particles:
341
k~
6rtrlr ~ 2 T
2At
Since the universal gas constant, R, can be determined from independent (macroscopic) experiments, the observation of the motion of individual (microscopic) dispersed particles revealed the possibility of a novel independent determination of Avogadro's number, N a = R/k. Such measurements, carried out by J. Perrin et al with gamboge suspensions, yielded N A= 5.6 - 9.4 x 10 23mol ~. Further studies with oil droplets dispersed in gases, carried out by Fletscher, yielded a value ofN A= 6.03 + 0.12 x 10 23 mol ~ which is close to the precise value. As mentioned above, these experiments allowed one to directly observe the thermal motion of particles and to determine its quantitative characteristics, and thus disproved Ostwald's previous statement on the impossibility of experimentally verifying the molecular kinetic hypothesis. For anisometric particles, along with translational Brownian motion one can also observe rotational Brownian motion. Studies of rotational Brownian motion indicate that the particle average square rotation angle, q)rot, is proportional to the observation time, At, i.e." Aq) 2rot -- 2Drot At
According to Einstein, the rotational diffusion coefficient, Drot, is given by the equation kT Dr~ =
8r~rlr
3 '
342 where r is the particle radius. From measurements of the rotational Brownian motion of particles with shape close to spherical, Perrin determined Avogadro's number. The value of latter was 6.5x 1023 tool-'. Rotational Brownian motion results in the disordering of anisometric particles previously oriented in some particular way owing to the flow of the dispersion medium (see Chapter IX) or the application of an electric field. From the time of the disordering one can determine the rotational diffusion coefficient, and, for known particle size and shape, also Avogadro's number. If the particles are able to undergo co-orientation, they usually are of substantially anisometric shape, and their translational and rotational diffusion coefficients differ from those obtained for spherical particles. For example, for prolate ellipsoids of revolution with a ratio of their main diameters d~ 9d 2 = 1 910, the diffusion coefficient, D, is about 2/3 of the value obtained for spherical particles of the same volume. Investigation of the Brownian motion of dispersed particles made it possible to experimentally verify the theory of fluctuations, also formulated by Einstein and Smoluchowski. Svedberg's observations of Brownian motion indicated that the number of particles confined within a small volume, ~ , changes continuously and deviates from the mean value, ,M'. Let us examine concentration fluctuations, Ac, in a volume V, small compared to the total volume occupied by a system. At equilibrium the free energy, g , has its minimum value, and its first derivative with respect to the concentration is zero. Consequently, the work required to cause fluctuations is determined by the second non-zero (positive) variation of the free energy, i.e."
343 1 d2~ 2 dc 2 (Ac) 2
For the probability of the concentration fluctuations one may write:
w(Ac) ~ e x p ( ~ )
- exp
Ac2
~
~
2kT/d2~dc 2 This means that the probability distribution with respect to the values of fluctuation is described by the Gaussian curve, in which the average square of fluctuation (the dispersion) is given by AC 2 =
kT d2 ~-/dc2 "
Analogously, for any other fluctuating value, the average square fluctuation equals the ratio of kT to the second derivative of work (free energy) of fluctuations with respect to the fluctuating parameter. We will utilize this approach in describing the optical properties of disperse systems (further down in this chapter), the electric properties of aerosols (see Chapter VIII), and the conditions of the formation of critical emulsions (see Chapter VI,2). Since
d~-/dc = (dg/dV)(dV/dc),
d2,~ dc 2
one may write that
d/d dV/ dc
dV
"
For a given number of particles the product of the volume within which these particles are distributed, V, and their concentration, c, is constant,
344 i.e.
Vc
= const. Consequently, d V / d c
finds that
d V/dc-
- - V/c.
If changes in c are small, one also
const. This allows one to write that d2~ dc2
where H = -d,r
V dH C
dc '
is the osmotic pressure. The above relationship is valid
for systems of different nature, such as lyosols, aerosols, polymer solutions. For ideal systems, using the van't Hoff law, i.e., H -
RTc; d H/dc =
RT, one obtains Ac 2 _
kTc RTV
_-
c2 N A Vc
_- c2 ..~
or AcI2 where ~
1
(V.19)
is the average number of particles in a volume, V, in which the
counting took place. Equation (V.19) is the fundamental relationship in the theory of fluctuations. According to Smoluchowski, the probability of counting the number of particles, ~ , different from the average number of particles, ~4P, is given by the Poisson distribution function: w(-4/') _ --~4/,~ exp ( - A/3_~. ,A/'!
(V.20)
345 The average square of the relative fluctuations is inversely proportional to the average number of observed particles, namely:
(
A/1/~) 2
1
,4 r
Smoluchowski also derived relationships for the frequency of the fluctuations, the average time of observation of fluctuation and the average of the square of the change in the number of particles between two observations. The experimental data reported by A. Westgren (Table V.2.) confirm the validity of eq. (V.20).
TABLE V.2. The experimental data reported by A.Westgren and evaluated from eq. (V.20)
( ..A/' =1.45)[61 Number of particles
0
1
2
3
4
5
6
7
Number of times the above number of particles was observed
381
568
357
175
67
28
5
2
Calculations according to eq. (V.20)
380
542
384
184
66
19
5
2
T.Graham was not able to establish the existence of osmotic pressure in colloidal solutions. Based on this fact he has classified solutions as either colloidal or true. Further studies, such as those carried out by A. Sabaneev with protein solutions, revealed that in fact osmotic pressure can be observed in colloidal systems, but its magnitude is very small due to the relatively large size of the particles in comparison with molecular dimensions, and, consequently, their low concentration. Indeed, in ideal systems the osmotic
346 pressure, H, is proportional to the particle concentration, c, i.e., to the number of particles per unit volume" FI - c R T -
nkT.
In true solutions at a concentration c - 102 mol m -3 (n - 6• 1025 particles per m -3 ) the osmotic pressure reaches
6 x 1 0 25 m -3 x
1.38x 1 0 -23
J x K -1 x
298 K -
2.4x 105 Pa (2.4 atm). For lyophobic colloidal systems the number of particles per m 3 does not as a rule exceed 102~, and hence the osmotic pressure is not greater than a fraction a millimeter of water column. Besides the experimental difficulties in measuring values that are so small, the presence of electrolytes introduces a considerable source of errors. The complete removal of electrolytes may lead to the destabilization of a colloidal system (see Chapter VIII). The presence of electrolytes leads to high osmotic pressures and in the presence of membranes results in the establishment of membrane (Donnan) equilibrium. In lyophilic colloidal systems, e.g. in polymer solutions, the disperse phase concentration may be sufficiently high, and the osmotic pressure reaches values that can be reliably measured. In this case osmotic pressure measurements, along with the application of osmosis-related phenomena, such as cryoscopy and ebullioscopy, provide methods for the study of such systems. For example, these methods allow one to determine the molecular weights of polymers. Osmotic phenomena related to the presence ofpolyelectrolytes govern to a significant extent the distribution of water and dissolved substances in living tissues, as well as the transport of these substances through various
347 semipermeable membranes (cell walls, veins, etc). These semipermeable membranes, permeable to dissolved molecules of the dispersion medium but impermeable to large molecules and colloidal particles, belong to colloidal systems and are the subjects of various studies. The methods of osmotic pressure determination are also based on the use of the semipermeable membranes. Let us now return to a detailed description of non-steady state diffusion.
To describe non-steady state diffusion let us take the derivative of Fick's first law with respect to coordinate x (assuming one-dimensional diffusion in a cell with a constant cross-section of unit area):
OJd - 8x 8x
(V.21)
D
The use of partial derivatives reflects the fact that the concentration, c, is a function of both coordinate and time. In the case of dilute systems, in which the diffusion coefficient is independent of concentration and consequently of the coordinate, one may write that
a ( a~X-X/ ~02C
8x
D
- Dsx 2 .
(V22)
Let us examine a cell volume element of length equal to Ax. At any given moment of time the quantity - A j d = Jd ( X ) -
Jd ( x + Ax)
describes the difference between fluxes from left to right (through the x plane) and from right to left (through the x+&x plane), respectively, i.e., it characterizes the rate of substance accumulation in a given volume element. Over a period of time equal to At the difference
348 between the substance fluxes to and from the volume element will result in a change in concentration, Ac, at a point x:
Ac - [Jd ( x ) - Jd (X + z50C)] A t , koc from which it follows that
Ac
AJd x= const
~
or
= - ~.
dt
t= const
(V.23)
dx
A combination of equations (V.21), (V.22), and (V.23) readily yields Fick's second law,
(v.13). The integration of eq. (V. 13), which is similar to the heat transfer equation (i.e. it is a parabolic-type equation), should yield the change in concentration distribution over time, i.e., c = c(x,t) function. To carry out the integration, one introduces a function defined as
y - x / x[2Dt
and Fick's second law can then be written in the form
_ym
dc
d2c
dy -
dy
2
9
The first integration of this equation yields
dc
dy
= lexp( /
where ~t~ is the integration constant, which gives the dc/dy value at y=0. The second integration yields
c ( x , t ) - cz 2 + cz 1
J
exp
dy,
0 where ~2 is the second integration constant. Let us assume that the concentration undergoes an abrupt change from Co to 0 at a point x = 0 at time t = 0, as shown in Fig. V-2. The boundary conditions at any finite time are c -- 0 when x -- % and c -~ co when x -. - oo. Since
349 iexp ( - - @ ) dY - @1:/2 , 0 one can write the solution as
co
c(x,t) - T
:XJ
1- x/2/r~
I
exp
o
[2] -
or
Co
-
In the above expression
-x/~/2Dt)].
ff2/~ Iexp -
(I)(y)-
dy
is the probability integral,
0
which is a tabulated function. Equation (V.22) shows that the constant value of x2/2Dt = k2 = const, corresponds to a constant concentration, c* = 1A Co [1 - ~(k)], i.e. the kinetics of the diffusion front migration, x(c*, t), is described by
x 2 (c*,t)- 2kZDt, which is equivalent to eq. (V. 14).
V.3. General Description of Electrokinetic Phenomena
The types of transfer processes occurring in disperse systems, referred to as electrokinetic phenomena, were first discovered in 1808 by Moscow University professor F.F. Reuss during his investigation of electrolysis. To
35O prevent the chemical reaction between the products of electrolysis, Reuss separated the space in a U-shaped tube between the electrodes (anode and cathode) with a porous diaphragm made of ground sand (Fig. V-5). Upon the application of an electric field, Reuss observed the transfer of liquid from the anode region to the cathode region, a phenomenon referred to as
electroosmosis. i-t--
.:-.....:
~i
Fig. V-5. A schematic representation of Reuss's experiment Electroosmosis causes a change in the level of the liquid in communicating vessels, i.e. in the anodic and cathodic parts of a U-shaped tube. This effect, referred to as the
electroosmoticrise,turns out to be very
strong; for example an applied voltage of 100 V may result in a change in liquid levels of up to 20 cm. Electroosmosis and the electroosmotic rise are thus related to the motion of the liquid with respect to the immobilized disperse phase (porous diaphragm). In the case of electroosmotic rise, at equilibrium the electroosmotic transfer of the liquid is compensated by its back flow due to the change in hydrostatic pressures in different arms of the U-shaped tube.
351 Reuss also discovered the phenomenon opposite to electroosmosis, i.e. the motion of dispersed particles due to an applied electric field,
in an
experiment similar to the one described above, in which the porous diaphragm was made of finely dispersed clay, rather than coarse sand. This phenomenon is referred to as electrophoresis. Upon placing two tubes with electrodes filled with water into a moistened piece of clay, and applying an electric field to the electrodes, Reuss discovered that in addition to the liquid rising in the vicinity of the cathode, a suspension of particles moving towards the anode appeared in the anodic space. In electrophoresis and electroosmosis the substance moves due to an applied electric field. The inverse phenomena in which an electric field is generated upon the motion of the dispersed phase or dispersion medium due to external mechanical forces were also discovered. The phenomenon inverse to electroosmosis, referred to as the streaming potential (current), was discovered by G. Quincke in 1859. In this phenomenon an electric current and a potential difference are generated during the forced flow of fluid through a porous diaphragm. The phenomenon inverse to electrophoresis, known as the
sedimentation potential (current), constitutes the appearance of a current and a potential difference when particles are settling in a gravity field. This phenomenon was discovered by Dorn in 1898 and is thus referred to as the Dorn effect. The group of effects that reveal the mutual relationship between the electric processes and the relative motion of the dispersed phase and dispersion medium are referred to as electrokinetic phenomena [7-9]. Electrokinetic effects are sensitive to electrolytes present in the dispersion medium. As a rule, the introduction of electrolytes into the system
352 leads to a decrease in the intensity of these effects (e.g., decrease in the rates of electrophoresis or electroosmosis and in the values of the streaming and sedimentation currents and potentials). Sometimes, introduced electrolytes cause moving phases to change the direction of their motion, or result in changes in the sign of the originating potentials due to surface recharging. Quincke was the first to explain the origin of electrokinetic phenomena as being due to the spatial distribution of charges in the vicinity of an interface. H. Helmholtz offered the first model for spacial charge distribution near the interface, and used this model for the quantitative description of the observed phenomena. These first model considerations originated in colloid science and were further developed in electrochemistry in the course of the analysis of the kinetics of processes at electrodes and of electrocapillary phenomena (see Chapter III, 3). Accounting to the simplest H e 1rn h o 1t z m o d e 1, the spatial distribution of charges in the vicinity of an interface can be viewed as an electrical (ionic) double layer, represented by two parallel plates of a charged electric capacitor separated by a layer of a dispersion medium of some average (effective) thickness 5. One of the plates is formed by the potential-determining ions (PDI), bound to the surface, while the other one is formed by counter-ions present in solution. The ions that bear a charge of the same sign as that at the surface, the so-called co-ions, are forced away from the surface into the solution bulk. This distribution of charges causes a potential difference between the phases in contact, Aq~, which in terms of the capacitor model corresponds to a linear potential drop between the plates (Fig. V-6). The charge density at capacitor plates bearing a potential difference of Aq~ and
353 separated by a medium with a dielectric constant of e is given by ee0Aqo / 15 ( ee0 / 6 is the double layer capacitance). ~ -~ T~
|
()
e
(')
9
() ( ()
| @ e
| @
() 0
8
x
Fig. V-6. The simplest scheme of electroosmosis
The spatial charge distribution in the electrical double layer is exactly what causes the electrokinetic phenomena, namely the mutual displacement of the phases in contact in an applied external electric field (electrophoresis and electroosmosis) or the charge transfer that occurs upon the mutual motion of phases (streaming and sedimentation potentials and currents). The following consideration, the simplest consistent with the Helmholtz model, establishes the relationship between the rate of the phase shift, e.g. that of electroosmosis, and the strength of the external electric field, E, directed along the surface 3.
3 It is worth remembering that the electric field strength equals the gradient of the external field potential taken with the opposite sign. To distinguish between the external and internal (in the EDL) electric fields, we will further describe the double layer field only in terms of potential gradient, dq~/dx, while the electric field strength, E, and the potential difference, A~, will only be used with respect to the external electric field.
354 The external electric field, parallel to the plates, generates a shear stress, zE, (see Chapter IX), i.e. it results in a pair of forces acting per unit area of capacitor plates, and directed along the plates (Fig. V-6). The force that acts on the plate with cations has the same direction as the external electric field (i.e the same as the E vector). The force that acts on the plate with anions has the same magnitude but the opposite direction. The shear stress equals ~oAq~E ZE=
~
"
The rate of mutual phase displacement is determined by the condition that zE is equal to the viscous resistance of the medium, zn ,and is found from Newton's equation (see Chapter IX)"
zE - z n
-
do
~ .
(V.24)
q dx
In the above expression 1"1is the viscosity of the dispersion medium, and d~)/dx is the gradient of the rate of displacement of the dispersion medium with respect to the surface of the solid phase. Assuming d~)/dx to be constant across the entire gap of thickness 8 between the capacitor plates, one can write that
do dx
o0 8
where % is the macroscopically observed rate of mutual phase displacement. Then, using the condition that zE- zn, one obtains ~oAcpE 5
oo 5
355 or
u0 =
eeoAq>E 9 q
(V.25)
Expression (V.25), referred to as the Helmholtz-Smoluchowski equation, relates the rate of relative phase displacement to some potential difference, Aq0, within the electrical double layer. In order to understand the nature of this quantity, let us examine in detail the mutual phase displacement due to the external electric field acting parallel to the surface, taking into account the electrical double layer structure. Let us assume that the solid phase surface is stationary. Figure V-7 shows the distributions of the potential, q>(x) (line 1 ), the rate of displacement of the liquid layers relative to the surface in the Helmholtz model, ao(x) (line 1 "), and the true distribution of the potential in the double layer (curve 2).
-D
I
I
I
_1.)0"
-loo
_
,,j_
. . . . . .
72
Iv" I / I d
A
s 6
x
Fig. V-7. The potential distribution, % and the displacement velocity, u, in the Helmholtz model (curves 1 and 1' ), in the diffuse layer model (curves 2 and 2' ), and the effect ofthe structure of the water within the near-wall layer on the rate of displacement, u=u(x) (curve 3)
356 One has to establish to what extent the difference in potential distribution influences the fluid velocity distribution, l)(x), and the displacement velocity, ~)0 ,(the displacement velocity is a limit of 1)(x) function at x -* ~). It is also important to consider such two features of the behavior of the solution in the vicinity of the solid surface, as the diffusive nature of the layer containing an excessive amount of counter-ions, and possible changes in the properties of the liquid phase in the vicinity of the solid surface, related to the action of adhesion forces. One can expect (see fine print further) that the greater diffusivity of the counter-ion layer as compared to that established in Helmholtz model, would only affect the velocity distribution profile of the displacement of individual fluid layers in the direct vicinity of the solid surface. The experimentally observed velocity of the mutual motion of the phases with respect to each other, v0, determined, as in Helmholtz model, by the potential %, will not change significantly (curve 2 ~approaches the same limiting value as curve
1').
This is also confirmed by the fact that the distance between the capacitor plates, 8, which is the only parameter defining the geometry of the system in the Helmholtz model, is not present in the final expression. 4 The thickness of the ionic atmosphere, K-1, may be used as the parameter closest to the distance 8, i.e. 8=l/K. The change in properties of the dispersion medium in the direct vicinity of the solid surface is of great importance. The existence in the
In general, if some parameter is excluded from the final relationship, the property of the system related to this parameter usually has no effect on the phenomenon described by the relationship 4
357 vicinity of the surface of a structured (bound or barely mobile) layer of the dispersion medium of some thickness A , formed owing to the action of adhesion forces, is the reason why only some portion of the electrical double layer, not the entire layer, is involved in motion relative to the surface (Fig. V7, curve 3). As a result, the potential difference in the Helmholtz Smoluchowski relationship (V.25), Aq0, is not equal to the thermodynamic potential, q00, and is determined by another (usually lower) value, ~, referred to as the electrokinetic, or ~-potential:
~;~;0YvE
u0 = ~ . n
(V.26)
To illustrate this, let us isolate in a double layer (Fig. V-8) a flat volume element, dV, of a unit area and of thickness dx, parallel to the surface (dV = dx). The charge in this volume element is given by pv(x)dx, while the force, d~E, acting on it is Egv(x)dx. Due to viscous friction in the moving fluid, the action of all elementary forces, d~E, is transferred to layers of the liquid located closer to the surface. Hence, the net shear stress, ~E (x), created by the external field in some plane x is oo
T,E (X ) - -J dT"E dx . dx ......
X
By equating this shear stress to the force of the viscous resistance in this section x, defined by Newton's equation (V.24), one obtains the expression for the velocity gradient: oo
at) - "cE (x) - I EOv (x)dx. rl--~ X
358 By applying the Poisson equation (III.9) for charge density, Or(X) = -ggod2q)/dx 2, and carrying out the second integration between the limits of 0 and x, one obtains the equation for the velocity distribution within the double layer: X
D(X)
--
X
gO g f l y o
d2q) 2 ~; dx 2 (dx) oo
%
dx
E
0
-Voi
X
Fig. V-8. The scheme ofelectroosmosis, with the changes in the properties of the water in the near-surface layer taken into account In order to determine the macroscopic phase displacement velocity, one has to carry out the integration over the entire double layer, setting the integration limits to x=0 and x = % i.e."
u o - ~o E
i l i d2q) (dX)2 ~- ~ dx 2 0 oo
In order to integrate the above expression, one needs to know how the viscosity, 11, and the dielectric constant, e, change within the double layer. Had these two quantities maintained their bulk values all the way up to the x=0 surface, the macroscopic phase displacement velocity, u0, would have been determined solely by the surface thermodynamic potential, qo0, regardless of the potential distribution in the double layer. Experimental results
359 contradict this argument, since the rate of the electrokinetic phenomena is strongly influenced by electrolytes, among which are those capable of causing only the contraction of the diffuse part of the double layer without affecting the value of the % potential. One may thus assume that the viscosity within the thin near-surface layer is much higher in the bulk due to the "structuring" of the liquid phase. At the same time one can also expect to see substantial lowering of the dielectric constant of the medium. According to existing data, the latter decreases within the near-surface layer from 81 (common value for water) to values close to the dielectric constant of ice (3.1). Let us assume that at distances from the surface less than some characteristic distance, A, the e/r I ratio is very small (i.e., close to zero), while outside this region both the viscosity and the dielectric constant maintain their bulk values (Fig. V-8). The integration of the expression for v0 then readily yields the Helmholtz - Smoluchowski equation, and the electrokinetic potential, ~ = q0~=Ahas the meaning of the potential at the plane where the e/rl ratio experiences a sharp change from 0 to a value typical for the bulk of the phases.
The electrokinetic potential may thus be viewed as the potential at some plane, referred to as the plane of shear (the slipping plane), located within the limits of the diffuse part of the double layer. The plane of shear separates the immobilized part of the liquid phase bound to the solid surface from the remaining mobile part in which the displacement takes place. The curve describing the change in the displacement velocity of the layers of liquid as a function of the distance from the wall, u(x), matches the x axis up to the plane of shear, and at x>A has the same shape as the function showing the change in the potential as a function of distance (see Figs. V-7 and V-8). It is worth remembering here that an increase in electrolyte concentration results in compression of the diffuse counter-ion atmosphere, and the greater "portion" of a decay is attributed to the immobilized layer of the dispersion medium, i.e. at x_
360 ~-potential. The establishment of an exact quantitative relationship between the thermodynamic potential, %, or the potential of the adsorption layer (the Stern layer) potential, q0d, and the electrokinetic potential, ~, is an important and at present unsolved problem. Depending on the thickness of the layer with increased viscosity near the solid surface, the electrokinetic potential may either approach the value of the Stern layer potential or be lower than the latter. In some cases (e.g. for quartz), as shown in studies by D.A. Fridrikhsberg and M.P. Sidorova [10,11], the difference between the electrokinetic and thermodynamic potentials may be related to the hydration (swelling) of the solid surface and the formation of a gel-like layer resistant to deformation, within which a partial potential drop takes place. The difference between q0dand ~ may also be related to microscopic surface roughness of the solids, i.e. to the presence of growth steps, dislocations and other defects (see Chapter IV). There are, therefore, theories that relate the q)d and % potentials, but no analogous theories exist for the q)d and ~ potentials. At the same time, in contrast to the % and q)d potentials, the absolute values of which can not be experimentally measured, the ~-potential is a directly measurable quantity, which, along with the thickness of the ionic atmosphere, is an important characteristic of the diffuse part of the electrical double layer. Based on the Helmholtz-Smoluchowski equation, let us further investigate different electrokinetic phenomena, taking into account the geometric features of real systems. Finally, we will also briefly address other transfer processes that take place in disperse systems.
361
V.4. Transfer Processes in Free Disperse Systems
Let us continue the discussion of transfer processes taking place in the free disperse systems. We will primarily focus on the role that electrokinetic phenomena and electrical double layer play in these processes. The influence of electrical double layer on the transport of current and particles in free disperse systems is related to mutual displacement of portions of electrical double layer separated by the plane of shear. The separation of counterions present around the particle into two portions; - the one moving together with the particle, and that "breaking off" from the latter under the influence of the outer field, allows one to write the peculiar chemical formulas, reflecting the structure of particles surrounded with the double layer. Such formations, referred to as the micelles ofhydrophobic sols, include the a g g r e g a t e (e.g. m molecules of AgI) which together with the layer of potential determining ions (e.g. n Ag § ) form the n u c 1e us which together with the portion ofcounterions located beneath the slipping plane, (n-x)N0~, form the p a r t i c l e . The remaining x N0~ ions, which do not move together with the particle during the electrophoresis form the outer part of a m i cell e" particle a~ggregate
-'~
{m[AgI] nAg+(n-x)NO3}xN03 v._ nucleus micelle In the above formula x defines the effective charge of particles, q = ex. Along
with
dispersed
inorganic
particles,
charged
m a c r o m o 1ec ul es and their aggregates, in particular protein molecules, may
362 also undergo electrophoresis. Depending on the composition of the medium (primarily, on the pH), the magnitude and sign of charge may be different (the same is also true for sols of inorganic amphotheric hydroxides). The charge, in turn, affects the shape of a macromolecule. If macromolecule forms a random coil in which the distance between ions is comparable to the thickness of ionic atmosphere, its motion may be accompanied by the penetration of dispersion medium through the coil. The electrophoretic properties of macromolecules (and their aggregates) that form compact globules are quite similar to those of "regular" colloidal particles. Following the methodology of irreversible thermodynamics, let us discuss the motion of particles and transport of the electric current that occur due to the action of an outer electric field with strength E and gravitational force, Fg. Let us examine the flux of particles and the electric current, remembering that the particle flux is given by their concentration, n, times their velocity, Up; while the electric current is given by the product of charge carriers concentration, n, their velocity, ~, and their charge, q. The action of external electric field on the free disperse system results in particle motion (electrophoresis). The electrophoretic velocity, rE, is not a function of ~-potential only, but also depends on the particle radius, r, and the type of electrolyte present in the system. However, it turns out (see fine print further down) that all of these factors can be simultaneously accounted for by the numerical coefficient, k~, introduced into the Helmholtz-Smoluchowski equation (V.26). If the particles are spherical, k~ changes from 2/3 for particles smaller compared to the ionic atmosphere thickness ( ~ ~ 1) to 1 for large particles ( Kr )) 1). Consequently, the particle flux due to the applied electric
363 field of strength E is given by JE - kl ee~162E ;~x12 - kl ee~162, 1"1 1"1
(V.27)
where (112is the phenomenological cross-coefficient. Along with electrophoresis, the electric field applied to free disperse system causes the flow of electric current, related to both the motion of ions in dispersion medium and charge transfer by particles moving with velocity t)E. The specific electric conductance of the free disperse system, ~,v, is equal to phenomenological coefficient (122 and includes the specific electric conductance of the dispersion medium, ?~0, and the additional conductance caused by moving charged particles. A more detailed consideration shows that for free disperse system (~;~;0q0)2 rn
~22
-- ~ V -- ~ 0 -t-4rck 2
q
where k2 = 2/3 when ~ <<1, and k 2 = 1 when ~:r ~ 1. The above expression includes the square of electric potential, qo, since the latter determines both the charge and velocity of particles. During sedimentation under gravity, particle velocity is given by the ratio of force, Fg, to friction coefficient, B. For spherical particles B is given by the Stokes equation, B=6~qr, and hence the sedimentation flux and corresponding phenomenological coefficient, a~, are given by
n
Jg
- ~ F g
6~rlr
n
= ~ApgV
6rcqr
n
;
all =
6rcqr
364 The sedimentation of charged particles results in the generation of sedimentation current. In agreement with the Onsager reciprocity relationship, the proportionality coefficient between the sedimentation current and the force Fg equals to the phenomenological coefficient a~2 given by eq. (V.26). The separation of charges along the heigh, that occurs during sedimentation of charged particles results in the appearance of potential difference, referred to as the sedimentation potential. This potential difference causes the current flow in the opposite direction. Under the conditions corresponding to dynamic equilibrium between sedimentation current and counter-current (Chapter V, 1), the strength of sedimentational electric field, Eg, can be obtained as:
klee~ ApgV. Eg = )~orl + 47ckz~2e2~2rn
(V.28)
The so-called suspension effect [12] originating from the differences in composition of dispersion medium within the diffuse layer and far away the particle surface, is another peculiarity associated with the role diffuse layers of ions play in sedimentation of dispersed particles. Sedimentation results in the concentration of disperse phase" the particles of density greater than that of dispersion medium get accumulated at the bottom, while those of lower density gather at the top. As a result of such segregation, the particles are separated from each other by distances comparable to the double layer thickness, and thus in the sediment (or in the "cream") the dispersion medium consists mainly of diffuse layers of ions. This leads to the average composition of dispersion medium being different in different parts of the system. Consequently, if the diffuse layer contains the excess of H + or OH ions, the
365 pH of dispersion medium in the sediment differs from that in the medium above it. The ion entrapment during sedimentation of precipitates takes place in geological processes. For instance, this phenomenon may be responsible for the formation of some mineral deposits. Among various electrokinetic phenomena that take place in free disperse systems, electrophoresis is the most significant one for the scientific studies and practical applications. A number of methods for the determination ofelectrophoretic velocity and electrokinetic potential of particles have been developed. These methods include the moving boundary method (a direct study of motion of the boundary between the disperse system and the free dispersion medium due to the applied potential difference), microelectrophoresis (a direct observation of moving particles using a microscope or ultramicroscope), electrophoresis in gels, paper electrophoresis, etc [ 13]. These methods are broadly used to study disperse systems formed with low molecular weight substances, as well as polymers, especially those of natural origin. Electrophoretic methods allow one to separate and analyze mixtures of proteins, and thus are effectively used in scientific research and medical diagnostic applications. The use of electrophoretic methods allows one to coat the surface of electrodes (of cathodes as well as of anodes) with films of various composition. Electrophoretic deposition is more economical than electrolysis, and allows one to apply coatings of complex composition, as well as to carry out deposition in non-aqueous media. The latter is especially beneficial in cases when it is not desirable to carry out electrolysis in water due to the saturation of material with hydrogen gas (the so-called "hydrogenation",
366 which makes some materials more brittle). Electrophoretic method is used to coat cathodes in vacuum tubes with a layer of metal oxide. Let us now discuss in some detail the peculiarities of particle motion during electrophoresis and some other electrical properties of free disperse systems. Electrophoresis usually takes place in a stationary liquid. In a moving fluid the motion of particles occurs only in thin flat gaps and capillaries (microelectrophoresis), where the fluid motion is caused by electroosmosis. If fairly large n o n - c o n d u c t i n g p a r t i c l e s are dispersed in a rather dilute electrolyte solution, the ratio of particle radius to the double layer thickness may be substantially greater than 1, i.e., r/6 - ~ ~ 1. The streamlines of outer electric field surround the particle and are parallel to most of its surface, as shown in Fig. V-9. In this case the particle velocity, v0, can be with good precision described by Helmholtz-Smoluchowski equation. In the case when c o n d u c t i n g parti c 1es undergo electrophoresis the current may pass through the particles, causing a significant distortion in the shape of streamlines near the particle surface (Fig. V- 10). The latter, however, is accompanied by polarization effects in double layers near the particle surface (the generation of overvoltage), and these particles, especially if they are sufficiently small, may behave as non-conducting ones.
E
~././.w/~//& U//_Z////////A
---v~///,y.//Z//~_~/
-
6 = 1/K Fig. V-9. The streamlines of external electric field around a non-conducting particle
Fig. V-10. The streamlines of the outer electric field around conducting particles
367 The double layer of particle that is large compared to the thickness of ionic atmosphere, may be considered to be flat. The motion of particle is related to the transfer of charge, q'~ which is approximately equal to the product of the area of plane of shear, 4~(r+A) 2, and the surface charge density, PA, in the part of double layer separated by the plane of shear, i.e., the plane at which q~(x)=~. The value of PAcan be defined, if one replaces in eq. (III-12) % with ~. Taking this into account, one obtains q~ - 4rc(r + A) 2 x/8~;~;0kTn0 sinh
ze~ ~
4rtr2~:~;0K~"
(V.29)
2kT This treatment is no longer valid in the case of electrophoresis of small particles surrounded by a thick diffuse layer of counterions, i.e. when r/8<
(V.30)
acting on such particle from the side of
external electric field of strength E, is given by F E - 4roe, e, org_,jE.
When the particle moves with a steady state velocity, ~0, the
FE
force is
balanced by viscous force, F n , given by the Stokes equation (eq. (V.7))" Fq - 6rtrlru 0
368 Consequently, the electrophoretic velocity at r,r <<1 equals 2 8~ 0(_y . Oo = -~- q
(V.31)
A comparison of eq. (V.31) with the Helmholtz-Smoluchowski equation (V.26) for the flat surface shows that the only difference between these two equations is in the numerical coefficient 2/3. It was shown by D. Henry that for particles of arbitrary shape at any ratio of particle size to the ionic atmosphere thickness, 8 = l/K, the HelmholtzSmoluchowski equation can be written in a generalized form as: O0
880~-~ -
k 1
where the numerical coefficient k~ is dependent on particle shape and on the ratio of particle size to the thickness of ionic atmosphere. In agreement with eqs. (V.31) and (V.26), the numerical coefficient k~ changes from 2/3 to 1 as the value o f ~ increases (Fig. V-11, curve 1) [8]. For anisometric particles the dimension along the field is the most significant one. Hence, if long rod-like particles are oriented along the streamlines of electric field, the characteristic dimension that is to be compared with the double layer thickness is the particle length, l. Alternatively, if they are oriented across the streamlines, the characteristic dimension is particle radius, r. In the first case the particle length is, as a rule, substantially larger than the double layer thickness, and the streamlines of external field are parallel to the particle surface, and k~ - 1, regardless of the value of particle radius (curve 2). For particles oriented perpendicular to the electric field streamlines, k~ = 1/2 at small values of Kr,
369 and k~ = 1 when ~ is large (curve 3). Large electrically conducting particles and thin rods oriented along the external field are affected by the current passing through them, and thus k~ decreases (curve 4).
-N/
2/3 1/2
I
0.01
0.10
,,
!
J,
1
1
10
100
_
_
Fig. V-11. The coefficient k~ as a function of v,r for spherical particles (curve 1), for nonconducting rod-like particles oriented parallel (curve 2) and perpendicular (curve 3) to the electric field, and for conducting rod-like particles oriented along the electric field (curve 4)
In further studies J. Overbeek, F. Booth, D. Henry, and S. Dukhin examined how the deformation of double layer that occurred due to the applied electric field influenced the electrophoretic velocity of particles (the so-called electrophoretic relaxation effect). For instance, it turned out that at ~1 in the presence of a trivalent counter-ion the deformation of the electrical double layer caused a decrease in the value of k~ by about 1/4. One must account for all these corrections if ~-potential is determined by microelectrophoresis. The contribution of electrophoretic motion of particles into electric conductance of the disperse system, )~v, can be accounted for by introducing a term proportional to particle concentration, n"
370 O0 )~v - )~o + ql n ~ . E In the above equation )~v and
)~0 are, respectively, the specific electric
conductance of the disperse system as a whole, and of the dispersion medium; 1)0is the particle motion velocity; uo/E isparticle electrophoretic mobility, and q~ is the effective charge related to the separation of particles from portion of the diffuse layer of counterions. Assuming that such separation takes place at the plane of shear and using eqs. (V.30) and (V.31), for particles smaller than l/K, one may write that 8g g282r~2 Xv -)~0 +
3
11
n.
Correspondingly, for Kr >>1, using eqs. (V.26) and (V.29) one obtains
~,v - )~0 + 4~
8280rK~ 1"1
n.
Experimental studies by Dukhin et al [14] showed that the specific electric conductance of disperse system depends on the frequency of applied field. These findings can be explained by changes in polarization effects at high frequencies. The presence of dispersed particles may significantly affect the value of d i e l e c t r i c c o n s t a n t of disperse system. In some cases, e.g. in nonaggregated (non-flocculated) inverse emulsions (Chapter VIII,3), the dielectric constant is related to the volume fraction of droplets in the emulsion, V~e~, by the Bruggerman relationship
371
~;v = (1 _ Vrel)3
'
where e is the dielectric constant of dispersion medium. Dukhin has shown that the flocculation (aggregation) of emulsion droplets results in an increase in the dielectric constant to values determined by the volume fraction of flocs as a whole, i.e. together with incorporated dispersion medium. In aqueous systems in which particles are surrounded by a welldeveloped double layer, such as in sols and emulsions, sharp increase in dielectric constant is observed at particular frequencies of external field. The observed unusually high values of dielectric constants typical for such systems (Fig. V- 12) are due to the fact that particles move relatively to the surrounding ionic atmosphere as charges of high magnitude. At high frequencies of external field such motion becomes impossible, and dielectric constant assumes its "normal" values. The studies of such trends in dielectric constant are in the basis of dielectric spectroscopy, which is an effective method for investigation of disperse systems, and in particular of emulsions [ 15].
2000
1000 0
I
0.1
_
I
|
1
10
.,
I
100 ~, kHz
Fig. V-12. Dielectric constant of sols and emulsions as a function of frequency of the outer electric field
372 Now we would like to briefly describe other transfer processes that may occur in free disperse systems. It was already stated at the beginning of this chapter that directed motion of particles may be caused by the action of forces other than those originated from the applied electric field. For example, the existing temperature gradient results in the motion of colloidal particles referred to as the
thermophoresis. In aerosols thermophoresis occurs due to a
higher average momentum of molecules striking the particle on a warmer side as compared to that of those hitting it on a cooler side. The net effect is that the particles translate towards the region containing cooler air. This phenomenon explains the deposition of dust on walls near the cold air outlets. Another phenomenon that may have the same nature is photophoresis: the particles may move due to the action of luminous flux which heats up their surface. This process is different from the one taking place in outer space, where the motion of interstellar dust particles can be caused by a direct action of the light pressure. The gradient in the concentration of substance dissolved in dispersion medium may lead to
diffusiophoresis of dispersed colloidal particles. The
theory of diffusiophoresis was developed by B.V. Derjaguin and his collaborators [ 16]. According to the concepts discussed in their studies, there are two major causes for diffusiophoresis. First, the presence of diffusion adsorption layer (containing ions and uncharged molecules) in a vicinity of the surface and the existence of external concentration gradient of solute result is a complex osmotic pressure distribution near the surface, which causes particle motion. In electrolyte solutions the particle velocity due to diffusiophoresis is proportional to the square of ~-potential. Second, changes
373 in the structure of the electric double layer along the particle surface (EDL polarization) result in the generation of a potential difference, Aqo. In this case the rate of diffusiophoresis is proportional to the first power of (-potential. Diffusiophoresis plays a role in life of microorganisms, allowing them to move towards the sources of substances that are vital for their existence.
V.5. Transfer Processes in Structured Disperse Systems (in Porous Diaphragms and Membranes)
In structured disperse systems, where particles of the dispersed phase form united spatial networks, as well as in porous media with open porosity, the existence of double layers at
interfacial boundaries results in some
peculiarities in the processes of substance transfer and electric current transport. We will devote most of our attention to the discussion of transfer phenomena in an individual capillary, which is the simplest element of any structured disperse system, and then only qualitatively address the peculiarities related to complex structure of porous medium. During filtration the laminar flow of dispersion medium with viscosity 11through a capillary of radius r and length l under the pressure gradient, Ap, is described by the P o i s e u i 11e e q u at i o n: gr 4 Ap
Op- 8n 1 where
Qpis a volume of liquid passing through capillary per unit time; Ap/l
is the pressure gradient in capillary. Under these conditions the flow profile,
374 i.e. the cross-sectional distribution of fluid velocities, is parabolic, as shown in Fig. V-13, a. ~NN\\\\\\\\\\\\\\\\\\\
a
2
~\\\\\\X\\\\\\\\\~,
,\xz
~\NN\\\\\\\\\\\\4\\\\\
b
_
~
\
1/4
\
~
\
\
\
\
1uo ~ \ N N N \ \ N N \ X
"k \ \ \ " '4 . ~ \ \ - < / \_\ \ \ \ \ \ \ \
\
C
Fig. V- 13. The fluid velocity distribution profile in a capillary: a - during filtration, b - during electroosmotic transfer, c - during electroosmotic rise Another example of a direct transfer process is the generation of electric current, ID between two ends of the capillary under the applied potential difference, A~. In this case the strength of the outer electric field in the capillary, E = -grad 9 =
AU?/l, while the magnitude of current, I E, is
determined by the capillary cross-sectional area, g r 2, and by the average electric conductivity of the medium in it, X" At high electrolyte concentration and rather large radius of a capillary, when
I E - g r 2 ~ AttJ
(v.32)
l m
>> 1, the value of X is essentially the same as X0, the conductivity of the dispersion medium. If this condition is invalid one must also account for the current transfer by ions of electrical double layer, where the net ion
375 concentration is higher than in the bulk (see Chapter III, Fig III-12). This contribution of electrical double layer may be taken into account if one introduces the correction for the surface conductivity, )~s, which is excessive electric conductivity of the near-surface layers of the dispersion medium. The average electrical conductivity of the dispersion medium in the capillary can be written as -
9~ -
2 9~0 + - - ) ~ s ,
F
where 2/r is the surface to volume ratio of the capillary. As we turn to the discussion of the cross-processes, it would be worth pointing out that when r,r >>1, the mutual displacement of dispersion medium layers occurs only within a thin layer of liquid in a direct vicinity to the wall. Consequently, the velocity distribution in the medium inside the capillary has the profile shown in Fig. V-13, b. The electroosmotic flux ofthe medium, QE, is thus equal to the product between the capillary cross-section and the net electrioosmotic phase displacement velocity, v0, described by the HelmholtzSmoluchowski equation (V.26), i.e.: 2 EEOC AtIJ ~ E -- 1rr2 D 0 -- 71:r
q
l
(V.33)
In agreement with the Onsager reciprocity relationship, the streaming
current, Ip, generated in the capillary due to external pressure drop, Ap, is given by I p - rtr 2 ggO~ A p .
q
376 The flow of medium leads to the appearance of difference in fluid levels in vessels attached to the capillary. The resulting pressure drop,
Ap=ggAH, causes the counter-flow of dispersion medium, and the flow profile in the capillary is such as that shown in Fig. (V-13, c), i.e., near the walls and in the center of a capillary the medium moves in opposite directions. Under the steady-state conditions, when the net flux of medium is zero (QE + Qp=0), the height of electroosmotic rise, HE, is given by
9g An inverse phenomenon, i.e., the appearance of a steady-state potential difference, A~ E, due to the action of the pressure gradient, Ap, (the streaming
potential) is described by the condition IE + Ip - O, and consequently A ~ E _ ~ 0 ~ Ap nX
(V.34)
In a transition from an individual capillary to a real structured disperse system (membrane or diaphragm), one faces complications related to the actual structure of porous medium, in which the transfer of substance and electric current take place. In such systems all previously described basic relationships remain valid, but the radius and length of a single capillary are replaced with coefficients having particular dimensions, referred to as the "structure parameters". In general, the determination of these "structure parameters" is a rather difficult task, but one may expect that in the description of electroosmotic transfer and the electric conductivity of the structured disperse systems these parameters are included in an identical way, similar to the identical dependence of I E and Q~ on r and l, as shown in eqs (V.32) and
377 (V.33). This allows one to determine the electrokinetic potential of disperse system with an unknown structure. By determining the electroosmotic flux and current passing through the investigated system (provided that the additional amount of electrolyte is added to satisfy the 9~~)~0condition) at some particular value of the potential difference, Aqj, one may estimate the electrokinetic potential from the equation rlL0 QE ee0 IE
Many directions of practical use of structured disperse systems (such as of porous diaphragms and membranes) are related to peculiarities of substance transfer through these systems. In addition to the appearance of streaming currents and potentials, generated during filtration, the c h an g e in the c o m p o s i t i o n of the d i s p e r s i o n m e d i u m occurs. Indeed, since the concentration of co-ions in thin channels is substantially lowered, their transport through these channels is impeded. As the flowing fluid tends to restore its electroneutrality, the counterions also become trapped by these fine porous membranes. The process of removing electrolyte from dispersion medium by filtration through membranes with fine pore size is referred to as the reverse osmosis, or ultrafiltration [ 17,18]. This process is used to remove dissolved salts from water and to purify liquids from impurities, such as, heavy metal salts. To facilitate sufficiently high rate of reverse osmosis, one needs to apply a large pressure gradient to the membranes, which requires the use of highly durable membranes.
378 The reverse osmosis takes place during ultrafiltration of sols - the process of the separation of dispersion medium on the fine porous filter under the applied pressure gradient. The resultant ultrafiltrate may have a substantially different composition from that of initial dispersion medium. Interesting peculiarities of mass transfer processes are observed in fine membranes permeable to ions but impermeable to colloidal particles (semipermeable membranes, e.g. collodium film). If such a membrane separates colloidal system or polyelectrolyte solution from pure dispersion medium, some ions pass through the membrane into the dispersion medium. Under the steady-state conditions the so-called D o n n a n e q u i l i b r i u m is established. By repeatedly replacing the dispersion medium behind the membrane, one can remove electrolytes from a disperse system. This method of purifying disperse systems and polymer solutions from
dissolved
electrolytes is referred to as the dialysis. Let us now look at what happens when the unit volume of disperse system containing n charged particles (or n /NA moles of particles) and c moles of electrolyte (e.g., NaC1) 5 is separated from a unit volume of pure distilled water by semipermeable membrane (Fig. V-14). If the effective charge of the particle is q~ (let's assume that q~>0), the diffuse layers of counterions contain ql n / eN A moles of anions (C1 ions in the present example).
5 It is implied that the concentration c is that in the bulk of a solution at distances R significantly greater than the diffuse double layer thickness, 6=1/~:, i.e. at R )) 5=1/~
379 ,~ ~
n
+
-~-~--AqI +
,e.,
nql
CI" eN A
.-'."
-.-
xNa
+
.,.
,..-
".'" xCI" ~o e. :-5
cNaCl
:..'. ,~ , ~
Fig. V-14. Transport of electrolyte through the membrane
The necessary condition for an equilibrium in the system that is close to an ideal solution is that product of concentrations of ions capable of passing through the membrane has to be the same for solutions on both sides of the membrane (in the case of concentrated solutions one has to account for the activity coefficients of ions). For this equilibrium to be reached, x moles of NaC1 have to diffuse through the membrane into pure dispersion medium. The value of x is thus determined by
c + qan - x l ( c - x ) - x 2, eN a from which it follows that x -
c + [ql n / (eN A )] 2c + [q l n / (eN A )]
c.
(v.35)
When the electrolyte concentration, c, is low, while the concentration of colloidal particles, n, and their effective charge are high, i.e. when c <<
q~n/eNA, the value ofx is close to the initial electrolyte concentration. In the other words, under these conditions essentially all of the electrolyte should transfer into pure dispersion medium. This means that in the case of highly developed diffuse layers of ions and rather compact arrangement of particles, when the ionic atmospheres come into contact, the co-ions (Na + in the present
380 example) are almost completely removed into the distilled water through the semipermeable membrane (these ions are, apparently, removed together with equivalent number of ions of opposite sign). Consequently, when the concentrated colloidal system is separated from the electrolyte solution by semipermeable membrane, the electrolyte will not enter the disperse system if c << q~n/eN A. These phenomena, also taking place in solutions of polyelectrolytes and proteins (for which membranes are impermeable), are important for proper functioning of cells of living organisms. If c >>q~n/eNA, then in agreement with the eq. (V.35) x=c/2, i.e., diffusion results in equal electrolyte concentrations in both parts of the system, the disperse system and initially pure solvent. A more detailed consideration 6 shows that
ql n x - ~ + 4 eN A c
1
(V.36)
Estimating the net equilibrium concentrations of particles and ions in both parts of the system, one obtains for colloidal system (see Fig. V-14, at the left from diaphragm) n+ q__2_ln + 2 N A(c
x)-n+cN
e
1 ql A+-;L 8
To obtain eq. (V.36), let us multiply and divide eq. (V.35) by the 2c -(q~n/eNA) and then neglect the quadratic terms small compared to the quantity q~n/eNA. This results in
6
2c + c[qln / (eNA)]}c
c
qln
4c 2
2
4eN A
x
381 and for the dispersion medium (at the right side of the diaphragm) 1 q~ 2 N A x - CNA + ~2 ~ e~
o
The difference between these net equilibrium concentrations, responsible for osmotic pressure, is n. Thus, the osmotic pressure seems to be caused only by the particles that can't pass through the membrane, while the ions that can pass freely do not contribute to it. Some interesting specifics are observed when the electric current passes through the structured disperse system with channel thickness comparable to the double layer thickness, 6=1/~:. It is worth reminding one here that there is an excess of counterions and a lack of co-ions within the double layer in the vicinity of the surface. The electric conductivity is thus primarily stipulated by the motion of ions of one sign, i.e. the change in the transport numbers, indicating the fraction of current transferred by each ion, takes place. The diaphragms that conduct current predominantly due to the motion of ions of one type are referred to as ion-selective, namely, cationic (diffuse layer is enriched with cations) and anionic (diffuse layer consists primarily of anions). The solid phase surfaces bear negative charge in the case of cationic membranes, and are positively charged in the case of anionic ones. An important application of ion-selective membranes is deionization of water and of colloidal solutions. If the vessel containing saline solution is separated into three parts by anionic (Fig. V-15, left side) and cationic (Fig. V-15, right side ) diaphragms, then by placing anode into the left compartment, cathode into the right compartment, and by passing the electric current, one may remove ions from solution in the middle compartment.
382 Indeed, the cations from the middle compartment will migrate through cationic diaphragm into the right side compartment, while the anionic diaphragm will not permit cations to enter the middle compartment from the left one. In the same way, the anions will migrate through anionic diaphragm towards an anode, and will not be able to enter the middle compartment from the side of a cathode. This method allows one achieve desalinization and purification of water without using pure water, as required in the dialysis and electrodialysis through non-ion-selective diaphragms. Another peculiarity of electrical conductivity of diaphragms with thin capillaries is related to an increased net concentration of ions in the electrical double layer in agreement with eq. (III.20). The concentration of counter-ions causes the electrical conductivity in thin capillaries to increase. Such an increase may be so strong that upon insertion of a diaphragm into solution the current not only will not decrease, but may also somewhat increase. This phenomenon of "capillary superconductivity" was studied by I.I. Zhukov and D.A. Fridrikhsberg in St. Petersburg State University [19].
@
e " An"
An Cat +
t§ ..~ anionic
diaphragm
Cat* "~ :., ii
~i~
A
1.1"~--" cationic diaphragm
Fig. V-15. Deionization of water by elcctrodialysis using ion-selective diaphragms
Our above discussion shows that clcctrokinetic phenomena play an important role in the studies of the electrical double layers. At the same time,
383 these phenomena are also important in nature and in technological applications. The studies of streaming potentials and electric conductivity of rocks allows one to find the location of mineral resources. A method for predicting earthquakes that is based on a steep rise in conductivity of rocks before the beginning of a quake has been recently developed. A number of complex technological problems are associated with the generation of voltage during the transport of non-conducting liquids, especially of crude oil and oil-based products. In agreement with eq. (V.31), low conductivity of hydrocarbon media results in the generation of high potential difference, which is of immediate danger due to inflammability of transported liquids. The fires in tankers and oil storage tanks were sometimes related to the streaming potentials. The major action that one can take to substantiallyreduce the risk of fire is to increase the conductivity of the medium by introducing oil soluble ionic surfactants. Electroosmosis finds practical use in drying the ground, dam beds and building walls. This is achieved by the accumulation of water at catodes. The studies of electrophoresis and other electrokinetic phenomena as well as the investigation of ion exchange (Chapter III), have shown a strong influence of electrolyte composition on the structure of electrical double layer and intensity of electrokinetic phenomena. One may subdivide electrolytes capable of causing such an influence into the following groups [13]:
I. Indifferent electrolytes are the ones that do not change the %potential but influence the ~-potential. Such electrolytes do not contain ions present in the lattice of solid phase, or ions isomorphic to them. Depending on the ratio between the magnitude of charges of counterions present in the
384 original double layer and that of ions of corresponding charge originated from introduced electrolyte, one may identify the following cases. A. The electrolyte containing the same ions as the counterions of original double layer. The addition of these ions causes the double layer thickness to decrease (increase in K).The double layer compression leads to a decrease in ~-potential and results in weakening of electrokinetic effects, up to their complete vanishing. In this case there is no exchange between ions of the original double layer and those of introduced electrolyte. B. The electrolytes that contain ions identically charged with counterions of the original double layer. The ability of ions to enter the dense (Stern) layer is determined by the adsorption potential of counterions" the higher the latter is, the more drastic is the lowering of q)d and ~ potentials by the introduced electrolyte, and the more pronounced is the decrease in intensity of electrokinetic effects. Correspondingly, the ions of the same charge can be arranged in series according to their ability to influence electrokinetic phenomena, their tendency towards mutual displacement out of electrical double layer, and their ability to cause coagulation (see Chapter VIII,5). The series of ions with decreasing adsorption activity are referred to as
lyotropic.
Monovalent cations form the following lyotropic series: Cs +> Rb + > K +>Na +>Li +, i.e., adsorption activity of counterions increases with increase in their size. Similar lyotropic series exist for the bivalent cations: Ba 2+> Sr2+ > Ca 2+> Mg 2+ , and for monovalent anions: .
.
.
.
.
CNS > I >NO3 >Br >C1 .
385 The greater adsorption activity of larger ions is related to their higher polarizability and lower hydration in aqueous solutions, which allows them to approach the solid surface more closely. A peculiar effect that strongly adsorbing ions make on the structure of electrical double layer is that q)d and ~ potentials may not only decrease but also increase; the latter occurs if high adsorption potential is associated with co-ions of introduced electrolyte. In addition to that, strongly adsorbing counterions may cause surface "recharging"" if upon increasing electrolyte concentration, one reaches the point at which the charge of the Stern layer becomes equal to the charge of the surface, the strong adsorption interaction may result in an additional (greater than the equivalent) adsorption of counterions, so that both q~a and ~ simultaneously change their sign. Indeed, the studies of electrokinetic phenomena, and of electrophoresis in particular, show that ~ potential decreases with increasing in electrolyte concentration, and at certain concentration, referred to as the isoelectric point (i.e.p.), becomes equal to zero (Fig. V-16, curve 1). No electrokinetic phenomena take place at the isoelectric point. Further increase in electrolyte concentration results in the reversal of direction of electrokinetic effects. The latter can be observed as the change of the direction of particle motion in electrophoresis. One should emphasize that in the present case the reversal of direction of electrokinetic effects is related to the change of sign of % and ~ potentials at constant % potential. As the electrolyte concentration increases the rate of electrophoresis first increases (due to an increase in the absolute value of q)d and ~ potentials), and then starts to decrease again, which is now due to the ~-potential decrease caused by the compression of counterion diffuse layer.
386
i.e.p. tl 0
Fig. V- 16. The ~-potentialas a function of electrolyte concentration: 1- indifferent electrolytes containing specificallyadsorbed counterions and non-indifferent electrolytescausing change of the sign of % potential; 2 - non-indifferent electrolytes containing ions bearing the charge of the same sign as that of potential determining ions C. Electrolytes containing specifically adsorbed counterions with charge different from that of counterions in the original electrical double layer. In this case, similarly to the previously described one, the recharging of the surface due to change in the sign of the Stern layer potential, tpo, at constant interracial potential, qo0, may take place. This phenomenon of "superequivalent" adsorption of ions with high adsorption potential is most typical for strongly adsorbing organic ions and for large and strongly polarizable polyvalent ions. II. Non-indifferent electrolytes are capable of changing the value of the surface potential, %. These electrolytes usually contain ions that are capable of entering the crystal lattice of the solid, e.g. by isomorphic substitution of ions forming the solid phase. The following characteristic cases can be outlined. A. The ion that is capable of penetrating into the lattice bears charge of the same sign as the potential determining ions. An increase in electrolyte
387 concentration results in an increase of the absolute value of the q00-potential. At the same time, the addition of electrolyte results in the double layer compression. Accordingly, at low electrolyte concentrations the absolute value of electrokinetic potential first increases, and then starts to decrease (Fig. V16, curve 2). B. The ion that is capable of penetrating into the lattice bears charge of the sign opposite to that of potential determining ions. The addition of such electrolytes results in a complete rearrangement of electrical double layer, i.e., in gradual decrease of the absolute value of %-potential down to zero (the isoelectric point), followed by recharging of the surface due to the change in the sign of %. The absolute value of electrokinetic potential first decreases, passes through zero, and then, after ~ - potential changes sign, increases. Further addition of electrolyte results in a decrease in ~- potential due to the double layer compression (Fig. V-16, curve I). Both cases showing the influence of non-indifferent electrolytes on % and ~ values are illustrated in Fig. V-17 by the experimental data acquired by H.R. Kruyt et al for silver iodide [20]. Since for slightly soluble salt the product of concentrations of ions forming the salt is related to the solubility product, Ksp, one only needs one quantity, e.g. pI (the negative logarithm of the concentration of iodide ions in solution), to describe the equilibrium in the system, i.e." [ag § [I-] = Ksp ; pI = -log [I-] = log [Ag+] - log K,p. If the starting pI is greater thanl 0, then moving further to the right (i.e., by adding KI) one obtains case II, A, while moving to the left (by adding soluble silver salt) one obtains case II, B. The isoelectric point, i.e. the point of
388 intersection ofq~0_-%(pI) and ~= ~(pI) curves with the x-axis occurs at pI=l 0.6, which corresponds to the concentration of Ag + ions of 10.5.5 tool dm -3. This indicates that the iodide ions interact with the solid phase stronger than the silver ions. mV
100 / 50
|
_
14 / l O 50
11
,
,
I
9
6
pl
-J/
+; Fig. V-17. ~- and % - potentials of AgI particles as a function of pI
For amphoteric solids and macromolecules containing different ionic groups (proteins, nucleic acids, etc.) the absolute value and sign of thermodynamic surface potential depend on pH of the medium, and some particular pH value corresponds to the i.e.p. For am p h o t e r i c hydroxides the pH corresponding to isoelectric point is determined by the correlation between their acid and base dissociation constants. P r o t e i n m o 1e c u 1e s contain a large number of different acidic and basic groups, present in the side chains of constituent aminoacids, and terminal -NH 2 and -COOH groups having different dissociation constants. The ionic state of a protein molecule in a solution with some given pH is thus determined by a complex equilibrium between different ionogenic groups.
389 Along with the i s o e 1e c t r i c p o i nt determined from electrokinetics, one can also distinguish the isoionic point, which is defined as the pH corresponding to equal number of ionized acidic and basic groups. The position of isoionic point, pHi, is primarily influenced by the strongest acidic and basic groups with the dissociation constants of K a and K b, respectively. For a 1:1 electrolyte the Michaelis equation, pH i - l o g K ~ / 2 + l o g K ~ 2 - l o g K ~ / 2 ' is valid to a good degree of approximation. In the above equation K w is the ionic product of water. If dispersion medium contains no added electrolyte, the isoelectric and isoionic points are the same. In the case when added electrolyte contains ions capable of adsorption, the positions of isoelectric and isoionic points shift in comparison with the value established for pure dispersion medium. The direction in which the isoelectric point shifts is opposite to the isoionic point shift. Indeed, due to the adsorption of e.g. cations macromolecule, previously present in isoelectric (isoionic) state, acquires an excessive positive charge, which perturbs the equilibrium in dissociation of acidic and basic groups of macromolecule. That is, the repulsion between H § ions and positively charged macromolecule results in a higher degree of dissociation of acidic groups, while the attraction between macromolecule and OH- ions suppresses the dissociation of basic groups. Such change in the degree of dissociation of acidic and basic groups often compensates the excessive positive charge acquired as a result of cation adsorption. In order to facilitate the return of
390 macromolecule into the isoelectric state, one must increase the concentration of OH- ions, which through adsorption will fully compensate the excessive positive. Consequently, if specific adsorption of cations takes place, the i s o e l e c t r i c point shifts into the alkaline region.
V.6. Optical Properties of Disperse Systems" Transfer of Radiation
The penetration of different kinds of radiation, namely of the visible light, X-rays, and neutrons through disperse systems represents a special kind of transfer phenomena. The propagation of radiation throughout the disperse system is to a large extent governed by the ratio of radiation wavelength to linear parameters of the system, such as the particle size, dispersion medium layer thickness, and intermolecular distance. Most of our discussion will cover the systems containing particles with sizes much smaller than the wavelength of light (the Rayleigh scattering). We will address more complex phenomenon of light scattering in systems containing large particles and other phenomena related to the transfer of light through disperse systems on a qualitative level only.
V.6.1. Light Scattering by Small Particles (Rayleigh Scattering) When light passes though medium, the polarization of the latter occurs due to the action of the electric vector, E , of the light wave. The atoms (molecules) of the medium acquire an alternating dipole moment which oscillates with the frequency of light wave, v-co/2~-c/~,
391 where co is the angular frequency, )~ is the wavelength, and c is the speed of light in a given medium. In agreement with laws ofelectrodynamics, the oscillating elementary dipoles are the sources of secondary waves with the same angular frequency, co. According to the Huygens-Fresnel principle, in homogeneous and isotropic medium with polarizability a0 the interference of secondary waves causes the light to propagate only in the direction of a primary (incident) wave. If particles or other unhomogeneous species (macromolecules, fluctuational regions) with polarizability a, different from polarizability of the medium, a0, are present, a complete cancellation of waves propagating in directions other than the direction of primary wave does not take place. This results in the diffraction of light on these non-isotropic species and gives rise to
opalescence, i.e. to the scattering of light by small particles. A direct result of opalescence is the visibility of light beam passing through a disperse system, known as
Tyndall effect.
The laws governing the scattering of light can be stated in their simplest form under the following" 1) the scattering particles are small and their shape is nearly isometric, so the largest particle dimension is significantly smaller than the wavelength of incident light beam, i.e. r
392 wavelength of incident beam; and the volume of disperse system through which the scattered light passes is small, so one does not need to account for secondary scattering [21,22]. If all the above conditions are maintained, the action of the oscillating electric vector of the incident polarized 7 light wave with the amplitude E0a, E 0 (t) - E 0acos ~ t , on the dispersed particle (Fig. V-18) results in the appearance of the excessive (uncompensated) in comparison with the dispersion medium dipole moment, given by lad(t)-
4n~oEo(t)Ac~V-47t~:0Ao;VE0a coso~t,
where Aa is the difference in polarizabilities between dispersed phase and dispersion medium; e0 is the dielectric constant. This u n c o m p e n s a t e d oscillating dipole serves as the source of the scattered light. It is known from electrodynamics that an oscillating dipole emits electromagnetic waves. The dipole radiation profile has a cylindrical symmetry with respect to the dipole axis. For this reason the strength of electric field in the secondary wave is determined by the angle q0between the direction of wave propagation and the dipole axis. The oscillating dipole is projected onto a plane perpendicular to the direction of secondary wave propagation (see Fig. V-18), so that the electric field strength, E~c, is
It is worth recalling that the polarization plane is the one in which oscillations of the electric vector of light wave take place. The magnetic vector oscillates in a plane perpendicular to the latter
7
393 proportional to sin q~. At the same time, the electric field strength of the secondary wave decreases with increasing distance from its source (dipole).
I
.,"x &,(0
&(t)
~~
"
E~
Fig. V-18. The incident wave, Eo, the oscillating dipole, g(t), the scattered wave, Esc(t), and the distribution of intensity of the scattered light, Isc The p r i n c i p a l f e a t u r e o f l i g h t s c a t t e r i n g is t h a t the s t r e n g t h o f s e c o n d a r y w a v e is p r o p o r t i o n a l to the a c c e l e r a t i o n of oscillating charges, i.e., proportional to the s e c o n d d e r i v a t i v e of dipole moment with respect to time: 1 Esc (R, q~) -
sinqo d2[.t
4 ~ 0 Rc 2 dt 2 '
while d 2 ~t
d 2 cosmt
dt 2
dt 2
2
1 )~2 "
Hence, the a m p l i t u d e of electric field in the scattered wave is given by
394
Esc (R, q~) -
sing)Acorn 2V E0a . Rc 2
(V 37)
The intensity of luminous flux, I, defined as the energy carried by light per unit time per unit surface area, illuminated at a right angle, is proportional to the e l e c t r i c field a m p l i t u d e in the second power, i.e. I ~ Ea 2. The relationship between the intensities of incident, I0, and scattered, Isc, light can be obtained by raising eq. (V.37) into the second power, i.e." Isc (R, q~) - sin2
q)(Au~)2 RZc 4 034V2 I0 - 16n: 4 sin2 q)(au~)2 V 2 R2)~4
Io"
The intensity of light scattered by particle that is a Rayleigh scatterer is thus proportional to the square of particle volume and inversely proportional to the fourth power of wavelength of an incident beam, i.e.
Isc
~ V2 / )4 .
It is implied here that the angular frequency, m, is far from the resonance frequencies of electronic oscillations (atomic absorbance lines) and that Aa is independent of m. It is important to emphasize that the described processes of
opalescence represent a case of"elastic" scattering, i.e., there is no change in the wavelength of light, in contrast to such phenomena as luminescence
(fluorescence, phosphorescence) and Raman scattering. Thus, when the system is illuminated with a monochromatic light, the opalescence has the same color as the incident beam. When the system is illuminated with white light, the preferential scattering of short wavelengths predicted by the Rayleigh
395 equation results in a blue color of opalescence. The blue color of sky is thus caused by the scattering of light on heterogenities present in the atmosphere. When the concentration of scattering centers
is small, the light
scattered by different particles does not undergo an interference, and the s u m m a t i o n of intensities of light scattered by all particles takes place. As a result, the intensity of light scattered in a given direction is proportional to the number of particles. The intensity of light scattered by a unit volume, Iv, is proportional to the particle concentration, n"
IV
-
16~r4 sin2 (D(Ao~) 2 n V 2 R2)~4
I0"
The above equation is known as Rayleigh's Law. In order to replace polarizabilities with
refractive indexes, one uses the Lorenz - Lorentz
relationship, namely 4
n 2 -1
3
n2+2
Rayleigh's Law can then be written as
I v _ 9g 2
n _n2 .) 2 n V 2 ~sin n 2 + 2n2o R 2)k,4
2 qolo,
(V.38)
where no is the refractive index of the medium. An important characteristic that describes the ability of system to scatter light is the t u r b i d i t y , ~ (m-~),
396 given by
z-
24~ 3
n
-n~ n 2 + 2n 2
.)
nV
2
)~4 "
(V.39)
The spacial distribution of intensities of light scattered by disperse system (Fig. V-19) can thus be described by the surface of rotation of sinZq~ function around the q~=0axis (bagel without a hole). The cross-sections of this surface by planes yield indicatrixes of the scattered light. y
I
x
a
b
Fig. V-19. The indicatrixes of light scattered in planes parallel (a) and perpendicular (b) to the axis of oscillating dipole One usually considers the cross-sections by xOz plane, which coincides with the polarization plane of the incident beam, and by xOy plane, which is perpendicular to the former plane and in which oscillations of magnetic vector of primary wave lie. The scattered light intensity is usually viewed as a function of angle 0 between the directions of incident and scattered waves. For an indicatrix in the plane of polarization xOz, Ill, the angle 0 is related to the angle q~ as 0 - 900 - cp. Consequently, taking into account eq. (V.39) one obtains
397 3 "c III - 871; R 2 I~ cOS2 ~"
(V.40)
In xOy plane, perpendicular to the plane of polarization, the angle qo=90 o at all values of 0, and the corresponding cross-section I• of the surface
Iv is a circle (Fig. V-19, b)" 3
I• -
T ~ I
(V.41)
8~ R 2 0-
The total luminous flux, ~ scattered in all directions by a unit volume of disperse system (the total energy scattered by a unit volume of disperse system per unit time) can be determined by integration ofeq.(V.38) around the sphere" 7~
9g - I Iv ((P )2~ R sin q~Rdq) . 0
This yields 8
n 2 + 2n 2
)24
I 0 - zI 0.
In the case when incident wave is c o m p l e t e l y p o l a r i z e d in xOz plane, the scattered light is completely polarized as well. However, the
8 The independence of 9g upon R follows from the conservation of energy law applied to light wave coming from the scattering region. This is exactly why the intensity of scattered light is inversely proportional to the square of distance from the scattering region, while the strength of electric field of the wave is inversely proportional to the distance, R
398 orientation of polarization plane in the scattered light changes depending on the direction in which light propagates" the polarization plane always coincides with the direction of propagation and is always perpendicular to the horizontal plane xOy. The scattering of unpolarized light is more complex. In the latter case the incident beam can be decomposed into two components with equal intensities that are polarized in horizontal and vertical planes, respectively. Both of these components are scattered independently of each other and have intensities of I0/2. Let us look at the result of scattering ofunpolarized light in the horizontal plane xOy (similar treatment is valid for any other plane). The light beam polarized in vertical xOzplane yields in xOyplane the scattered light with same intensity in all scattering directions. The intensity of scattered light is 1/2 ofthe intensity I. identified by eq. (V.41). This portion of scattered light is vertically polarized and its intensity, IvERw,can be conveniently represented as
Z_L
IvEav - 2 (sin2 0 + cos 2 0). The light polarized in a horizontal plane yields in that plane the scattered light that is horizontally polarized as well. In agreement with eq. (V.40), the scattering intensity, IHOR, equals 1/2IH, i.e., it is proportional to cos20. The total intensity of light scattered in a given direction in the case of unpolarized incident beam is thus given by half the sum of eqs. (V.40) and (V.41):
12 =
3 1: 16rt R 2 I~ (1 + cos 2 0).
399
It follows from expressions for IVERTand IHORthat at all angles different from 0 ~and 180 o IVERV>IHoR, i.e., the scattered light is either partially or completely polarized. The difference between IVERVand IHoR is, therefore, the intensity of completely polarized portion of scattered light, I p" 3
T
Ip = 16re R 2 I~ sin2 0.
The intensity of unpolarized portion of scattered light, I u, is twice the intensity of horizontally polarized light, i.e." 3
"~
Iu - 2IHoR - I I I = 87t R 2 I~ c~
0.
The degree of polarization of the scattered light changes from 0 (at angle 0 - 0 ~ or 180 ~ to 1 (at angle 0 = 90 o or 270~ Figure V-20 shows the indicatrixes of horizontally and vertically polarized light (in xOy plane), IHoR = I1/2 and IVERV= I~/2, respectively, the intensity of polarized light, I p, that
of unpolarized light, I ,, and the total intensity of scattered light, I z, as a function of angle 0. The polarization of light is the second important feature that allows one to differentiate between opalescence and luminescence. In the case of the latter light is not polarized. The expression (V.39) for turbidity does not contain any information on polarization of light, and hence it can be applied to the scattering of unpolarized light as well.
400
&
h HOR
Fig. V-20. The indicatrixes of Rayleigh light scattering: Ia is the total intensity ofthe scattered light; /HOP. and IVERT are the intensity of horizontally and vertically polarized light, respectively; I. and l p are the intensities of unpolarized and polarized light, respectively It follows from the energy conservation law that due to scattering the intensity of transmitted light beam decreases by the amount corresponding to the total intensity of light scattered in all directions (provided that no light is absorbed). The intensity of light transmitted in the direction of propagation of primary wave, x, should be thus viewed as being a function of coordinate, i.e. I =
I(x).
One can then write a relationship between the intensity of
transmitted light, I, at a point x and the decrease in this intensity due to scattering by a volume with unit cross-sectional area and thickness dx: dI - -~dx
- -zldx.
The integration of this equation yields the Beer-Lambert law: I - I 0 exp(-l:x). In the above expression I0 is the intensity of luminous flux at a point where x - O, and z is the turbidity. The quantity z is also referred to as the imaginary absorption coefficient. It is worth reminding here that 1:includes the
401 concentration of scatterers as a multiplying factor. The quantity ~ is the inverse value of distance at which the intensity of transmitted light beam decreases by a factor of e. The quantity ~ = log (I0 / / ) ~ 0.43 ~x is referred to as the extinction or the optical density of the system.
In agreement with eq. (V.39), in the case of Rayleigh scattering the turbidity is proportional to concentration and squared volume of dispersed particles and inversely proportional to the wavelength of light in the fourth power. Since the turbidity during Rayleigh scattering experiences a sharp decrease with increase in wavelength, the illumination of the system with white light results in a red color of transmitted beam. In nature this phenomenon can be observed during the sunrise and sunset when the sunlight that passes through a thick layer of atmosphere lacks blue component. The scattering of red light by clouds makes the usual picture that one can see during sunrise or sunset. The turbidity also increases with the increase in the difference between the refractive index of dispersed phase, n, and that of dispersion medium, no. In the case when An - n - no <
(n 2-n2) 2 4(_~o) 2An n2 + 2n 2
~ -~
.
(V.42)
If the weight concentration of the dispersed substance is constant, Vn = = const, the turbidity of the system is proportional to the volume of dispersed particles and increases when particle size becomes larger due to coagulation or Ostwald ripening (see Chapter VII). It is, however, important to realize that if the particle size increases to the extent when it becomes comparable with
402 the wavelength of incident beam, the system no longer obeys the Rayleigh equation.
V.6.2. Optical Properties of Disperse Systems Containing Larger Particles The particles behave as Rayleigh scatterers if their radius, r, is less than 0.1 - 0.05)~. In the latter case all molecules that make the particle are polarized in the same phase, and the entire particle in a luminous flux may be viewed as one oscillating dipole. For particles with sizes comparable to the wavelength of incident beam, the polarization of molecules does not occur in the same phase, and the resulting dipole moment, ~td,is not proportional to the particle volume. As a result, the intensity of scattered light and the turbidity of the system are no longer linear functions of particle volume at constant volume fraction (i.e., concentration) of dispersed phase. The graph of ~(r) passes through a maximum when particle size is ~ V3 (Fig. V-21). However, with respect to the light scattered in directions close to the direction of the transmitted light beam, the oscillations of molecular dipoles are in closer phases and consequently are summed. Oppositely, for back scattered light these oscillations may occur in opposite phases, which results in a substantial decrease in the intensity of back scattered light (Fig. V-22). In addition to that, in particles that are comparable to the wavelength of light, the electric quadrupoles are induced along with dipoles, while in even larger particles the multipoles of higher order may be generated (Fig. V-23). It was shown by G. Mie, G. Blummer, R. Gans, et al that for such large particles the scattering diagrams are substantially different as compared to those observed with the smaller ones. The diagrams contain clear minima and maxima the number of
403
I.oA xl
0.1
0.2
0.3
0.4
0
r / ~.
Fig. V-21. The turbidity, ~, as a function of the r/)~ ratio
Fig. V-22. Light scattering indicatrixes for a system containing particles with size not sufficiently small as compared to the wavelength of incident beam
Fig. V-23. A scheme of quadrupole generation
which increases with increasing size of dispersed particles (Fig. V-24). This phenomenon, predicted by Mie theory, was observed by Victor La Mer with fairly monodispersed sulfur sols [23].
The color of these sols
upon
illumination with white light depended on the angle of observation, yielding the Tyndall spectra of different orders. Depending on particle size, different number of spectra orders was observed; as particles grew larger the polarization picture became more and more complicated. In polydisperse systems in which particles even slightly differ in size, the maxima due to the particles of different sizes overlap, and the scattering indicatrixes appear smoother.
404
,4
II
i
f ~
r
"4
v
Fig. V-24. The scattering indicatrixes for monodisperse systems containing large particles For polydisperse systems the dependence of scattered luminous flux on light wavelength is given by ~~
)C x
(V.43)
where power x is a function of particle radius, r. Theoretical and experimental studies indicate that the type of x(r) function is dependent on the relation between refractive indexes of dispersed phase and dispersion medium. Figure V-25 shows types ofx(r) curves obtained for large (a) and small (b) difference
2.6 2.4 2.2 n/no = 1 . 5
2.0 n/n o = 1 . 1
1.8
0
40
80 a
120
r, am
0
i
i
2
4
,
1
1
6
8
1
L
2nr/~
b
Fig. V-25. The power x as a function of particle radius, r, for large (a) and small (b) difference in refractive indexes
405 between refractive indexes. Using such graphs as calibration curves one may estimate the particle size from experimentally determined 9g(;~) dependence. A number of modern methods utilized in the studies of disperse systems and polymer solutions are based on this principle. The example of such methods include the "turbidity spectrum" method and spectroturbidimetric titration. The theory developed by Mie and his successors also explains the phenomena of light scattering and light absorption by the e l e c t r i c a l l y c o n d u c t i n g p a r t i c l e s and particles that s p e c i f i c a l l y absorb light due to their own color. In the latter case a decrease in luminous flux propagating through the system is caused not only by imaginary absorption due to scattering, but also by true light absorption by particles, followed by the conversion of luminous energy into thermal energy. In the case of electrically conducting particles the curves showing light absorbance as a function of wavelength contain maxima, which position is influenced by the particle size as well. The experimental studies carried out with the gold sols containing particles of different sizes showed that the maximum of light absorbance shifted towards lower wavelengths as the particle size decreased. This finding agreed well with the theoretical treatment. For relatively coarse gold dispersions the scattering of light and fairly weak true absorption occurs within the orange region of the spectrum. This results in blue or violet color of such sols in the transmitted light and in the dark red opalescence due to some change in color upon partial absorption of light. As dispersed particles become smaller, the maximum of true absorption shifts into the yellow-green part of the spectrum gradually approaching the
406 m
yellow absorption band of AuC14 ions. Consequently, the color of such sols in transmitted light changes to red (for particles ~40 nm) and further to green and yellow (for much smaller particles). In finely dispersed sols blue opalescence is observed. For a long time the sols of various colors with opalescence enhanced by multiple scattering, have been used in preparation of pigments and colored glasses. For instance, ruby red glasses are the colloidal solutions of gold in glass. The weight fraction of gold in such solutions is ~10 .4 %. In an analogous way the natural and synthetic precious stones and gems may acquire various colors. The systems containing dispersed particles with a n i s o t r o p i c p o l a r i z a b i l i t y possess some unique optical properties. The axis of dipole induced by primary wave in such a particle does not coincide with the direction of electric vector in incident light wave. As a result, upon irradiation with polarized light, the dipole moments of chaotically distributed particles form different angles with respect to the initial direction of polarization. This leads to the appearance of perpendicularly polarized components in the scattered wave, i.e., partial
depolarization of light occurs (Fig. V-26) [21 ].
( Fig. V-26. A scheme of partial light depolarization
407 Krishnan [24] studied other causes for partial light depolarization during scattering, related to nonobservance of r
408 refraction and to obtain information on particle shape, one measures double refraction, n ~ - %, as a function of refractive index of the medium. Birefringence has a minimum for rod-like particles (Fig. V-27), and a maximum for flat ones. The double refraction that is observed at the point of extremum is caused solely by particles' own anisotropy. n~- n~o
I ,
_
(n~-n.,) of particles
J
no = n
,
,
,
~
no
Fig. V-27. Birefringence in disperse system as a function of the refractive index of dispersion medium Birefringence may occur in combination with preferential absorption of one of the refracted beams, referred to as the dichroism. In this case the disperse system may serve as an effective light polarizer. The action of polarizing films, such as those containing co-oriented crystals of quinine iodine sulfate, is based on this phenomenon.
V.7. Transfer of Ultrasonic Waves in Disperse Systems. Acoustic and Electroacoustic Phenomena
Over the last 50 years the transfer of ultrasound through disperse systems has been the subject of numerous theoretical and experimental studies. Depending on the power level of the ultrasound, its applications can
409 be subdivided into high and low intensity ultrasonics. The power levels associated with the propagation of high intensity ultrasound are so high that physical changes in the material through which ultrasound is propagated take place. In colloid science the use of high intensity ultrasound is common in dispersing aggregates, emulsifying liquids, forcing the detachment of adhered particles, defoaming, etc. Leaving these applications of ultrasound outside of the scope of this book, we will focus our attention on low intensity ultrasound applications that are non-destructive to colloidal systems. Special interest in these applications is related to the possibility of using ultrasound for the characterization of concentrated disperse systems.
V.7.1. Theoretical Principles of Ultrasound Propagation Through Disperse Systems (Acoustics) Let us begin our discussion with a description of a plane ultrasonic wave in terms of the displacement of a particle, ~, from its equilibrium position as a function of the distance that wave has traveled. Let us assume that the wave propagates along the x coordinate. We can then write the general wave equation as
C~2~_(k) 2 6~2~ c3t2 ~X2'
(V.44)
where m is the circular frequency and k is the wave number. When acoustic waves pass through heterogeneous systems they undergo attenuation, which results in a decrease in the wave's amplitude, and is related to different forms of energy loss or to the deviation of the propagation direction due to
410 scattering. With attenuation taken into an account, the wave number can be written as k - co / c -
ic~,
(V.45)
where c is the velocity of ultrasonic wave propagation through disperse system, a is the attenuation coefficient (or simply attenuation), and i is the imaginary unit, i.e. i2 - -1. From expression (V.45) it is evident that the ultrasound velocity c - c0/Re[k], and attenuation a = = Im[k]. A convenient form of the solution of eq. (V.44) can be written as r - r
expi (cot- kx),
where ~0 is the initial amplitude of the particle displacement. The attenuation, a, is expressed in either Napers per meter, Np m l , or in decibels per meter, dB m -~, where 1Np - 8.686 dB. The amplitude of the wave undergoes an exponential decrease as the wave propagates through the system, namely A - A 0 exp (-u~x), where x is the distance the wave has traveled in the disperse system; A0 is the initial amplitude of the ultrasonic wave, and A is the amplitude at position x. The attenuation a includes contributions from losses of different types, each of which is described by a partial attenuation coefficient. Six mechanisms of the interaction of ultrasound with a disperse system leading to attenuation of the incident ultrasonic wave are summarized in recent reviews by Dukhin [26,27]. These are viscous ((lvisc), thermal ((lth), scattering (asc), intrinsic (l~int) ,
411 structural,
(~str),
and electrokinetic (%~) attenuations. Expressions for the
attenuations associated with different types of energy losses have been derived in numerous studies, among which the theory of sound attenuation developed by Epstein and Carhart [28] and Allegra and Howley [29], further referred to as the E C A H theory,
is the most well known. The ECAH theory yields
expressions for the first four types of attenuation mechanisms. The theoretical treatment of acoustics is greatly simplified in the so-called long-wavelength regime, in which the particle radius, rp, is much smaller than the ultrasound wavelength, )~. In the long-wavelength regime the attenuations due to viscous, thermal, intrinsic and scattering losses may be treated independently from each other, and one may thus write for the total attenuation, a:
-- ~ v i s c + ~ t h + ~ s c + ~ i n t
9
Following Dukhin and McClements [26,27,30,31 ], let us give a brief description of each of these mechanisms of acoustic energy loss. A detailed theoretical treatment of acoustics is rather cumbersome, and will not be discussed here. We, however, will present the principal results of the ECAH theory when discussing different types of ultrasound attenuation below. Viscous losses of acoustic energy occur due to the oscillation of particles in an acoustic pressure field. The oscillations of particles are caused by the difference in density of the particles and the dispersion medium. As a result of these oscillations the liquid layers in the vicinity of the particles are involved in non-stationary sliding motion, referred to as a shear wave. The shear waves generated by the oscillating particles dampen exponentially in the vicinity of the particles. The extent of penetration of the shear wave into the dispersion medium is characterized by the viscous depth, 8v~, which is the distance at which the amplitude of the shear wave decreases by the factor of exp. The viscous depth is given by
412
visc
;
I Jm '
where v k is the kinematic viscosity of the dispersion medium. According to the ECAH theory, in dilute suspensions of solid particles the attenuation due to viscous losses can be written as [29]
visc -=~
18(m/c)(l_Pm/Pp)2y2 (y + 1)
4y4(m+2]
/
2
pp
(V.46)
2
2'
k,.ppJ
where Pm and pp are the densities of the dispersion medium and the dispersed particles, respectively, ~ is the volume fraction of the dispersed phase, and Y = rp / 8v~sc. This expression indicates that viscous attenuation is a function of the ultrasound frequency, the ratio of the particle radius to the viscous depth and the density gradient between the dispersed particles and the dispersion medium. The higher the density gradient, the more pronounced the viscous losses. This mechanism of acoustic energy loss is typical for small rigid particles with sizes below several microns, such as many pigments and mineral oxide powders. The above expression exhibits the strong dependence of%sc on the parameter Y, i.e., on whether the viscous depth is small or large compared to the particle radius. In these two cases substantially different types of dependence of aviscon ultrasound frequency, m, and on the particle radius are obtained. Indeed, for the case when 8visc>>rp, i.e., when Y -~0, which corresponds to the low-frequency limit, eq. (V-46) becomes
_ ~012 2 Pm Otvisc--9 rP7
/ 02 OP -
pm
((3visc > > r p )
,
and, consequently, when ~visc << rp ( Y -~ ~), corresponding to the high-frequency limit,
413 eq. (V-43) can be written as
__ 9* (1)1/2 1] (l--Om/Pp) 2 (Zvisc -- ~ /"pC (2+Om/Pp
(~visc <<rp)
Thermal losses occur due to the temperature gradients near the particle surface. Thermal losses arise due to the thermodynamic temperature-pressure coupling and are typical in the systems in which particles are involved in pulsating motion caused by the ultrasonic wave. Pressure-temperature coupling means that the temperature of the pulsating particle fluctuates relative to the dispersion medium, and thus heat flows across the interface. In contrast to the density gradient, responsible for oscillatory motion leading to viscous losses, pulsating motion is attributed to the difference between the adiabatic 9 compressibility of the particle and the surrounding medium. Consequently, this mechanism is typical for soft particles, such as emulsion droplets and polymer latex beads. Like shear waves, thermal waves also dampen exponentially in the vicinity of the particle. By analogy, one can define the thermal depth, fith, as the distance from the particle surface at which the amplitude of the thermal wave decreases by a factor of e. Thermal depth is defined as
m -- I ' 2"Cm th -- ~ th ~PmCp m
where Tmis the heat conductance of the dispersion medium, and Cp is the specific heat of the medium at constant pressure. Since the dispersion medium in the vicinity ofthe interface with the pulsating particle, also undergoes pulsating motion, a similar quantity, ~t~ can also be defined for the dispersed phase, in which case in the above expression the indices "m" will P
be replaced with "p". The 5th characterizes the distance inside the d i s p e r s e d phase at which the amplitude of the thermal wave decreases by a factor of e. Like viscous depth, thermal depth is a function of the ultrasound frequency, co. The
9 The term "adiabatic" emphasizes that strain variations due to acoustic pressure occur so rapidly that the system does not have enough time to retum to a state of thermal equilibrium.
414 expression for the ultrasound attenuation due to viscous losses (in the long-wavelength regime) can be written as [28,29]'
Or
3(~TCPmq:O = 24
(
[~m
[~P
PmCpm
pC p
t2
Re[H]
(V.47)
where, as before, the indices "m" and "p" refer to the medium and the particle, respectively, 13is the thermal expansion coefficient, Cp is the specific heat, T is the temperature, and the function H is defined as
[1 + ( 1 - i)X m ] [ ( 1 - i)Xp - t a n h ( 1 - i)Xp] H~
[(1 - i)Xp - tanh(1 - i)Xp ]+ [1 + (1 - i)X m ](1: m/T p ) tanh(1 - i)Xp ]
where functions Xp and X mare analogous to the function Y in eq. (V.46), i.e., Xp=/"p/StPh,and m
Xm=fp/~th. Depending on the magnitude of the thermal depths in comparison with the particle radius, one can obtain a simplified expression for 0~thcorresponding to the cases of high and low frequencies. In the case of low frequencies (Xp<<1, Xm<
~)(0 2 (Xth =
CTPmPp(C p) 6Tp
/
m PmC?
_
--)2I 1
P ppC p
"c P 5'1; m
rp >> ;5th m ,t'p > > and in the case of high frequencies (Xp>>1, Xm))l)'
~Ph ) ,
415
~ / P m C ? l ; mPpCPl: p ~th -- 2x/2-rp
cTPm m _
PmC?
p
Pp Cp
m + ppC %p (rp < < S mth ,rp << ~3Ph) .
The above two expressions indicate that there is a similar type of dependence of the attenuation coefficient on the ultrasound frequency and the particle size at low and high frequencies for thermal and viscous losses. Scattering losses occur by a mechanism different from those of thermal losses and viscous losses. In fact, the mechanism of acoustic scattering is similar to the scattering of light. Acoustic scattering does not result in any dissipation of acoustic energy; the attenuation caused by scattering losses occurs due to partial redirection of the wave by particles, as a result of which a portion of the ultrasound does not reach the sound transducer. Depending on whether the particles are involved in oscillatory or pulsating motion (for emulsion droplets or soft solid particles), or both, one distinguishes between the dipole and monopole contributions to scattering. As shown in Fig. V-28, pulsation leads to the generation of a monopole scattered wave, while oscillation results in dipole wave generation. Soft particle /
Incident wave .....~[[~.~[,...::..... Attenuatedwave
1
Dipole scattering (oscillation)
Monopole scattering (pulsation)
Fig. V-28. The generation of a monopole scattered wave by pulsation and a dipole scattered wave by oscillation [32]. The physical reasons for the oscillatory and pulsating motions are the same as those described above: oscillations are caused by the density gradient between dispersed phase particles and
416 the dispersion medium, while pulsations are governed by the difference in adiabatic compressibilities. The waves scattered from individual particles are characterized by socalled scattering coefficients, whose values depend on the thermophysical properties, ultrasonic frequency and size of the dispersed phase particles. In the long-wavelength regime there is no coupling between the pulsating and oscillating motions, and hence scattering can be described by only two scattering coefficients,
Apuls and Aosc" m
Apuls ~-, ( T m - T P - 1 ~m
," / o s c --~
(2+9
m /lOp )2 "
The corresponding expression for attenuation due to scattering can then be written as [26,32]
43(t2(t2
co rp ~)
~
2c 4
1 Ym 7 3 7m -
p
+
Pp -Pm 2pp+gm
,
(V.48)
where 7mand lip are the adiabatic compressibilities for the dispersion medium and the disperse phase, respectively, and Cmis the velocity of ultrasound in the dispersion medium. It is worth emphasizing here that the attenuation due to scattering is a strong function of frequency. Scattering losses are typical for large particles (>3 gm) and high frequencies (> 10MHz) [29]. This mechanism of energy loss is predominant in emulsions. Scattering of ultrasonic waves dominates over other types of losses in the intermediate wavelength regime ()~-rp), which represents the case of a system with rather coarse particles (> 10 gm) subjected to rather high frequencies (>10 MHz). In the IWR the interaction of ultrasound with the disperse system results in the generation of many types of different waves, and thus many more scattering coefficients are required for the evaluation of the attenuation due to scattering. Intrinsic losses are related to the molecular-level interaction of ultrasound with the material of the homogeneous phases making up the disperse system. Ultrasonic waves undergo partial attenuation when they propagate through any homogeneous system. For example, in water this attenuation is very low, - 20 dB/cm at 100 MHz [33]. In disperse
417 systems the intrinsic losses are dependent on the concentration of the dispersed phase:
(Xint --
(1-~)(Xm
+ (~@p
,
(V.49)
where am and ~p are the attenuations in the homogeneous phases making up the disperse system. Both % and % are frequency-dependent; they are approximately proportional to c02. Structural losses are characteristic of structured disperse systems. In free disperse
systems these losses play a role at high concentration of dispersed phase. Structural losses are related to the oscillations of the network of particles, which may be viewed as a number of interconnected oscillators. As pointed out by Dukhin [26,27], structural losses provide a link that bridges acoustics with rheology. At this time there is no adequate theoretical description of this energy loss mechanism. Eleetrokinetie losses are associated with the presence of the electrical double layer.
Oscillations of charged particles in the field of the ultrasonic wave result in the generation of an alternating electric field, and thus a portion of the acoustic energy becomes converted into electric energy and then irreversibly to heat. Although the contribution of electrokinetic losses to the total ultrasound attenuation is negligible in comparison with attenuations due to other mechanisms, the appearance of the alternating electric field is the basis for the so-called
electroacoustic phenomena. The theoretical description of electroacoustic phenomena is mathematically complex and far beyond the scope of this book. Thus, we will here only briefly explain the origin of these phenomena and emphasize their significance.
V.7.2. Electroacoustic Phenomena
In 1933 Debye published a theoretical study in which he predicted the origination of an electric field, the so-called ionic vibration potential (IVP), upon the passage of ultrasonic waves through electrolyte solutions [34]. Debye outlined that ultrasonic waves should cause the separation of charges due to the differences in the effective masses and friction coefficients of the solvated anions and cations, and suggested that such an effect might serve as a means
418 for determining the of masses of the electrolyte ions. It took sixteen years to experimentally confirm the Debye effect. In the meantime, however, a similar effect, referred to as the colloid vibration potential (CVP) was discovered in colloidal suspensions.
The intensity
of the CVP is several orders of
magnitude greater than the IVP, and thus it is not surprising that it was discovered earlier than the IVP. The nature of the CVP is very similar to the nature of sedimentation potential (the Dorn effect) and relaxation effects in electrophoresis (See Chapter V,4), i.e. CVP originates from the polarization of the electrical double layer due to the relative motion between the charged particle and the electrolyte solution in the field of the ultrasonic wave. In the case of the sedimentation potential, the ion atmosphere lags behind the settling particle, while in the case of the CVP it runs ahead of the particle due to higher density of the latter, Pp > Pm" The ion transport in the Dorn effect produces a fixed electric field, and in this sense the polarization of the double layer is static. In the case of the CVP one is dealing with the d y n a m i c p o l a r i z a t i o n of the electrical double layer, i.e. the electric field constantly changes in magnitude and direction, as shown in Fig V-29. +
+@+ +
+
+
r
E
-I-
+ +
o (3.
.~_ a
13 ave
propagation direction r 12. +
+ +
Fig. V-29. Schematic representation of dynamic polarizationof the electrical double layer in the field of acoustic wave, leadingto the appearance of colloid vibrationpotential (CVP) [35]
419 The alternating potential has the same frequency as the original ultrasonic wave. Due to the opposite directions of the vibration-induced electric fields, the phase of the CVP generated in a suspension of positively charged particles is shifted with respect to the CVP in a suspension of negatively charged particles by the angle n.
According to O'Brien [36,37], in dilute dispersion the CVP is related to the dynamic
electrophoretic mobility, gdyn,as r (Op-Pm) CVP
=
~ dyn (cO) Vp
,
(V.50)
Pm)~0 where ;~0 is the electric conductance of the background electrolyte solution, and Vp is the pressure gradient generated by the ultrasonic wave. The frequency-dependent dynamic electrophoretic mobility is defined similarly to the ordinary electrophoretic mobility as the ratio of particle oscillation velocity, Up, to the strength of alternating electric field, E, i.e.
Up(O) E(c0)
~tdyn(03)-
"
The dynamic electrophoretic mobility is a useful system characteristic. For instance, it provides a means for determining the isoelectric point from electroacoustic measurements. Other properties of colloidal dispersions, namely ~-potential and particle size, can also be obtained from measurements of dynamic electrophoretic mobility. For a dilute dispersion (less then a few weight percent) of spherical particles, the dynamic electrophoretic mobility is given by a Smoluchowski-type relationship [37,38]:
~dyn -- 3~ G
n
1
where the parameter G describes the effect of inertia forces, and the (1 +J) factor accounts
420 for the tangential electric field at the particle surface. The quantity )~s is the surface conductance of the double layer, representing the enhanced conductance of the particle surface due to the presence of the electrical double layer. Lyklema [9] refers to the ~s/)~0rp ratio as the Dukhin number. Both G and fare complex functions, so the dynamic electrophoretic mobility is a complex function as well, i.e. one can write that ~ d y n - - gdyn [(COS ~ + i s i n ' 3 ) 9 Thus, by plotting the real part of [adyn, Re[gdyn] versus the imaginary part, |m[gdyn], one can obtain the
magnitude, I~tdyn[, and the phase angle, i~, of the dynamic electrophoretic
mobility. By fitting gdynto eq. (V-5 l), and then extracting real and imaginary parts, one can estimate the electrokinetic potential, ~ (from [gdy~[), and the particle size, rp (from I~). In recent years a lot of attention has been devoted to the application of electroacoustics for the characterization of concentrated disperse systems. As pointed out by Dukhin [26,27], equation (V-5 l) is not valid in such systems because it does not account for hydrodynamic and electrostatic interactions between particles. These interactions can typically be accounted for by the introduction of the so-called cell model, which represents an approach used to model concentrated disperse systems. According to the cell model concept, each particle in the disperse system is inclosed in the spherical cell of surrounding liquid associated only with that individual particle. The particle-particle interactions are then accounted for by proper boundary conditions imposed on the outer boundary of the cell. The cell model provides a relationship between the macroscopic (experimentally measured) and local (i.e. within a cell) hydrodynamic and electric properties of the system. By employing a cell model it is also possible to account for polydispersity. Different cell models were described in the literature [26,27]. In each case different expressions for the CVP were obtained. It was argued that some models were more successful than the others for characterization of concentrated disperse systems. Nowadays further development of the theoretical description of electroacoustic phenomena is a rapidly growing area. An interesting practical application of electroacoustic phenomena is the recent development of a so-called zeta potential probe, which, essentially, is the "electroacoustic
421 electrode" that can be used to measure ~-potential in concentrated and dilute disperse systems [39]. The probe consists of a piezoelectric transducer that generates ultrasonic pulses and a detector which measures the colloid vibration current (directly proportional to CVP) resulting from the interaction of the colloidal system with the pressure field of the ultrasonic wave. The use of such an instrumental device is especially advantageous for the proper characterization of concentrated disperse systems, the properties of which can change upon dilution, which one has to carry out in order to apply other methods, e.g. microelectrophoresis. Similarly to the reciprocity that exists between electroosmosis and streaming potential (See Chapter V,5), there is a reciprocity between the ultrasonic waves and the alternating electric field generated by them. That is, the application of the alternating electric field to disperse systems results in the generation of an ultrasound signal, referred to as the electrosonic amplitude (ESA). The ESA is related to the dynamic electrophoretic mobility via
a relationship similar to eq. (V.6), namely
ESA
-
~(Pp -Pm)
gd (co) E
PmZO where E is the strength of the applied electric field. A comparison of the above expression with eq. (V-50) indicates that either ESA or CVP can be measured for determining the dynamic electrophoretic mobility.
V.8. M e t h o d s o f P a r t i c l e Size A n a l y s i s
Many properties of disperse systems, such as those described above and those analyzed in subsequent chapters, are defined by the degree to which substance is dispersed and by the type of particle size distribution. Various methods for the investigation of dispersion composition of dispersed phase
(dispersion analysis) have been developed. Many of these methods utilize the concepts of transfer phenomena discussed in previous sections of this chapter.
422 A great variety of existing methods is primarily explained by the restriction of certain methods to particular particle size ranges, by complexity and cost of the required equipment, by the quality of acquired data and other factors. The most complete information that can be obtained in dispersion analysis comes from the determination ofparticle size distribution function (in some cases one may be interested in obtaining particle shape distribution). Some methods yield only the information on the average particle size, which in some cases may be accompanied by some conditional distribution width. These terms require a more detailed discussion, as different methods may yield different size distribution functions and average sizes for the same disperse system. One can distinguish between the differential and integral (or cumulative) particle size distribution functions. These two types of functions are related to each other by the differentiation and integration operations, respectively. The adequate description of distribution function must include two parameters: the object of the distribution (i.e. "what" is distributed), and the parameter with respect to which the distribution is done. The first parameter may be represented by the number of particles, their net weight or volume ~~ their net surface area or contour length(in some rear cases). The second parameter typically characterizes particle size. It can be represented as a particle radius, volume, weight, or, rarely, surface area. Consequently, the differential function of the particle number distribution with respect to their
~0 It usually makes no difference whether weight or volume is used, as the normalization over the total volume or weight of all particles is typically introduced
423 radii, f,(r), gives the ratio of the number of particles, An, with radii in the range between r and r + Ar to the total number of particles of all sizes,n0, and the value of Ar, i.e., the differential function gives the fraction of total number of particles that have radii within the given size range:
An -
noAr The integral distribution function, q~, gives the fraction of total number of particles that represent particles with radius greater than r ~ : oo
q, (r) - I f, (r)dr r
The use of integral distribution functions is common for two reasons: first, these distribution functions have simpler shape and thus are more suitable for curve smoothing; second, they allow for an easier determination of the fraction of particles belonging to a particular finite size range, Ar, as a simple difference between corresponding values, q(r + Ar) - q(r). Different methods of particle size analysis yield different distribution functions as primary information, depending on what parameters are measured in the course of experiment. These parameters may be converted into different ones. It is, however, important for one to realize that such a conversion may yield errors of different magnitudes in different size ranges.
~ This definition of integral distribution function is common in colloid science, while in polymer science the molecular weight distributions are typically evaluated by the summation of molecular weight s m al 1e r than the current value
424 Since one uses a number of different distribution functions, the average particle size may also be defined in more than one way [40]. In general one may write
irf(r)dr --
r =
0
ir'f(r)dr or
if(r)& 0
F=
0
i r'-lf(r)dr 0
where the power, n, is defined by the type of the distribution function. We will further discuss basic methods used to describe the disperse systems with various degrees of dispersion. Some methods will be discussed in detail, while the others will be mentioned briefly. The degree of dispersion in coarse disperse systems, e.g. coarse powders used in such applications as mining and material science, is commonly characterized by the
sieve analysis. In this method the studied
system is passed through a sequence of sieves with gradually decreasing pore sizes. By weighing the powder retained by sieves of different sizes one obtains the weight distribution histogram with respect to sieve sizes. The number of holes per inch (mesh) in sieves is often used as a particle size characteristic. The sieve analysis of dry powder is suitable for particles greater than 30 gm. This size limit can be lowered if liquid suspensions are used instead of dry powders. A lower size limit can in this case be achieved due to weaker attraction forces between suspended particles and inner walls of sieve openings. A rapid development of automated imaging techniques gave one the
425 opportunity to enhance substantially the possibilities of optical and opticalelectronic methods allowing for broader particle ranges to be covered. Optical microscopy allows one to analyze particles of sizes down to fractions of a micron; scanning electron microscopy (SEM) has a lower size limit of several nanometers, while the transmission electron microscopy (TEM) allows one to reach molecular and even atomic size levels. Electron microscopy is sometimes combined with advanced methods of elemental analysis at the microscopic level, such as, e.g., the energy dispersive X-ray diffraction spectroscopy (EDX). Obvious disadvantages of electron microscopy are the requirement of costly equipment and difficulties associated with the preparation of representative samples. It is often difficult, if not impossible, to achieve a monolayer arrangement of particles on a sample pad. The conductometric particle counter (Coulter Counter) is widely used for particle size analysis, e.g. in the clinical analysis of blood. In Coulter Counter the dispersion of particles in electrically conducting liquid is forced to through the opening in the wall of a test tube between a system of two electrodes [40]. One of the electrodes is mounted inside the test tube while the other is in the outer vessel. The instrument monitors the conductivity (resistance) of fluid passing through the opening. When particles pass through the opening, the peaks due to changes in the conductivity are recorded. The amplitude of these peaks is proportional to the particle volume. When the signal is treated by amplitude analyzer, one obtains the histogram of particle volume distribution. The lower size limit of particles that can be analyzed by this method is determined by the diameter of the opening, and is on the order of fractions of a micron.
426 Let us now turn to a detailed discussion of some widely used methods of particle size analysis.
V.8.1. Sedimentation Analysis Sedimentation analysis is commonly used and rather simple method of determining the size and size distribution function, based on the difference in particle settling rates in a gravity field. In this method a pan is placed into a homogeneously mixed disperse system, and the weight of particles, P, that accumulate on the pan, is monitored as a function of time, t (Fig. V-30).
t"
Fig. V-30. The sedimentation analysis experiment
Initially, at t=0, particles of all sizes are equally distributed along the entire height, H, of the disperse system volume. For monodisperse system the weight of settled particles as a function of time is given by a straight line (Fig. V-31), the slope of which is proportional to the concentration, c, of the dispersed particles (mass of particles per unit volume of dispersion), the particle sedimentation rate, ~, the
427 relative density gradient between dispersed particles and dispersion medium, (P - Po)/P, and the area of a pan, S, on which the settling particles accumulate"
dPP=__=c(p-po)usg. dt
t
p
P Pmax
I
_
tr
_
t
Fig. V-31. The accumulation of sediment in a monodisperse system
Apparently, change in the weight of sediment, dP / dt, is related to the change in its mass,
m=cvSt,as
(tam
dP_ dt
P
Po g ~
p
dt
In agreement with eq. (V-7), the rate of sediment accumulation is constant up to time
tr,
H tr --~.--
9rlH 2r2(p-po)g
which corresponds to a complete settling of all particles with the radius r. The weight of sediment accumulated on a pan is then
428
9 while the mass is given by mmax -- c S H . By the time
tr a
complete settling of particles that were initially present at the
top layer of suspension at distance H from the pan also took place. At any intermediate time, t <
tr,
the fraction of sediment accumulated on the pan is
given by
P(t)
m(t)
t
Pmax
mmax
tr
while the relative sediment accumulation rate is
d(P/Pmax) dt
d(m/mmax) dt
1 t
Obviously, at times longer than tr the weight of accumulated sediment no longer changes, and hence the break point on the P(t) curve appears at t =
tr .
One can then estimate the velocity of particles that traveled distance H over the time period tr: ~= H~ tr, and therefore, the particle size, r:
F --
I
91"1
H.
2 ( p - po)g t~
429 In a real polydisperse system the particle radii, r, are distributed within some size range between rmin and rmax, and the fractional composition can be described by an appropriate mass distribution function, fir)"
f(r)
1
dm(r)
mma x
dF
-
which represents the fraction of mass of particles with sizes in the range between r and r + dr. It is usually assumed that in polydisperse system particles of different sizes settle independently from each other with the velocities individual for each size, t)(r). Thus, sedimentation in such systems occurs with a continuously changing velocity ( opposite to settling with constant velocity that occurs in monodisperse systems) and, consequently, the weight of sediment plotted as a function of time is given by a smooth curve (Fig. V-32). The curve shown in Fig. V-32 contains an initial linear portion corresponding to t < tmi n,
and a limiting portion corresponding to constant weight of
precipitate at t > tmax.
Pnllx f
0
t~in
dP t- tant9 - t - ~
tmx
Fig. V-32. The sediment accumulation curve in polydisperse system
430 The treatment of data acquired in sedimentation analysis usually involves graphical differentiating of the sediment accumulation curve. This method of obtaining particle size distribution is based on the Svedberg - Oden equation" P=q+t~,
dP dt
in which q stands for the weight of particles with sizes greater than r, = r(t), which complete their settling by the time t, i.e. the particles of all fractions that settled by the time t. This equation has a simple physical meaning, since at any given moment, t, the sediment weight increases with the rate dP/dt due to settling of particles with sizes smaller than r, = r(t). Since prior to time t the sedimentation occurred with constant rate, the product t (dP/dt) represents the weight of particles with size r < r, that have settled onto the sedimentation pan by the time t. The value of q - P- t(dP / dt) gives the weight of larger particles that have already completed their settling. The value of q is given by an intercept of the line tangent to the P(t) curve, (Fig. V-33). By plotting such tangent lines and determining the corresponding values of q and the size of particles, r(t), that complete sedimentation by time t, one obtains an integral distribution curve, q(r) / q
Pmax
-Pmax"
,f I ! ! I
i 0
rmin
q
f
/ rms x
r
Fig. V-33. Integral and differential particle size distribution curves
431
f(r)
The differentiation of this curve yields differential distribution curve, dq(r) / Pmax dr , also shown in Fig. V-33. The values ofrmi . and rmax
are determined from the times/max and train, respectively (see Fig. V-32). Sedimentation analysis can be successfully used in systems containing particles with radii in the range between 1 and 100 ~tm. When larger particles settle in a low viscosity medium, such as water, one has to account for the deviations from the Stokes equation due to turbulent flow of medium around the particles, and introduce correction factors accounting for the acceleration of particles at the beginning of sedimentation. Sedimentation of particles with sizes on the order of fractions of a micron and those of smaller sizes is influenced by the diffusion phenomena to a significant extent (see Chapter V, 2.3).
V.8.2. Sedimentation Analysis in the Centrifugal Force Field When particle with radius r settles in centrifugal force field, its velocity,
dR/dt,
is determined by centrifugal acceleration, co2R,where o3 is the
angular velocity of the centrifuge rotor and R is the distance between the particle and the axis of revolution. The particle velocity is given by dR
~3 ~:P3 ( P - Po) r176
dt
B
where B is the friction coefficient. Consequently, one can write ln(R / R 0 )
4/3np3 ( p - P o )
Atr 2
B
m(1-9~ B O
= S,
(V.52)
432 where R 0 and R are the distances between the particle and the axis of revolution at the beginning of sedimentation and after the period of time, At, had elapsed, respectively; m is the mass of particle. The quantity S is referred as the sedimentation coefficient, or the sedimentation constant. If AR = R - R 0 <
AR
S~
RoAto3 For spherical particles B -
(v.53) 2
"
6rtrlr (see eq.(V.11)), and hence the
sedimentation coefficient is related to particle radius, r, as
S - 2r2 (19-19 0) _9q
AR RoAteo 2 "
Independent measurements of sizes of spherical particles
by diffusional
methods (r o~ D-l), and by sedimentation methods (r o~ S ~/2) usually yield results that are in good agreement with each other. For non-spherical particles the friction coefficient, B, depends on both particle size and shape, for which reason the use of any one, either diffusional or sedimentation, method yield only some conditional particle radius corresponding to that of a spherical particle with the same sedimentation or diffusion coefficient. The values of such equivalent radii may differ, depending on which experimental method was used for their assessment. To determine the true particle size or mass, m, of non-spherical particles, and to obtain the information regarding particle shape, one must perform an independent determination of the sedimentation coefficient and friction factor
433 by using two principally different, usually sedimentation and diffusion, methods. The product of sedimentation coefficient and friction factor are independent of particle shape and proportional to the particle mass"
SB-m(1-P0).
(V.54)
P
If the mass of particles is known, the friction factor, B, characterizes particle shape. Such investigations were developed in conjunction with the analysis of structure of macromolecules in polymer solutions. In the centrifugal sedimentation in systems consisting fairly coarse particles one sometimes employs analysis by weight (the centrifugal balance). For the analysis of
finely disperse systems and polymer solutions
ultracentrifuges with co2Rvalues as high as 106 g are used. The use of optical system, such as schlieren optics, to monitor sedimentation process allows one to gain information about the concentration distribution, c = c(R, At) [41 ]. One often employs a deposition method, in which the disperse system or polymer solution is superposed on top of pure dispersion medium, where the motion of particles is then followed. If under the conditions of experiment the rates of sedimentation and diffusion are comparable, the blurriness of the boundary occurs even during the analysis of monodisperse system (Fig. V-34). One may assume that sedimentation and diffusion of dispersed particles occur independently of each other and their rates can be summed up. This means that the curves shown in Fig. V-34 can be obtained from those shown in Fig. V-2 by shifting the origin, x = 0, which corresponds to the concentration c - Co/ 2. The origin moves with the rate determined by the
434
tl
Ro
R =Ro+&R
Fig. V-34. The influence of diffusion on the boundary position during the centrifugation of system containing monodispersedparticles
&R/At Fig. V-35. The effect of diffusion on the shape of sedimentation curves plotted in c - AR / At coordinates
sedimentation coefficient, defined by eq. (V.53). Exactly for this reason the sedimentation coefficient can be determined from the rate with which the point corresponding to half the concentration, R c0/2 (At), moves. By comparing the shape of the c(x) curves (where x - R - R co/ 2) with eq. (V.23) one can determine the diffusion coefficient and then estimate the friction factor, B, from it. From such "single" experiment one may thus estimate the mass of settling particles from the SB product using eq. (V.54), and retrieve the information about particle shape. In po l y d i s p e r s e systems the blurriness of sedimentation boundary is related to both the diffusion and the differences in the sedimentation rates of particles having different sizes. In cases when diffusion is negligible, the
c(R) dependence represents the shape of integral particle size distribution curve at any moment of time. To obtain the q(r) /
Dmax
curve
from sedimentation curve, c (R,
At=const.), one can plot the relative concentration, C/Co, as a function of particle radius obtained from particle displacement, AR, that occurred over the time, At, using eq. (V.53). If the diffusion rate is negligibly small, the c = c (AR / At) curves match each other at all times, At. The latter allows one to separate sedimentation and diffusion in polydisperse systems as well. To
435 achieve this, the sedimentation curve is plotted in c - (AR/At) coordinates, as shown in Fig. V-35. The difference between positions of these curves at different At values is caused solely by the contribution from diffusion processes into sedimentation boundary blurriness. Therefore, by analyzing changes in the shape of c ( A R / A t ) curves with increasing sedimentation time, one may determine the average value of diffusion coefficient, D. A more detailed analysis allows one to obtain the values of D corresponding to individual fractions. Since the diffusional displacement is proportional to (At) ~/2,the c (AR / At) curves can be reduced to zero sedimentation time. As a result, one obtains true sedimentation curve that does not include any contributions from diffusion. From this curve one gets the integral distribution curve of particle sedimentation coefficients. It is then possible for one to convert sedimentation coefficients into particle size by using the value of friction factor, B, estimated from the diffusion coefficient. It is worth mentioning that in most high speed centrifuges there is a special optical system that allows one to register the refractive index gradient, which is proportional to the concentration gradient of dispersed particles. The so-obtained experimental curves, resembling by their shape the differential particle size distribution curves, are then numerically integrated to yield c(AR) curves, from which the values of D, S, B and particle size can be evaluated.
V.8.3. Nephelometry. Ultramicroscopy The scattering of light by disperse systems is one of the most universal, effective, and commonly used methods of dispersion analysis utilized for the investigation of disperse systems and polymer solutions. For systems that
436 follow the Rayleigh equation, the methods based on the turbidity measurements, i.e. measurements of a decrease in intensity of the transmitted light (turbidimetry), and those involving measurement of the intensity of light scattered at different angles (nephelometry) are quite equivalent. A direct estimation of particle size from Rayleigh's equation is rarely performed. The relative method in which optical properties of investigated system are compared to those of systems with known particle size and concentration is often employed. Using the condition V2n - const, one can estimate the volume of particles, or substance concentration in the studied system, if the particle size is known. These methods are highly sensitive: for instance, noticeable turbidity in arsenic sulfide sols can be detected at the particle concentration of 10.3 %, while in gold sols at concentration as low as 10.5 %. In order to achieve the necessary precision in nephelometry, one must use dilute systems in which secondary scattering is negligible. Nephelometry can be used in a broad concentration range, including very low particle concentrations, while turbidimetry requires the use of a lot higher concentrations. When carrying out nephelometric measurements with extremely dilute systems, one must work in a dust free environment and account for the scattering of light on fluctuations of density and concentration. Along with the methods based on investigation of the scattering of light by disperse system as a whole, there are also methods based on the scattering (the diffraction) of light on individual particles. One of such methods, ultramicroscopy, played an important role in the development of colloid science. Dark field optical systems, ultramicroscopes, and dark field condensers, such as those utilized in optical microscopes for side illumination,
437 are used for the observation of scattering of light by individual particles. The glowing dots resulting from the light scattered by individual particles are very well visible on dark background, and thus allow one to determine the concentration of the dispersed particles, as well as to study concentration fluctuations and Brownian motion. Such experiments yielded experimental confirmation of both the theory of Brownian motion (see Chapter V, 2), and of molecular kinetic concept as a whole. The experimental method based on a photographic imaging of dispersed particles participating in the Brownian motion was developed by S. Vavilov. Due to Brownian motion, the particles appeared on photographic images as a number of blurred spots. In a complete agreement with the theory of Brownian motion, the average area of these blurred spots was proportional to the exposition time. This method allows one to register several particles simultaneously, thus making it easier for one to obtain large amount of data needed for a representative statistical averaging. To determine the concentration of the dispersed particles, one often replaces ordinary ultramicroscope with fl o w u 1t r a m i c r o s c o p e designed by B. Derjaguin and G. Vlasenko [42]. In the flowing suspension this device registers the number of particles that travel per unit time across the microscope field, allowing one to rapidly determine the particle concentration in sols. The use of optical electronic devices for the measurement of intensity of light scattered by individual particles makes it possible for one to obtain also the particle size distribution curves. Along with optical methods, the methods based on the use of X-rays are employed in the analysis of the disperse systems. A major difference between these methods and "regular" light scattering is in the small, compared
438 to particle size, wavelength of X-ray radiation. X-ray diffraction (XRD) is commonly used for the investigation of internal structure (e.g., crystallinity, molecular packing, etc.) of dispersed particles. It is possible for one to determine the particle size by examining the shape of lines in the diffraction pattern. X-ray diffraction on small crystals produces blurred diffraction maxima, from the width of which one can estimate the particle size (the size of areas with the perfect crystal lattice, to be more precise). It is well known that amorphous particles do not yield any diffraction pattern that contain maxima. The size of such particles can be estimated by the analysis of diffuse X-ray scattering at low angles with respect to the primary beam (the so-called low angle X-ray scattering). The theory of this method used to obtain the size of amorphous particles resembles the theory that describes the scattering of light by large particles. The methods of scanning and transmission electron microscopy are broadly used for the investigation of various objects of colloidal nature [40]. It is worth mentioning here the technique of the preparation of replica of rapidly frozen sols, which allows one to "freeze" the system at a given moment of time. The surface structure can be effectively analyzed by such methods as Auger Electron Spectroscopy (AES), Low Energy Electron Diffraction (LEED), Secondary Ion Mass Spectroscopy (SIMS), and others.
V.8.4. Light Scattering by Concentration Fluctuations In agreement with the Rayleigh equation, the scattering of light by homogeneous systems, such as pure liquids and true solutions, should be very small due to a small size of scattering centers. In reality, however, there is a
439 significant scattering from such systems due to fluctuations in density and concentration, which serve as the scattering centers. Several commonly used methods of dispersion analysis and investigation of interactions between molecules in solutions are based on the studies of light scattering by such systems. The scattering of light is especially strong by systems in near-critical state (See Chapter VI, 2) in which linear dimensions of fluctuations are large and are comparable to the wavelength of light. Let us now discuss the scattering of light on the concentration fluctuations following the treatment originally established by L. Mandelshtam and P. Debye. In agreement with the Einstein- Smoluchowski general theory of fluctuations (see Chapter V, 2) the average square of concentration fluctuations, Ac 2, in a volume V is given by the first derivative of osmotic pressure, l-I, with respect to concentration, c, i.e."
ckT
AC 2 =
V dH~dc
(V.55)
Assuming that the refractive index fluctuation, An, is proportional to the concentration fluctuation, Ac, i.e."
dn A n
--
m
dc
one obtains
Ac;
dn dc
= const,
440
ckT
An 2 = n~
(d_d__~_c/2 1
Vd~dc
n2"~
(V.56)
The unit volume of the disperse system can be imaginary divided into n equal parts with volumes V = 1/n, such that V1/3(( ~. Let us assume that each of these microvolumes scatters the light, i.e. is identical (in this respect) to a particle of dispersed phase. It is then possible to write in agreement with eqs.(V. 39), (V.42), and (V.56) that 32
3
ckT
(dn.) 2
m
3
4n dUdc T c
Since kn0 is the wavelength of used light source in vacuum,
)Lvac, the above
expression can also be written as
32 3 n~ - ~ ~ 3 ~4 vac
ckT (dn:/2 -@c "
~d
-Ydc
(V.57)
This expression for turbidity thus does not contain a volume of fluctuation (i.e., the imaginary microvolume) which can be viewed as some average quantity. One may also say that the fluctuations in different volumes contribute equally to the net turbidity of the system. Virial expansion of osmotic pressure (the system is assumed to be nonideal!) yields
441 dH
RT = ~ + 2B2c+... dc M
(V.58)
where M is the molar mass of dissolved substance; B2 is the second virial coefficient, and c is the concentration expressed in kg m -3. Using eq. (V.58), one can write the expression (V.57) as Hc
dH
"c
1
1
2B2c
dc R T
M
RT
+,,.~
where H (m 2 kg l ) is the cluster of constants with units given by
H
32 --
3 ~
3
n~
(d') 2
4~ [k~C) )~vacNA
By experimentally measuring the turbidity as a function of concentration, one can obtain the molecular weight of solute and the second virial coefficient, B2, which characterizes interactions between solute molecules (or particles). In systems containing charged particles, such as e.g. micelles of ionic surfactants, the second virial coefficient describes the effective charge of particles. The molecular weight and the second virial coefficient can be determined by plotting the quantity Hc/~ as a function of concentration, c. A more general description of light scattering, based on Mie theory, shows that for large particles that do not obey Rayleigh equation the determination of molecular weight and second virial coefficient, as well as obtaining information regarding the molecular structure (conformation), is also possible. To determine these, one has to study the angular dependence of
442 the scattered light, I(0), at different concentrations, along with the concentration dependence, I (c). The experimental data at low concentrations can be approximately described by the equation
3Hch
1
=
16rt I ( 0 ) / I o
2B2c
+~
M
+ k 0 sin 2
(O /
RT
where h is the instrument constant, k0 is a function of the size of a macromolecule in solution (the average square end-to-end distance of polymer coil). A more precise determination of M, B2, and k0 is possible with the double extrapolation method using Zimm plots. The experimental data are plotted with 3Hchlo/16rtI(0) as the ordinate and sin2(0/2) + Ac as the abscissa. An arbitrary constant, A, is chosen in such a way that its multiplication by the highest solution concentration yields a value close to 1 (Fig. V-36) 3
Hch
16rr I(0) / 10 C2
r
c=O
3
Oz
0=0
r
C5
C3
0
sin 2 -0+ A c 2
Fig. V-36. Zimm plot
V.8.5. Photon Correlation Spectroscopy (Dynamic Light Scattering)
The experimental points in the Zimm plot form two series of lines with c = const and 0 - const. The slope of 0 = const lines yields B2, while the slope of c = const lines yields the value of k0. By extrapolation one obtains the 0 =
443 0 and c = 0 lines, the intercept of which yields the value of M. Another technique of particle size analysis that has been widely used over the last two decades is Photon Correlation Spectroscopy (PCS), also known as dynamic light scattering, or quasi-elastic light scattering [43,44]. This method is based on the measurement of fluctuations in the intensity of light scattered by a small volume of disperse system (the scattering volume) as a function of time. As the particles due to the Brownian motion move randomly through the solution, their positions relative to each other change, and the light scattered by them forms an interference pattern. This interference pattern results in intensity fluctuations occurring on a microsecond and millisecond scale that are picked by the photo multiplying tube (PMT) detector. Changes in the intensity of light scattered by the particles are related to the particle diffusion coefficient (or coefficients, in case of polydisperse system), from which particle size can be estimated on the basis of EinsteinSmoluchowski theory. It is clear to one that under the influence of Brownian motion larger particles change their positions slowly, and cause lower fluctuations in the intensity of scattered light than the smaller, rapidly moving particles. Thus, measuring the intensity of the scattered light at time t, I (t) at a given scattering angle, 0, and correlating it to the intensity measured at time t + ~r
I(t+Zco~), allows one to obtain information on how far the particles
have diffused during the period of time between the measurements, i.e. Z~o~, which is referred to as the correlation delay time. Mathematically, this can be expressed in terms of the normalized intensity autocorrelationfunction (ACF), given by [44]
444
g(2) (z c o r r ,
0 ) --
(I(t ,0) I(t + "Ccorr
,0>
(;(,,o)> where the averaging is performed over the entire measurement time. High value of ACF at two different times, i.e. high correlation between intensities, indicates that the particles have not diffused very far during the time period between the measurements. If ACF maintains high value over a long time interval, "~corr,it is an indication of the presence of large, slowly moving particles. Obviously, the choice of proper correlation delay time is critical for accurate measurements. The information on the approximate particle size range obtained by independent methods can substantially simplify the procedure of choosing suitable Zco~. For a Gaussian distribution of the scattered light intensity profile, the intensity ACF, g(2)(Zcorr,0), is related to the electric field ACF, g(~)(Zco~r,0), via the Siegert relation:
g(2) (,r corr , 0) - 1 + 13 g(1) (Tcorr , 0)[ 2 where 13is an instrument efficiency constant, 0<13___1. The autocorrelation function is determined by digital autocorrelator. Depending on the type of the latter, the values of "Cco~at which ACF is determined can be spaced either linearly or logarithmically (Fig. V-37). The ACF presented on a logarithmic scale of Z~o~rallows one to obtain maximum information about the system from scattered light intensity measurements.By computing ACF at each angle 0 for a large range of correlation delay times,
445 1.0
1.0
0.8
0.8
=,
\
w
\
0.6-
.,, !
0.6 =
e.t
0.4
"'e~o 0.4
0.0
\
0.2
0.2,
84
0.0 2.0x10 4
2.4x10 8
|
,
6.3x10 -~
i
,
"~
corr
~corr (S)
a
(s)
|
==
|
2.oxlo-'
b
Fig. V-37. Autocorrelation function with linearly (a) and logarithmically (b) spaced correlation delay times, ~r one obtains a quantitative measure for the time scale of fluctuations, F(0), a characteristic of the rapidity of the light fluctuations, and, consequently, of the rapidity of changes in particle configuration. The quantity F(O) is referred to as the decay constant; for a dilute dispersion ofmonodispersed non-interacting particles one can write that
g(1) (Zcorr , 0) I - exp (-F(0)Zcorr)"
(V.59)
The decay constant, F(0), is related to the translational and rotational motions of particles in a given medium; for a macromolecule chain F(0) is primarily determined by the chain length and conformation in the medium. For solid particle F(0) describes the shape and dimension. In the absence of rotational motion, the decay constant is directly related to the translational diffusion coefficient, DT: r(0)-
K2DT,
(V.60)
446 where K is the quantity referred to as the scattering vector, and is given by
K
m
4 rm sin(0 / 2) )~0
In the above expression n is the refractive index of the medium, and )~0is the wavelength of light. Using eq. (V.60) in combination with the Stokes-Einstein equation (V.11), one can relate the decay constant, F(0), to the radius of spherical particles" kT ( 4~nsin(0/2) t 2 )~o 6z~rlr
F(0) -
where all symbols have their usual meaning. Equation (V.59) thus summarizes the main objective of dynamic light scattering: the extraction of decay constant, F(0), from ACF. This, in general, may not be an easy task. In polydisperse systems, to which most of real systems belong, the light scattered by individual particles is added, and ACF is the sum of decaying exponentials corresponding to particles of different size. Consequently, one must replace eq.(V.59) with the following expression:
00
]g(1) (~corr , 0)]- I G(F, 0)exp (-F(0)~ corr)dF,
(v.61)
0
in which the distribution function of decay constants, G(F,0), is used instead of a single decay constant, F(0). This distribution of decay constants, which
447 is to be extracted from the measured ACF, can be continuous, discrete, or be represented by some combination of the two. Apparently, ifG(F, 0) is discrete, the
integration in
eq. (V.61) is replaced with summation. In eq.(V.61)
G(F,0)dF represents a fraction of the total intensity of light scattered, on average, by particles (or macromolecules) for which eq. (V.60) holds within dF. Thus,
i
G(F,O)dF-
1.
o
A number of methods for obtaining G(F, 0) are available. One, for example, may assume that G(F,0) obeys certain distribution with a few parameters (such as, e.g., Gaussian), and calculate the parameters that generate the best fit to the data. The outcome of such approach, apparently, depends on the type of assumed distribution function. This method is thus suitable only when there is a reasonable basis to believe that the chosen distribution function adequately represents the studied sample. In most cases, however, one has a limiting a priori knowledge on the type of actual particle size distribution. For this reason the mentioned method is used mostly for theoretical modeling and simulation. A very simple and widely used method, known as the method of cumulants, was described by D. Koppel [45]. This method allows one to obtain mean value ofF(0) and the width of distribution, characterized by the polydispersity index, PI. In the method of cumulants the natural logarithm of ]g(1)(~corr,0)] is expanded into polynomial series, in which usually only the first two terms are retained, i.e.:
In g(1) (~corr , O) - - a ( O ) ' c
+
b(0)
2
1: + . . .
2 The coefficients a(0) and b(0) represent the first two cumulants (moments) of the distribution, and are related to mean decay constant, (F(0)), and polydispersity index, PI, as follows:
448
oo
a(0)-
(F(0))-
IF(0)G(F,0)dF, 0 oo
b(0)
I
P I - [a(0)]-------T:
(r(0))2 0 [r(o)-
(r(o))]2G(r,o)dr.
Method of cumulants is suitable for characterizing a reasonably narrow u n i m o d a l distribution. When G(F, 0) is broad or can not be adequately represented by unimodal distribution, the method ofcumulants fails. Indeed, two very different distributions may have the same first two moments, and thus can not be distinguished from each other based on the method of cumulants [46]. A method for obtaining G(F, 0) from ACF, g(1)(lZcorr,0),suitable for treating broad and multi-modal distributions, was developed by S. Provencher [47,48]. This method, based on
regularizednon-negative least-squares technique, also known as the CONTIN algorithm, has won common acceptance and is utilized in commercial PCS instruments. While the detailed mathematical description ofthis method is rather cumbersome and is beyond the scope of this book, we will still briefly explain the essence of it. In his studies Provencher addressed a frequently encountered general problem of retrieving the desired function from a noisy set of data. This problem can be formulated by the following equation: b ID
Yk - j Fk (Z,) s(~,) dZ, + ~ k ,
(V.62)
a
where Yk are the observed data, s(k) are the sought functions, Fk(~.) are known functions representing linear integral operators relating Yk to s(~), and ek are the unknown noise components. One can see that eq.(V.61) for ACF is the specialized case of general eq. (V.62) with the noise components included into the ACF. Mathematical process of extracting individual contributions from particles of
449 different sizes from the ACF, is known as deconvolution, or inversion [49]. Direct inversion of eq.(V.61), which is the Laplace integral equation, requires highly precise ACF data collected over an extraordinary range. Basically, the inversion is an ill-posed problem because of the noise in g_0. Same constraint can be successfully transferred to G(F, 0), since this quantity represents the particle size distribution, and can't thus be negative. The first step in finding a solution of eq. (V.62) is converting it into a system of linear algebraic equations. In CONTIN this is done automatically by numerical integration eq. (V.62). When applied to eq. (V.61) such integration yields
[g(1)('Cc~
ai(O)exp(-Vi)'c corr ,
2 i
where at are the amplitudes of each bin size in the particle size histogram. In agreement with previously imposed constraint, none of at are negative. Obtaining amplitude factors, a;, from the ACF is the standard problem in curve fitting. The CONTIN algorithm performs the leastsquared fitting, i.e. it varies at until the sum of squared differences of the ACF from the fit is minimized. The minimization, however, yields a solution that is just one m e m b e r of a series of possible solutions of eq. (V.61) remained after the non-negativity constraint had
450 been imposed. This solution contains an arbitrary amplitude of the "noise", A, characteristic of large variations between possible solutions. Due to the random nature of A, it is highly unlikely that the obtained solution will be close to the true one. One, thus, needs to take further steps in sorting out unsuitable solutions, in order to find an optimal one. To do so, one needs to perform regularization, which establishes a strategy of further elimination of unsuitable solutions. Mathematically, regularization is equivalent to minimizing a comb ination ofthe sum of squared differences ofthe ACF from the fit and some function, referred to as the regularizor. Proper choice ofregularizor is a very important step in finding the optimal solution. This function is chosen either on the basis of prior statistical knowledge (when available), or onparismony. The concept ofparismony, as explained by Provencher, states that among the solutions that still have not been eliminated by imposing constraints or by prior knowledge, one needs to chose the "simplest" solution, i.e. the one that reveals the least amount of information that was not already known or expected. With respect to the fluctuational nature of the ACF, parismony means the choice of a solution that is smoothest and contains the minimum number of peaks.
In our discussion of the PCS we indicated that the autocorrelation function and the distribution of decay constants are both the functions of scattering angle, 0. We, however, have not yet elaborated on the use of multiangle dynamic light scattering. In monodisperse systems of rather narrow size distribution in the absence of motions other than translational one, there is only one characteristic decay constant, and it is sufficient for one to perform measurements at only one scattering angle. The scattering angle is often set at 90 o which allows one to minimize the scattering from dust particles. In polydisperse systems the particles of different sizes have different scattering intensities and different decay constants at a given scattering angle, and hence it is insufficient for one to perform measurements at one scattering angle only. In such systems measurements at several scattering angles provide certain
451 advantages and are often necessary for one to be able to obtain correct results. The angular scattering intensity patterns per unit volume are different for particles of different sizes. As the particle size increases, the absolute scattering intensities per unit volume at low angles become higher. This may create a"blind spot"effect: at certain angles scattering from particles of certain size present in the system may not be detected at all. One thus needs to perform measurements at a variety of angles, in order to get results representative of the studied system. For smaller particles the angular pattern becomes smoother and less angle dependent. Generally, there will be higher input from larger particles in measurements performed at lower angles, and higher input from smaller particles at higher scattering angles. Another advantage of multi-angle light scattering is the flexibility in dealing with the concentration effects. Proper sample preparation is critical for obtaining reliable and representative data. Typically, the studied suspension should be dilute to the extent when multiple scattering and interparticle interactions in Brownian motion can be avoided, so the saturation of the detection device does not occur. At the same time, the dilution should not be too low, i.e. there should be sufficient scattering to produce a good signal-tonoise ratio. In many cases, however, one is not free to vary the concentration to fit experimental requirements. Some of the typical examples include biological samples in which the concentration is very low to start with, surfactant and polymer samples in which the scattering is still weak at rather high concentrations, and concentrated systems that undergo changes upon dilution. In all these cases changing the scattering angle is the most feasible and convenient approach.
452
V.8.6. Particle Size Analysis by Acoustic Spectroscopy The method of particle size analysis based on the measurement of ultrasound attenuation is referred to as acoustic spectroscopy. The acoustic spectrometer generates ultrasound pulses that undergo attenuation in disperse systems due to interaction with dispersed particles and the dispersion medium. The experimentally obtained value of attenuation can then be fitted to a particular theoretical model for different attenuation mechanisms, and the particle size can be evaluated from the fit, for instance by using eqs. (V.46 V.49). Depending on the properties of particular disperse systems, different types of the acoustic energy losses may prevail. For example, in suspensions of mineral oxides the attenuation of acoustic signal occurs predominantly due to viscous losses, while in emulsions thermal and (at high frequencies) scattering losses prevail. In such cases it is possible to simplify the theoretical treatment by considering only the factors that make a significant contribution to the attenuation, while neglecting others. An important advantage of the ultrasonic method of particle size analysis over other methods is its applicability to systems that are concentrated, electrically non-conductive and optically opaque. Equations (V.46 - V.49) indicate that attenuations due to different mechanisms of acoustic losses are proportional to the volume fraction of the dispersed phase. This dependance becomes critical for the evaluation of particle size in concentrated dispersions. A number of studies
(see [26,27], and references
therein) have showed that such proportionality does not hold over the entire range of volume fractions, which indicates that eqs. (V.46) - (V.49) are not suitable for the characterization of concentrated disperse systems.
453 Figure V-38 shows the attenuation measured (normalized by frequency, co= 15 MHz) as a function of weight and volume fractions for suspensions of rutile (TiO2) (a), and neoprene latex (b) particles. As one can see, the linear relationship between ~/o3 and the volume fraction, ~, for rutile dispersion holds up to only -~ 10 %, and for latex particles up to ~ - 32%. 10-
6-
RUTILE
LATEX
17"27 24.71
9 8-
N
7-
E
6-
~
37.4
59.81
E
N
~
32.4
N
-
;14.92 m
4 2I.S D
-
5
O
4,95
3
i11,9
21-91
< ~
2
~4
o.~
"
I.I s
o
.........
0.
i .......
10.
' 1 ~'* . . . . . . .
20.
i ........
30.
i
40.
" ....
i ........
50.
It .....
60.
Weight fraction, %
a
P*'"I
....
70.
32 i
. . . . . . . . .
0.
! . . . . . . . . .
10.
i . . . . . . . . .
20.
! . . . . . . . . .
30.
i . . . .
40.
Weight fraction, %
b
Fig. V-38. The experimentally measured attenuation to frequency ratio, a/o3 (co= 15 MHz), as a function of weight and volume fractions of (a) - YiO2 (rutile), and (b) - neoprene latex suspensions in water. The frequency was kept constant at 15 MHz. Volume fractions, ~, are shown by numbers over the data points. (From ref. [26], with permission) According to Dukhin [27], such a remarkable difference in the validity of eqs. (V.46) (V.49), representing the ECAH theory (see Chapter V, 7), for latex and rutile dispersions can be explained by the fact that in the case of rutile viscous losses of acoustic energy play a predominant role, while in the case of latex thermal losses are prevalent. The decay of the amplitude of the ultrasonic signal in the case of thermal losses is determined by the thermal depth, while in the case of viscous losses by the viscous depth (see Chapter V,7). It was shown in [33] that for aqueous systems, 5visc/Sth = 2.6, which means that the shear waves propagate into the dispersion medium to a greater extent than the thermal waves. At higher volume fractions of the dispersed phase the deviation of a/co versus ~ dependence from linearity occurs due to the appearance of particle-particle interactions. This means that at certain volume fractions the decaying shear and thermal waves from a given particle start to
454 interact with the boundary layers of the neighboring particles. Since shear waves propagate into the dispersion medium deeper than the thermal waves, the shear waves start to "sense" the presence of other particles at lower volume fractions than the thermal waves. The illustrated example clearly shows the limitations of ECAH theory for describing the attenuation in concentrated systems. A considerable effort has been made in recent years to alleviate the deficiencies of this theory and to develop theoretical relationships linking the attenuation with the particle size that take into account particle-particle interactions and can therefore be used for the analysis of concentrated disperse systems. In a number of studies [26,27,31,32] the expressions for attenuation coefficients, more complex than eqs.(V.46)(V.49), suitable for the analysis of concentrated systems were derived. As in the case of electroacoustics, the modeling of particle-particle interactions was done by employing the appropriate cell models. The polydispersity of colloidal systems was also taken into account in these studies by solving the equations governing the propagation of ultrasonic waves in disperse systems for each particle size fraction individually. Good agreement between the particle size evaluated from attenuation measurements and measured by other methods was observed [27]. It is also worth mentioning that particle size can also be estimated from electroacoustic measurements of the phase angle of CVP. In this case measurements done at a single frequency yield mean particle size, while in the case of multiple frequencies particle size distribution can be obtained. Acoustic spectroscopy is, however, more practical for particle size analysis than for electroacoustic CVP measurement.
References 9
0
9
0
Forland, K.S., Forland, T., Ratkj e, S.K., Irreversible Thermodynamics: Theory and Applications, Wiley, Chichester, 1988 Lyklema, J., Fundamentals of Interface and Colloid Science, vol.1, Academic Press, London, 1991 Neimark, A.V., Percolation and Fractalos in Colloid and Interface Science, World Scientific Publishing Co., Inc., 1999 Svedberg, T., and Pederson, K.O., The Ultracentrifuge, Oxford University Press, London, 1940
455 .
,
~
~
0
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
van de Ven, T.G.M., Colloidal Hydrodynamics, Academic Press, London, 1989 Shchukin, E.D., Pertsov, A.V., and Amelina, E.A., Colloid Chemistry, 2nded., Visshaya Shkola, Moscow, 1992 (in Russian) Dukhin, S.S., in "Surface and Colloid Science", vol.7, E.Matijevid (Editor), Wiley, New York, 1974 Dukhin, S.S., and Derjaguin, B.V., in "Surface and Colloid Science", vol.7, E.Matijevid (Editor), Wiley, New York, 1974 Lyklema, J., Fundamentals of Interface and Colloid Science, vol.2, Academic Press, London, 1995 Sidorova, M.P., Dmitrieva, I.B., and Fridrikhsberg, D.A., Kolloidn. Zh., 34 (1972) 640 Zhukov, A.N., Kibirova, N.A., Sidorova, M.P., and Fridrikhsberg, D.A., Dokl. Akad. Nauk SSSR, 194 (1970) 130 Chernoberezhskii, Yu. M., in "Surface and Colloid Science", vol. 12, E.Matijevid (Editor), Plenum Press, New York, 1982 Hunter, R.J., Zeta Potential in Colloid Science. Principles and Applications, Academic Press, London, 1981 Dukhin, S.S., Adv. Colloid Interface Sci., 44 (1993) 1 Sjoblom, J., Fordedal, H., Skodvin, T., Gestblom, B., J. Dispersion Sci. Technol., 20(3) (1999) 921 Derjaguin, B.V., and Dukhin, S.S., in "Surface and Colloid Science", vol.7, E.Matijevid (Editor), Wiley, New York, 1974 Souririjan, S., Reverse Osmosis, Academic Press, New York, 1970 Zeman, L.J., Zydney, A.L., Microfiltration and Ultrafiltration, Dekker, New York, 1996 Fridrkhsberg, D.A., A Course in Colloid Chemistry, Mir PUN., Moscow, 1986 Troelstra, S.A., and Kruyt, H.R., Kolloid-Z., 101 (1942) 182 Kerker, M.J., The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York, 1969 van de Hulst, H.C., Light Scattering by Small Particles, Dower, New York, 1981 Barnes, M.D., and La Mer, V.K., J. Colloid Science, 1 (1946) 79 Krishnan, R.S., Kolloid-Z., 84 (1938) 2, 18 Lyklema, J., Fundamentals of Interface and Colloid Science, vol.1, Academic Press, London, 1991
456 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.
Dukhin, A.S., Goetz, P.J., Wines, T.H., and Somasundaran, P., Colloids Surf., A 173 (2000) 127 Dukhin, A.S., Goetz, P.J., Adv. Colloid Interface Sci, in the press Epstein, P.S., Carhart, R.R., J. Acoust. Soc. Am., 25 (1953) 553 Allegra J. R., and Hawley, S.A., J. Acoust. Soc. Am., 51 (1972) 1545 McClements, D.J., Colloids Surf. 90 (1994) 25 McClements, D.J., Adv. Colloid Interface Sci. 37 (1991) 33 McClements, D.J., Langmuir 12 (1996) 3454-3461 Dukhin, A.S., Goetz, P.J., Langmuir 12 (1996) 4998 Debye, P., J. Chem. Phys. 1(1933) 13 Marlow, B.J., and Fainhurst, D., Langmuir 4 (1988) 611 O'Brien, R.W., J. Fluid. Mech. 212 (1990) 81 O'Brien, R.W., J. Fluid. Mech. 190 (1988) 71 Hunter, R.J., Colloids Surf., A141 (1998) 37 Dukhin, A.S., Goetz, P.J., and Truesdail, S., Langmuir 17 (2001) 964 Allen, T., Particle Size Measurement, 4th ed., Chapman and Hall, London, 1990 Hiemenz, P.C., Principles of Colloid and Surface Chemistry, 2nded., Dekker, New York, 1986 Derjaguin, B.V., Vlasenko, G. Ja., Storozhilova, A.I.,and Kudrjavtseva, N.M., J. Colloid Sci., 17 (1962) 605 Chu, B., Laser Light Scattering" Basic Principles and Practices, 2nded., Academic Press, New York, 1991 Mt~ller, R.H., Mehnert, W., Particle Size and Surface Characterization Methods, Medpharm Scientific Publishers, Stuttgart, 1997 Koppel, D.E., J. Chem. Phys., 57 (1972) 4814 Provencher, S.W., Makromol. Chem., 180 (1979) 201 Provencher, S.W., Comput. Phys. Commun., 27 (1982) 213 Provencher, S.W., Comput. Phys. Commun., 27 (1982) 229 Jansson, P.A., Deconvolution" with Applications in Spectroscopy, Academic Press, Orlando, 1984
List of symbols
Roman symbols dg A,A
work needed to induce fluctuations amplitude
457 ultrasound scattering coefficient (for pulsation) ultrasound scattering coefficient (for oscillation) first moment (cumulant) of distribution amplitude of i-th bin in particle size distribution histogram ai B friction coefficient B: second virial coefficient second moment (cumulant) of distribution b specific heat at constant pressure Cp CVP colloid vibration potential concentration C speed of light C c velocity of propagation of ultrasound wave (section V.7) diffusion coefficient D translational diffusion coefficient DT rotational diffusion coefficient Drot g extinction (optical density) strength of external electric field E electric vector of the light wave (section V.6) E ESA electrosonic amplitude strength of sedimentational electric field (section V.4) electric field strength of scattered wave (section V.6) Esc g free energy gravitational force F, thermodynamic force viscous force differential function of the particle number distribution f. g acceleration of gravity g(1),g(2) autocorrelation function distribution function of decay constants G H height of the disperse system volume (in sedimentation analysis) height of electroosmotic rise I intensity of transmitted light intensity of incident beam /0 Ill, L indicatrixes of the scattered light electric current (section V.5) IHOR intensity of horizontally polarized scattered light intensity of completely polarized scattered light streaming current Apuls A OSC a
458 intensity of scattered light intensity of unpolarised portion of scattered light intensity of light, scattered by a unit volume IVERT intensity of vertically polarized scattered light total intensity of scattered light imaginary unit, (- 1)~/2 i flux J diffusional particle flux particle flux due to electric field gravity sedimentation flux L sedimentation flux Js scattering vector K solubility product k Boltzmann constant wave number k parameter describing the size of a macromolecule in solution ko length of a capillary 1 molar mass M m mass effective particle mass m e mass of a molecule mM Avogadro's number NA number of particles X number of particles per unit volume refractive index of the medium no refractive index of dispersed phase n refractive index of extraordinary beam n E refractive index of ordinary beam no~ PI polydispersity index weight P pressure P electroosmotic flux Qp fluid flux integral function of the particle number distribution (p 114) q. q effective charge universal gas constant R R distance total luminous flux sc
/u
459 radius of particle average particle radius S sedimentation coefficient cross-sectional area S sedimentation pan surface area (section V.8) S absolute temperature T t time t~ time of complete settling of all particles with radius r sedimentation time tsed V volume volume fraction of droplets in the emulsion particle velocity D electophoretic velocity 1)E W probability of concentration fluctuations x, y, z Cartesian coordinates F~ Fp
r
Greek symbols 0~ 0~ ~0
%. ,y F A, 8 (~visc (~th
n o K
kv ko ks
fraction of"active" grain boundaries (section V. 1) polarizability of particles (section V.6) ultrasound attenuation coefficient (section V.7) polarizability Onsager phenomenological coefficients thermal expansion coefficient adiabatic compressibility decay constant (in dynamic light scattering) small distance viscous depth thermal depth electrokinetic potential dynamic viscosity scattering angle phase angle inverse thickness of electrical double layer (Debye-Htickel parameter) wavelength of light, also ultrasound wavelength specific conductance of the free disperse system (section V.4) specific conductance of the dispersion medium (section V.4) surface conductivity (section V.5)
460
~d ~dyn V Vk
I--I %
P Pv P~ ~gb ~SL T
~COl*r I*m Tp TE Tq
(I) O~ ~o
q~ q) q)d q)rot
average conductivity of the medium (section V.5) chemical potential dipole moment dynamic electrophoretic mobility radiation frequency (section V.6) kinematic viscosity particle displacement osmotic pressure 3.14159... density volume charge density surface charge density surface energy of grain boundary surface energy at solid/liquid interface turbidity correlation delay time heat conductance of dispersion medium heat conductance of dispersed phase shear stress viscous resistance of the medium probability integral streaming potential angular frequency (velocity) dielectric constant electrical constant electric potential angle between the direction of wave propagation and dipole axis (section V.6) Stern layer potential particle rotation angle volume fraction
461 VI. LYOPHILIC COLLOIDAL SYSTEMS
Thermodynamic description presented in Chapter IV allowed us to subdivide all colloidal systems into two large classes" thermodynamically stable systems, referred to as lyophilic and those characterized by kinetic stability only, referred to as lyophobic systems. Detailed description of properties and stability of lyophobic systems is presented in chapters that follow, while in the present chapter we will focus on the properties, structure and formation conditions of the lyophilic colloidal systems. The possibility of existence of lyophilic systems in equilibrium with macroscopic phases is determined by the nature of dispersed phase and its interaction with dispersion medium. In systems consisting of simple molecules without strong diphilic features the formation of equilibrium colloidal systems occurs only within narrow temperature range in a direct vicinity of the critical point. These systems are referred to as critical emulsions. The formation oflyophilic colloidal systems in broad temperature and concentration ranges is typical if molecules of one of the components are of a strong diphilic nature. Such are the surfactant molecules with either ionic or large non-ionic polar group and long hydrocarbon chain. The ability of these surfactants to spontaneously form equilibrium colloid systems, referred to as micelles ~, stipulates their broad use in various applications [ 1].
This term should not be confused with the one introduced earlier for sols (see Chapter V.4)
462 VI.1. The Conditions of Formation and Thermodynamic Stability of Lyophilic Colloidal Systems
Lyophilic colloidal systems are ultramicroheterogeneous systems formed by spontaneous dispersion of macroscopic phases. These systems are thermodynamically stable with respect to both the growth of dispersed particles and their further refining to molecular level sizes. The equilibrium particle size distribution that does not change with time is characteristic of these systems. It was shown in Chapter IV that the spontaneous formation of lyophilic colloidal systems occurs because the free surface energy increase associated with bringing the macroscopic phase into the dispersed state is compensated by the gain in free energy due to participation of the formed particles in the Brownian motion, i.e by the gain in entropy. This is possible only when the specific interfacial energy does not exceed the critical value, %:
]3'kT Cy _< ~ c r =
Let us
now
examine
d2
(gI.1)
9
in detail
the
equilibrium
in
such
thermodynamically stable system. We will base our discussion on the analysis of the change in free energy, A g , of idealized monodisperse system of constant composition, formed by dispersing a known volume of continuous phase 1 in another continuous phase 2 (the dispersion medium). Depending on the particle size, expressed either as radius or diameter, the number of particles, JV'~, in the newly formed dispersed phase changes. If the total volume of substance forming the dispersed phase is constant, one may write
463
4 -
3
-- n r
rt JV~l -
3
-- d
3 ~4~1-
c o n st
6
In agreement with eqs. (IV.2) and (IV.9), in dispersion process in which monodispersity is maintained, the change in the free energy, A ~ can be approximately written as
A g-(d)
.
otd .2ey ~
.
[3kT,~l . .
oil d
13 kT Of, 2
d3
where 13is logarithmic function of the ratio of the number of molecules in the dispersion medium, At2, to the number of dispersed colloidal particles, jlr~, and the numerical coefficients, ~ and ~2 are defined by the particle shape. While looking at change in the free energy of such monodisperse system as a function of particle size, A g (d), or more conveniently as a function of the logarithm of size, A g ( l o g d), one needs to account for the influence of degree of dispersion on the value of specific surface free energy (interfacial tension), rs, of dispersed particles. In the simplest case, when the latter is independent of particle size, i.e., ey(d)=const, the curve describing A g (d) function contains a maximum only (Fig. VI- 1, curve 1) [2]. This maximum is of no particular interest, since it can not be viewed as the height of the real energy barrier in the dispersion process. Indeed, the maximum was obtained under the conditions of idealized dispersion, when the system remained in monodisperse state. The point that is of more interest is the intersection of A g (log d) curve with the x axis, i.e. the point at which A g - 0 at d - d* - ([3kT /(I(Y) 1/2. The region to the right of this point corresponds to
464 thermodynamically unfavorable states of the system, while that to the left corresponds to thermodynamically favorable ones. A3
I
12
I ~ 9 1o
Io~b
-8
-7
-6
log d, m
jl
J/
zl
Fig. VI. 1. The change in free energy of monodisperse system as a function of logarithm of the particle size, d [2] The formation of this equilibrium colloidal system may take place if the value of d* lies within the region where the particle size, d, is significantly larger than molecular dimensions, b: d >>b, for instance in the region where d >~ (5 - 10)b. Then the necessary condition of spontaneous formation of lyophilic colloidal system, and, consequently, the condition under which this system may be in equilibrium with macroscopic phase, can be expressed by the Rehbinder- Shchukin criterion [3,4]:
RS-
b
=
~b2~
=
b2~
>> 1.
This criterion is equivalent to the condition of
(VI.2)
macroscopic phase
465 spontaneous dispersion, given by eq. (VI. 1). Thus, thermodynamically equilibrium lyophilic colloidal systems consisting of particles significantly larger than those of molecular dimensions may form by the spontaneous dispersion of macroscopic phase at sufficiently low, but finite positive values of u, i.e., when u ~(3cr (RS ~ 5 - 10). For example, lyophilic colloidal system containing particles with diameter, d ~ 10.8 m, may form when o does not exceed several hundredths of mJ/m 2. Since the critical value, %, is a function of particle size, d, (see eq. (VI. 1)) the formation of colloidal system consisting of particles with larger size is possible only at lower values of cy, and vice a versa. It is necessary for one to point out that the above estimates are related to the case when particle concentration in the formed lyophilic colloidal system, A/~, is much smaller than the number of molecules in the disperse phase, JV"2, and when [3~ 15 - 30. In agreement with eqs. (VI. 1) and (VI.2), the values of % and RS criterion, which determine the equilibrium conditions of lyophilic colloidal system and the conditions necessary for its spontaneous formation from a macroscopic phase, depend on the value of [3, which decreases with increasing particle concentration.
The necessary condition for spontaneous formation of disperse system and condition of its equilibrium with a macroscopic phase can be also obtained by utilizing the concepts of theory of fluctuations. This may be conveniemly illustrated by the example of a highly mobile interface, such as liquid - liquid or liquid-vapor. The surface of liquid is not completely flat: thermal fluctuations result in the appearance of capillary waves. It was shown by L. Mandelshtam (1914) that in the vicinity of critical point, e.g. around the temperature corresponding to a complete mixing of two liquids, the interface acquires substantial
466 roughness which can be easily observed by a significant enhancement in scattering of reflected light (Fig. VI- 2).
Fig. VI - 2. The reflection of light from an interface in the vicinity of critical temperature corresponding to the total mixing of two phases The work of fluctuational formation of a single "bump" (that, for example, can be represented by a hemisphere with a radius r which will further separate as a separate droplet) on such an interface can be expressed as
W(r) ~
27tr2cy.
In agreement with the general theory of fluctuations (see Chapter V), the average value for the squared radius of such fluctuations, r 2 , is determined by the second derivative of the work of fluctuation with respect to the fluctuating parameter, i.e., with respect to radius in the present case:
-~-
kT
kT
d2 W ( r ) / d r 2
7tcy
from where it follows that
kT (Ycr '~
~" 2 7ZF
(VI.3)
Equation (VI. 3) differs from eq. (VI. 1) only by the numerical coefficient. The estimates for % based on eq. (VI. 3) yield, however, the values lower than expected, because this equation does not account for a number of other factors, such as the frequency of fluctuations of a given size and, consequently, the number of particles of this size.
467 The conditions of stability of colloidal particles with respect to further dispersion down to molecular sizes can be found by analyzing the A ~r (d) dependence at d -- b. If the value of G does not change with the decrease in d down to the molecular dimensions, and if (~ can be used to describe the work of dispersing, further dispersion of particles down to molecular sizes is thermodynamically favorable. In a real polydisperse system the dispersed particles of colloidal range with some defined particle size distribution may, however, also fluctuationally form at (~ = const. In described monodisperse colloidal systems the dependence of the surface energy on the particle size may have substantially different shape if a sharp increase in surface free energy, (~, occurs during the dispersion process. In this case the A g (6/) curve may contain a minimum at dm > b. This minimum corresponds to thermodynamically more stable particle size (see Fig. VI-1, curve 2). The existence of a minimum indicates that microheterogeneous disperse system with average particle size on the order of dm is thermodynamically most favorable one. While leaving the discussion of reasons for this sharp increase in o upon decreasing the size of dispersed particles until further sections, it would be worth mentioning here that in a two component system ~ can sharply increase if dispersed phase consists of asymmetric, i.e. diphilic, molecules. This is especially typical in systems that contain micelle-forming surfactants (see Chapter VI, 3). In multicomponent systems a sharp increase in the surface energy occurring upon dispersion, i.e., upon increasing the interfacial area, may be related to a decreased adsorption of surfactants due to their lower content in the system.
468 Vl.2. Critical Emulsions as Lyophilic Colloidal Systems
The unique properties that substances reveal in their critical state, i.e. in the vicinity of critical temperature, Tcr,corresponding to a complete mutual miscibility of two phases in the liquid - vapor and liquid - liquid systems have attracted the attention of scientists for a long time. A strong scattering of light by such systems suggested that they were of colloidal nature. Theoretical concepts of the existence of thermodynamically stable two-phase systems (critical emulsions) at temperatures slightly below the critical point, Tc~,were for the first time developed by M. Volmer, who assumed that critical systems were a peculiar colloidal solutions with certain size distribution of microdroplets, in contrast to true solution in which the size of all components was on molecular level [5,6]. By applying usual relationships, used in thermodynamic of true solutions, to the description of critical systems, Volmer obtained an exponential function describing the number distribution of droplets, n(r): 4=r2cy ) n(r)
~ r x
exp -
kT
where the power, x, depending on the model used, may vary within a broad range, e.g., from 2 to 12, which, however, does not have a significant influence on the results. The views on the nature of stability of critical emulsions as microheterogeneous systems were further developed in the works by Ya. I. Frenkel and J. Meier, and utilized in the theory describing thermodynamically stable colloidal systems formed by a spontaneous
469 dispersion of substance at low positive values of o (See Chapter IV, l) [7,8]. This theory, developed by E.D. Shchukin and P. A. Rehbinder, is based on quantitative analysis of changes in the entropy of system due to dispersion [9]. Using critical emulsions as an example, let us now discuss the formation of critical systems using the data collected by E.D. Shchukin, L.A. Kochanova et al [3,10-12]. The single upper critical point corresponding to particular values of temperature and system composition, is often present in two-component systems, such as tricosane - oxyquinoline. In this case one can approach the critical state from the side of two-phase system by changing the temperature in the system that has composition close to the critical one. The difference between the critical temperature and the temperature of experiment 9AT= T~-7', or the difference in the phase compositions, Ac, may be chosen as parameters that characterize the deviation of system from the critical state. As the system approaches the critical state, the interfacial tension, o, at the interface between coexisting phases (i.e. between liquid I and liquid II) decreases. Figure VI-3 shows the experimentally determined values of o as a function of T for tricosane - oxyquinoline system [ 11 ]. In the vicinity of the critical point the surface tension is on the order of hundredths of mJ m -2 or even smaller. The temperature dependence of emulsion stability, evaluated in terms of the emulsion
"life time", tf, was studied simultaneously with the
temperature dependence of the surface tension (Fig. VI-4). These studies showed that the stability of emulsions (of noticeable concentrations) in a system with nearly critical composition experienced a sharp increase and
470 tf
mJ 0.7
rain
mJ CY, m2
m
tf
ff
30 I 0.06
0.6 0.5 0.4
20
0.04
10
0.02
0.3 0.2
/
0.1 /1 75
80
85
90
95
~,
~~
Fig. VI-3. The interfacial tension, ~, as a function of temperature, T [3]
/ 1
L
t
2
3
4 AT
Fig. VI-4. The interfacial tension at the interface between the two liquids, ~, and the emulsion stability, tf, as a function of temperature [ 10,11 ]
became infinite when the surface tension was exactly on the order of a few hundredths of mJ m 2, in good agreement with theoretical predictions described earlier. In the temperature - composition coordinates, the region in which these critical emulsions are stable is shown ( Fig. VI-5) by a thin sickle in the vicinity of Tc~. From the top this region is limited by curve I corresponding to molecular solubility of liquids, (the binodal) while from the bottom it is limited by curve II, which one can view as the border line of colloidal solubility. The system within the sickle is stable, regardless whether it was formed by cooling the homogeneous solution (transition from above), or by heating the heterogeneous, i.e., two-phase, region (transition from below). The part of the sickle region that is enriched with the polar liquid corresponds to the direct emulsion of hydrocarbon in oxyquinoline, which is saturated with that hydrocarbon. The other part of the sickle corresponds to the inverse emulsion of oxyquinoline in hydrocarbon. In the temperature region between Tcr and Tk these two emulsions coexist simultaneously; a continuous
471 transition between direct and inverse emulsions occurs within this region. When T>T~ the usual continuous transition between two homogeneous solutions takes place.
r, oc
TCr
90
88
86
f
1
70
75
80
85
90
oxyquinoline, tool %
Fig. VI-5. The temperature - composition phase diagram of the tricosane - oxyquinoline system [11] Particle size analysis in the critical emulsions is a rather complex task, in part due to the high particle concentration. However, such studies were carried out and yielded the size of microdroplets on the order of tens of nm. Similar treatment can also be applied to three-component systems, in which two of the three components are immiscible with each other, but each of these components is infinitely miscible with the third one. The phase diagrams in such three-component systems contain a so-called line of the critical states, which shows the critical composition as a function of temperature. In such systems the critical state can be approached from the side of a two-phase system by both changing the temperature and altering the composition.
472 VI.3. Micellization in Surfactant Solutions
Micellar dispersions, which contain micelles along with individual surfactant molecules, are the typical examples of lyophilic colloidal systems. Micelles are the associates of surfactant molecules with the degree of association, represented by aggregation number, i.e. the number of molecules in associate, of 20 to 100 and even more [1,13,14]. When such micelles are formed in a polar solvent (e.g. water), the hydrocarbon chains of surfactant molecules combine into a compact hydrocarbon core, while the hydrated polar groups facing aqueous phase make the hydrophilic
shell. Due to the
hydrophilic nature of the outer shell that screens hydrocarbon core from contact with water, the surface tension at the micelle - dispersion medium interface is lowered to the values c~_
Fig. VI-6. The schematic illustration of surfactant micelle in aqueous solution Under particular conditions the thermodynamically stable micellar dispersions may form as a result of spontaneous dissolution of crystalline or liquid surfactant macroscopic phase. Even though the state of substance in a micelle is not always equivalent to that in a macrophase, a rather high degree
473 of association of surfactant molecules in micelles allows one to view them as particles of another phase, different from a molecular solution. Micellar dispersions of surfactants reveal properties that are typical for colloidal systems, namely they scatter light, have an increased viscosity, etc. Not all of the surfactants are capable of forming micelles. The appropriate ratio between the size of hydrophobic (hydrocarbon chains) and hydrophilic (polar group) parts ofsurfactant molecules, which determines their
hydrophile-lipophile balance (HLB, see Chapter VIII, 3), is necessary for the formation of miceUes to take place. Sodium and ammonium salts of C~2 - C20 fatty acids, alkylsulfates, alkylbenzenesulfonates, and other synthetic ionic and nonionic surfactants are the examples of micelle-forming surface active substances. The true solubility, i.e. the concentration of dissolved substance in its molecular or ionic form, of such surfactants is rather low: for ionic surfactants it is on the order of hundredths and thousandths ofkmol m -3, while for nonionic ones it can be even lower by one or two orders of magnitude. The ability of polar groups to effectively screen hydrocarbon core from contact with an outer aqueous phase determines, to a significant extent, whether or not the surfactants would typically form micelle solutions. This ability depends not only on the size of polar groups but also on their nature (i.e. if they are ionic or nonionic) and interactions with solvent (e.g. their ability to become hydrated). As the aggregation number, m, becomes smaller, the size of micelles decreases, and, hence, the fraction of substance at the interface between the solution and the micelle increases, resulting in a less effective shielding of hydrocarbon core by the polar groups [13]. The existence of spherical micelles with some optimal degree of surfactant
474 association, which usually corresponds to colloidal particles with radius, r, comparable
to
the
hydrocarbon
thermodynamically favorable.
chain
length,
lm, is, therefore,
For example, the diameter of the stable
micelles present in the sodium oleate solution is around 5 nm, which corresponds to the aggregation numbers on the order of several dosens. The formation of micelles with higher aggregation numbers (when r>/m) that maintain spherical symmetry is thermodynamically unfavorable, since it has to involve the inclusion of polar groups into the body of a micelle. For this reason the degree of association of molecules in micelles increases not due to the growth of spherical micelles, but due to changes in their shape, i.e. due to the transition to asymmetric structures. The formation of colloidal particles (surfactant micelles) in the disperse system either as a result of spontaneous dispersion of macroscopic phase, or by spontaneous association (condensation) of individual molecules upon the increase of surfactant concentration, corresponds to a qualitative change in the system. The latter undergoes transformation
from
macroheterogeneous or homogeneous state into microheterogeneous colloidal dispersion. Such qualitative change causes an abrupt experimentally observable change in physico-chemical properties, which in most cases represented by a characteristic break on the curves showing various physicochemical parameters as a function of surfactant concentration. As the surfactant concentration in solution increases above some critical concentration, Ccr,one can observe a noticeable increase in the intensity of scattered light, which is characteristic of the formation of a novel dispersed phase. Instead of their usual smooth behavior, described by the Szyszkowski
475 equation, the surface tension isotherms experience a break at c - c~. Further increase in surfactant concentration above c~ results in essentially constant values ofcy (Fig. VI-7). Similarly, the break at c = C~rappears also in the curves showing specific and equivalent (A) conductivities as a function of concentration of an ionic surfactant (Fig. VI-8). The surfactant concentration, c~, above which micellization begins (some experimentally detectable number of micelles form) is referred to as the critical micellization concentration (CMC). Abrupt changes in the properties of surfactant- water system that occur in the vicinity of the CMC, allow one to determine the latter with high precision from the break point in the curves showing various properties as a function of surfactant concentration. In the discussion of micellization we will primarily focus on features of this process that are common for both ionic and non-ionic surfactants. The ability of ionic surfactants to undergo ionization in aqueous solutions results in the generation of charge at the micellar surface, which stipulates some specific features of systems containing such surfactants. A
I
I
!
CMC
c
Fig. VI-7. The surface tension isotherm of aqueous solutions containing micelleforming surfactants
0
. . . . .
CMC
Fig. VI-8. The equivalent electric conductivity of aqueous solutions of ionic surfactants as a function of surfactant concentration
476
VI.3.1. Thermodynamics of Mieellization The equilibrium between dispersed phase (i.e., micelles) and molecular solution o f a surfactant (or the macroscopic phase, in case of saturation) exists in thermodynamically stable systems containing micelle-forming surfactants. One can, to a certain degree of approximation, describe the equilibrium between micelles consisting of m surfactant molecules and molecularly dissolved surfactant as a chemical reaction, namely [ 15,16] m[S] a (S)m, where S stands for surfactant molecules. In agreement with the law of mass action, one can write nmic/NA Kmic
-"
m
CM
where n mic is the number of micelles per 1 m3; cM is the concentration of molecularly dissolved surfactant in kmol m -3, and K mic is the equilibrium constant of micellization.
In systems containing ionic surfactants, the molecules of which undergo dissociation into ions with monovalent counterions, it is more proper to describe the formation ofmicelles with aggregation number m and effective charge q as nmic/NA Kmic -
m
~ q / e
(VI.4)
C M t; i
where ci is the concentration of counterions and e is the charge of electron. Since the CMC corresponds to some particular value of nm~c,determined by the precision of available experimental methods, for electrolyte that contains an ion identical to the one present in a surfactant molecule, eq. (VI.4) yields Ccr = CMC as a function of
477 electrolyte concentration, i.e 9
log C M C -
k~- k 2 log c~,
where k~ = (l/m) log (n ~c/NA Kmic),and k 2 = q / ( m e ) is the degree of dissociation of the ionic groups in a micelle. One can also obtain this expression by examining the work of micelle charging.
The experimental studies indicate that aggregation numbers of surfactant molecules in micelles increase from 20 to 100 or higher, as the surfactant hydrocarbon chain length grows longer. Consequently, the dependence of nmic on the total surfactant concentration in the system, Co, can be represented by a high order parabola, and may be viewed as the curve with inflection point corresponding to the CMC (Fig. VI-9). At low net surfactant concentrations, i.e. when c0 CMC can be readily understood, since the value of ~ is determined by the concentration of molecularly dissolved surfactant. Indeed, in agreement with the Gibbs equation, dc~= Fdg, the condition ofc~- const corresponds to an independence of chemical potential of concentration at Co> CMC, i.e., dg - 0. One can thus
478 say that the formation of micelles causes a characteristic non-ideality of solution above the CMC. f/mic
Cm
C'mi c
I I J
J CMC l
r
,,, ,
o Co Fig. VI-9. The micelle number concentration, r/mic,as a function of the total surfactant concentration in the system, Co
CMC
Co
Fig. VI-10. Changes in the surfactant content in the molecularily dissolved and micellar states as a function of increase in the total surfactant concentration, Co
The amount of substance present in the micellar state, Cmic mnmic /NA, -
-
may exceed the concentration of it in the molecular solution by several orders of magnitude. The micelles thus play a role of a "reservoir" (a depot) which allows one to keep the surfactant concentration (and chemical potential) in solution constant, in cases when surfactant is consumed, e.g. in the processes of sol, emulsion and suspension stabilization in detergent formulations, etc. (see Chapter VIII). A combination of high surface activity with the possibility for one to prepare micellar surfactant solutions with high substance content (despite the low true solubility of surfactants) allows for a the broad use of micelle-forming surfactants in various applications. Important information regarding the nature of the micellization process can be obtained from the studies on the temperature dependence of the CMC. It is worth reminding here that CMC corresponds to the state of
479 thermodynamic equilibrium between micelles and individual surfactant molecules. CMC is the concentration of true solution, C~r,at which a particular, experimentally detectable number ofmicelles per unit volume, nmic,is formed. If one assumes that this measurable micelle concentration, nm~, and the aggregation number of molecules in micelles, m, in the vicinity of the CMC remain constant within some temperatur range, and that the activity coefficient of molecular solution is 1, the thermodynamic expression for the enthalpy of micellization can be written as d In Ccr A ~rVrmic = - RT 2 m ~ dT Numerous experimental studies on micellization in various surfactant solutions indicated that the values of A J~mic are usually very small and often positive [1,15]. Since a spontaneous processes is accompanied by a decrease in the system free energy, small and, moreover, positive values of A ~ mic indicate that e n t r o p i c a l c h a n g e s play a significant role in spontaneous micellization process. Such changes are primarily related to the specific features in the structure of water as a solvent (see Chapter II,2). The driving force for the association of hydrocarbon chains into a micellar core is an increase in the entropy of the system, which occurs primarily due to destruction of iceberg structure present in water. Such structures are present around the hydrocarbon chains of dissolved surfactant molecules. The studies performed with aqueous dispersions of micelle-forming surfactants have shown that the micelle formation by both association of individual molecules and dispersion of macroscopic phase may occur only
480 above certain temperature, referred to as the Krafftpoint, TKr(Fig. VI- 11) [ 13]. Below the Kraffl point the surfactant solubility is small and its concentration is lower than the CMC. The equilibrium between the surfactant crystals and true surfactant solution (the concentration of which rises as the temperature increases) exists in this temperature range. Thus, in surfactant solutions, for which the Kraffl point is in the range of elevated temperatures, the formation of micelles does not occur under the normal conditions. I
I Micelles Crystals
+
+
solution
solution
CMC ution
TK~
T
Fig. VI-11. Phase diagram of a micelle-forming surfactant- water system
Due to micelle formation the total surfactant concentration undergoes an abrupt increase. Since true (molecular) solubility ofsurfactants, determined by the CMC, remains essentially constant, an increased surfactant concentration in solution is caused by an increase in a number of formed micelles. Micellar solubility increases with increase in temperature, and thus a continuous transition from pure solvent and true solution to micellar solution, and further to different liquid crystalline systems and swollen surfactant crystals (see below), may take place in the vicinity of the Krafft point.
481 The molecular solubility and the surface activity of micelle-forming surfactants, respectively, decrease and increase by a factor o f ~ 3 to 3.5 when the hydrocarbon chain is extended by one CH2 group (see Chapter II,2). Since in the vicinity of the Krafft point the value of the C M C differs little from the molecular solubility, C M C within the same homologous series also decreases by a factor of-- 3 to 3.5 upon the transition to each subsequent member. The highest possible lowering of the surface tension at the air - surfactant solution interface ofmicelle-forming surfactants, as well as of"regular" surfactants, is essentially constant within a given homologous series.
The CMC's of all micelle-forming surfactants are usually low (about 10-s to 10-2 kmol m-3), i.e. low concentrations of molecular solutions correspond to the micelles=solution equilibrium. This means that the existence of particles with sizes d, different from the size of micelles, dm, is thermodynamically unfavorable (see Chapter VI,1). The transition from particles of size dm to those with smaller sizes, hence, results in the increase in free energy of the system, and the A J-(log d) curve contains a minimum in the colloidal range at d=dm (see Fig. VI- 1). The increase if A J-occurring at d
Spg.
This area is dependent not only on the size of a polar
group, but also on its interactions with the solvent, i.e. on hydration. The value of ~ at micelle - medium interface is determined by the degree to which hydrocarbon core is shielded: the lower the degree of screening, the higher cy.
482 If one assumes that the hydrocarbon core of a micelle has density similar to that of the corresponding bulk hydrocarbon, the spherical micelle with a core of radius r I and surface area S~ contains
m
4 gr] 3 -
-
3Vl
NA
molecules (and, consequently, the same number of polar
groups), where m is the aggregation number of molecules in a micelle, and V~ is the molar volume of a corresponding hydrocarbon. The fraction of micelle surface within which the hydrocarbon part is screened from contact with water, would then be given by
mSpg 4~rl3NASpg
S1
3V14~rl2
f 1SpgNA 3V1
and will decrease as micelle becomes smaller. One can then estimate how c~would increase with the decrease in r~. Let us assume that the value of ~ at an unscreened portion of surface is close to that at the hydrocarbon - water interface (c~0 - 30 to 50 mJ m2), while at the "screened" portion is much less than that. The "mean" effective value of the specific particle surface energy, C~m~c,would then be given by CYmic ~
CYo/1-rlSpgNA-/ 3V1
'
and with the decrease in particle size may abruptly increase from small values (e.g. hundredths ofmJ m -2 for thermodynamically stable systems) to those comparable with % i.e. to tens of mJ m -2. Surfactants that are capable of forming micelles are, consequently, those that along with the well-developed hydrophobic chain contain one (or more) strong polar groups capable of screening a hydrocarbon core on a sufficiently large area, Spg.In this sense, the strong polar _
groups are those typically present in ionic surfactants, i.e. -COO-, -SO3, -OSO3, -NH3 + , which due to their strong hydration reveal high screening ability. Among nonionic surfactants those containing a substantial number of polar groups are the micelle-forming ones. Examples of surfactants belonging to this class include polyoxyethylenated compounds, the derivatives
483 ofsaccharides, glukozides and others. Nonionic surfactantswith one small polar group, such as fatty alcohols, are not able to form micellar dispersions: small size and relatively weak hydration of-OH group make the required screening of hydrocarbon core impossible. The surface energy of surfactant micelles is discussed in the works by C. Tanford, E. Ruckenstein, A. Rusanov, and others.
For many nonionic surfactants, that are liquid, the Kraffl point does not exist, and one uses the cloudpoint as a thermal characteristic. An increased cloudiness above the cloud point is related to the increased size of micelles and the separation of system into two phases due to the dehydration of polar groups at elevated temperatures.
VI.3.2. Concentrated Dispersions of Micelle-Forming Surfactants Within a broad range of concentrations above the CMC the surfactant molecules are associated into spherical micelles, the so-called Hartley-
Rehbinder micelles. The hydrocarbon core of such micelles remains liquid, even though its state
is different from that of the corresponding bulk
hydrocarbon, such as in the emulsion droplets. The formation of mixed micelles containing different additives (even when the constituent molecules substantially differ in size), as well as the dissolution of liquid hydrocarbons (otherwise insoluble in water) in hydrophobic cores (solubilization, see Chapter VI, 4) points to a liquid-like state of micellar hydrocarbon cores. As surfactant concentration in solutions with Co> CMC increases, not only the concentration of spherical micelles becomes higher, but also their shape changes. Spherical micelles turn into anisometric ellipsoidal and cylindrical micelles and further to rod-like, thread-like and lamellar micelles
484 [ 13, 17]. In these types of micelles the hydrocarbon chains assume more and more ordered arrangement. Each surfactant concentration corresponds to the thermodynamic equilibrium,
spherical micelles ~anisometric micelles ~ lamellar micelles
The formation of anisometric colloidal particles, the existence of which was first predicted by McBain, can be verified experimentally by optical, rheological, and radiological methods. For instance, the flow of surfactant solutions containing McBain micelles deviates from Newtonian behavior (see Chapter IX). The structure of lamellar and ribbon-like micelles formed with parallel-packed surfactant molecules is identical to bimolecular layer. The surface properties of anisometric, and in particular of ribbon-like, micelles are non-uniform within different regions of their structure. Within the flat regions where the concentration of polar groups is higher, the hydrocarbon core is more effectively screened from the aqueous phase, while the edge regions reveal lower hydrophilicity as compared to the flat ones. Further increase in the total surfactant concentration in the system (which is the same as the total decrease in water content) results in decrease in the mobility of micelles and their further connection with each other primarily via the endregions. These phenomena were studied by Z. Markina et al, who showed that three-dimensional network structures (gels) with typical mechanical properties, such as plasticity, durability and thixotropy, form under these conditions (See Chapter IX) [18,19]. Systems with ordered arrangement of molecules, characterized by
485 optical anisotropy and mechanical properties intermediate between those of true liquids and those of solids, are often referred to as
liquid crystals
[13,20,21]. Further removal of the dispersion medium results in a conversion of gel into a solid macroscopic phase, i.e. into the soap crystal. Based on the results of the X-ray diffraction analysis, soap crystals were shown to have a lamellar structure. The surfactant- water system can thus undergo transitions into various states, depending on the content of components" from a homogeneous system (surfactant molecular solution) to lyophilic colloidal state and further to macroscopic heterogeneous system (soap crystals in water). Different states of the system can be described by a particular thermodynamic equilibrium, i.e."
true solution
~
spherical micelles in solution
~
anisometric micelles
~
gel
~
crystals m solution
..,r
free (unstructured) colloidal system One may also consider a reverse transition from macroscopic heterogeneous system (surfactant crystals in water) to micellar solutions existing above the Kraffl point, via gel formation and its spontaneous dispersion stages. In this case, the swelling of soap upon the penetration of water between the lamella formed with polar ( strongly hydrated) groups occurs prior to the formation of colloidal solution. At sufficient dilutions separation of individual particles, e.g. lamella, from crystals occurs due to the
486 thermal motion. These separated platelets first form lamellar micelles, which upon the decrease of total surfactant content further transform into cylindrical, ellipsoidal and spherical micelles.
VI.3.3. Formation of Micelles in Non-Aqueous Systems in contrast to to aqueous surfactant solutions, in which micelles with the polar groups oriented towards the aqueous phase, micelles with inverse orientation of polar groups may form in the solutions of surfactants in hydrocarbons (Fig. VI- 12). Polar groups in such inverse micelles are combined into hydrophilic (oleophobic) core, while the hydrocarbon chains facing the non-polar medium form oleophilic shell, which screens an internal hydrophilic part of micelle from contact with hydrocarbon medium [1,13]. Surfactants that are capable of forming micelles in non-polar solvents are, as a rule, insoluble in water; the balance between hydrophilic and oleophilic properties of their molecules is shifted towards oleophilicity. The aggregation numbers of molecules in inverse micelles, m, are significantly lower than those in direct micelles, which is explained by molecular structure" in the present case the micellar core consists of polar groups that are rather small compared to hydrocarbon chains. Consequently, the screening required for thermodynamic stability of inverse micelles is reached at lower degrees of association, i.e. with lower number of oleophilic groups on micellar surface.
Fig. VI-12. The schematic drawing of an inverse micelle in non-aqueous medium
487 The solvent polarity (non-polarity), which determines the interactions of its molecules with polar and non-polar regions of surfactant molecules, plays an important role in the formation of micelles in non-aqueous medium. For micelle formation to take place, the medium has to be a"good solvent" for hydrocarbon chains only. Micelles do not form in the medium of nature similar to both parts of diphilic surfactant molecules; the surfactants reveal only true solubility in such medium. Low alcohols (less than C 5 ) which are good solvents for both polar and non-polar regions ofsurfactant molecules are the typical examples of such media. In contrast to aqueous systems, micelle formation in non-polar media is driven by the benefit in energy rather than by an increase in entropy. The replacement of polar group - hydrocarbon interaction (as in the case of dissolution) with the interaction between polar groups upon their association into micellar core is thermodynamically beneficial. The benefit in energy upon association of polar groups is so large, that even at low concentrations true surfactant solutions contain small pre-micellar associates rather than individual surfactant molecules.
VI.4. Solubilization
in Solutions of Micelle-Forming Surfactants.
Microemulsions
In the previous section we discussed two-component lyophilic colloidal systems, namely the dispersions of micelle-forming surfactants. The third component, when introduced into the system, depending on its nature can either retard the formation of micelles or, oppositely (which occurs more
488 often), favor it. The association of surfactant molecules into micelles is suppressed when significant amounts of polar organic substances, such as lower alcohols, are introduced into the aqueous surfactant solution. Such substances increase molecular solubility of surfactants, and hence retard the formation of micelles. When the same substances, especially non-polar hydrocarbons, are introduced in small quantities the CMC of a surfactant becomes somewhat lower, which allows for easier formation of micelles. The structure of micelles formed in the presence of an additional component is substantially different than in its absence: introduced additive becomes incorporated into the micelles. As a result, h y d r o c a r b o n s , practically insoluble in water, d i s s o l v e in m i c e l l a r s o l u t i o n s of s u r f a c t a n t s . Incorporation of a third component, insoluble or nearly insoluble in the dispersion medium, into micelles is referred to as solubilization [13]. One may distinguish between direct solubilization, occurring in aqueous surfactant solutions, and inverse solubilization, occurring in hydrocarbon systems. Let us now turn to the discussion of main trends in solubilization process using on the results of studies carried out by Z.N. Markina [ 19]. Let us use the direct solubilization of hydrocarbons and alcohols in aqueous surfactant solutions as an example. It is well known that the solubility of hydrocarbons in water is very small, e.g. for octane it is about 0.0015% by weight. At the same time, one may prepare ~2% solution of octane in 10% sodium oleate solution, i.e., the effective solubility of this hydrocarbon increases by more than three orders of magnitude. Solubilization can be described quantitatively by characteristic referred to as the relative
solubilization, s, given by the ratio of the number of moles of solubilized
489 substance, N~o~,to the number of moles of surfactant in micellar state, Nm~c,i.e." S-
Nso l / N m i c .
Figure VI-13 shows relative solubilization, s, of octane (curve 1) and cyclohexane (curve 2) as a function of surfactant concentration (sodium oleate) above the CMC. These data indicate that solubilization of hydrocarbons is constant at surfactant concentrations corresponding to the region in which spherical micelles exist (to the left from arrows). For instance, at temperatures between 6 and 20 ~C, solubilization of octane is ~ 0.5 moles, and that of cyclohexane is ~ 1.2 moles of hydrocarbon per mole of micelleforming surfactant. A sharp increase in relative solubilization occurs at higher surfactant concentrations, corresponding to the existence of anisometric micelles. This increase is followed by changes in the micellar structure" anisometric micelles undergo an inverse conversion into spherical micelles. This is also confirmed by rheological studies. Simultaneously with an increase in relative solubilization, a sharp decrease in viscosity, often by two orders of magnitude, occurs. The rheological behavior of the system becomes similar to that of Newtonian fluids. This is related to the fact that due to the penetration of hydrocarbon into the micellar core the diameter of spherical micelles may exceed twice the length of hydrocarbon chains in surfactant molecules, as shown in Fig. VI-14. Under the conditions when there is no excess of this solubilized substance (when there is no contact with bulk hydrocarbon phase), the distribution of solubilized substance between micelles and molecular solution is governed by the amount of work required to transfer hydrocarbon molecules
490
2.0
1.0
0
t 0.2
! 0.4
! 0.6 r tool / I
Fig. VI- 13. Relativesolubilizationof octane Fig. VI- 14. Solubilization of hydrocarbon (curve 1) and cyclohexane (curve 2) as a in a direct micelle function of sodium oleate concentration above the CMC from aqueous phase into the micellar core. Hydrophobic interactions act as a driving force of solubilization process (as well as of micelle formation). Equilibrium distribution of hydrocarbon between its macroscopic phase, true solution and micelles is established on contact ofmicellar solution with hydrocarbon macrophase, provided that there is an excess of solubilized substance. The temperature dependence of solubilization can in this case be described as d lns dT
A 5~sol RT 2 '
where A 5~so~is the solubilization enthalpy determined by energy of transition of hydrocarbon from macroscopic phase into micelles and by restructuring of micelles during solubilization. The incorporation of hydrocarbons into micellar hydrocarbon core, occurring during solubilization, may be confirmed by specialized studies involving spectroscopic and radiospectroscopic methods. For example, the state of water that is incorporated into the cores of inverse micelles(during inverse solubilization) can be monitored by the NMR, due to changes in the
491 mobility of protons. Some characteristic phenomena occur upon the solubilization of dyes, whose absorption spectra differ in polar and non-polar media. At small surfactant concentrations (below the CMC) the dye is dissolved in aqueous phase, and its optical absorbance spectrum is characteristic of an aqueous solution (for instance, the spectrum ofRhodamine 6G contains an absorption band with )~ma•= 590 rim). Above the CMC nearly all of the dye is solubilized, and its absorbance spectrum becomes characteristic of the dye solution in hydrocarbon (in the case of Rhodamine 6G, the band with
)~max-- 590
nlTl
dissapears, and the solution changes color from crimson to orange). This feature allows one to determine the CMC of a surfactant from changes in spectral properties of surfactant solutions due to the micelle formation. Changes in spectral properties of dye upon its solubilization prove that the dye becomes localized within a micellar hydrocarbon core. The solubilization of polar organic c o m p o u n d s , and particularly of surfactants that do not form micelles, has a somewhat different character. The presence of both polar and non-polar regions in the molecules of such substances results in an incorporation of these "solubilizing" molecules into micelles along with the "original" surfactant molecules forming the micelles. Micelles of mixed composition, shown in Fig. VI-15, are formed as a result [13,14, 17]. If a solubilized substance with high surface activity is present in the system as an admixture, the curve showing surface tension as a function of solution composition, may contain a minimum (Fig. VI-16). This minimum appears due to the solubilization of an admixture at concentrations of the main
492 component that are close to the CMC. A decrease in the concentration of an admixture due to solubilization causes a decrease in its adsorption, and, as a consequence, results in a rise of the surface tension upon an increase in the concentration of the main component above the CMC.
Fig. VI- 15. Schematic drawing of the mixed micelle
0 c Fig. VI-16. The surface tension isotherm of the solution of micelle-forming surfactant in the presence of solubilizing surface active admixtures
The orientation and concentration of molecules solubilized in micelles may lead to substantial changes in the kinetics of chemical interaction of solubilized molecules with each other and with other substances dissolved in the medium. In some cases solubilization causes a substantial increase in the rate of chemical interaction, which is the basis of micellar catalysis. Solubilization plays an important role in emulsion polymerization of unsaturated hydrocarbons, leading to the production of latexes, which are aqueous dispersions of synthetic rubber. The polymerization process primarily occurs in micelles containing soiubilized hydrocarbon, rather than in the hydrocarbon emulsion droplets. Emulsion polymerization usually yields spherical particles of uniform size [22]. As was discussed earlier, solubilization may be direct (penetration of
493 hydrocarbons into micelles in aqueous dispersions) and inverse (penetration of water into micelles in aqueous dispersions). In these cases, depending on the composition of the system and the temperature, both spherical micelles and more complex structures may form. The latter include cylindrical and lamellar micelles, as well as ordered liquid crystalline structures formed with such micelles, and so-called microemulsions [ 13,14,20,23]. This determines a great variety of structural transitions in systems with three or more components, containing micelle-forming surfactants. The ternary phase diagram of such a system may contain regions corresponding to phases of different composition. These phases may include micellar systems with micelles of different sizes and shapes and liquid crystalline phases formed with ordered direct and reverse spherical, cylindrical or reverse micelles. The formation of such phases of different structures is especially typical in the case of nonionic surfactants with large polar and non-polar regions. In the case of ionic surfactants the formation of these structures usually takes place in the presence of a fourth component, the so-called co-surfactant, which is usually a C5 - C~2 alcohol. In these cases one usually considers pseudo-ternary phase diagrams in which two corners of the triangle correspond to pure water and hydrocarbon (oil) phases and the third corner corresponds to a particular ratio of concentrations of ionic and nonionic components (i.e., surfactant and co-surfactant, respectively).
It is worth briefly recalling here that in a phase diagram of three components, A, B, and C at fixed temperature, each of the comers corresponds to 100% of the corresponding pure component (0% of the other components!). The amounts of different componentsat any given point in the diagram are givenby the distances from the correspondingbases. For more details see Fig. VI- 17 and [ 13, 24].
494 B~[ 0 o _rE/ "~.1.0
1~176176176 of component 13-] I no component A / __J
I no component C
0.3,/ : ~ 5.,~. N 07 o " 25% of component 75 goof component A 0.41/~-,:/"',~ ~ m n n"A 25% of component B ,/7, ~ ' - - , ' ~. ;--~u.{5- ~/o OTcon po e ! no component C _ 0-X..ii,{' I . } ' i ' / i } - - ~ ' ~ 0 " 5 ~ p ~
" < E . - i
....
ii"" A
1.0 00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C
Fig. VI-17. Composition of three-component system (A-B-C) at three characteristic points in the phase diagram
Numerous experimental and theoretical studies have been devoted to the investigation of structural transitions and processes taking place in such systems [13]. Among these systems microemulsions occupy a special place [23]. Let us examine the structure of a ternary water-hydrocarbon-nonionic surfactant system, or of a pseudo-ternary system consisting of hydrocarbon,
and
surfactant/co-surfactant
mixture
(at
water, variable
water/hydrocarbon ratio and constant surfactant concentration, given by horizontal lines in Fig. VI-18). At high surfactant content (line a) a m a c r o s c o p i c a 11y h o m o g e n e o u s system is formed. This system behaves as an insulator (i.e. does not conduct electric current) at high concentrations of hydrocarbon, and is electrically conductive at high contents of water. The appearance of electric conductivity at increasing water content is the percolation process related to the formation ofa bicontinuous system in which both hydrocarbon and water regions mutually penetrate each other. This is analogous to a moistened sponge with a structure that undergoes continuous (dynamic) changes with time.
495 SURFACTANT
WATER
WIII
OIL
Fig. VI-18. Phase diagrams of water- hydrocarbon(oil) - nonionic surfactant system at three different temperatures. Winsor equilibria. At lower surfactant concentrations (line b) regions of phase separation appear. In such a phase-separated state there is a sequence of equilibria between phases, commonly referred to as Winsor equilibria [ 13,23 ]. In point A two macroscopic phases are formed. These are a microemulsion of composition B and an aqueous solution containing dissolved surfactant and micelles with solubilized hydrocarbon. The volumes of these macroscopic phases can be estimated in the usual way by applying the lever rule with a correction for the densities of these phases. In such a state of separation into two macroscopic phases, the equilibrium between the microemulsion and the aqueous solution is referred to as the Winsor II (WII) equilibrium.
496 Analogously, at point C the phase separation into a microemulsion of the composition D and a micellar solution of inverse micelles containing solubilized water takes place. This is a Winsor I (WI) type equlibrium. At the intermediate point E the system consists of a single microemulsion phase (ME). At even lower surfactant concentrations (line c), depending on the water/hydrocarbon ratio, the system will either separate into one of the twophase systems (Winsor I or Winsor II), or a three-phase system may form at point F. This is a Winsor III (WIII) equilibrium with an aqueous phase at the bottom, a microemulsion phase in the middle and a hydrocarbon phase at the top. The shape of this phase diagram changes with temperature. In the case of nonionic surfactants this is related to the dehydration of their polar groups at elevated temperatures, which results in an increase in affinity of surfactant molecules towards hydrocarbons and a decrease in their affinity towards water. Due to this the elevation of temperature results in a decrease in the size of the separation region in the oil-rich area of the phase diagram and in an increase in the size of the separation region in the water-rich area (Fig. VI-18). Similar effects are caused by the addition of electrolytes. The formation of three different condensed phases in the waterhydrocarbon-surfactant system allows one to measure the surface tension at the three interfaces, and to study the c~(T) dependence at them (VI-19). Due to the dehydration of surfactant molecules, the interfacial tension at the aqueous solution - microemulsion interface, Ow-me,increases with temperature, while the interfacial tension at the microemulsion - oil interface, %me, drops until a complete vanishing of this interface occurs. For the hydrocarbon-water
497
T Fig. VI-19. Interfacial tension at water-oil (W-O), water-microemulsion (W-ME), and oilmicroemulsion (O-ME) interfaces as a function of temperature. interface, which forms upon the removal of the intermediate phase, the minimum of the interfacial tension, %-o, occurs at a particular temperature. The values of c~at this minimum may be rather low, e.g. tenths or hundredths of mJ m 2. This temperature, sometimes referred to as the HLB temperature, corresponds to the equal solubilization of oil in the water phase and of water in the oil phase. The appearance of ultra-low interfacial tension determines the use of such microemulsion systems for enhancing the degree of oil pool recovery. Microemulsion systems, sometimes also referred to as the micellar solutions, are pumped into the secondary satellite holes located at a certain distance from the production oil well. Water containing the required amount of electrolyte is pumped further into these satellite wells. While penetrating the oil pool, this microemulsion with substantial surfactant content, "washes off" the oil and forces it towards the production well [25,26]. In recent years there is also a great interest to the investigation of chemical reactions taking place in microemulsion systems. This is one of the methods used for the preparation of the nanometer-sized particles with narrow size distribution [21,25].
498
VI.5. Lyophilic Colloidal Systems in Polymer Dispersions
The ability to aggregate in solutions and form thermodynamically equilibrium lyophilic colloidal systems reveal not only low molecular weight surfactants with asymmetric structure, but also some polymers, especially those consisting of molecules with regions of substantially different polarity. The properties of systems formed by such polymers are close to those ofmicellar systems formed by low-molecular weight surfactants, despite the fact that in the case of polymers individual particles may form by the aggregation of just a few macromolecules. In many systems, such as in solutions of globular proteins, separate macromolecules behave as particles whose properties are close to those o fsurfactant mice lies. A complete description of properties of polymer solutions, including those of lyophilic colloidal systems formed in such solutions is a special topic of physical chemistry and is beyond the scope of this book. Nevertheless, it would be worth to give a brief description of structure, conditions of formation, and properties of such systems, and to compare them to colloidal systems formed by low-molecular weight substances. It is well known that polymers form thermodynamically stable molecular solutions with unique thermodynamic properties which originate from high flexibility of macromolecular chains that may exist in a large number ofpossible conformations [ 13,27,28]. Numerous studies have shown that the processes leading to the association of macromolecules are likely to take place in solutions. Depending on the solution concentration and on the nature of interactions of macromolecules with each other and with the solvent molecules, macromolecules may exist in the form of random coils, compact globules, or associates. Macromolecules may reveal a substantial degree of surface activity if the segments of constituent chains significantly differ in polarity. Polymers with these features show an increased tendency towards aggregation and globulization of molecules, along with an ability to solubilize substances insoluble in a given medium. According to the studies carried out by V. N. Izmailova et al, the molecules of some proteins and enzymes reveal high capability to solubilize hydrocarbon molecules, which can incorporate into different segments of macromolecules. The solubilization studies, and in particular those ofsolubilization ofhydrocarbons in aqueous solutions of proteins, allows one
499 to draw conclusions regarding the structure of such molecules in solution [29,30]. For example, an investigation of a relationship between the solubilization and the size of hydrocarbon molecules, allows one to estimate the number and size ofhydrophobic segments in the protein molecules. The studies of aggregation in polymer systems, association ofmacromolecules with other molecules, as well as ofsolubilization, are of extreme importance, since many processes that occur in living organisms (the formation of membranes and cellar structures, exchange processes, enzymatic catalysis, etc.) are all based on these phenomena. The formation oflyophilic colloidal systems may also take place via phase separation of polymer solutions. A typical phenomenon that occurs upon phase separation is the formation of the so-called coacervates, which are characteristic nuclei containing higher concentration of polymer, as compared to that in the medium surrounding them. It is speculated that coacervation was a second stage (after the formation of adsorption layers) in the ordered structuring of organic matter on Earth. One of the oldest problems in colloid science (that has not lost its importance today) is the problem of relationship between colloidal systems formed by low-molecular weight substances and solutions or dispersions of polymers. The term "colloid", introduced by T. Graham, was primarily related to the description of glue-like gelly dispersions of organic polymers and does not emphasize objectives and topics of modem colloid science. The investigation of physical properties of such gelly systems and those of dilute polymer solutions, referred by G. Freundlich as the "lyophilic coloids" (as a generalization of the term "hydrophilic colloids" introduced by J. Perrin), constituted for a long time the subject of colloid science. Differentiation between lyophilic and lyophobic colloids was primarily based on the ability of the former to form spontaneously, and the sensitivity of the latter to the addition of small amounts of electrolytes. Hydrophilic colloids are destroyed by the addition of only high electrolyte quantities due to the salting out effect. The properties of lyophilic and lyophobic colloids were differentiated on the basis of high ability of liophilic sols to solvate colloidal particles (micelles) with solvent molecules, while lyophobic sols always require a stabilizer in order to maintain their stability towards aggregation. These conservative views for a certain period of time impeded the development of
500 physical chemistry of systems containing polymers. In the meantime, V.A. Kargin, S.P. Papkov, and A. Mark proved that polymer solutions were thermodynamically equilibrium systems to which the Gibbs phase rule could be successfully applied. Later P. Flory and G. Staudinger developed a molecular-statistical theory describing the dissolution of flexible chains [27]. Since then physical chemistry of polymer solutions has separated from "general" colloid chemistry into an individual discipline. The molecular-statistical theory played in important role in the development of polymer solutions thermodynamics. At the same time M. Volmer, and subsequently P.A. Rehbinder introduced their concept of lyophilic colloidal systems (see Chapter IV) as thermodynamically stable microheterogeneous dispersions (the term "lyophilic colloids" is not used in order to avoid confusion). It occurred that the term "phase" extended onto microscopic objects has a rather conditional meaning. Because of this the main criterion used to separate colloidal systems and polymer solutions, i.e., viewing "true" solutions as thermodynamically equilibrium systems, was no longer significant. The studies on polymer solutions indicated that the latter often contained molecular aggregates and that ideal solutions obeying the statistical theory of dissolution of flexible chains, were rarely encountered. As a result of all that, there is a tendency in modem science to bring the theories of colloidal systems, polymer solutions and solid polymers close to each other. This process is still going on, encountering various difficulties on its way, including those of terminological nature. One must account for specific properties of large polymer molecules, the sizes of which are comparable with typical colloidal particles. For instance, even rather small unbalance in molecular forces results in a characteristic association process, that may take place within the same molecule (e.g. transition from a random coil to a compact globular state) or cover several molecules. In the latter case one molecule, using its different sites, may become incorporated into several aggregates, forming supramolecular structures or particles of macroscopic phase (Chapter IV). The entropy of polymer chains and the degree of association are, probably, the main factors that allow one to differentiate between polymer solutions and lyophilic colloidal systems. The association processes are most pronounced in substances that contain molecules
501 non-uniform ("mosaic-like") in their polarity. Many natural polymers, such as proteins, lipids, starch, cellulose and derivatized cellulose are the examples of such substances. Depending on conditions, and particularly on the pH, macromolecules may bear charges of different signs (see Chapter V), which in tum may influence the degree of internal and intermolecular association. These systems reveal many properties similar to those of lyophilic colloidal systems formed in dispersions of micelle-forming low molecular weight surfactants. These systems reveal the ability to solubilize molecules of hydrocarbons and other non-polar substances. The increase in the concentration ofbiopolymers leads to a transition to structured systems, such as gels, and in protein-lipid systems results in the formation of unique colloidal objects referred to as the cellar membranes which are vital in birth and functioning of living organisms. Structure and properties of such systems are beyond the scope of this book and are covered in the specialized literature on biochemistry and biophysics. Among the studies of colloid-chemical properties of polymer solutions one may outline three directions that have been under the most intense development in recent decades. First, it is the experimental and theoretical (including computer modeling) investigation of adsorption layers formed on solid surfaces by natural and synthetic polymers, especially by polyelectrolytes. Such studies, and in particular those involving the use of Atomic Force Microscopy (AFM, see Chapter VII), provide important information regarding the optimal conditions for the use of polymers for flocculation or stabilization of disperse systems (Chapter VII), and establish the theoretical basis for understanding the mechanism behind the action of structural-mechanical barrier. Another actively developing and interesting direction is the investigation of interactions of polyelectrolytes and other polymers with surfactants. At Moscow State University a significant contribution to this area has been made by the associates of the scientific school founded by V.A. Kargin and V.A. Kabanov. In their studies a number of systems with peculiar structures were described, such as those containing micellar "beads" attached to the macromolecular chains, or systems in which gels form at very low polymer and surfactant concentrations. Finally, the investigation of the specifics of the Rehbinder effect (see Chapter IX, 4) in polymer materials, e.g. the formation ofnanofibrous structures in the course of polymer
502 deformation in activemedia,has also attracteda considerable interest. All these topics are addressedin greater detail in PolymerChemistrycourses. References ~
~
3. ~
~
6. 7. ~
9. 10. 11. 12. 13.
14. 15. 16.
17.
Rusanov, A.I., Micellization in Surfactant Solutions, Harwood Academic Publishers, Reading, 1997 Shchukin, E.D., Rehbinder, P.A., Kolloid. Zh., 20(5) (1958) 645 Schukin, E.D., Fedoseeva, N.P., Kochanova, L.A., Rehbinder, P.A., Doklady Akad. Nauk SSSR, 189 (1969) 123 Shchukin, E.D., Amelina, E.A., Yaminskiy, V.V., J. Colloid Interface Sci., 90 (1982) 137 Volmer, M., Z. Phys. Chem., 125 (1927) 151 Volmer, M., Z. Phys. Chem., 155 (1931) 281 Frenkel, Ya. I., Critical Theory of Liquids, Izdatelstvo Akademii Nauk SSSR, Leningrad, 1959 Mayer, J., Harrison, S., J. Chem. Phys., 6 (1938) 87 Shchukin, E.D., Kochanova, L.A., Colloid J. 45 (1983) 637 Kochanova, L.A., Fedoseeva, N.P., Kuchumova, V.M., Pertsov, A.V., Rehbinder, P.A., Koloidn. Zh., 35 (1973) 839 Kochanova, L.A., Fedoseeva, N.P., Kuchumova, V.M., Pertsov, A.V., Rehbinder, P.A., Koloidn. Zh., 35 (1973) 843 Shchukin, E.D., Kochanova, L.A., Pertsov, A.V., Fiz. Khim. Meh. I Liofiln.Disp. Syst., 11 (1979) 15 J6nsson, B., Lindman, B., Holmberg, K., and Kronberg, B., Surfactants and Polymers in Aqueous Solution, Wiley, Chichester, 1998 Rosen, M.J., Surfactants and Interfacial Phenomena, 2nd ed., Wiley, New York, 19089 Rusanov, A.I., Adv. Colloid Interface Sci., 45 (1993) 1 Noskov, B.A., and Grigoriev, D.O., in "Studies in Interface Science", vol. 13, D. M6bius, R. Miller, and V.B. Fainerman (Editors), Elsevier, Amsterdam, 2001, in the press Lindman, B., Tiberg, F., Piculell, L., Olsson, U., Alexandridis, P., and Wennerstr6m, in"Micelles, Microemulsions, and Monolayers: Science and Technology", D.O. Shah (Editor), Dekker, New York, 1998
503 18.
19.
20. 21. 22. 23.
24. 25. 26.
27. 28.
29. 30.
Markina, Z.N., Bovkun, O.P., Rehbinder, P.A., in "Selected Works", vol. 1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Markina, Z.N., Bovkun, O.P., Levin, V.V, Rehbinder, P.A., in "Selected Works by P.A. Rehbinder", vol. 1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Friberg, S.E., and Lindman, B. (Editors), Organized Solutions, in "Surfactant Science Series", vol. 44, Dekker, New York, 1992 Laughlin, R.G., The Aqueous Phase Behavior of Surfactants, Academic Press, London, 1994 Asua, J.M. (Editor), Polymeric Dispersions" Principles and Applications, NATO ASI Series E, vol. 335, Kluwer, Dordrecht, 1997 Holmberg, K., in "Micelles, Microemulsions, and Monolayers: Science and Technology", D.O. Shah (Editor), Dekker, New York, 1998 Evans, D.F., Wennerstr6m, H., The Colloidal Domain: Where Physics, Chemistry, and Biology Meet, 2na ed., Wiley-VCH, New York, 1999 Shah, D.O., in "Micelles, Microemulsions, and Monolayers: Science and Technology", D.O. Shah (Editor), Dekker, New York, 1998 Nasr-E1-Din, H.A., and Taylor, K.C., in "Micelles, Microemulsions, and Monolayers: Science and Technology", D.O. Shah (Editor), Dekker, New York, 1998 Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953 Fleer, G.J., Cohen Stuart, M.A., Scheutjens, J.M.H.M., Cosgrove, T., and Vincent, B., Polymers at Interfaces, Chapman & Hall, London, 1993 Izmailova, V.N., Yampolskaya, G.P., Suture, B.D., Surface Phenomena in Protein Systems, Khimia, Moscow, 1988 Izmailova, V.N., Yampolskaya, G.P., in"Studies in Interface Science", vol.7, D. M6bius, R. Miller (Editors), Elsevier, Amsterdam, 1998
504
List of Symbols
Roman symbols b intermolecular distance CMC critical micellization concentration C concentration total surfactant concentration Co critical concentration Ccr concentration of counterions Ci concentration of molecularly dissolved surfactant CM d particle diameter dm size of micelles e elementary charge free energy A ~#m~centhalpy of micellization emhalpy of solubilization Kmic equilibrium constant of micellization k Boltzmann constant hydrocarbon chain length lm aggregation number m number of droplets in critical system number of micelles per 1 m 3 nmic number of particles Avogadro's number NA number of moles of surfactant in micellar state Nmi~ number of moles of solubilized substance Nsol q effective charge R universal gas constant Rehbinder-Shchukin criterion RS particle radius radius of spherical micelle F1 surface area of spherical micelle $1 relative solubilization S area of surfactant polar group Spg absolute temperature T critical temperature Tcr Krafft temperature
505 tf V
v, W
emulsion life time volume molar volume of hydrocarbon work of fluctuational formation of surface "bump"
Greek symbols (1
P 7I t5 (Ycr (Imic
~W-O O'o_me (Iw.me
A ~max
particle shape coefficient In ~ / j V ~ 3.14159... specific surface free energy (interfacial tension) critical value of interfacial tension effective value of the specific surface free energy of a micelle interfacial tension at water-oil interface interfacial tension at oil-microemulsion interface interfacial tension at water-microemulsion interface equivalent conductivity wavelength of maximum absorption
506 VII. G E N E R A L CAUSES FOR D E G R A D A T I O N AND R E L A T I V E
STABILITY OF LYOPHOBIC COLLOIDAL SYSTEMS
As discussed in Chapter IV, lyophobic colloidal systems are coarse and fine disperse systems that are thermodynamically unstable due to a substantial excess in surface free energy. The latter may be related both to the existence of highly developed interface between the dispersed phase and the dispersion medium and to relatively high values of interfacial tension, ~. This excess of the surface free energy is the reason for instability of such systems, in which various processes leading to a coarsening and finally to complete degradation of systems (i.e. separation into individual macroscopic phases) take place. General stability of lyophobic colloidal systems , as well as the rate of processes leading to degradation, are determined by nature, phase state, and composition of the dispersed phase and dispersion medium, and also by degree of dispersion and concentration of dispersed phase. Stability of disperse systems may vary in a very broad range from complete instability (when the life time of colloidal system is on the order of seconds or even fraction of a second) to nearly infinite stability (when changes taking place in the system may only be noticed on the geological time scale) This chapter is devoted to the discussion of fundamental theoretical principles regarding the nature and mechanisms of processes leading to degradation of disperse systems, and of factors that determine the rate of such processes, and hence the stability. In the chapter that follows we will illustrate the application of these general principles and considerations to particular systems of different nature.
507 VII.1. The Stability of Disperse Systems with Respect to Sedimentation and Aggregation. Role of Brownian Motion
One may define the stability of t h e r m o d y n a m i c a l l y
non-
e q u i l i b r i u m l y o p h o b i c d i s p e r s e s y s t e m s as their ability to resist changes in their structure, such as changes in the degree of dispersion, particle size distribution, and changes inside the volume of dispersed phase. It was proposed by N. Peskov to distinguish between sedimentation and aggregation stability of disperse systems [1]. Sedimentation stability is the stability of system against a decrease in potential energy of dispersed particles during their sedimentation under gravity. Aggregation stability can be viewed as the ability of systems to resist processes leading to a decrease in interfacial free energy at the interface between the dispersed phase and dispersion medium [2]. The examples of processes that lead to degradation of disperse systems due to a decrease in interfacial free energy include isothermal mass transfer from small particles to larger ones, coalescence (when one or more particles fuse into a single but larger particle), and coagulation (a combination of single particles into flocs) ~. During isothermal mass transfer, driven by higher
In various areas of science and applications, where these basic mechanisms of system destabilization play an important role, different terms, often specific to a particular area of application, are used as synonyms. For instance, the isothermal mass transfer taking place in solid materials is referred to as recrystallization, or "coalescence" of inclusions in the case of two-phase systems (we will further avoid the use of term "coalescence" in such sense). Such process taking place in precipitates is referred to as the Ostwald ripening. The fusion of solid particles (in many cases similar to coalescence) is known as sintering.
508 chemical potential of small particles as compared to that of the large ones (see Chapter 1,3), and during coalescence, the surface free energy, g s , decreases due to a decrease in interfacial area, which often takes place at constant specific free energy, cy, i.e., A,~-s - c~AS<0. In the process of coagulation, especially in the cases when the gaps between particles are filled with residual dispersion medium, the interfacial area remains essentially unchanged (or changes insignificantly), and lowering of g s is achieved mostly by partial saturation of uncompensated molecular forces at particle surface. This is equivalent to local lowering of interfacial tension in the zone of contact, i.e. for coagulation one can write that A g s = SetA~ < 0, where Sefcorresponds to a portion of interface at which partial compensation of molecular forces takes place. For the same initial degree of dispersion, coalescence and isothermal mass transfer result in a higher decrease in surface free energy as compared to coagulation. A decrease in free energy due to coagulation can be estimated in a more precise way. If aggregation results in formation of an aggregate containing ,4rparticles, and each of these particles interacts with Z neighbors, a total number of contacts between particles is 89Z~4/". If average energy of pairwise interactions in a contact (the energy of particle cohesion) is u, the net benefit in the free surface energy is 89ZJV" u~, where u~ is the absolute value
ofu, ul-lul. Depending on the aggregate state of dispersion medium, isothermal mass transfer, coagulation and coalescence may play different roles in the loss of stability of disperse systems towards aggregation. Coagulation, coalescence, and phase separation during sedimentation are typical in systems with liquid
509 or gaseous dispersion medium. Isothermal mass transfer may take place in systems with dispersion media in any aggregate state, including the solid one. In systems with solid dispersion medium isothermal mass transfer is the only way to change degree of dispersion. In systems with highly mobile dispersion media (i.e., liquid and gaseous) isothermal mass transfer does not usually play an important role in decreasing degree of dispersion. If, however, for some reasons coagulation and coalescence are retarded, or, especially, dispersed substance is well soluble in dispersion medium, it is the isothermal mass transfer that determines the rate of destabilization in such systems. In real systems which undergo thermal fluctuations, the rate of"recondensation" of substance from small particles to larger ones may increase significantly. The conditions under which disperse systems are stable, and the nature of processes leading to their degradation depends to a significant extent on dispersed phase concentration, interactions between particles and on other factors. In free
disperse
systems,
in particular those with low
concentration of dispersed phase, the nature of colloid stability and conditions under which the collapse occurs, are to a great extent dependent on thermal motion of dispersed particles, which may contribute to both stability and destabilization. For example, the necessary condition for sedimentation stability is sufficiently small particle size, so that the tendency of particles to distribute within the entire volume of disperse system due to the Brownian motion (an increase in entropy) would not be affected by gravity. As a quantitative criterion for the presence of noticeable amount of dispersed particles in equilibrium with a sediment one, for instance, may use the
510 condition of HI/e / d ~) 1, where d is particle diameter and H~/e is the height of particle "atmosphere", described by eq. (V. 18). One may thus write that
H~/~/d- kT/m'gd )~ 1, where m'g - 3/4 zc93(9 - 90)g is the weight of particles of density 9 in the medium of density 9o. It is also worth mentioning that convective streams, which always exist in real systems, favor sedimentation stability. Brownian motion of dispersed particles may also play a leading role in aggregation stability of free disperse systems. When interfacial tension at the particle - medium interface is very small (this corresponds to the condition of formation oflyophilic colloidal systems, ~ _<13kT/ad 2 (see Chapter IV, 1)), the involvement of particles in Brownian motion makes an increase in particle size (during isothermal mass transfer and coalescence) thermodynamically unfavorable, i.e. stipulates a complete thermodynamic stability. As emphasized above, the change in surface free energy during coagulation is smaller than that during coalescence and isothermal mass transfer, and hence the stabilizing role of Brownian motion during coagulation may be quite significant even at rather high values of interfacial tension. As a result, under certain conditions Brownian motion can ensure thermodynamic stability of disperse systems with respect to particle aggregation (only!), and, consequently, the reverse process of spontaneous particle disaggregation. The process of spontaneous disintegration of aggregates and the transition from structured disperse systems to the free disperse ones is referred to as
peptization. In Chapter IV,1 we used the approach developed by Rehbinder and Shchukin to evaluate change in free energy during dispersion of macroscopic
511 phase into colloidal particles. Let us now use the same approach to examine change in free energy of the system during degradation of aggregate composed of Jl/'colloidal particles. As described above, an increase in free energy of the system upon a complete dispersion of an aggregate can be written as a , , ~ -s - 1 Z . A r 2
Ul .
At the same time, the transfer of particles from structured disperse system (the aggregate) into free disperse system (the sol) is followed by an involvement of freed particles into Brownian motion, leading to entropy increase. The gain in entropy, A J, may be described by expression, similar to eq. (IV.6), i.e." k J " ~ [3' ~4"k. Since initial aggregate is already impregnated by the molecules of dispersion medium, the meaning of numerical coefficient [3" in the above expression is different from that given by eq. (IV.7)" [3* represents here the ratio of particle concentrations in aggregated and peptized state, na and np, respectively, i.e
- In
(na/np).
For common dilute sols one may safely
assume that 13" is between 10 and 20. The net change in free energy of the system is, consequently, given by
1 A g - - - - Z ./V" u 1
2
- [3*.~kT.
For friable aggregates the coordination number, Z, usually does not exceed 3-4.
512 Consequently, spontaneous degradation of aggregate into a sol (spontaneous dispersion) becomes thermodynamically favorable, when
U1 <
*kr = 1/2 Z
(7 to 15) kT.
(VII. 1)
Such process (peptization) yields colloidal system, thermodynamically stable towards coagulation. Oppositely, when
[3*kr b/1 >
1/2Z
peptization is thermodynamically unfavorable, and the system with such u~ value becomes unstable towards coagulation, i.e. behaves as a typical lyophobic one. The equilibrium between aggregation and peptization of dispersed particles is given by u~
kT/1/2 Z condition, which corresponds to a
particular particle concentration in free disperse system, equilibrium with respect the to sediment (aggregate): P'/p - /%
exp (- 1/2 Z/'/1/iT).
(VII.2)
These deaggregated particles are in the state of partial equilibrium" their aggregation is thermodynamically unfavorable, while isothermal mass transfer and coalescence, leading to a decrease in degree of dispersion, are possible. At the same time, if isothermal mass transfer does not occur within realistic observation time periods, the equilibrium from partial turns into complete" the
513 system becomes stable. In free disperse systems coagulation is the first stage of coalescence, i.e. individual particles, prior to coalescing into a single particle, must approach each other at a close distance, that is, they have to aggregate. Under conditions corresponding to a weak interaction between dispersed particles, when coagulation is thermodynamically unfavorable, coalescence becomes impossible. The properties of these (strictly speaking, lyophobic)
systems, in which at low solubility of dispersed phase the
interaction energy between particles in contact, u~, is also low, are very similar to those of lyophilic ones. Consequently, such systems may be referred to as "pseudolyophilic". The analysis of how various factors, including the adsorption of surfactant, influence the contact interactions between particles, is the basis of theoretical principles for controlling stability in lyophobic disperse systems. In free disperse systems Brownian motion, along with stabilizing action, may also reveal a destabilizing one. Such destabilizing action is typical for "truly" lyophobic systems, i.e. the systems that are unstable with respect to aggregation and do not belong to the class ofpseudolyophilic ones. We will further show that in these systems Brownian motion is indeed the mechanism responsible for particle coagulation. In systems that initially reveal sedimentation stability, particle coagulation, as well as particle growth due to subsequent coalescence or isothermal mass transfer, may result in a loss of sedimentation stability. At the same time, coagulation may not necessarily be followed by phase separation due to sedimentation, as in the case when particle aggregation results in formation of continuous three-dimensional networks of particles that fill an
514 entire volume of disperse system, i.e. in the formation of s t r u c t u r e d d i s p e r s e s y s t e m , referred to as gel (see Chapter IX). At the same time, particle settling in systems unstable towards sedimentation may significantly increase the rate of particle aggregation during the so-called orthokinetic coagulation (see Chapter VII, 7). Analogously, increase in coagulation and subsequent coalescence rates may be achieved during settling of particles in centrifugal gravity field. In s t r u c t u r e d d i s p e r s e s y s t e m s (close in their properties to the systems with high concentration of dispersed phase, where particles are forced to stay close to each other) the particles may remain separated by layers of dispersion medium, or the latter may be completely squeezed out from the interparticle gap. The collapse of the dispersion medium layer results in a direct contact between solid particles and in coalescence of drops and bubbles. Consequently, the rate of system degradation is strongly dependent on the stability of such dispersion medium layers and on their resistance towards squeezing out from between the particles. The stability of dispersion medium layers, especially of liquid ones, may be significantly influenced by the presence ofsurfactants. Foam films and emulsi on films, which constitute separate fragments of corresponding disperse systems, as well as t h i n fi 1m s o f we tt i n g 1i quid (present on the solid surface), represent characteristic model examples for the investigation of properties of dispersion system layers [3-7]. Let us examine properties of a thin film that is formed when two volumes of phase 1 are brought together in the medium 2, as shown in Fig. VII-1.
515
Fig. VII-1. Disjoining pressure in thin films Let us assume that the surfaces between which the film is formed are flat and parallel to each other. One should also keep in mind that this film of thickness h contacts with the macroscopic volume of phase 2 via its edges. Such layer of dispersion medium separating two identical phases is commonly referred to as two-sided symmetric film. The thinning of such film results in the flow of a portion of dispersion medium from the gap into the bulk of phase 2. The rate of film thinning, and consequently its stability, is determined by both thermodynamic factors, dependent on the film thickness only, and the kinetic ones, governed by both thickness and time [3,6]. According to B.V. Derj aguin [4], thermodynamic stability of such film is related to the overlap between discontinuity surfaces of individual phases, i.e. the overlap between transition zones of thickness 5 near the border between phases 1 and 2, within which the densities of free energy and other extensive parameters undergo changes from values characteristic of phase 1 to those characteristic of phase 2 (see Chapter I, 1). If the distance between the volumes of phase 1 is large compared to the thickness of transition zones, h ~ 25, the free energy per unit area of two-sided film, 5vf, is simply twice the value of the specific interfacial energy, cy, at interface between phases 1 and 2, i.e. g f -
2c~.
The quantity ,~f is also commonly known as the film tension, % By
516 analogy with an interface, the specific energy of film is numerically equal to its tension, ~ - f - % In the case when volumes of phase I are brought together so close, that film thickness becomes comparable to the thickness of transition zone, 5, i.e., h z 25, further thinning of film leads to a greater and greater overlap of discontinuity surfaces, and the work, A W, is performed as a result. In isothermal process the work performed o n t h e fi l m is stored in the form of excessive free energy of the film, A , ~ - f - kW. At h < 25 the specific free energy of film is given by cyf - ~ff (h) - 2cy + A ~ (h) - 2~ + Acyf . The difference Acyf= cyf- 2~ characterizes the excess energy in the film and is referred to as the excessive film tension, or the specific free energy of
interaction. The change in system's energy due to film thinning may be viewed as a result of action of some excessive pressure, which is referred to as the
disjoining pressure, according to definition given by Derjaguin [4,8,9]. Disjoining pressure, H = 1-I(h), is the excessive pressure that one has to apply to the surfaces confining the thin film, for the film thickness to either change reversibly or remain constant, i.e., for the system to maintain thermodynamic equilibrium. Disjoining pressure may be viewed as the excessive (relative to that in the bulk of phases) pressure acting from the side of the gap that tends to push the surfaces apart. In order to decrease the thickness of the gap by dh in a reversible process, one has to perform the work d A W -
-II(h)dh.
Consequently, one may write the relationship between disjoining pressure, H, the free energy of interaction and the thickness of the film as
517
1-I-
d~f dh
d A ~ f (h) = -
dh
,
(VII
3)
which is analogous to p = -0 g / c ? V. This links together the "regular" 3dimensional pressure, free energy and volume of a bulk phase. Consequently, h
A ~f(h)-
(VII.4)
- III(h)dh. oo
The quantity A g r -
~r expressed in J per
m 2
is the excess (in
comparison with the bulk) of free energy per unit area of film 2. Consequently, pressure II can be also viewed as the excessive density per unit volume of the film free energy, in J m -3. Both A g f and I-I are of the same sign, and can be either positive (real repulsion) or negative (attraction). Positive disjoining pressure prevents film thinning, while negative FI favors it. If dispersion medium is air (or vacuum, to be more precise), disjoining pressure is negative. Disjoining pressure may
2
The film tension, cyf -
, is the specific work of film area
c3S T,h
expansion at constant temperature, T, and thickness, h. Another quantity, referred to as the fullfilm tension, and defined as 7 -
- cyf + FIh
aS T,V
is also frequently used. This quantity represents the work of film area expansion at constant temperature and volume, i.e., at variable thickness h. For films of equilibrium thickness, the values of Hh are usually small compared to Acyr, and consequently 7~Gf
518 be stipulated by factors of different nature. According to Derjaguin, one can recognize several constituents, or parts, of the disjoining pressure [4]. The molecular component of disjoining pressure (see Chapter VII, 2) is characteristic of intermolecular attraction; this component is usually negative, i.e., it favors the particles coming closely to each other and thus destabilizes the system. The ionic-electrostatic component of disjoining pressure (see Chapter VII,4) may serve as an example of the most theoretically developed factor responsible for stability of disperse systems. Disjoining pressure may sometimes be regarded as the action of c a p i l l a r y e f f e c t s of t h e s e c o n d k i n d, related to the dependence of cy on geometric parameters characterizing the phase (gap thickness, h, in the present case) [10]. In concentrated systems with highly mobile interfaces (foams and emulsions) capillary phenomena of the first kind, related to the surface curvature in regions of film - macroscopic phase contact or in the regions where three films come into contact, may play a significant role in the energy and dynamics of film thinning. As shown in Fig. VII-2, a concave surface is formed in these types of regions. Under this surface the pressure is lowered by the
amount
Pc, - ~
equal +
< O,
to
capillary
pressure
(see
Chapter
I,
3),
where r~ and r 2 are the principal curvature radii of
Fig. VII-2. Capillary pressure in the Gibbs - Plateau channels
519 meniscus surrounding the film. For emulsions and foams this meniscus is referred to as the Gibbs - Plateau channel, or Plateau border [10,11 ]. If a small individual film is surrounded by broad Gibbs-Plateau channel, one may assume that channel surfaces closely resemble those of spherical shape, and hence r~ - r 2 - r, and ]p~ I - 2cy / r. In the case when large film is surrounded by a narrow channel, the shape of channel surface is close to cylindrical, i.e. r z - r, r 2 = o~ and
Pol
- ~ / r. The film is in equilibrium
with channels surrounding it when absolute values of capillary pressure, po, and disjoining pressure, H, are equal. When ~f < 2~(negative excessive film tension, A~f < 0), stable films equilibrium with macroscopic phase form. In this case contact angle, 0, exists between the film and the Gibbs-Plateau channel (see Fig. VII-3) [12]. The value of this contact angle is given by cyf - 2(y c o s 0 , from where it follows that - A c y f - 2cy(1 - cos0 ). Since contact angles, 0, are usually very small, one can write - A c ~ f ~ ~0 2 . Measurements of contact angles, 0, and film thicknesses,
h, are the main
approaches to study films and their thermodynamic properties. Studies of film structure and measurements of its thickness are usually carried out by optical methods, in particular by interferometry.
520
G 2
0
Fig. VII-3. Interferometricpicture of film surrounded by the Gibbs-Plateau channel, and the scheme of obtaining contact angle, 0, from such picture [12] It is well known that due to interference the intensity of light reflected by film is a complex function of the ratio of film thickness to the wavelength of light (Fig. VII-4). When "thick" films are illuminated by monochromatic light, several maxima of intensity I appear. These maxima correspond to film thickness,
h- (k + 1/2))~
, where k (the order of interference) is an integer,
2n and n is the refractive index of film. When these films are viewed in white light, they appear colored into different colors, depending on their thickness. Thin films with thickness h zVlO appear in the reflected light as grey, and those even thinner as black. For gray and black films measurements of intensity of reflected light, I, reveal film thickness, while the study of intensity of reflected light as a function oftime gives information on the kinetics of film thinning.
521 I
1~. 4n
3~. 4n
5~. 4n
7~. 4n
h
Fig. VII-4. Intensity of reflected monochromatic light, I, as a function of film thickness, h
Along with film thickness, interferometric methods also allow one to measure the value of contact angle, 0, and the film tension, cyf. Interferometric determination of contact angle is based on the measurement of distance between Newton's rings, i.e. between maxima of intensity of light reflected in the area of Gibbs-Plateau channel (Fig. VII-3). Since film thicknesses corresponding to intensity maxima are known (Fig. VII-4), these measurements allow one to estimate the profile in the Gibbs-Plateau channel, and thus determine both the contact angle and the tension of film.
VII.2. Molecular Interactions in Disperse Systems
An important feature of dispersion intermolecular forces (see Chapter I, 1) is their additivity: interaction between two volumes of condensed phases separated by a gap is the result of summed attraction between all molecules making these volumes. For non-polar phases in the absence of non-dispersion forces the interaction energy,
U(h), is almost entirely determined by dispersion
forces. The role of dispersion interactions is especially important in disperse systems in which each particle represents a microscopic volume of condensed
522 phase with dimensions large compared to molecular ones. In this case partial compensation of dispersion interactions (partial saturation of surface forces) may take place at noticeable distances (larger than molecular dimensions, but still comparable to the size of particles), corresponding to the attraction forces between dispersed particles. For two particles separated by a thin flat gap of thickness h the attraction energy per unit area of a gap, Umo~,is given by (see Chapter 1,2):
UmoI (h) -
A 12~:h2 ,
(VII. 5)
where the Hamaker constant, A, is determined by a number of molecules per unit volume of interacting phases, n, polarizability of molecules, aM, and some 2
2
energy quant approximated with ionization energy, hv0: A - 3/4 nhv0 aM n . By taking a derivative of this expression with respect to gap thickness, one can obtain the molecular component of disjoining pressure, acting between two condensed phases, separated by vacuum, namely:
1-1m~ =
dUmo 1 dh
A 6~h 3
(VII.6)
The negative sign ofdisj oining pressure corresponds to the tendency of phases to approach each other due to intermolecular attraction forces. Equations (VII.5) and (VII.6) are also valid for the case of a free film of condensed phase, i.e. for the case of a symmetric film that is in contact with gas phase (or vacuum, to be more precise) on both sides. It is worth
523 mentioning, however, that such a match between energies and disjoining pressures in films and in gaps occurs only if h is small compared to the characteristic wavelength of dispersion interaction, )~0 - C/Vo, where c is the speed of light. If thicknesses of films (gaps) become comparable to ?~0,the so-called
electromagnetic retardation effect, related to a finite speed of propagation of electromagnetic waves starts to play a role [6],. It was shown by G. Kasimir and D. Polder [ 13] that at such large film thicknesses, in equation describing the attraction potential between molecules, eq. (I.8), the power n -
7.
Consequently, the film energy and disjoining pressure become inversely proportional to third and forth power of the gap thickness, h, namely Umo~~ h -3, and
1-Imo I ~
-
h -4.
For gaps and films with such a large width, the
proportionality coefficient in eq.(I.9) reveals the dependence on refractive index of the medium through which interactions between molecules take place. This results in different values of Umo~and II for films and gaps between condensed phases. We will focus our subsequent discussion on a simple case of London interactions between condensed phases, disregarding the electromagnetic retardation effect. Hamaker constant in the case of interaction between two d iffe re n t p h a s e s in contact is defined by polarizability and density of both phases; A~2
~(A1A2) 1/2. In t h r e e - p h a s e s y s t e m s in which all three phases have significant concentration of molecules, one has to account for interactions of phases with each other and for those inside an intermediate phase, i.e. three Hamaker constants, Ay, are needed; here the i andj indices are related to the corresponding phases. The decrease in gap thickness results in phases 1 and
524 3 getting closer to each other and in stronger interaction between them, as well as in the flow of medium 2 out of the gap into the bulk phase. The terms describing this should be included into the final expression with a negative sign. At the same time, the decrease in the gap thickness leads to the separation of phase 2 from phases 1 and 3, and hence the corresponding terms must be included into final expressions with the "+" sign. In line with these rather qualitative considerations a more strict treatment reveals that the expressions for the interaction energy, Umo~,and disjoining pressure, FImo~, can be written as Umol
_
_ A13
+ A22 -
A12 -
A 2 3 __ _
12~h 2
l--[mo1 = _
A13 + A22 - A12 - A23 = 6~h 3
A . 12~h 2 '
(VII.7)
A* 6~h 3 '
(VII.8)
where the value A* is complex Hamaker constant (see Chapter III, 1). Depending on the values of Hamaker constants of interacting phases, A ~3, A22, A12 ,
and
A23 ,
disjoining pressure in three-phase systems may not only be
positive, but also negative (in some special cases). For symmetric films of 121 type, expressions (VII.7) and (VII.8) can be written as
Umol
-_
A _ _ All 127~h 2
4- A 2 2 -
127~h 2
2A12 . '
(VII.9)
525
1-Imo 1 -
A* = 6~h 3
All +
A22 - 2A12 6rch 3
'
( V I I . 1 O)
where complex Hamaker constant, A*, is approximately given by ( Av~-~- A,/~-22)2, in agreement with eq. (III.3). Consequently, for s y m m e t r i c films the molecular component of disjoining pressure is always negative, which corresponds to a tendency of dispersion medium layer separating identical phases to decrease its thickness. At the same time, one should emphasize that in such systems in the absence of non-dispersion interactions the lower the value of complex Hamaker constant is, the more similar in nature the interacting phases (dispersed phase and dispersion medium) are. If contacting phases are essentially similar in structure and chemical composition, the value of A* may be as low as 10 -2~ J or even much lower. The so low Hamaker constants result in changes in the nature of colloidal stability. Qualitatively the same result may be obtained if one utilizes more strict treatment of molecular interactions in disperse systems. This approach is based on the so-called m a c r o s c o p i c theory of van der Waals forces developed by E.M. Lifshitz, I.E. Dzyaloshinski and L.P. Pitaevski [14]. In contrast to Hamaker's microscopic theory, the macroscopic theory does not use a simplified assumption of additivity of interactions between molecules, on which their summation is based (see Chapter I, 2). Mutual influence of molecules in condensed phases on each other may alter polarizabilities and ionization energies, making them different from those established for isolated molecules, which results in molecular interactions being not fully additive.
526 The basic principle of macroscopic theory is the idea that fluctuations of electromagnetic field, existing in condensed phases and propagating beyond the limits of these phases, interact in the gap between phases and create forces of intermolecular attraction. The quantum nature of such fluctuations results in the main contribution into interaction being made from the so-called zero oscillations,
which are temperature
independent.
Only at very high
temperatures one has to account for the thermal nature of fluctuations. Characteristic frequencies of fluctuations of electromagnetic field may be obtained from optical properties of a condensed phase, namely from the relationship between true (i.e., not related to scattering, see Chapter V,1) coefficient of light absorbance by contacting phases, k, and the frequency, co.
Without going into a detailed discussion of rather cumbersome macroscopic theory, let us present a frequently used result of this theory, namely the expression for complex Hamaker constant related to the most general case of two semi-infinite phases 1 and 3 separated by a film consisting of phase 2; 03
A*=
3h I (~31-g2)(~33-g2)dY,. 16rt
0
(el + g2)(g2 -I- g3)
In the above expression the values of e; are the functions of variable { and are determined by relationships existing between the circular frequency, co, and absorbance coefficients, k, of corresponding phases" 03
~;i - 1 + - -
2 I
0
ki(c0)c de0 , (032 + ~2)
where c is the speed of light. It is worth noting that in the limit of {=0 the values of e, are equal to dielectric constants of the corresponding phases.
527 In order to estimate the values of Hamaker constants from Lifshitz's theory, one needs to know optical characteristics of condensed phases. Calculations of this type were carried out for a number of primarily simple systems, including two identical phases separated by vacuum. For instance, such calculations yielded the values of Hamaker constants for water and for quartz of 5.13x 10.20 and 5.47• 10.20 J, respectively. In the case of symmetrical film ( el = e3), the numerator of expression for A* contains (el - e2)2, which agrees with eq. (VII.9).
While moving from the discussion of molecular interaction between condensed phases separated by a gap filled with dispersion medium to the analysis of molecular interactions between dispersed particles, it is necessary to outline that the interaction energy and force should be related to a pair of particles as whole, and not to the unit area of intermediate layer, as was done above. The interaction energy and force are not only the functions of distance between particles and the value of complex Hamaker constant, but also depend on size and shape of interacting particles.
For two spherical particles with equal radii, r, whose centers are located at distance R from each other, and if the smallest possible gap between their surfaces is h = R - 2r, the integration of molecular interactions, carried out by Hamaker, yielded the general expression for the attraction energy between particles:
U~ph =
6
R 2 - 4r 2 + 2 R-T + In 1 - 4
It is worth emphasizing that the value of/'/sph ,
as
.
(VII.11)
well as A, has the units of energy.
At large distance between centers, i.e. when h )) r, the terms in the above expression
528 can be expanded into series 3 , and equation (VII. 11) may be written in a form similar to the expression for interaction between molecules (see Chapter 1,2)"
16
r6
4
Usph ~ - ~ A ~ = -
9
3
aL
71;r /71
R 6
R
6 '
where nl is the number of molecules per particle unit volume, and ac is the London constant. In another limiting case when the particle size is significantly larger than the distance between interacting particles, i.e. when r >>h = R - 2r (Fig. VII-5), the major contribution into Usphcomes from the first term in parenthesis in eq. VII-11, which can be written as
2r 2
2r 2
2r 2
r
R2 -4r 2
( R - 2 r ) ( R + 2r)
h(h + 4r)
2h
Consequently,
Ar b/sph
12h
Fig. VII-5. Two particles separated by a thin gap
The energy of molecular interaction (attraction) between two particles each or radius r separated by a thin gap filled with dispersion medium is given by the expression
r2)
In 1 - 4 - ~
r2
r4
~-4-R--T-8- ~
64
r6 R6
~
2r 2 R2 _ 4 r 2
r2
r4
6
~ 2 - ~ + 8 7 - a - + 3 2 Rr 6
9
529
A*r Usph ~
12h = =hrUmo 1
,
(VII.12)
in which eq.(VII.9) is taken into consideration in the right hale
The
interparticle interaction force in this case is given by
A*r
F ~
12h 2 9
(VII. 13)
According to eqs.(VII. 12) and (VII. 13), the molecular attraction force between two identical spherical particles may be written as F(h)
-
71;rUmoI (h).
Derj aguin obtained an analogous expression valid for any interaction potential, U(h) between curved interfaces of various shape 4 [15]" F ( h ) - rckU(h) - ~kA ,~-ff(h).
(VII.14)
In this equation k is the linear parameter related to geometry and determined by the curvature of surfaces in contact. For two spherical particles of different radii, r' and r", k = 2r'r"/(r' + r"); for two cylindrical surfaces positioned at a right angle with respect to each other, k - 2 (r' r") ~/2. For two p l a n e - p a r a l l e l s u r f a c e s separated by an equilibrium distance h0, there is a minimum in specific free energy of interaction (free energy of film), U(ho) = A g f (h0). This minimum (Fig. VII-6) is referred to as the near potential energy minimum, or simply the primary minimum. The
4
Derjaguin's expression is valid only for the surfaces of second order
530 values of h 0 are approximately equal to intermolecular distance in the bulk of condensed phase (or the size of dispersion medium molecules in the residual adsorption layer in the gap). The attraction forces between the surfaces (to make things simpler, we will further consider dispersion interactions only) predominantly act when h > h0, while Born repulsion becomes significant at h
ho=b
/ / Umol
Fig. VII-6. The surface free energy of film, Agf, as a function of film thickness, ho One must realize that when the distance between interacting surfaces becomes smaller (the film becomes thinner) the attraction forces increase rather slowly; for dispersion interactions this increase is described by
1/h 3
power law. Oppositely, repulsion forces, down to some rather small distance h, remain essentially unnoticeable, but experience a sharp increase as h decreases further, e.g. according to 1/h~ power law, where n ~ 10. As a result, when surfaces are brought together to an
equilibrium distance, h0, the
prevailing amount of work has been performed by attraction forces, and the depth of primary minimum is close to the absolute value of work performed by attraction forces, i.e to the molecular component of free energy A , ~ f (h0), see Fig. VII-6.
531 The interaction energy between particles, b/sph (h), also has a minimum at distances close to molecular dimensions, i.e. h0 = b. For sol particles with radius r the absolute value of the depth of this minimum is the energy of cohesion, u~, in the contact between such particles in a coagulate"
Ar
u~ ~ ~ , 12h o
(VII. 15)
from which, using eq. (VII. 12), one obtains bl 1 ~
--
nrh o Umo1(h0).
It is worth reminding here that particularly this value of u~ was introduced earlier (see Chapter VII, 1) in the criterion describing equilibrium between peptization and coagulation and thermodynamic stability of system towards coagulation. In agreement with eq. (VII. 1), a disperse system is stable when Ul _<_
kT/
89 Z, which at
~15 to 20 and Z ~ 3 to 4 constitutes
approximately 7 to 15 kT. The energy of molecular interaction between particles is dictated by complex Hamaker constant, A*, eq. (VII. 15), which in turn is determined by the nature of both the dispersed phase and dispersion medium. The condition which reflects stability of system towards coagulation can be expressed as
[3*kT > 1~2ZA*r
(VII.16)
12h o Since [3*- In (ha/r/p) , the equilibrium concentration ofpeptized particles above the aggregate, np, is given by
532
np- r/a exp - i ~ 0 ~
(VII.17)
,
which corresponds to expression (VII.2) For typically lyophobic systems (e.g. aqueous dispersions of solid hydrocarbons) with a characteristic value of A* ~ 5 x 10.2o J at r = 10 -s m and h0 =
2x10 -1~ m the interaction energy between particles in a contact, in
agreement with eq. (VII. 15), is not less than 5 x 10 -20 910 -s H1 =
1 2 . 2 . 1 0 -l~
2
x 1 0 -19
J ~ 50kT,
which is significantly higher than the critical value of u~ ~ 10 to 15 kT in such systems, and corresponds to a complete thermodynamic instability. For this system to become stable towards aggregation, the value of complex Hamaker constant, A*, should be decreased to 10 .20 J and lower, or stabilizing factors of different nature (corresponding to other components of disjoining pressure), which decrease cohesion between particles in a contact, must be present in the system.
The nature of these stability factors of disperse systems will be
discussed further in this chapter. Expression (VII. 13) is a key relationship leading to one of direct ways to estimate Hamaker constant, A, and to study forces of intermolecular attraction that exist between condensed phases. It particularly allows to carry out the direct measurements of attraction forces between surfaces of known curvature as a function of distance between them, h. For the first time such measurements were carried out by B.V. Derjaguin and I.I. Abrikosova using
533 quartz lenses with radii of curvature of 10 and 25 cm, respectively, and lense and flat quartz surface as model systems [ 15-19]. The experiments were carried out with the help of specially constructed balance that allowed to automatically set a given distance between the flat plate and the lens, and to measure the force required to maintain the set distance. These measurements yielded the interaction parameters and confirmed the law stating how force changes with distance. In most cases the distance corresponded to a molecular interaction with a delay. In subsequent studies various methods were used to determine disjoining pressure in different systems. It is necessary to point out that the properties and behavior of dilute (free) disperse systems are in most cases viewed from the standpoint of their stability with respect to coagulation and sedimentation, expressed in terms of a relationship between the p o t e n t i a l energy of particle interaction, u, and thermal energy, kT. Measurements of interaction energy require the evaluation of both long range and short range forces, F(h), and their integration, i.e. one faces a complicated problem of simultaneous measurement of forces and distances [16-32]. To accomplish this, one needs highly precise and sophisticated experimental techniques, many of which employ hard dynamometric devices, special negative feed-back systems, etc. Among these techniques are the Surface Forces Apparatus (SFA) measurements, Atomic Force Microscopy (AFM), Interfacial Force Microscopy, Total Internal Reflection Microscopy, Colloidal Particle Scattering [24,25,28,31-34]. The Surface Force Apparatus (SFA), designed by J. Israelachvili is a commonly used instrument for the measurements of surface forces. The
534 technique utilizes two molecularly-smooth mica sheets, glued to two transparent (glass) half-cylinders. The instrument allows one to measure simultaneously the interaction force between these two crossed cylindrical surfaces, and the distance (interferometrically), i.e. to obtain F(h) dependence. Surface forces can be measured in the air as well as in liquid media. Modification of mica sheets by applying adsorption layers (including Langmuir-Blodgett films), depositing polymers and coating with fine metal films, extensively practiced over the last decade, yields an important information regarding the surface forces and their changes in various media. The measurement of interaction force between a nanosize sphere and a macroscopic ("flat") surface has been utilized in the Atomic Force Microscope, used nowadays in various scientific and technological applications. In the AFM the surface is probed by a spherical tip of a tiny cantilever, and the displacement, h, of the tip necessary to maintain constant attractive force, F, is measured (i.e., in the AFM one fixes the force and varies distance). Scanning of the surface produces digitized images of surface profile. In contrast to Scanning Electron Microscopy (SEM) the AFM allows one to carry out studies without special sample treatment (e.g. drying, gold sputtering, etc), and in liquid medium. A complicated nature of described instrumental devices and techniques often imposes a restriction on a variety of objects and experimental conditions that one can use in the studies. To overcome these obstacles and to significantly broaden the number of solids and conditions that can be used, E.Shchukin et al proposed a technique that utilizes an electrical capacitor as a distance measuring element [35-37]. This was later integrated into the SFA
535 by J. Klein et al. In the case of concentrated (structured) disperse systems the essential features that are usually of interest are their mechanical and rheological properties and behavior. The main parameter describing these features is the
cohesive force, F~. ( p~ in Chapter IX), or the strength of immediate contacts between particles. Stability is manifested as a correlation between the applied mechanical stress, P, and the sum of strengths of individual contacts, i.e. as the zF~ product, where Z is the number of contacts per unit area [35].In this case one only needs to evaluate the force in the immediate contact, without distance measurements. The experimental devices for such measurements may be extremely soft (pliable),which makes them very sensitive towards the measured forces. Corresponding methods and highly sensitive instruments for direct measurements of cohesive forces between individual particles of any nature in any media were developed by the authors and their co-workers [3841 ], and are described in Chapter IX. The force of cohesion, i.e. the maximum value of attractive force between the particles, may be determined by a direct measurement of force, F 1, required to separate macroscopic (sufficiently large) particles of radius r~ brought into a contact with each other. Such a measurement yields the free energy of interaction (cohesion) in a direct contact, A 5r(h0) = F~ / ~r~. Due to linear dependence of F on r, one can then use F~ to evaluate the cohesive force F 2 = (r2/r~)F~, acting between particles in real dispersions consisting of particles with the same physico-chemical properties but of much smaller size, e.g. with r 2 ~ 10 -s m (i.e. in the cases when direct force measurements can not be carried out). At the same time, in agreement with the Derjaguin equation
536 (VII. 14), from cohesion force, F, one can determine the primary minimum depth, U(ho), and from eqs (VII.12) and (VII.13) estimate the energy of molecular cohesion in a contact, i.e. -
F, ho.
VII.3. Factors Governing the Colloid Stability
It has been repeatedly emphasized that lyophobic disperse systems are thermodynamically unstable as compared to macroheterogeneous systems. The cause for this instability is a high excess of surface free energy at the interfaces. At the same time, many lyophobic colloids are stable towards aggregation and may maintain such stability for infinite periods of time. Let us now discuss the basic thermodynamic and kinetic factors that favor stabilization in disperse systems. In this sections we will restrict ourselves to just naming some of these factors, and will return to their detailed discussion later on. 1. The effective elasticity of films with surfactant adsorption layers. An increase in film size related, for instance, to film deformation (flexure, stretching) due to the action of external force, leads to changes in equilibrium between adsorption layer and surfactant solution in the volume of film. If deformation occurs slowly, and the film thickness is small, the stretching causes some of surfactant molecules in the film to move from the depth onto the surface. As a result, the surfactant concentration in the bulk of film decreases, leading to a decrease in equilibrium adsorption. Consequently, the surface tension increases (the Gibbs effect) [6] . The dependence of surface
537 tension on the film area, arising from the Gibbs effect, is equivalent to the existence of the elasticity modulus, EsE:
EsE - 2
Ac~
d~ =2 ~ AS / S As-~0 d InS
The effective film elasticity is especially important in emulsions in which the interfacial tension is small and can not ensure the stability of surfaces against the deformation due to random causes. The Gibbs effect is a t h e r m o d y n a m i c factor ofcolloid stability (this emphasizes only the nature of the effect, and one need not assume that this factor can ensure high stability of disperse systems). If the rate of film stretching becomes so high that there is not enough time for the equilibrium between the adsorption layer and inner (bulk) portion of film to be established during film deformation, the effective elasticity modulus assumes higher values. This results in higher (compared to equilibrium Gibbs effect), increase in the film stability, and hence in higher stability of disperse system. The extent to which equilibrium is established between adsorption layer and inner part of film, and consequently the effective elasticity modulus, are determined by the rate of diffusion of surfactant molecules from the bulk of film to its surface, and depend on a type of surfactant. Fast and, particularly, localized film deformation also disrupts equilibrium distribution of substance at the film surface, which also leads to an increase in the effective elasticity modulus. In the latter case surface migration of surfactant molecules from regions with high adsorption (undistorted portion of film) to areas with lowered F (stretched portion of
538 film) plays an important role. This stability factor characterized by the absence of equilibrium at the film surface as well as between the adsorption layer and inner portion of the film is referred to as the Marangoni-Gibbs effect [43,44]. 2.The electrostatic repulsion between diffuse parts of electrical double layers (ion-electrostatic component of disjoining pressure), which according to Derjaguin, Landau, Vervey and Overbeek [4,15,44] can be classified as thermodynamic factor of stability of disperse systems towards aggregation (see Chapter VII,4). Along with this factor, Derjaguin introduced other positive components of disjoining pressure, pertinent to equilibrium conditions, such as the s t r u c t u r e and a d s o r p t i o n components. The first of these two components is related to the overlap between boundary layers of dispersion medium that have structure different from that of bulk liquid. Such layers form at solid surfaces that are lyophilic with respect to dispersion medium. The adsorption component is caused by the overlap between the diffuse adsorption layers, occurring when the surfaces are brought closer together. 3. The hydrodynamic resistance of dispersion medium in the gap between particles against flowing out is one of the kinetic stability factors. The decrease in thickness of fluid layer between the particles during coagulation is related to viscous flow of liquid out of a narrow gap between the particles. For solid particles the liquid flow velocity is zero at the interface and highest in the center of a gap. The rate with which the gap between two circular planeparallel plates of radius r (Fig. VII-7) shrinks, dh/dt, is related to the volume of liquid that flows per second across the side surface of cylindrical gap,
dV/dt, via the following relationship:
539 F
Fig.VII-7. A decrease in gap thickness due to escape of liquid
dh
1 dV
dt
~r 2 dt
If Ap is some mean excessive pressure acting on the liquid in the gap, the value of d V/dt, in agreement with Newton's equation (see Chapter IX), should be inversely proportional to fluid viscosity, rl, and directly proportional to the pressure gradient of the order of Ap/r, to the perimeter of a gap through which the fluid flows out, 2~r, and to some power of the gap thickness, h ~ (by analogy with the Poiseuille equation), i.e." dh
oc
dt
1 Ap
2~rh n .
~r 2 fir
Dimensional analysis of the above equation reveals that n=3. Strict mathematical treatment yields dh
2 Aph 3
2 "
--
dt
(VII. 18)
3 fir
At high viscosity of dispersion medium the resistance of gap to thinning may be the reason for very high, or even infinite (such as in glasses) stability of
540 system towards coagulation, and, consequently towards coalescence. Depending on the type of films (liquid interlayers between solid surfaces, wetting films on solid supports, free symmetric foam and emulsion films, etc.), the type of boundary conditions in the region where film comes in contact with macroscopic phase, and on the degree of deviation from equilibrium state, the pressure, Ap, and the total force compressing the film surfaces, F - ~r2Ap, may be of different nature. In all of these cases disjoining pressure, H, plays an important role. In thin films distant from the state of thermodynamic equilibrium, the value of Ap may almost completely be determined by II. In systems with highly mobile interfaces between dispersed phase and dispersion medium (foam and emulsion films), the capillary pressure may play a role of Ap. Capillary pressure is especially important in relatively thick films, as well as in thin films that approach thermodynamic equilibrium. The solid particles separated by a gap filled with dispersion medium may approach each other due to an external force, F, such as gravity. The investigation of kinetics of thickness decrease in thin films carried out experimentally (see Chapter VIII, 2) allows one to obtain important information regarding the nature of forces acting in such fielms, and, consequently, regarding the stability of disperse systems. Expression (VII. 18), referred to as the Reynolds equation, is often written as d(1/h 2)
4 Ap
dt
3 fir 2
"
Experimental data may be conveniently presented in a form of l/h 2 as a function of t. This allows one to determine the value of Ap from the slope of
541 experimental curve, provided that the values of 1"1 and r are known (see Chapter VIII). The so-obtained value of Ap characterizes viscous resistance of medium to film thinning that occurs when particles are coming closer to each other, and may be viewed as being analogous to the positive disjoining pressure, i.e. as viscous component of the latter [4]. Thus, the concept of disjoining pressure introduced by Derjaguin and pertaining to equilibrium conditions, i.e. those corresponding to the absence of energy dissipation and to time independence, may also be extended to non-equilibrium processes accompanied by the dissipation of energy, the parameters of which depend on time. In the latter case one considers kinetic factors of stability rather than thermodynamic ones. If film thickness decreases solely due to the action of a molecular component of disjoining pressure, Ap = -II(h). This allows one to determine the molecular component of disjoining pressure from experimental data on film thinning kinetics, i.e. from the 1/h 2 as a function of t dependence.
In the case of highly mobile interface between dispersed phase and dispersion medium (as in foams and emulsions) the condition of zero fluid flow velocity at interface (non-slip condition), determining the validity of Reynolds equation, may not be obeyed. In this case the decrease in the film thickness occurs at a greater rate. However, in foam and emulsion films stabilized by surfactant adsorption layers the conditions of fluid outflow from an interlayer are close to those of outflow from a gap between solid surfaces even in cases when surfactant molecules do not form a continuous solid-like film. This is the case because at surfactant adsorption below Urea x the motion of fluid surface leads to the transfer of some portions of surfactant adsorption layer from central regions of film to peripherical ones, adjacent to the Gibbs-Plateau channels. As a result, the value of adsorption decreases in the center of film, but increases at the periphery, which stipulates the appearance of the surface
542 tension gradient (or the gradient of two-dimensional pressure) along the film surface, i.e. of the above mentioned Marangoni-Gibbs effect. This surface tension gradient to significant degree may balance the tendency of the border regions of liquid film towards flowing out. The surface acquires solid-like properties, and the flow regime described by Reynolds equation (VII. 18) establishes. The described condition of dynamic equilibrium between viscous forces acting during film thinning, the driving force of thinning, zXp, and the gradient of surface tension that appears during fluid outflow, indicate that "solidification" of surface occurs when the surface tension gradient between the central and peripherical parts of film, A~, satisfies the condition 2 Acy - h A p .
(VII. 19)
In thick films (h >_1 lam) the fluid outflow is primarily caused by the capillary pressure, i.e. Ap = [po[. For not too expanded films the capillary pressure is ]Pal "~ (y/r,, where r, is the average curvature radius of the Gibbs-Plateau channel. If ~ -- 70 mN/m, and r -- 1 mm, one has for Ap -- po-- 1400 mN/m. In this case for h ~ 10 to 20 gm a small value of Ao .~ 0.35 to 0.70 mN/m corresponds to eq.(VII. 19). Therefore, even very small amounts of surfactants are capable of causing the film surface "solidification" retarding the outflow of dispersion medium and thinning of film.
4. S t r u c t u r a l - m e c h a n i c a l barrier (after Rehbinder) is a factor o f strong stabilization, c a p a b l e of p r o m o t i n g essentially unlimited stability t o w a r d s a g g r e g a t i o n and coalescence o f disperse systems, including the c o n c e n t r a t e d ones (see C h a p t e r VII,5).
543 VII.4. Electrostatic Component of Disjoining Pressure and its Role in Colloid Stability. Principles of DLVO Theory
The electrostatic interaction between diffuse layers of ions surrounding particles is one of the most thoroughly theoretically developed factors of colloid stability. The theory of electrostatic factor is, essentially, the basis for the quantitative description of coagulation by electrolytes. This theory was developed in the Soviet Union by B.V. Derjaguin and L.D. Landau in 1935 1941, and independently by the Dutch scientists E.Verwey and T. Overbeek, and is presently known by the initial letters of their names as the DL VO theory [44,45]. The DLVO theory is based on comparison of molecular interaction between the dispersed particles in dispersion medium and the electrostatic interaction between diffuse layers of ions, with Brownian motion of particles taken into account (in the simplest version of theory this is done on a qualitative level). In order to obtain the expression for electrostatic components of disjoining pressure, I-Iel and free energy of interaction in the film, A ~r~f, let us examine the distribution of potential between two charged parallel surfaces separated by distance h (Fig. VII-8) in a rather dilute solution of electrolyte. In the vicinity of charged surface (see Chapter III) there is a diffuse part of electrical double layer containing an excess of counter-ions, in which the potential drops from the value q~d at the Stern layer to 0 at infinitely long distance away from the surface (Fig. VII-8, dashed lines). To simplify things, we will further use the surface potential, % in place of q~d (if one needs to account for the Stern layer, in treatment that follows the potential % should
544 q~
q0
-
.
h/2 Fig. VII-8. The distribution of electrostatic potential between two charged surfaces separated by distance h in a solution of electrolyte be replaced with q0d, and the coordinate x with x - d). When charged surfaces are brought together to distances comparable with the thickness of ionic atmosphere, 6 = 1/~: (see Chapter III), the potential distribution in a gap between surfaces changes, and the minimum in potential distribution appears in the middle of gap (Fig. VII-8, full line). Similar to the case of a single diffuse double layer in a vicinity of one surface, in order to obtain the potential distribution, one has to solve the P o i s s o n - Boltzmann equation. The appropriate boundary conditions in the present case are, however, different: dqo/dx = 0 not at x -~ ~, but at x - hi2, where q~ (h/2) ~0. Such change in boundary conditions causes mathematical complications (the appearance of elliptical integrals). At sufficiently long distances from both surfaces, one may successfully use an approximation of summation (superposition) of potentials. The potential in the center of the gap, q~ (h/2), within such an approximation is roughly twice the value of a potential of a single diffuse double layer taken at the same distance from the surface, q0~ (h/2): q~(h / 2) - 2q~l (h / 2).
545 Since at such distances from the surface eq. (III. 17) is valid, one obtains 8kT q~(h / 2) - ~ 7
exp(-Kh / 2),
ze
where 7 - tanh
zeq)o 4kT
; ze is the charge of counter-ions.
In agreement with eq. (III.10), the excessive charge density corresponding to the potential at the center of gap, Pv (h/2), is given by
9v ( h / 2 ) - - 2 z e n o
sinhlZe(p(h/2)]~kT 2
(VII.20)
,~ - 2 z e Znoq~(h/2 ) . kT The product 9v (h/2)q~(h/2) characterizes the density of electrostatic energy at the central point between the surfaces, i.e. the work consumed due to increase in counter-ion concentration upon bringing the charge surfaces close to each other. At the same time, in agreement with eq. (VII.3), the density of excessive free energy in the film represents the disjoining pressure, and, in this case, the electrostatic component of the latter, YIel.A more detailed consideration reveals an additional numerical coefficient of 1/2. As a result, the electrostatic component of disjoining pressure is given by
1 z2e2no(p2(h/2) Ilei ~ --~gv (h/Z)q~(h/2) ~ = kT
= 64nokT72 exp(-~:h).
(VII.21)
546 In the case of a more strict approach to evaluation of change in energy with decreasing distance between charged surfaces, the expression for electrostatic component of disjoining pressure has to be written as q~(h/2) l--[e1 = - -
I
pv(h/2)dg
.
o
Using eq. (VII.20), this expression can be integrated to yield
l--[el-2n~176176 The above relationship for electrostatic component of disjoining pressure has the following meaning: the first term, in agreement with eq. (III.20), represents the osmotic pressure of ions in the center of a gap, while the second term is the osmotic pressure in the bulk of dispersion medium. One may thus say that the electrostatic component of disjoining pressure equals to the difference in osmotic pressure between the gap and the bulk, that forces the dispersion medium to flow into the gap between surfaces causing a"disjoining" action. For small values of q0(h/2) the expansion of hyperbolic cosine into series as cosh (y) = 1 + 89(y)2 readily yields eq. (VII.21).
Thus, the properties of diffuse part of electrical double layer determine the dependence of electrostatic component of disjoining pressure on the thickness of film [15]. The screening of the charged surface with a layer of counterions results in a sharp decrease of the electrostatic component of disjoining pressure with a corresponding increase in film thickness. For the surface bearing low charge, when 4kT
q)o < ze
,
i zego 1 zeg~ 4kT
and 3 ' - t a n h L4kT
547 the expression (VII.21) can be written as 4z2e2q~ 2 l-[el ~
0 ?/0
kT
exp(-~:h),
i.e., the electrostatic component of disjoining pressure is proportional to the square of surface potential, %. For a strongly charged surface, for which
4kT q~0 >
ze
, and7 ~ 1,
the disjoining pressure is given by Hel ~ 6 4 n 0 k T e x p ( - n h ) . In this case IIe~ is independent of the surface potential due to a significant screening of surface with counterions. The dependence of disjoining pressure on % at some constant value of h, shown in Fig. VII-9, has two characteristic regions: a sharp change in gle~at low surface potentials, and constant value of IIe~ at high %. The increase in electrolyte concentration, which causes ~: to increase proportionally to (no) ~/2(see Chapter III, 3), leads to a decrease in the value of disjoining pressure at a given distance between surfaces.
IIel ........
~
,
0
~
2
,
%
Fig. VII-9. The electrostatic component of disjoining pressure, 1-Ie~,as a function of surface potential, %
548 Integration of expression (VII.20) with respect to the gap thickness in agreement with eq. (VII.4), allows one to estimate change in the film energy, A ~ . At constant potential
A '~f(el) ~
64n0kT 7 2
exp(-tch).
(VII.22)
K
The electrostatic components of disjoining pressure and free energy of interaction in the film, given by eqs. (VII.21) and (VII.22), are positive, i.e. represent repulsion. These quantities may be compared with corresponding molecular components that are negative and describe attraction. This allows one to analyze according to the DLVO theory the stability of thin films, and consequently of disperse systems stabilized by adsorption layers. Carrying out summation ofeqs. (VII.21) and (VII.22) with expressions (VII.9) and (VII. 10) one obtains: H - 64n0kT 7 2 exp(-Kh)
A~
- 64n okT 7 2 K:
exp(-tch) -
A 6~rh 3 ' 9 A 127rh 2 "
(VII.23)
(VII.24)
Typical curves corresponding to eqs. (VII.23) and (VII.24) are shown in Fig.VII-10. The appearance of the so-called secondary minimum at film thickness h ~ 1/~:is related to the fact that electrostatic repulsion decreases with distance faster than the molecular attraction. Molecular interaction also prevails at low film thickness; change in the sign of derivatives causes a maximum to appear in the II (h) and A gf(h) curves. One has to remember
549 that in real systems at shortest distances, h - b, there are repulsion forces of different nature, namely, the Born repulsion as well as other components of disjoining pressure related to, e.g., the solvation of surface with dispersion medium or to the formation of strong adsorption layers. Due to these factors H and A gf(h) do not experience infinite drop when the film thickness is decreased to zero, but revealprimarypotential minimum, which may be rather deep. In agreement with eqs. (VII.3) and (VII.4) the extrema in the A gf(h) curve correspond to the points at which I-I= H(h) intersects the x axis (Fig. VII10).
A {h)
0
II(h)
I !
I I 1 I i I
I I I I I I
N
Fig. VII-10. The excessive free energy, A~ , and disjoining pressure, H, as a function of film thickness, h The decrease in thickness of foam and emulsion films may result in a metastable equilibrium described by the H =-po condition (Fig. VII-10, point A). In this case film thickness depends on electrolyte concentration: the increase in the latter results in a decrease of electrostatic component of
550 disjoining pressure and in decrease of equilibrium film thickness. Until the film is not too thin (h ~ 1gm) it reveals colored bands which appear due to the interference of light. At high electrolyte concentrations the films become so thin that they loose ability to reflect light; there are the so-called common black films. In addition to that, an increase in electrolyte concentration results in a decrease of the height of potential barrier which preserves the film in the state of this metastable equilibrium, i.e., film stability decreases. Thermal oscillations of interface, i.e., the Mandel'shtam waves (See Chapter VI,1), help the system to overcome a potential barrier. If other stabilizing factors are absent, such (local) overcoming of potential barrier results in film rupture. In the case of films with high stability the overcoming of potential barrier does not result in a rupture of film, but leads to another metastable state corresponding to the primary minimum (Fig. VII-10, point B). This results in the formation of a rather stable, very thin Newtonian black films [ 15]. The investigation of the nature of stability of black films is one of the central problems in colloid science; nevertheless, at present there is no commonly accepted opinion concerning the nature of forces that are responsible for high stability of black films (see Chapter VIII). It follows from eqs. (VII.23) and (VII.24) that values of 1-I and A 5c~ depend
on
electrolyte
concentration.
Concequently,
the
electrolyte
concentration defines the height as well as position of potential barrier (see Fig. VII-10), which characterize film stability. The addition of electrolyte to colloidal system results in compression of electrical double layer, and, hence, in compression of the region of the effective action of electrostatic repulsion
551 forces. As a result, the maximum in the A ~ (h) curve decreases (or even completely vanishes in some cases) and shifts towards smaller values of h. The lowering of maximum and the shift in its position reflect an increasing role of molecular attraction, which leads to decrease in film stability. The maximum in A ~ (h) curve characterizes the potential barrier that one has to overcome for the film thickness to start decreasing spontaneously, all the way down to the film rupture. When applied to disperse system this situation corresponds to coagulation in the primary minimum. In order to describe the stability of fine disperse systems stabilized by diffuse ionic layers, one has to use the total free energy of interaction between particles, 5c, instead of the energy per unit film area, and compare the barrier height, gm~x, to the thermal energy, kT. For us to be able to use the solution derived for the case of plane-parallel surfaces, let us introduce some effective area of particle contact, Sef. Then the potential barrier height for the particles can be expressed as '-~max -- A ~ f , max Sef. When diffuse part of electrical double layer is strongly developed, this maximum may be quite high compared to kT, and the energy barrier, G-m,x, becomes essentially insurmountable. Increase in electrolyte concentration results in gradual decrease and finally in total disappearance of potential barrier. The condition corresponding to the loss of aggregative colloidal stability can be written as A ~ ;
max S e f ~<
kT. The critical
condition corresponding to a complete loss of stability is identical to the total vanishing of positive barrier, A 5r-f. max
Sef -
0. The latter means that A ,,~-f =
= 0 and d( A,~-f)/dh - 0, i.e. the maximum lies on the h axis (Fig. VII-11). One can thus write
552
A $
64ncr 7 2 kT exp(-K crhcr ) - 6~h3cr '
(VII.25)
and
64ncr72kTexp(-Kcrhcr)
A*
Kcr
12 ~hc2r "
(VII.26)
Division of eq. (VII.25) by (VII.26) yields K:cr - 2 / hcr , where indices "cr" is attributed to the critical conditions corresponding to the disappearance of potential barrier. Substitution of hcr into either (VII.25) or (VII.26), yields
64ncr7 2kT e x p ( - 2 ) -
A*K
3 cr
48~
Raising the above equation into the second power, and realizing that 2 K cr - -
2z 2e2ncr
one obtains the expression for critical electrolyte
~:okT
concentration at which the potential barrier dissappears, also known as the
critical coagulation concentration (c.c.c.), namely (~;g0)3 (kT)57 4 nor - k 1
(A*)2z6e6
553 where k~ = 2x 105. This is the main relationship of the DLVO theory, which establishes the connection between system properties, charge of counterions, and the value of ncr, corresponding to a full loss of stability. I I
Ii
h~r= 2/Kcr h
Fig. VII-11. A condition corresponding to a complete loss of colloid stability at electrolyte concentration n = n c r If dispersed particles are s t r o n g l y c h a r g e d , 7 = 1, and the critical concentration of electrolyte is inversely proportional to the charge of counterion in the sixth power (the z -6 rule). In this case the coagulation occurs due to diffuse double layer compression upon introducing high concentration of electrolyte into the system (concentration coagulation). For electrolytes containing mono-, di- and trivalent counter-ions, the values inverse to ncr (coagulation abilities) form 1 : 64 : 729 series, which agrees well with the Schulze - Hardy rule (See Chapter VIII). In the other limiting case o f w e a k l y c h a r g e d c o l l o i d a l p a r t i c l e s 7 ~ ze%
/ 4kT (see Chapter III,3). In this case the critical coagulation
concentration of electrolyte is a weaker function of counter-ion charge, given by the equation originally derived by Derjaguin"
554 (880 )3 kTq~40 2 '
ncr - k2
(A*)2zZe
(VII.27)
where k2 ~ 800. This situation occurs when introduced electrolyte is capable of decreasing the surface potential, %, or (and) the Stern layer potential, q~j, reaching charge neutralization on colloidal particles (coagulation by neutralization) and further the reverse of particle charge. This agrees with earlier views by Freundlich on the determining role of ion adsorption in lowering electrokinetic potential and facilitating coagulation. The replacement of ss0kT/2z2e2ncr with 1/K:c2rin eq. (VII.27), and raising the latter in the power of 1/2, results in SSoq) 2 / A
* K cr
-
k 3,
(VII.28)
where k3 ~ 0.024. In the case of low surface potential, q~0, and low electrolyte concentrations, typical for the case of coagulation by charge neutralization, the values of %, qod, and ~ are close to each other (see Chapter V,3), and hence eq. (VII.27) may also be written as
880~ 2 -
A*K
cr
k 3
.
555 The above expression corresponds to the empirical coagulation criterion introduced by H. Eilers and J. Korff [4]. The quantity ee0~2 / ~:~r describes the electrostatic repulsion energy between diffuse layers of ions, while the Hamaker constant, A*, is related to the attraction energy. The ratio of these two characteristic interaction energies thus determines the stability of a colloidal system under a given set of conditions. It is apparent that in the case of coagulation the energy of molecular attraction prevails over the energy of electrostatic repulsion. Many systems obey both the z 6 rule and Eilers Korff criterion. DLVO theory explained major principles of coagulation of hydrosols by electrolytes and brought to common grounds all previous observations (primarily of qualitative nature) that related to individual cases and often seemed to be contradictory. In years that followed further extensions of DLVO theory that took into account the possibility of reversible particle aggregation were developed. At very small distances between particles in addition to the "usual" long-range interaction, molecular attraction and electrostatic repulsion, one must account for other factors that play role at a direct particle contact. The formation of peculiarly structured hydration layers in the vicinity of solid surface, the appearance of elastic forces that are responsible for the Born repulsion between surface atoms at the point of contact, the repulsion between the adsorbed surfactant molecules in contact zone between two particles, all represent the so-called "non-DLVO stability factors". This means that more or less deep primary minimum remains finite. The exact evaluation of the shape of this potential energy minimum is rather difficult. Some of these difficulties are related to the integration of
556 Poisson-Boltzmann equation at such small distances where properties of dispersion medium are significantly different from those in the bulk. It is, however, evident that the depth of a potential energy minimum is affected by particle size and charge; the larger are the particles, and the lower their charge, the deeper the primary minimum. In the case when the depth of potential minimum is smaller than several kT, the coagulation (i.e., the combination of two particles) becomes thermodynamically unfavorable even at low height of the potential barrier (Chapter VII, 1), and the stability of colloidal system towards coagulation is of thermodynamic nature. This is confirmed by observed peptization of coagulated precipitates upon washing out the excess of coagulating electrolyte and by stabilization of sols by specifically adsorbed ions.
VII.5. Structural-Mechanical Barrier
The term
"structural-mechanical barrier" was for the first time
introduced by P.A. Rehbinder [2,46-48]. This is a strong factor of stabilization of colloidal systems related to the formation ofinterfacial adsorption layers of low and high molecular weight surfactants which lyophilize interfaces. The structure and mechanical properties of such adsorption layers are able to ensure very high stability of dispersion medium interlayers between dispersed particles. According to Rehbinder, the structural-mechanical barrier appears due to the adsorption ofsurfactant molecules that are capable of forming g e 1-1 ike s t r u c t u r e d layer at the interface, but are not necessarily highly surface
557 active with respect to a given interface. These are the surfactants belonging to the
3 rd and 4 th
groups of surfactants, as classified in Chapter II,3. Such layer
is similar to a three-dimensional structure, i.e., to gel, which may form in sufficiently concentrated solutions of some substances. These substances include glukozides, proteins, cellulose derivatives (carboxymethylcellulose) and other so-called protective colloids, which are the polymers with complex molecular structure that have the regions of higher and lower hydrophilicity within the same molecule. With respect to dispersions ofhydrophilic powders in non-polar liquids, oil-soluble surfactants capable of strong chemisorption on the surface ofhydrophilic particles, act as very effective stabilizing agents. Lyophobic systems stabilized in such a way acquire properties of dispersion of a given stabilizer, i.e. they become lyophilized. According to Rehbinder, the following factors determine high effectiveness of structural-mechanical barrier: 1. An increased viscosity and mechanical strength of stabilizer adsorption and interfacial layers; the capability of such layers to resist deformation and collapse in combination with sufficient mobility allows for healing of random defects. In systems containing solid particles, strong attachment of stabilizer molecules to particle surface (i.e., high interaction energy between stabilizer molecules and the solid surface of particles) may be sufficient for effective stabilization. Under these conditions the requirement of mechanical strength of the adsorption layer due to the interactions between the adsorbed molecules themselves may be of less significance. 2. The lyophilicity of outer portion of adsorption or interfacial layer, i.e. the high degree of its similarity (affinity) to the dispersion medium, which
558 ensures the "smoothness" of transition from dispersed phase to the dispersion medium. Thus, the important features of the structural-mechanical barrier are the rheological
properties
(See Chapter IX, l,3) of interfacial layers
responsible for thermodynamic (elastic) and hydrodynamic (increased viscosity) effects during stabilization. The elasticity of interfacial layers is determined by forces of different nature. For dense adsorption layers this may indeed be the "true" elasticity typical for the solid phase and stipulated by high resistance of surfactant molecules towards deformation due to changes in interatomic distances and angles in hydrocarbon chains.
In unsaturated
(diffuse) layers such forces may be of an entropic nature, i.e., they may originate from the decrease in the number of possible conformations of macromolecules in the zone of contact or may be caused by an increase in osmotic pressure in this zone due to the overlap between adsorption layers (i.e., caused by a decrease in the concentration of dispersion medium in the zone of contact). Often one relates this type of stabilization to the so-called stericfactor [48-51 ], the notion of which was introduced much later than Rehbinder's concept of lyophilic structural-mechanical barrier. Steric factor primarily reflects
configurational
elasticity
of tails
and
loops
of adsorbed
macromolecules as well as osmotic effects. Steric factor represents only an entropic ( generally speaking, small) contribution to the elastic resistance of film, and by itself can not account for the strong stabilization. For the interfacial layer to be able to protect dispersed particles and to prevent their cohesion and coalescence, it should be able to withstand the
559 stress from particle collisions and be resistant towards collapse and displacement out of the zone of contact. This means that for a reliable stabilization, the interfacial layer must either be strongly attached to the particle surface, or must form the structure with an increased viscosity and particularly mechanical strength. It must also be able to restore its original configuration quickly. The relaxation processes resulting in the elimination of stresses (see Chapter IX, 1,3) are thus also of importance for this factor of colloid stability. The existence of a firmly fixed adsorption layer or of a rather thick interfacial layer with sufficient mechanical strength at the particle surface is sufficient for preventing the coalescence but may not necessarily ensure the stability of system towards coagulation. If there is no close similarity between the layer of stabilizer and the dispersion medium, the action of molecular forces will result in the attraction between stabilizing layers themselves. Such "close similarity"(affinity) is characterized by low values of A*. Indeed, the formation of strongly solvated
stabilizing layer (consisting primarily of
solvent molecules, e.g. the layer of gelatin at the droplet surface in oil-in-water emulsion) at the particle surface, results in the volumes of the layer in direct vicinity to the contact to be composed mainly from solvated molecules (Fig. VII-12). These particular volumes contribute the most into the attraction energy between particles. If the Hamaker constant of solvated layer of stabilizer, A3, is close to the Hamaker constant of the medium, A2, the complex Hamaker constant, A 23 --( A4~2- A4-3~-3 )2, may be by 1- 2 (or even more) orders of magnitude less than the value of A*~2, characteristic of the particle medium system in the
560 L2
Fig. VII-12. The stabilization of emulsion droplets by structural-mechanical barrier
absence of a stabilizer. For a majority of common lyophobic systems with aqueous or hydrocarbon continuous phase, the values ofA ~2~ ( ~
Av~2)2~
10 -19 to 10 -2~ J (here A~ is the Hamaker constant of dispersed phase).
According to eqs. (VII. 16) and (VII. 17) and in accordance with quantitative estimates obtained on the basis of these equations, it is sufficient for one to lower the complex Hamaker constant by two orders of magnitude, in order to achieve a high degree of lyophilization of system and to convert unstable lyophobic system into the one thermodynamically stable towards coagulation (pseudolyophilic). From the same standpoint one may view the stabilizing action of surfactant adsorption layers, including those formed by low molecular weight ones, at surfaces of solid (primarily hydrophilic) particles dispersed in liquid hydrocarbons. A firm fixation of molecules at the solid surface, especially by chemisorption, determines a high strength of adsorption layers, i.e., their ability to resist deformation and collapse when particles come into contact with each other. At the same time high lyophilicity of the adsorption layer results in lowering of interaction energy between such lyophilized particles to the level corresponding to an infinite stability of system towards coagulation.
561 As one can see, the structural-mechanical barrier is a complex factor of colloid stability, which includes the contribution from a number of different thermodynamic, kinetic and structural-rheological (i.e., related to peculiarities in structure of adsorption layers) factors.
VII.6. Coagulation Kinetics
The expressions for collision frequencies of particles participating in Brownian motion (Chapter V, 1) and coagulation rate ofunstabilized colloidal system in which every collision leads to aggregation ( r a p i d c o a g u l a t i o n ) were derived by M.Smoluchowski for the case of initially monodisperse system [52,53]. In a dilute colloidal system the probability of simultaneous collision of three particles is very low, so in reality one has to account only for pairwise collisions and describe coagulation in terms of consecutive bimolecular reactions involving collisions between two single particles, a single particle and a doublet, and so on. The collision frequency of m-dimensional aggregate (i.e., of the aggregate consisting of m primary particles) with the ndimensional particle is determined by their concentrations, n m and n,: J-
(VII.29)
kmnnmm n .
The rate constant of this process is not very sensitive to particle size, and in the case of particles that have more or less similar sizes can be written as 8kT k-
kmn "~
3q
,
(VII.30)
562 where r I is the viscosity of dispersion medium. Each collision results in a combination of two aggregates into a single one. Thus, the change in total number of aggregates of all sizes (from m - 1 to m = 30), as a function of time, n~= fit), is described by the following differential equation" dnz _ _kn 2" dt by solving which one obtains the Smoluchowski equation, i.e." no
no
1 + knot
1 + t/t c
n E =
In the above equation the time o f coagulation, t c - 1/kn o, is the time corresponding to a two time decrease in the number of aggregates. It is determined by initial concentration of particles in the system, n o, the dispersion medium viscosity, and the temperature" 3n tc = ~ . 8n0kT
(VII.31)
In the case of slow coagulation the number of collisions between particles leading to their aggregation decreases due to the presence of a potential barrier, which prevents the particles from approaching each other. The fraction of"successful" (i.e. leading to aggregation) collisions is referred to as the collision efficiency, a. Inverse of collision efficiency is the stability ratio, W=I/a, which is equal to the ratio of true coagulation rate constant to
that predicted by the Smoluchowski equation (VII.29). The stability ratio
563 depends on the height of potential barrier, Umax, and on the double layer thickness, 1/~. The decrease in b/max caused by introduced electrolytes leads to a decrease in stability ratio, i.e., results in an increase in observed coagulation rate up to the values predicted by the Smoluchowski theory (or even slightly higher ones due to the action of long-range interparticle attraction forces). In recent years a great deal of attention has been devoted to the computer modeling of aggregation phenomena and the use of
fractals
theory of
(fractal geometry) for the analysis of the structure of aggregates,
recognized as fractal objects. Such studies have indicated that if one particle is firmly deposited at the surface of another particle already included into the aggregate, the particle density in the aggregate decreases as the latter grows bigger. This occurs because the particle interacts with individual particles in the aggregate, rather than with the entire surface of an aggregate (as was implied in the derivation of Smoluchowski equation). The probability of particle attachment to the particles located at the surface of an aggregate is higher than the probability of their attachment to coagulation centers located at depth. This results in the formation of particle chains, which block the access of new particles to the internal region of an aggregate, where, as a result, only little branching occurs. Consequently, the average particle coordination number in these loose aggregates is between 2 and 3, as has been already pointed out (Chapter VII, 1). The structure of particle aggregate consisting of 100 particles is shown in Fig. VII-13. This figure represents the results of computer modeling of a two-dimensional coagulation.
Computer modeling was done in the following way. First, the central particle was chosen. One after another, new particles were then randomly placed into a circle with a
564 radius several times the final radius of an aggregate. Random displacements were assigned to the newly introduced particle, and at each step the distances between the particle and all particles in the aggregate were evaluated. In the event the distance between the newly entered particle and any particle in the aggregate was lesser than or equal to twice the particle radius, the particle was considered to be included into the aggregate, and the next particle was introduced into the circle. To prevent particles from leaving the screen area, a "reflecting wall" was set.
O
Fig. VII-13. Structure of particle aggregate obtained by computer modeling of twodimensional aggregation
It was shown [53-55] that the total number of particles in the aggregate, Jg', is related to its radius of gyration, Rg, via relationship of the type ~4/'- R ~ , where the power D is a non-integer number equal to 1.72 in the case when coagulation is modeled in two-dimensional space, and 2.5 in the case of 3D modeling. Ii is worth mentioning that the mathematical principles of the theory of fractals are similar to those of the theory of percolations (see Chapter V, 1). If the energy of a contact is not too high, the bonds between particles in the aggregate are weak, and particle rearrangement in the aggregate takes place along with coagulation. Such rearrangement of particles results in a
565 gradual increase of coordination number, Z, up to Z= 12, which is characteristic of a dense packing of uniform spheres. This dense packing is typical in the case of aggregation of inverse emulsions and is sometimes encountered during the coagulation of monodispersed latexes. Such aggregation processes lead to the formation of ordered structures known as "colloidal crystals". Along with the case of perikinetic coagulation, when particle aggregation results from collision between particles involved in Brownian motion, one should not overlook the
importance of the orthokinetic
coagulation and differential sedimentation, when aggregation is caused by gradients in fluid flow or particle velocity [53]. For example, an increase in particle aggregation rate is frequently observed when suspensions are subjected to shear, e.g. by stirring or flow. During sedimentation, the larger particles that settle faster may catch up with the slower moving ones and entrap them. The probability of such entrapment depends on the ratio of sedimentation velocities as well as the conditions governing the adhesion of smaller particles at the surface of larger ones. Orthokinetic coagulation and differential sedimentation play an important role in such processes as flotation, water treatment, dust entrapment and natural precipitation from the atmosphere. Let us now examine the theory of coagulation in a greater detail. In agreement with the theory of random processes, one of the two particles involved in Brownian motion may be viewed as stationary. It is thus possible for the one to bind the coordinate system origin to this stationary ndimensional particle and say that the diffusion coefficient of the second particles is given by the mutual diffusion coefficient, D~ = D~ + D~ (see
566 Chapter V,2). The coagulation occurs when the particles approach each other at distance
Rm,,which is the sum of their radii. Due to coagulation the particles
assume a new n + m-dimensional state. One thus can state that the concentration of m-dimensional particles at distance
Rm, from the center of n-
dimensional particle is equal to zero. Under such boundary conditions, assuming that particles are spherical, one can write eq. (IV.20) as
Jmn - 47r,RmnOmnnm, where Jm~is diffusional flux of m-dimensional particles towards the fixed ndimensional particle. Multiplying the above expression by the concentration of "central"
n-dimensional particles,
n , , and comparing the result to eq.
(VII.29) one obtains
kmn - 4ZtRmnOmn. The Einstein eq. (V.11) establishes the relationship between the diffusion coefficient, D, and the radius of diffusing particle, r:
O
kT 6Ztrlr
where rl is the viscosity of the medium. Consequently,
k
= ~2kTRmnl ~1--~nl + mn
31]
rm
2kT(rm+rn)2 3~ rmr~
.
(VII.32)
567
(rm+ r.) 2 Figure VII-14 shows
as a function of the radii ratio of diffusing
rmr,
particles, rm / r~. In the vicinity of a minimum (where r~ / r, z 3), the curve has a nearly flat region in which
(r m +rn) 2
(rm + r.) = rmr.
4
9The number of primary
rmrn
10 8 6 4 20
I
I
I
I
2
4
6
8
9
10 rr.
Fig. VII- 14. The value of (rm+r,,)2/rm r,, as a function of r,, / r,,
particles in the aggregate of size rm is proportional to (rm/r~)3, where r~ is the radius of a primary single particle, n=l. Consequently, the ratio of r m / r , - 3 corresponds to approximately 30-time increase in the volume of aggregate, as compared to that of primary particles, i.e. this value corresponds to a rather deep stage of coagulation. In agreement with Eq. (VII.32) in the range where
(rm+rn) 2
is constant, km, reveals very little dependence on the size of
rmr~ colliding particles"
k
8kT - kmn
3q
568 Change in the concentration of m-dimensional particles with time is a function of the rate of formation of such particles via all possible routes of pairwise combination of smaller particles minus the rate of disappearance of these particles due to collisions with various other particles:
-
dt
(lmZ
rlinm_ i - nmn z
2- i-1
/
,
where the coefficient 89takes into account the fact that each collision is counted twice. A subsequent solution of such equations for single particles, particle doublets, triplets, etc., yields a series of the following equations"
no(t/tc) (1 +
nm
t/tc)
m+l "
Figure VII-15 shows the values of n~, nz and concentration of some small aggregates (m = 2 to 4).
n/no[ 1 r
3/4
1/2
1/4
0
n ffno
ln4/n ~ 2
3
4 t/tr
Fig. VII-15. The concentration of single particles, small aggregates(m = 2 to 4), and the totgal number of aggregates, n~, as a function of time
569 A coagulation process in the system consisting of particles with substantially different sizes was described by Miller. In Fig. VII-14, this case corresponds to a steeply rising portion of
(F m Jr- ?'n) 2
as a function of r ~ / r ,
rmr, curve. In agreement with eqs. (VII.31) and (VII.32), a sharp increase in coagulation rate is observed in this region. The rate of coagulation is affected by interaction between particles separated by distances greater than twice their radius. According to N.A. Fuchs [56], one has to look at the diffusion of particles of radius r in their interaction field, described as a function of energy (negative!) on distance, u(R). Fick's equation in this case can be written as
J _ 4rcR2[ D ~ + 0 n n 0u(R)] OR 67rrlr OR ' and the stability ratio,/IV, is given by the expression
W-2r
exp k T
R 2"
(VII.33)
2r
The true coagulation rate constants are by a factor of W lower than the values predicted by eq. (VII.30), i.e. k~
8~tkT 3rlW
Consequently, real coagulation time is by a factor of W greater than the time of rapid coagulation. Because the function u(R) is rather complex, the solution
570 of eq. (VII.33) requires numerical integration, and thus for qualitative analysis one assumes that the stability ratio approximately given by W~
2Kr
exp
max ( / kT
,
i.e., is determined by the potential barrier height, b/max,and by the ratio of the double layer thickness, 1/~:, to particle radius, r. A large settling particle is capable of entrapping smaller particles, whose centers are within the volume of some vertical round cylinder with the axis passing through the center of a larger particle. The base of such cylinder is referred to as the entrapment cross-section. Since dispersion medium together with smaller particles flows around the larger particle, the entrapment cross-section is less than the sum of radii of small and large particles and depends on the conditions under which the smaller particle approaches the larger one. The smallest particles d i f f u s e towards the surface of a larger as the fluid flows around the latter, and the diffusion rate is increased due to the action of molecular attraction forces. Particles with size close to that of a large particles, are primarily affected by for c e s o f i n e rt i a, especially important at the beginning of the flow process when fluid flow is directed away from the axis along which the large particle settles (Fig. VII- 16). These forces cause the particles to approach each other, while molecular (or electrostatic) forces, responsible for particle deposition, act only at very short distances. Theoretical and experimental studies indicated that upon sedimentation of very coarse particles, the particles that are either close to them in size, or those much smaller are the ones that are most effectively entrapped [53,57]. For the former class of particles the
571 forces of inertia are rather large, while for the latter ones the diffusion is fast. In contrast, the entrapment of particles with intermediate sizes is not very effective, since for such particles the forces of inertia and the diffusion rates are both rather small.
capture cross-section
Fig. VII- 16. The entrapment of smaller particles by a settling large particle due to the action of forces of inertia.
VII.7. The Influence oflsothermal Mass Transfer (Ostwald Ripening) on the Decrease in Degree of Dispersion
In disperse systems where coagulation and coalescence occur with very low rates, and under the conditions of substantial solubility of dispersed matter, the decrease in degree of dispersion may be caused by the matter transfer from smaller particles to the larger ones. These processes are quite common in nature and may take place in a variety of disperse systems, such as lyosols, suspensions, emulsions, foams, aerosols, in the systems with solid
572 continuous phase, as well as in minerals and alloys. Main laws that govern isothermal mass transfer in various media are rather close due to the same nature of driving forces and mechanisms of these processes. In each case isothermal mass transfer is driven by diffusional substance transfer under the influence of chemical potential gradients, generated by the difference in surface curvature of particles with different sizes. In agreement with the Thomson (Kelvin) law (Gibbs-FreundlichOstwald law for solutions, see Chapter 1,3), the increase in chemical potential of substance in small particles in comparison with its value in the bulk phase equals Ag = 2cyVm/r. This leads to the dependence of the vapor pressure, p, and solubility, c, on particle size, given by:
p(r)-
Po exp
c(r) - co exp
I2Vmt rRT
rRT
'
"
(VII.34)
In the above expressions P0 and Co are, respectively, the saturated vapor pressure and solution concentration that are in equilibrium with a flat surface of macroscopic phase; Vmis the molar volume of dispersed substance. Due to obvious similarity between expressions for vapor pressure and concentration, we will focus our discussion on the analysis of eq. (VII.34) only. For particles whose size is significantly larger compared to molecular dimensions, b, i.e. when r ~ b, the exponent in eq.(VII.34) may be expanded in series, which yields
573
c(r) ~ c o 1 +
2crVm .
(VII.35)
Since the solubility of small particles turns out to be higher than that of large particles, the transfer of substance from small particles to the larger ones should take place in a real polydisperse system. As a result, small particles may entirely disappear from the system. Large particles, in turn, will be consumed by even larger ones, and such process will continue further on, until the system turns into rather coarse disperse one. In agreement with eq. (VII.35), the difference in solubilities of particles with different sizes in such system is no longer significant, and the rate of Ostwald ripening is negligibly small. Exact solution to the problem of change of the average particle size and of particle size distribution with time was given by E.M. Lifshitz, V.P. Slezov and C. Wagner [58-61 ]. We will present their treatment with some simplifications. The growth rate of large particles in diffusion regime (Chapter IV,4) is given by dr
AcDVm
dt
5
where 5 is effective diffusional path; Ac is solution supersaturation, and D is diffusion coefficient. In dilute solutions the ~5is determined by particle size, and usually ~5 = r. Supersaturation, responsible for growth of large particles, depends on the difference in solubilities of small and large particles.
574 According to Lifshitz and Slezov [58,59], the value of Ac is a function of the difference between the radius of particles, r, and the average radius, f 9
A c - 2cYVm Co RT
--
. r
Consequently, particles with a radius greater than the average radius grow, while those with a radius smaller than the average one dissolve. Hence, the average particle radius increases with time. There is a steady-state stage of the "recondensation" process during which no changes in the shape of particle size distribution function occur, and all changes that the system undergoes with time are described by the change in the average radius. Under these conditions the largest particles have radii that are 1.5 times greater than the average one, i.e. rmax - 1.5 r. Consequently, the ratio between the volumes of the largest, Vma• and the average, 4/3~ f3, particles is (3/2) 3 - 27/8. Taking this into account, one can write that
dvmax
dt
=47cr 2 max
drmax = 4 dt
2cYVmcoD (rma____~X_ l) _ 4~cYVmcoD RT \ ) F RT
Using this, Lifshitz and Slezov showed that dF 3 dt
3
8 drmax
4~ 27
dt
8CYVmcoD 9RT
575
Thus during the steady-state stage of isothermal mass transfer the cube of average radius grows linearly with time; the growth rate is a function of interfacial tension, as well as of the solubility and diffusion coefficient of dispersed substance. If an admixture that is nearly insoluble in a continuous phase is introduced into the dispersion phase, a sharp decrease in recondensation rate, as well as changes in laws describing the process, may take place. In concentrated systems with liquid dispersion medium there may be an equilibrium interlayer of thickness 8 through which the recondensation takes place. Under such conditions the surface area of growing particles is a linear function of time, i.e.
dS dt
d7 2 CYcoDVm 7~ = = const. dt 8RT
(VII.36)
This case is most typical for foams and emulsions. Thus we have seen that the rate of change in degree of dispersion in a colloidal system is governed by the solubility and the diffusion coefficient of dispersed substance and by the interfacial tension. The diffusion coefficient, D, in turn, depends to a significant extent on the aggregate state of dispersion medium (very small values of D are typical for solid dispersion media)and, to a lesser extent on the size of molecules of dispersed substance, and, as a rule, can not be altered by introducing any kind of admixtures into the system. At the same time, the presence of adsorption layers at the particle surface (particularly in concentrated dispersions, where such layers fill most of the
576 space between the particles) may significantly retard Ostwald ripening. This retardation is caused by low penetration of dispersion medium into adsorption layers due to a small diffusion coefficient and low solubility of substance in such layers. Decrease in the rate of particle growth during Ostwald ripening may also be achieved by lowering the surface tension. In lyophilic colloidal systems isothermal mass transfer does not occur at all. The solubility of dispersed matter is not significantly dependent on the presence of foreign admixtures, but strongly depends on the nature of phases. In a majority of lyophobic systems resistant towards Ostwald ripening, the dispersed phases consist of substances that are essentially insoluble in dispersion medium. The incorporation of additive that is practically insoluble in dispersion medium into the dispersed phase (still somewhat soluble), as e.g. incorporation of perfluorobutylamine into perfluorodecaline, may drastically decrease the rate of recondensation, as shown by Pertsov et al [62-65]. Since with an increase in temperature the solubility and the rate of diffusion typically increase, one can effectively control Ostwald ripening by monitoring the temperature. This is utilized in recovery of metals after they were subjected to a mechanical treatment (work hardening). Annealing leads to a collective recrystallization of grains to sizes at which the material, while retaining sufficient hardness, acquires ductility that prevents brittle fracture. In natural minerals collective recrystallization is one of the major mechanisms of metamorphism, the process leading to changes in structure and mineralogical composition of minerals. In chemical technology Ostwald ripening is broadly used for the separation of precipitates by precipitation and filtration. The processes based
577 on isothermal mass transfer are of great importance in highly concentrated emulsions and foams. If these systems are finely disperse, the isothermal mass transfer may be the main mechanism by which degradation of these systems Occurs.
Isothermal mass transfer plays an important role in the transition of substance from convex surfaces to concave ones; it stipulates the concretion, as well as sintering of solid particles that are in direct contact with each other. The transfer of matter may proceed via different mechanisms, which may include bulk diffusion of dispersed substance either through the dispersion medium (if dispersed substance is sufficiently soluble) or through dispersed phase itself, as well as the surface diffusion along the interface. The kinetics ofsintering process, involving all of these mechanisms, was described in detail by Ya.E. Geguzin [66].
References o
~
~
~
~
,
Peskov, N.P., Physico-Chemical Principles of Colloid Science, 2"%d., Goskhimizdat, Moscow- Leningrad, 1934 (in Russian) Rehbinder, P.A., "Selected Works", vol.1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Mysels, K.J., Shinoda, K., and Frankel, S., Soap Films. Studies of Their Thinning, Pergamon Press, London, 1959 Derjaguin, B.V., Churaev, N.V., and Muller, V.M., Surface Forces, Consultants Bureau, New York, 1987 Exerowa, D., Kruglyakov, P.M., Foam and Foam Films, in "Studies in Interface Science", vol.5, D. M6bius, R. Miller (Editors), Elsevier, Amsterdam, 1998 Kralchevsky, P.A., and Nagayama, K., Particles at Fluid Interfaces and Membranes, in "Studies in Interface Science", vol. 10, D. M6bius, R. Miller (Editors), Elsevier, Amsterdam, 2001
578 0
~
Ivanov, I,B., Dimitrov, D.S., Thin Film Drainage, in "Thin Liquid Films", I.B. Ivanov (Editor), Dekker, New York, 1988 Derjaguin, B.V., and Obukhov, E., Acta Physicochim. URSS, 5 (1936) 1
0
10 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
Derjaguin, B.V., and Kusakov, M.M., Izv. Akad. Nauk SSSR, Ser. Khim., 5 (1936) 741 Krotov, V.V., Rusanov, A.I., Physico-Chemical Hydrodynamics of Capillary Systems, Imperial College Press, London, 1999 Akers, R.J., Foams, Academic Press, London, 1975 Scheludko, A., Radoev, B., Kolaroc, T., Trans. Faraday Soc., 64 (1968) 2213 Casimir, H.R., Polder, D., Phys. Rev., 73 (1948) 360 Dzyaloshinskii, I.E., Lifshitz, E.M., Pitaevskii, L.P., Zh. Eksp. Teor. Fiz., 37 (1959) 229 Derjaguin, B.V., Theory of Stability of Colloids and Thin Films, Consultants Bureau, New York, 1989 Derj aguin, B.V., Abrikossova, I.I., Zh. Eksp. Teor. Fiz., 21 (1951 ) 945 Derjaguin, B.V., Abrikossova, I.I., Zh. Eksp. Teor. Fiz., 30 (1956) 993 Derjaguin, B.V., Abrikossova, I.I., Zh. Eksp. Teor. Fiz., 31 (1956) 3 Derjaguin, B.V., Abrikossova, I.I., Disc. Faraday Soc., 18 (1954) 182 Tabor, D., and Wintertorn, R.H.S., Proc. Royal Soc. London, A312 (1969) 435 Peschel, A., and Adlfinger, K.H., Ber. Bunsenges. Phys. Chem., 74 (1970) 351 Derjaguin, B.V., and Churaev, N.V., J. Colloid Interface Sci., 49 (1974) 249 Rabinovich, Ya. I., Colloid J., 39 (1074) 1094 Israelachvili, J.N., Intermolecular and Surface Forces, 2 "d ed., Academic Press, London, 1991 Israelachvili, J.N., and Ninham, B.M., J. Colloid Interface Sci., 58 (1977) 14 Pashley, R., and Israelachvili, J., Colloids Surf., 2 (1981) 169 Claesson, P., Ederth, T., Bergeron, V., Rutland, M.W., Adv. Colloid Interface Sci., 67 (1996) 119 Israelachvili, J., and Pashley, R., Nature, 300 (1982) 341 Churaev, N.V., Colloid J., 46 (1984) 302 Christenson, H.K., Claesson, P.M., Adv. Colloid Interface Sci., 91 (2001) 391
579 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.
44. 45. 46. 47. 48. 49. 50. 51.
52. 53.
Prieve, D.C., Bike, S.G., and Frej, N.A., Disc. Faraday Soc., 90 (1990) 209 van de Ven, T.G.M., Warszynski, P., Wu, X., and Dabros, T., Langmuir, 10 (1994) 3046 Nardo, D., and John, N., Nanoscale Characterization of Surfaces and Interfaces, Weinheim, New York, 1994 Houston, J.E., and Michalske, T.A., Nature, 356 (1992) 266 Shchukin, E.D., Izv. Akad. Nauk SSSR, Ser. Khim., 10 (1990) 2424 Steblin, V.N., Shchukin, E.D., Yaminsky, V.V., Yaminsky, I.V., Colloid J., 53 (1991) 684 Yamisky, V.V., Steblin, V.N., Shchukin, E.D., Pure Appl. Chem., 64 (1992) 1725 Shchukin, E.D., Yusupov, R.K., Amel,ina, E.A., and Rehbinder, P.A., Colloid J., 31 (1969) 913 Shchukin, E.D., and Amelina, E.A., Adv. Colloid Interface Sci., 11 (1979) 235 Shchukin, E.D., in "Fine Particles Science and Technology", E. Pelizzetti (Editor), Kluwer, Dordrecht, 1996 Shchukin, E.D., J. Colloid Interface Sci., in the press Edwards, D.A., Brenner, H., and Wasan, D.T., Interfacial Transport Processes and Rheology, Butterworth-Heineman, Oxford, 1991 Dukhin, S.S., Kretzschmar, G., Miller, R., Dynamics of Adsorption at Liquid Interfaces, in "Studies in Interface Science", vol. 1, D. M6bius, R. Miller (Editors), Elsevier, Amsterdam, 1995 Verwey, E.J.W., and Overbeek, J.Th.G., Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948 Derj aguin, B.V., Landau, L., Acta Physicochem. URSS, 14 (1941) 633 Rehbinder, P.A., Wenstr6m, E.K., Zh. Phys. Khim., 1 (1930) 533 Wenstr6m, E.K., Rehbinder, P.A., Zh. Phys. Khim., 2 (1931) 754 Shchukin, E.D., Colloid J., 59(2) (1997) 270 Lankveld, J.M.G., Lyklema, J., J. Colloid Interface Sci., 41 (1972) 454 Lankveld, J.M.G., Lyklema, J., J. Colloid Interface Sci., 41 (1972) 466 Sato, T., Ruch, R., Stabilization of Colloidal Dispersions by Polymer Adsorption, in "Surfactant Science Series", vol.9, Dekker, New York, 1980 Smoluchowski, M., Z. Phys. Chem., 92 (1917) 129 Elimelech, M., Gregory, J., Jia, X., Williams, R., Particle Deposition and Aggregation, Butterworth-Heinemann, London, 1995
580 54. 55.
56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66.
Meakin, P., Adv. Colloid Interface Sci., 28 (1988) 249 Meakin, P., in "On Growth and Formation of Fractal Patterns in Physics", NATO ASI Series, Ser. E, H.E. Stanley, N. Ostrowsky (Editors), Martinus Nijhoff Publ., 1986 Fuchs, N., Z. Physik., 89 (1934) 736 van de Ven, T.G.M., Colloidal Hydrodynamics, Academic Press, London, 1989 Lifshitz, I.M., Slezov, V.V., Zh. Eksp. Teor. Fiz., 35 (1959) 331 Lifshitz, I.M., Slezov, V.V., J. Phys. Chem. Solids, 19 (1961) 35 Wagner, C., Ber. Bunsenges. Phys. Xchem.,, 65 (1961) 581 Taylor, P., Adv. Colloid Interface Sci., 75 (1998) 107 Kabal'nov, A.S., Pertsov, A.V., Shchukin, E.D., Colloid J., 46 (1984) 1108 Pertsov, A.V., Kabal'nov, A.S., Shchukin, E.D., Colloid J., 46 (1984) 1172 Kabal'nov, A.S., Pertsov, A.V., Aprosin, Yu. D., Shchukin, E.D., Colloid J., 47 (1985) 1048 Kabal'nov, A.S., Pertsov, A.V., Shchukin, E.D., Colloids Surf., 24 (1987) 19 Geguzin, Ya.Ye., Physics of Caking, Nauka, Moscow, 1984 (in Russian)
List of Symbols
Roman symbols
A A* aL
b r C
D D d ESE e
Hamaker constant complex Hamaker constant London constant intermolecular distance concentration speed of light diffusion coefficient fractal dimension particle diameter elasticity modulus elementary charge
581 F
force surface free energy free energy per unit area of two-sided film free energy of interaction in the film g acceleration of gravity height of dispersion medium with dispersed particles over the sediment h film thickness I intensity of light J collision frequency diffusional flux Jm, k Boltzmann constant 2r ~ H/(r "+ r H) k coagulation rate constant portion of interracial area where molecular forces are partially compensated interracial area S Ar number of particles n refractive index m,n number of particles in aggregate n number of molecules per unit volume na particle concentration in aggregated state critical coagulation concentration ncr nm, n n concentration of m-and n- dimensional aggregates np particle concentration in peptized state total number of aggregates of all sizes n2 pressure P capillary pressure Po R universal gas constant R distance between centers of particles Rg radius of gyration r, r ,r particle radius r~, r 2 principal curvature radii Sef effective area of particle contact
/i
.,.
582 attraction energy per unit area of a gap energy of particle cohesion absolute value of u b/1 height of the potential barrier ]"/max attraction energy between spheres Z/sph V volume molar volume Vm W = 1/a stability ratio x, y, z Cartesian coordinates Z coordination number mol
?2
Greek symbols (~ ~M
7 8 q 0 K
g H I/el I-Imo 1 7~
P P0 Pv (5 (If 03 E0
% q0d
collision efficiency polarizability In (na/np) full film tension small distance electrokinetic potential fluid viscosity contact angle inverse thickness of electrical double layer (Debye-Htickel parameter) wavelength chemical potential disjoining pressure electrostatic component of disjoining pressure molecular component of disjoining pressure 3.14159... density of particles density of medium charge density specific surface free energy, surface tension film tension circular frequency dielectric constant electrical constant surface potential Stern layer potential
583 VIII. STRUCTURE, STABILITY AND DEGRADATION OF VARIOUS LYOPHOBIC DISPERSE SYSTEMS
This Chapter describes preparation, structure, and properties of different colloidal systems. A lot of attention will be devoted to the connection between particular properties of disperse systems (and possible ways that can be used to monitor colloid stability) and the a g g r e g a t e states of both the dispersed matter and dispersion medium. Among all disperse systems the aerosols, in which the dispersion medium is a gas, are unique. These systems are principally lyophobic, and their stabilization, e.g. by introduced surfactants, is ineffective. Aerosols reveal also some specific electric properties. In systems with liquid dispersion medium, i.e. in foams, emulsions, sols and suspensions, there is a broad variety of means to control colloid stability. In these systems the nature of colloid stability depends to a great extent on the aggregate state of dispersed phase. Similar to aerosols, foams are lyophobic, but in contrast to them can be effectively stabilized by surfactants. Properties of emulsions, and, to some extent, those of sols may be quite close to the properties of thermodynamically stable lyophilic colloidal systems. In such systems a high degree of stability may be achieved with the help of surfactants. In colloidal systems where the dispersion medium is solid all processes aimed at changing the degree of dispersion are retarded due to high viscosity of dispersion medium and small diffusion coefficients of components.
584 VIII.1. Aerosols
Disperse systems with gaseous dispersion medium, regardless of the aggregate state of dispersed phase, are referred to as aerosols. The systems in which the dispersed phase is liquid are referred to as fogs; those in which it is solid are called smokes in the case of finely dispersed matter or dusts and
powders in the case of coarsely dispersed matter. Aerosols in which the liquid phase is present due to vapor condensation at the surface of solid particles along with the solid phase, are commonly referred as "smogs". Aerosols of this type are usually present in the atmosphere of large industrialized cities. Aerosols play an important role in nature and have various applications. They are the "key objects" in all meteorological phenomena, including thunderstorms. In agriculture aerosols are widely used for overhead irrigation and application of pesticides. In technology one encounters aerosols in the
cleaning of air and various gaseous mixtures before these are
introduced into reactors. Aerosols are also used for entrapment of various valuable materials from gas waste stream. In medical applications some drugs are delivered in the form of aerosols. In everyday life one frequently encounters aerosols in a form of various sprays. Prevention from being exposed to aerosols that form during mining, combustion of fuels etc, is an important aspect of health and environmental protection. For all of the above mentioned reasons a lot of attention is drawn to the investigation of properties and stability of aerosols. Studies of aerosols represent a well developed and important area of modern colloid and surface science [1-4]. Gaseous dispersion medium introduces a number of important features
585 into the properties of aerosols. Among the latter principal hydrophobicity and unavailability of effective means of stabilization are the most important ones. Degradation time of aerosols is solely determined by the rate of sedimentation or aggregation, i.e., aerosols with noticeable content of the dispersed phase reveal kinetic stability only. Another feature of aerosols is related to the fact that the particle size of dispersed matter, r, is comparable to the mean free path of molecules in a gas, A M. The choice of a particular method used to adequately describe the particle motion depends on the ratio between r and A M, which is given by the
Knudsen number, Kn=AM/2r [1-4]. When Kn 10-2, one can apply the relationships established in the molecular kinetic theory of gases, according to which the resistance to the particle motion is proportional to the cross-sectional area of particles, and the velocity of their motion, 1), due to the applied force, F, is given by F
Ar
F
m
2mM UM
2~r2mMUM NAc
In the above expression mMis the mass of a molecule of gas; ~M=(8kT/71;mM)1/2 is the average velocity of gas molecules; Ar is the mean free path of particle, defined
as
A r -
1/~r2NAc, where c is the concentration of gas. Retardation in
586 particle motion occurs because the rate of collision of molecules with the front side of particles is higher than with the rear side. For a majority of most important aerosol systems the Knudsen number has intermediate values, i.e 10-2
+ 1.26Kn) -
F (l+0.63AM]. 6rcrlr r
For r ~)A Mthe above equation becomes identical to the Stokes law, while for r ~ A M it yields a quadratic type dependence of the friction force (and consequently of the velocity) on the particle radius, i.e. ~) ~ F / r 2. Many specific properties of aerosols are related to such features of dispersion medium, i.e., of air, as its low viscosity and low electric conductivity. Lyophobicity of aerosols and high diffusion coefficients of particles in gaseous phase are the factors responsible for high rates of coagulation and isothermal mass transfer, resulting in the system being unstable towards aggregation. Low viscosity of dispersion medium promotes rapid settling of particles, and hence aerosol system fractures at much smaller particle or aggregate sizes than lyosols. As a result, the concentration and degree of dispersion in aerosols experience a rapid decrease. In real aerosols the concentration of dispersed phase hardly exceeds 106 - 108 particles per cm 3, which is substantially lower than the concentration of lyosols, where it reaches 10 ~5particles per cm 3. Particle size in most aerosol systems falls into the range between 10.5 and 10.3 cm; the larger particles typically settle, while
587 the smaller ones disappear from the system due to coagulation. In contrast to sols in electrolyte solutions, the charge on particles in aerosol is a random value determined by chaotic collisions between particles and ions in gases. The work of charging of a particle, represented by a spherical capacitor with capacitance, C, proportional to particle radius,
r,
is
given by q w1-
2
q
2
q
2
_
8=egor
8=gor
where e 0 = 8.85x 10 -~2F/m is the electric constant in SI units. Since dielectric constant of air (e= 1) is much smaller than that of water (e~80), charging particles to the same charge in aerosols requires more work than in hydrosols. Because of this, an average charge on particles in aerosols is weaker and is subject to higher fluctuations than in hydrosols. According to the theory of fluctuations, the average charge is given by the expression: 2
q
kT
- d 2 W e l / d q 2 - 4~gorkT.
For particles with radius r = 10 .6 m at T =300K, and kT~4.2xl0 -2~ J, one obtains
-
q2 z 4.7 x 10 . 3 7 (Coulomb)2 '
1/2
7 x 10-~9Coulomb.
It is worth reminding one here that the elementary charge, e equals to 1.6x 10 -19 Coulomb, i.e., on average the charge of such particle corresponds to only a
588 few (around 4) elementary charges. By observing the electroforetic motion of oil droplets (along with their Brownian motion) during experiments on electrophoresis of oil fog, Millikan showed that the charge on particles was always a multiple of 1.6x10 -~9 Coulomb. These experiments proved the discrete nature of electric charge and allowed one to determine the value of elementary charge. The charge at the surface of aerosol particles is the source of high potentials that originate during settling of aerosols with particles that bear charges of the same sign. These high potentials cause thunderstorms and interference in radio signals. Settling of aerosol particles, e.g., of the fog droplets, with radius r, average charge q, and concentration n generates the electric current and the counter electric field with the strength E, (sedimentation potential). In agreement with eq. (V.28) the strength of electric field that originates during sedimentation in a medium with electric conductivity, X0, is given by
E =
2~~176
nmg(q/r)
-
3X oq + 8roe.2r(p 2n
( )2,
6=;L oq + nr q/r
where the surface potential, %, which for aerosol is equal to electrokinetic potential, (, is related to the particle surface charge as
e0- =
q 4=Sor
A.N. Frumkin showed that due to high dipole moment of water molecules electric potential at the surface of droplets in a fog, q)0, may be as
589 high as -250 mV. In agreement with the above expression for E, the settling of water droplets with r = 10.5 m and m = 5 x 10.9 g at n = 10 ~~particles per
m 3
(these parameters are characteristic of cumulus clouds), 11 ~ 1.7x 10.5 Pa s, )~0 =4 x 10-~4S m -~may generate electric fields with strength up to 105 V/m. Under non-steady state conditions, i.e., in the presence of convection, E may assume even higher values resulting in an electric discharge through the air in the form of lightning. Similarly to all other disperse systems, aerosols may be obtained (see Chapter IV) by both dispersion of macroscopic phases and condensation [5]. Aerosols generated as a result of macroscopic phase dispersion are, as a rule, characterized by a rather low degree of dispersion and are more polydispersed as compared to aerosols prepared by condensation. Dispersion methods of aerosol generation are used to prepare various important materials. These methods include the preparation of powders by grinding of solid materials, nebulizing liquid fuel ( in order to achieve a more efficient combustion), spraying pesticides and paints, etc. In nature dispersion processes lead to the formation of dust. Condensation is the main route leading to the formation of finely dispersed aerosols in nature and industry. The formation of cumulus (composed of water droplets) and cirrus (composed of ice crystals) clouds mainly starts with heterogeneous nucleation on fine dusts and microcrystals of salt. These microcrystals form when splashes of sea water are dried and raised to high layers of atmosphere by convection air streams. The formation of aerosols by condensation is substantially influenced by electric charge. Charge generation at the surface of aerosol particles, which
590 requires that the work of charging is consumed, may lead to a significant lowering of interfacial tension at the particle-medium interface. This is of great importance for the formation of nuclei (see Chapter IV). Surface tension lowering of particle with radius r bearing the charge q may be determined by integrating the Lippmann equation (III.21)" dG dqoo
where 9~ -
4nr
2
=Ps'
is the surface charge density, and % is the potential of a
droplet. Integration of this equation yields q2 O" 0 - - 0 "
-
-
32n2r3~ o
i.e., the lowering of (y at the droplet- vapor interface upon droplet charging is proportional to the second power of charge. The equilibrium vapor pressure as a function of particle size in aerosol is given by Kelvin (Thomson) equation, which for particles bearing the charge q can be written as
p ( r ) - Po
exp
r
2 4 16~on r
"
The above relationship can be represented by a curve with a maximum (Fig. VIII-l); for % ~70 mJ
m 2
and the charge, q, equal to elementary charge, the
position of maximum corresponds to particle radius of 1 - 2 nm.
591
P
P
P0
.
0 r0
.
.
rl
.
.
.
.
rm
.
r2
r
Fig. VIII-1. The equilibrium pressure, p, as a function of radius of charged aerosol particles
Formation of stable nuclei in the process of homogeneous nucleation (see Chapter IV) requires a fluctuation the work of which is determined by the degree of supersaturation in mother liquor. In the presence of charges (for instance of free ions) in the atmosphere of vapor, the pressure of which is greater than Pm, corresponding to the maximum in the p(r) curve, the formation of nuclei of radius rm does not require fluctuations: drops of liquid phase form as a result of condensation on ions, serving as condensation centers. The drops then spontaneously grow within the entire size region up to r - ~ . When p = p* < Pro, droplet nuclei of radius r~ are generated. Their further growth requires fluctuation. As a result of fluctuation nuclei reach the size r 2, and spontaneously grow further. The work of fluctuation in this case is significantly smaller than in the case of homogeneous formation of uncharged particles. Even at pressure p = P0 (i.e. in the absence of supersaturation) the formation of droplets with a radius r0 occurs. The ability of electric charges to decrease work required for the formation of new phase nuclei is utilized in such devices as the Wilson chamber and bubble chamber.
592 In the Wilson chamber intensive condensation of vapor is caused by the presence of ions, which are generated by charged particles passing through the chamber. The trajectories of these particles become visible due to the scattering of light, which allows one to identify the presence of elementary particles and to determine their origin. In the bubble chamber the "stretched" liquid (i.e. the liquid exposed to negative pressure) serves as the working body. The problem of controlling the stability of aerosols is of importance in various practical applications. In some cases, e.g. in the generation of smokescreens, one is interested in maintaining the stability of aerosol system, while in other instances it is important to either prevent the formation of aerosols or to ensure their fast degradation. For example, one needs to prevent the formation of fine airborne dusts that are always present during grinding and milling of solid materials. Such dusts are often harmful and constitute serious threats to ones health, as they penetrate human lungs and cause lung diseases (e.g. silicosis and anthracosis). Many organic substances, when present in the state of fine aerosols, are explosive: upon ignition, the burning zone covers a very large surface area and results in a sharp volume increase. This relates to such common substances as flour, coal dust, sugar, dust-like polymer waste products, etc. High amounts of technogenic aerosols forming upon combustion, when discharged into the atmosphere, may influence the conditions of cloud formation and affect the climate on the Planet. An increase in the amount of anthropogenic aerosols, constituting about 20% of all aerosols present in nature, also poses an ecological threat: these aerosols tend to decrease the
593 transparency of Earth's atmosphere. There are data indicating that large volcano eruptions lead to an increase in the amount of aerosols present in atmosphere, which in turn may affect the climate. Catastrophic eruption of the Krakatau volcano in Indonesia in 1883 resulted in a discharge into the atmosphere of ~18 km 3 of ash consisting of solid particles of different sizes. The finest ashes remained suspended for several years. Most of the methods used for aerosol degradation are based on intensifying the processes of coagulation, coalescence, adhesion of aerosol particles on different surfaces (on solid walls of filters, or water drops, as in artificial irrigation), and sedimentation (by changing the velocity and direction of aerosol streams during the inertial settling e.g. in so called cyclones). In a confined volume, e.g. in the pore of diameter d, the degradation of aerosol may occur due to sedimentational transport of larger particles and diffusional transport of smaller ones to the pore walls and subsequent adhesion on them. The time of aerosol degradation due to sedimentation is where u =
mg
tse d ~
d/u,
is the transport velocity of particle with a radius r in a
6~rlr medium with viscosity r I. The degradation time due to diffusion is tdir
~--
d 2/D,
where D = kT/6rtrlr is particle diffusion coefficient. Competition between these two processes results in higher stability of particles of medium size range, i.e. between 10 .5 and 10.4 cm in filters with pore radius of 10.3 to 10.2 cm. For these particles tse d ~/dif, i.e. mgd = 10 to 15 kT. Entrapment of such particles is a rather difficult task. One may reach higher levels of entrapment effectiveness by using filters with extremely tortuous pores (so-called
594 Petryanov' s filters). One may effectively control the stability of atmospheric aerosols by spraying concentrated solutions of hygroscopic substances, such as calcium chloride, or solid substances, such as silver iodide and solid carbon dioxide. These substances cause condensation of water vapor (or the formation of small ice crystals in supercooled clouds), and result in precipitation. Analogous means can be used to dissipate fog. The so-called Cottrell method, based on the use of electric field, is commonly practiced in industry for purification of gaseous mixtures. Aerosol particles, when passed through Cottrell's electrostatic smoke precipitator, acquire electric charge. The charging of particles due to adsorption of ions (primarily of negatively charged ones), originated from air ionization by corona discharge, ensures electrophoresis and particle precipitation on the anode.
An important application of aerosols that deserves addressing here is their use for synthesis of m o n o d i s p e r s e d colloidal particles [6-12].
Main principles behind the
preparation of uniform particles by aerosols are either physical changes in the state of matter forming aerosol droplets or chemical reactions between the dispersed droplets of one reactant and the vaporized second reactant. Uniform particles can be prepared by nebulizing various solutions and then evaporating water. This process yields solid particles dispersed in the carrier gas [8]. Alternatively, one can obtain monodispersed particles of such such substances as NaC1, AgC1, V205 by vaporization and subsequent condensation of the corresponding solids [6]. Powders of metal oxides, sulfides, chlorides, etc., containing uniform particles can be prepared by chemical reactions in aerosols. For example, uniform powders of titanium dioxide are obtained when the aerosol droplets of titanium ethoxide, Ti(OC2Hs)4 or titanium
595 isopropoxide, Ti(i-OC3H7)4, are interacted with water vapor [9], namely Ti(OCzHs)4 +
2H20 =
YiO2 + 4CzHsOH.
When aerosol droplets of organic monomers are exposed to the vapor containing polymerization initiator, the uniform particles of polymer latexes, such as polystyrene and polydivinylbenzene, are obtained [6]. The size distribution of the reactant aerosol droplets is directly related to the particle size distribution in the resulting powders. Consequently, the generation of aerosols with droplets of narrow size distribution is the critical step in the preparation of monodispersed colloids by the aerosol method. Such aerosols, suitable for the preparation of uniform powders, can be generated either by mechanical dispersion of liquid stream or by the condensation of vapors. A variety of aerosol generators suitable for the preparation of monodispersed particles were described in the literature [6,7]. The oldest generator utilizing the vapor condensation method was designed by Sinclair and La Mer [ 10]. In their technique the gas carrying nuclei (prepared by heating various solid salts, such as NaC1, NaF, AgC1, in a stream of carrier gas) was brought into contact with a hot liquid which saturated the gas phase. When the gas saturated with liquid was then cooled down, condensation on the nuclei resulted in the formation of aerosol. Another generator utilizing the same principle was proposed by Nicolaon, Matijevid, et el [11,12], who investigated the formation of dibutyl phthalate aerosols. In their setup a film of heated dibutyl phthalate was flowing down along the walls of a vertical tube (this explains the name of this technique), through which the carrier gas containing NaC1 nuclei was passed. The aerosol was formed by the condensation of dibutyl phthalate on NaC1 nuclei as the nuclei-laden gas saturated with dibytyl phthalate vapor cooled upon leaving the tube. The "falling film" generator allowed to improve the stability of output during prolonged operation and reproducibility of aerosols, which were the two major problems encountered with the original La Mer's setup. Both LaMer's and the "falling film" aerosol generators yield only small quantities of products. Much larger amounts of aerosols can be produced by dispersing liquids with the help of various mechanical devices, e.g. rotating disks or ultrasonic nozzles [6]. These techniques, however, usually yield aerosols with broad distribution of droplet sizes and thus lead to polydisperse systems. The dispersion aerosol generators can, consequently, be used
596 when it is not essential for one to produce systems consisting of uniform particles. Generation of solid colloidal particles in aerosols has certain advantages over precipitation from homogeneous solutions described in Chapter IV. During precipitation from solutions it is usually impossible to predict a priori the shape of the resulting particles, while particles prepared by the aerosol methods are usually spherical because of the natural shape of liquid droplets dispersed in gas. Also, it was pointed out earlier (see Chapter IV) that in the case of particles of internally mixed composition, the molar ratio of constituents in the solid phase differs from that in solution [13], while in the case of aerosol technique the content of resulting solid particles is determined by the molar ratio of components in solution that is dispersed in the gas phase to form aerosol droplets.
V l l I . 2 . F o a m s and Foam Films 1
Foams are the dispersions of gases (of air in most cases) in liquid dispersion medium. They represent a typical lyophobic colloidal system. One distinguishes between dilute dispersions of gases in liquids, which due to their similarity with emulsions are commonly referred to as the gaseous emulsions, and "true" foams with the gas phase content of 70% by volume or higher. The ratio of the volume of foam to the volume of constituent liquid phase is often used as a characteristic value of foam concentration and referred to as thefoam
number. Due to sedimentational instability of majority of gaseous emulsions, bubbles that float atop (undergo reverse sedimentation) form a layer of concentrated foam in which further disintegration of system occurs. In foams the gas-filled areas are separated by thin films of dispersion medium. A characteristic "idealized" example of foam cells (Fig. VIII-2) is the
1
See also Volume 5 of the series [14]
597 pentagonal dodecahedron, a polyhedron with 12 faces, 30 edges and 20 vertices (nodes). These polyhedra, however, may not fill the space continuously, and the average number of films surrounding each cell in real foams is close to 14.
Channels
Fig. VIII-2. The structure of foam cell
The edges of foam cells are the Gibbs-Plateau borders (channels) filled with dispersion medium (see Chapter VII, 1). It was shown by Plateau that only three films may be joined by one border, and that the films must meet at 120 ~ The surface of Gibbs- Plateau border has a complex concave shape dictated by the condition requiring that the sum of two main curvatures remains constant. The capillary pressure under a concave surface is the reason for the lowered pressure in the Gibbs-Plateau border. In foams with high foam numbers the surface in Gibbs- Plateau borders is close to cylindrical, i.e. has a constant triangular-shaped crosssection with concave sides. The pressure in such cross-section is lowered in comparison with the pressure in foam cells by the amount of~/rc~v, where rc,~ is the curvature radius of the border surface (i.e., of the side of triangle).
598 Plateau showed that the vertices of the neighboring pentagonal dodecahedrons form junction corner in which four borders join. As a result, four vertices closest to the reference one form a tetrahedron. This arrangement is similar to that of atoms in the nearest coordination sphere in the diamond crystal lattice. The borders and nodes form a united branched network along which the transfer of dispersion medium, including drainage under gravity, may take place. In foams with foam numbers between 100 and 1000 (depending on the degree of dispersion in the system and film thickness) most of the dispersion medium is present in the borders and only little portion of it stays in the films. Liquid content in junction nodes is the highest in films with low foam numbers (close to 3). Real foams are typically polydisperse 2, which results in changes in the shape of foam cells. The Plateau rules (three films form a border; four borders meet at a node) remain valid in all cases. The degree of dispersion in foam may be characterized by its specific surface area. One, however, usually measures some mean values of foam cell geometrical parameters, such as the average number of cells per unit volume, n, or the mean equivalent radius,--r-, 3~
related to-ff via 4/37(?- n - 1. Film thickness, h, mean equivalent cell radius, 7 (or the number of cells per unit volume), the average foam number, K, and the height of foam column, Hm, are the main geometrical parameters that, to a certain approximation, characterize structure of foams. In a more detailed consideration, one must account for changes in the first three parameters along
One may also obtain a monodisperse foam by bubbling gas through a single capillary. This foam will become polydisperse upon further film rupture
2
599 the foam column height. Foams and isolated foam films are convenient objects for the studies of colloid stability as well as of the mechanism and kinetics of disintegration in lyophobic disperse systems. At the same time, foams are widely used in various areas of technology, e.g. in fire extinguishing, flotation, bakery (bread represents a solidified foam), manufacturing of heat and sound insulating materials (styrofoam, foam concrete, microporous rubber), etc. Foams are commonly made by dispersing air (or another gas in more rare cases) in a liquid containing a foaming surfactant (foamer). Sometimes foam-stabilizing additives (the surfactants that enhance the action of a foamer) are introduced into the system. One can disperse gas in a liquid by passing air through the layer of liquid or by operating stirrers in a fluid bulk. Foam generators of various designs are also broadly used. [14-20] In many such devices foam formation takes place on a wire gauze (Fig VIII-3). By controlling the flows of air and of foam-making solution one may obtain foam of desired foam numbers. To maintain the desired degree of dispersion, one may pass foam through a series of wire gauzes on which the dispersion of foam cells takes place. Such foam generators are capable of producing large amounts of foam in short periods of time, which makes them useful in fire extinguishing, in particular under the circumstances when one deals with fires caused by gasoline fuel or flammable organic liquids. Foams in which solidification of dispersion medium takes place (bread, solid foam materials, etc.), are usually prepared by condensation methods based on the gas evolution (CO2 in most cases) due to chemical
6OO reaction or biochemical process. The processes of f o a m b r e a k i n g
are accompanied by changes in
time in the parameters that characterize foams, and are related to a decrease in film thickness, film rupture, isothermal mass transfer resulting in the transfer of gas from small cells to larger ones (see Chapter VII). Foam disintegration may also occur due to syneresis, i.e. draining of the dispersion medium from Gibbs - Plateau borders under the influence of gravity.
Solution
~
~L
Air
o,o o l [ o Gauze
Gauze
Fig. VIII-3. Schematic drawing of a gauze-based foam generator
Changes in thickness of foam films with time were studied in the works by K.J. Mysels, J. Perrin, A. Sheludko, B.V. Deryaguin, H. Zonntag and others. [21-25] Such studies are typically carried out with individual films or microfilms using an experimental cell designed by K.J. Mysels and A. Sheludko et al. (Fig. VIII-4) [26-29]. The cell is first filled with foamer solution. Before the droplet appears at the bottom of the cell, the solution is sucked out with a micropump connected to a side outlet. As a result, meniscuses are formed
601 at the top and the bottom parts of the cell, as shown in Fig. VIII-4. As the liquid is removed, meniscuses approach each other. When meniscuses come into contact, a microscopic film forms in their centers. This film is surrounded by broad Gibbs - Plateau border, the pressure in which (see Chapter VIII, 1) is lowered in comparison with the atmospheric pressure by 2O/ro, where ro is the radius of a cell. Further removal of liquid leads to an expansion of film, narrowing of the border and increase in its curvature radius, which results in pressure being lowered further.
Fig. VIII-4. The cell for the experimental studies of individual films The drainage of fluid from films in general follows Reynold's equation (Chapter VII, eq. (VII. 25)), see Fig. VIII-5. This means that the presence of surfactant layers at film surfaces may cause their solidification due to Gibbs-Marangoni effect (see Chapter VII). At the same time, one may encounter deviations from Reynolds regime of film thinning. In some cases these deviations are caused by volume and surface diffusion of surfactant molecules. In other cases, particularly in the case of large films, one may observe more rapid thinning at peripheral regions of film, while its thicker portion may still remain present in the center (a so-called "dimple"). This central part may further merge with the Gibbs-Plateau border. 1
7
1
ho 0
t
Fig. VIII-5. Kinetics of thinning. Change in thickness, h, during liquid drainage from foams, plotted in 1 / h 2 - t coordinates
602 The decrease in film thickness may end in film rupture or lead to the establishment of metastable equilibrium state, in which disjoining pressure in the film equals (by absolute volume) the capillary pressure, determined by surface curvature of meniscus surrounding the film. The pressure can be changed by sucking liquid out from the Gibbs-Plateau border. The appearance of positive disjoining pressure in foam films may be stipulated by electrostatic component of disjoining pressure. In the region of relatively thick films (see Chapter VII), disjoining pressure in the film is determined by its molecular and electrostatic components, and, consequently, one can write that Pry + l-[el + l'-Imol - - 0
(VIII.l)
This corresponds to a decrease in the equilibrium thickness of films, which is sometimes observed, when an electrolyte that causes a decrease in I-Ie~is introduced into dispersion medium along with the surfactant. It was established that the dependence of film thicknesses on the electrolyte concentration is described by the DLVO theory. Moreover, by varying the capillary pressure in the Gibbs-Plateau border and measuring the film thicknesses, one can obtain, in agreement with eq. (VIII.l), the dependence of disjoining pressure on film thickness at constant electrolyte concentration. Figure VIII-6 shows the regions of disjoining pressure isotherms for "gray" (curve 1), common black (curve 2), and Newtonian black (curve 3) films, established by this method by Kh. Khristov, D. Exerowa, and P. Kruglyakov [30,14]. This figure indicates high compressibility (decrease in thickness with increasing disjoining pressure) of
"grey" films, low compressibility of common black films, and
essentially complete incompressibility of Newtonian black films [14]. 1-I, bl/m z l0 s 104 10 3
10 z 10 0
x 20
~ 40
60 h, nm
Fig. VIII-6. The disjoining pressure isotherms, I-l(h), for films obtained from 10 -3 mol d m -3 solutions of sodium dodecylsulfate with added NaC1 in the amounts of 1 - 10 -3 ; 2 - 0.1 ; 3 0.4 mol d m -3 (After Khristov, et al., [30])
603 In his studies Sheludko established that lower values (hundredths of mN/m) of excessive pressure (see Chapter VII, 1), Act, were typical for the common black films, as compared to the Newtonian black films (tenths of mN/m) [27,31,32]. The thickness of common film exceeds 7 nm (Fig. VIII-7). One can observe reversible transitions between these two types of films upon changes in the capillary pressure. The relationship between the type of films and the stability (i.e. foam life time, t~) of foam formed from a solution of the same composition was established: stable foams form at surfactant concentrations higher than the concentration Cb~,corresponding to the formation of black films (Fig. VIII-8). h~ a m
40-
1
302010I
1
,I
I
2
,
-log CN,c~
Fig. VIII-7. The thickness of primary (curve 1) and Newtonian (curve 2) black films as a function of electrolyte concentration
Cbl
Csurfactant
Fig. VIII-8. Film lifetime, tf, as a function of surfactant concentration
Changes with time in the degree of d i s p e r s i o n of the foam may be related to both the process of isothermal transfer of gas through the films and
film rupture.
Measurements of the degree of dispersion and its change with time are usually performed by counting the number of cells (shown in micrographs) in contact with the wall of the vessel containing foam. Isothermal transfer of gas from smaller cells with higher air pressure to the larger ones with lower air pressure is of primary importance in fine polydispersed foams, i.e. at the initial stages of foam breaking. Since in foams that are close to the meta-stable equilibrium, film thickness does not change with time, the kinetics of growth of foam cell due to isothermal transfer of gas is described by eq. (VII.36), as was outlined by de Vries [33]. In foams the rupture of film is a random process; the probability of film rupture as well as of merging of the neighboring cells is proportional to the number of films, and,
604 consequently to the number of cells, n, at a given time:
dn
kn
dt
t~
where n is the number of cells per foam unit volume; t~ is the average lifetime of a single foam cell; the coefficient, k, is determined by the average number of films per one cell (k = 6 - 7). If tl is independent of film size, integration of the above equation yields
n - n o exp (-t / t f). In the above expression no is the starting number of cells per unit volume of foam; tr = t~/k is the time during which the number of cells decreases by a factor of e. Change with time ofthe fo am n um ber, K, is related to gravity drainage (syneresis) of dispersion medium along the network of Gibbs-Plateau borders. The drainage is resisted by the capillary pressure in Gibbs-Plateau borders. Consequently, hydrostatic equilibrium corresponding to the condition of 9gz + po = const, may be established, i.e.
-gg-
dpc'-d rd ( c - u~ / r- v - -cY, dz dz rcurv rc2ur~ dz
(VIII.2)
where rcu~vis the curvature radius of Gibbs-Plateau borders and z is vertical coordinate [34]. In the state of hydrostatic equilibrium, corresponding to foams with high foam numbers, the curvature radius of Gibbs-Plateau borders decreases with height, z, in agreement with eq. (VIII.2). Foam number, thus, increases with height. Under the equilibrium conditions a froth column of a given height, Hm, with a given cell size distribution, may contain only some particular amount of liquid phase, corresponding to an equilibrium mean foam number. In finely dispersed foams with high foam numbers, the initial foam number may exceed this mean value. Not only there is no drainage of liquid from such foams during initial periods of time (Fig, VIII-9, curve 1), but these "dry" foams are capable of sucking in liquid, similarly to paper towels. Further decrease in the degree of dispersion leads to a decrease in the amount of liquid that the foam may contain, which promotes drainage and, consequently, results in a foams with higher foam numbers. In this case an increase in foam number with time
605 depends solely on the decrease of the degree of dispersion: film rupture results in a lower number of borders per unit volume of foam, while the cross-sectional area of borders (at a given height) remains unchanged. In foams with low foam numbers fast gravity drainage of dispersion medium and increase in foam number (Fig. VIII-9, curve 2) occur particularly during
initial periods of time. These processes may continue until the state close to
hydrostatic equilibrium is reached. Further change in foam number is similar to that occurring in foams with high foam number. A convenient method of determining foam number is based on the measurements of foam electric conductance, which is inversely proportional to the foam number. log K
1
0
t
Fig. VIII-9. Change in the foam number, K, as a function of time for foams with high (curve 1) and low (curve 2) foam numbers The change with time ofthe height of froth column is caused by an increased rate of rupturing of upper foam films that are in contact with outer environment. This process may be related to the rate at which evaporation of dispersion medium occurs, and hence the rate of froth column breaking to a significant extent depends on the humidity of the surrounding atmosphere. An accelerated breaking of the foam layer may also occur if foam is placed on the top of liquid hydrocarbon. In this case breaking occurs in the zone of contact, where cavities are formed, while the height of froth layer may still remain unchanged for some time. Foam breaking is especially fast when foams are brought in contact with polar organic liquids. This makes difficult the use of foams in extinguishing of fires caused by burning polar solvents.
606 Depending on the area of application, different levels of foam stability are desired. For instance, in flotation, when large quantities of air are bubbled through a layer of aqueous phase carrying ore, the formation of rich foams with high stability is undesirable: it makes further extraction of valuable minerals from foam difficult. In this case one employs surfactants belonging to the first group (according to Rehbinder's classification, Chapter II), which are weak foamers. For these surfactants the lifetime of individual bubbles does not exceed tens of seconds. A typical examples of such weak foaming agents are medium chain length (C~0- C~2)alcohols and products obtained from wood during the pulping process, e.g. pine oil. The foam containing floating particles (the so-called three-phase foam) is more stable than the foam containing no solid particles, and hence a rather thin layer of"cream" enriched with valuable mineral is formed in the flotation apparatus at the surface of working fluid. One obtains a concentrate of the desired mineral by periodical removal of this layer of "cream" and foam breaking. Highly stable foams stabilized by surfactants belonging to the third and fourth groups are used in fire fighting, in particular to extinguish flames of burning oil and liquid fuels. In this application the essential characteristics of foams are the rate of their spreading and their isolating ability, i.e. the time during which foams can prevent the exit of flammable vapors. For the preparation of such highly stable foams, one uses complex formulations, which in addition to the main foaming agent also include other surfactants serving as stabilizers. Fluorinated surfactants are especially effective in stabilization of foams. Their use is however limited due to high cost and poor biodegradability.
607 In many industrial applications it is important to either break the foam or completely
prevent its formation. For instance, in microbiological
production excessive foaming causes difficulties in successfully carrying out the manufacturing process. Another area where foam formation causes problems is papermaking. Foam causes uneven drainage of pulp and results in poor sheet properties. Excessive foam formation also prevents normal functioning of laundry machines; for this reason laundry detergent formulations contain large amounts ofnonionic surfactants, which are weaker foamers than alkylsulfates. Foam killing may be achieved by treatment with overheated vapor, by ultrasound or by using special substances, referred to as
defoamers [35,36].
According to Rehbinder, defoamers are the substances with higher surface activity than foaming agents, but without structure-forming ability. Because of higher surface activity, they displace foaming agents from the surface of bubbles but at the same time they are unable to effectively stabilize the foam. Common defoamers include low alcohols (C5 - C8) and silicon-organic compounds, which are especially effective in a complete prevention of foam formation.
VIII.3. Emulsions and Emulsion Films
Dispersions of liquids in liquid dispersion media, referred to as emulsions, are in general similar to foams but reveal some important distinct features. Stabilization of foams with surfactants does not affect lyophobic nature of foams, while emulsions may reveal properties that make them
608 similar to lyophilic thermodynamically stable colloidal systems, and it is sometimes difficult, if not impossible for one to clearly draw a borderline between lyophilic and lyophobic emulsions. This constitutes the most significant difference between foams and emulsions. Lyopilic critical emulsions and other similar systems were discussed in Chapter VII,2, so here we will focus on the discussion of lyophobic emulsions. Another important feature of emulsions is their ability to form systems of two types: direct, in which phase with higher polarity (usually water), forms the dispersion medium, and inverse, in which more polar liquid forms the dispersed phase 3. Under certain conditions emulsion inversion may occur. Emulsion inversion is the process of conversion of emulsion of one type into an emulsion of the other type upon addition of some chemical substances or change in conditions. One can distinguish between emulsions of different kinds by performing the conductivity measurements (the electric conductivity of aqueous phase is higher than that of dispersion medium in the inverse emulsions by several orders of magnitude), by observing the ability of an emulsion to mix with polar or non-polar solvents or to dissolve polar and nonpolar dyes, or to wet non-polar surfaces. One distinguishes between dilute emulsions, in which dispersed phase occupies a few percent by volume, and all others, including concentrated (and
highly concentrated) ones, in which dispersed phase occupies 70% by volume or more. The structure and properties of the latter are close to those of foams,
3 One also distinguishes the so-called "multiple emulsions" in which the dispersion medium is partially dispersed in the droplets of dispersed phase (see Chapter VII) [36,37]
609 for which reason such emulsions are referred to as the spumoidal, or foam-like emulsions. The investigation of emulsion stability [36,38,39-41] allows one to understand how the structure of surfactant molecules influences their ability to stabilize direct and inverse emulsions. Isolated films of inverse emulsions (hydrocarbon films in aqueous media stabilized by surfactant adsorption layers of various nature) serve as model systems for understanding structure and action mechanism of different cellar membranes. Crude oil recovered by collector flooding is the most important representative of natural inverse emulsions, containing up to 50 - 60 % of highly saline water and strongly stabilized by natural surfactants and resins. Disintegration of this emulsion is the first and the most difficult step in the processes of oil recovery and treatment. Emulsions are widely used in various areas of chemical industry: in the processes of chemical treatment involving the use of cooling and lubricating fluids, emulsion polymerization, the production and treatment of food items (milk, butter, margarine, etc.) and drugs. Lyophobic
emulsions
are
generally
obtained
by
dispersion
(emulsification) of one liquid in another in the presence of surfactants. Surfactants used in this application are referred to as emulsifiers; these are typically the surfactants belonging to the third and fourth groups (see Chapter II). Only a few types of usually dilute emulsions can be formed by condensation. These include an oil emulsions formed in steam engines. Emulsification is performed with the help of devices that generate ultrasound, vibration, use high shear gradient (the so-called colloidal mills), collision of two thin jets of liquids, etc.
610 In some cases under the conditions similar to those corresponding to the formation of lyophilic colloidal systems, a spontaneous formation of emulsions, the so-called self-emulsification, may take place. This is possible e.g. when two substances, each of which is soluble in one of the contacting phases, react at the interface to form a highly surface active compound. The adsorption of the formed substance under such highly non-equilibrium conditions may lead to a sharp decrease in the surface tension and spontaneous dispersion (see, Chapter III, 3), as was shown by A.A. Zhukhovitsky [42,43]. After the surface active substance has formed, its adsorption decreases as the system reaches equilibrium conditions. The surface tension may then again rise above the critical value, %. Similar process of emulsification, which is an effective method for preparation of stable emulsions, may take place if a surfactant soluble in both dispersion medium and dispersed liquid is present. If solution of such a surfactant in the dispersion medium is intensively mixed with pure dispersion medium, the transfer of surfactant across the low surface tension interface occurs (Fig. VIII- 10). This causes "turbulization" of interface
(
L2
0 )
eol Q
ooi e GO
Fig. VIII-10. The formation of microemulsion and its stabilizing action
611 and results in formation of a large number of tiny microemulsion droplets along with larger emulsion drops. Such tiny microemulsion droplets may stabilize the system. A type of emulsion formed by mechanical emulsification is largely determined by volume ratio of liquids: the liquid present in a much greater quantity usually forms a continuous phase, i.e. dispersion medium. In the case of approximately equal volumes of two liquids both direct and inverse emulsions are formed; among these two the emulsion that survives is the one with higher stability against droplet coalescence and further phase separation. The stability relationship between direct and inverse emulsions depends in this case on the nature of stabilizer, which, consequently, determines the "final" type of emulsion that forms. The ability of emulsifier to stabilize emulsions of particular type is governed by the structure of surfactant molecules and interaction of these molecules with polar and non-polar media [39,44]. The so-called "wedge theory" was one of the first attempts to describe on a qualitative level the selectivity of surfactants towards stabilization of emulsions of different kinds. According to this theory, the direct oil-in-water emulsions are stabilized by surfactants that consist of highly hydrated (large) polar head and moderately developed hydrophobic part, such as in sodium oleate. Inverse emulsions are stabilized by surfactant molecules with weakly hydrated (small) polar group and strongly developed, preferably branched, hydrocarbon tail containing 2 or 3 hydrocarbon chains, such as in soaps of polyvalent metals (for instance, in calcium oleate). The role that the ratio between geometric sizes of polar head and hydrocarbon chains plays in controlling the ability of surfactants to
612 stabilize emulsions of different types is by all means indisputable, but at the same time it is obvious that this "geometrical" scheme is rather oversimplified [45]. The role of the interaction energy between surfactant molecules and liquids in the stabilization of emulsions is reflected in so-called Bancroft's rule [36,46]. This rule states that in the emulsification, the liquid in which the
emulsifying agent is more soluble becomes the dispersion medium. Thus, water soluble surfactants stabilize direct oil-in-water emulsions, while oil soluble surfactants stabilize inverse water-in-oil emulsions. Among various ways used to describe the ability of surfactants to stabilize direct and inverse emulsions, the semi-empirical characteristic of hydrophile - lipophile balance (HLB) of surfactant molecules is the one that
is most commonly used [46,47]. Various attempts to use a single HLB number to describe the extent to which polar and non-polar groups of surfactant molecules are developed and to relate the properties of surfactant molecules to the properties of emulsions stabilized by these compounds were made. Parameters that were used include the number of ethylene oxide segments in molecules of nonionic surfactants, hydration heat of polar groups, chromatographic characteristics of surfactant molecules, etc. The HLB numbers for surfactant molecules may, e.g. (according to Davies4), be determined by summing additively the empirically established increments -"group numbers", Bi, of all groups present in the
The HLB concept was originally introduced in 1949 by Griffin specifically for nonionic surfactants. It was later extended by Davies who introduced a scheme to assign HLB group numbers to chemical groups composing a surfactant. For details see [46] 4
613 molecule, i.e. HLB- Z
B~ + 7 . i
Group numbers for some groups are summarized in Table VIII. 1. The HLB numbers of some surfactants estimated by the addition of the group numbers are given in Table VIII.2.
TABLE VIII. 1. Group numbers of some groups [1,48] Group
Group number
Group
Group number
-SO3Na
38.7
-OH
1.9
-COOK
21.1
-O-
1.3
-COONa
19.1
-(C2H40 )-
0.33
-N
9.4
-(C3H60 )-
-0.15
-COOH
2.1
=CH-, -CH2-, -CH 3
-0.475
TABLE VIII.2. HLB numbers of some surfactants estimated from the corresponding group numbers Surfactant
HLB
Sodium dodecylsulfate
40
Potassium oleate
20
Sodium oleate
18
C,8H37N (C2H4OH)(C2H4OC2H4OH)
10
Butanol
7.0
Glycerine monostearate
3.8
Oleic acid
1.0
614 High HLB numbers are characteristic of hydrophilic surfactants which stabilize direct emulsions; the highest numbers correspond to micelle-forming surfactants. Oppositely, low HLB numbers are typical for oleophilic surfactants which act as stabilizers of inverse emulsions. In agreement with the data presented in Tables VIII. 1 and VIII.2, large variation in HLB numbers may be achieved with pluronics (see Chapter II,3), which are the block co-polymers of ethylene oxide (group number of 0.33) and propylene oxide (group number of-0.15). Here it is worth pointing out that the HLB number primarily reflects the difference between hydrophobic nature of hydrocarbon chain and hydrophilic nature of polar head of surfactant molecules. For surfactants to reveal stabilizing properties, both of these factors have to be strongly expressed. It was shown by Rusanov et al. that the empirical HLB scale is supported by thermodynamics: the analysis of work of transfer of surfactant molecules from aqueous phase to hydrocarbon phase revealed that group numbers, B,, are proportional to the work of transfer of individual groups present in the molecule, and that the work of transfer of the entire molecule is given by summation of works of transfer of individual groups. The following discussion illustrates the reasons for the influence of HLB on the ability of surfactants to stabilize direct and inverse emulsions. Let us imaginarily divide the discontinuity surface between phases and, consequently, the interfacial tension of emulsion droplets, ~, into two parts" - ~ + %, where G~ is related to the region of contact between polar groups and water, and % corresponds to the region of contact between the hydrocarbon chains of surfactant molecules and non-polar phase. These two
615 components of interfacial tension may be assigned to two dividing surfaces which in an emulsion droplet have curvature radii of r~ and r2, respectively, while the outer surface has a greater area. In the case of an emulsion stabilized by surfactant molecules with high HLB numbers, o~ < o 2, and the surface of contact between polar groups and water (r~ > r2) becomes more developed, i.e. this surface becomes an outer one, corresponding to a direct emulsion (Fig. VIII- 11, a). Consequently, for surfactant molecules with low HLB number o2< < o~, and r 2 > rl, and the low-energy surface of contact between hydrocarbon chains and oil phase becomes more developed, leading to the formation of an inverse emulsion (Fig. VIII-11, b). ~
r
i J a
b
Fig. VIII-11. The stabilization of direct (a) and inverse (b) emulsions with surfactants
The phase that is similar to the stabilizing reagent tends to become a dispersion medium, as clearly seen in the example of emulsions stabilized by finely dispersed powders. Such stabilization is possible under the condition of a finite selective wetting of powder, i.e. at finite values of contact angle, 0 ~ < 0<180 ~ The powders are able to stabilize the phase that poorer wets the particles, while the liquid that is more similar to the powder becomes the dispersion medium. The reasons of such behavior are apparent from Fig. VIII12. If water droplets covered by hydrophobic powder, such as e.g. carbon black, are placed into hydrocarbon phase (oil), the layer of carbon black due
616 to its hydrophobic nature prevents a direct contact between water droplets upon their collision. Oppositely, hydrophilic powder, such as calcium carbonate, creates a protective shell around the oil phase and does not allow the droplets of oil dispersed in aqueous phase to come into direct contact with each other. The affinity of powder towards the outer phase can be characterized in terms of the selective wetting contact angle or by the ratio of heats of wetting of a given solid phase by two liquids (see Chapter III, 4). These quantities are analogous to the HLB numbers of surfactant molecules.
Fig. VIII-12. Stabilization of emulsions by powders Stabilization of emulsions by powders can be viewed as a simple example of structural- mechanical barrier, which is a strong factor of stabilization of colloid dispersions (see Chapter VIII, 5). The stabilization of relatively large droplets by microemulsions, which can be formed upon the transfer of surfactant molecules through the interface with low cy(Fig. VII- 10), is a phenomenon of similar nature. The surfactant adsorption layers, especially those of surface active polymers, are also capable of generating strong structural mechanical barrier at interfaces in emulsions. Many natural polymers, such as gelatin, proteins, saccharides and their derivatives, are all effective emulsifiers for direct emulsions. It was shown by Izmailova et al [4952]. that the gel-alike structured layer that is formed by these substances at the surface of droplets may completely prevent coalescence of emulsion drops.
617 This can be illustrated by demonstration experiment designed by Rehbinder and Wenstr6m [43,54].
If a solution of surfactant capable of forming
mechanically strong adsorption layer (e.g. saponin) is placed on top of the layer of mercury of 0.5-1 mm thickness, one may successfully cut the mercury pool in two portions with a glass rod. The mercury pool remains separated in two portions for fairly long time, despite of the existing compressive stresses of hydrostatic nature. The use of substances that due to their ability to form structuralmechanical barrier are capable of very strong stabilization of emulsions (and especially of concentrated ones), allows one to prepare many commercial emulsions that are used e.g. in emulsion polymerization [55], lubricantcooling liquids, etc. Such surfactants, and especially natural ones, are widely used in food and pharmaceutical applications [56-58]. These surfactants are, for instance, formed as a result of chemical reaction between dextrins and their derivatives (generated by thermal decomposition and partial oxidation of starch) and oils. The processes of emulsion b re akin g are similar in their nature and mechanisms to those of foam breaking [59]. Sedimentation of droplets takes place in dilute emulsions. Depending on the difference in densities of dispersed phase and dispersion medium, the sedimentation may proceed either in downward or upward directions. At lower density of dispersed phase the emulsion droplets float in an upward direction (the so-called creaming,typical for most direct emulsions [1]), while at higher density of dispersed phase sedimentation proceeds in a downward direction. In order to decrease sedimentation rate, one has to achieve the highest
618 possible degree of droplet dispersion, which can be achieved in homogenizers of various types. Homogenization is used to inhibit the separation of fat from milk. Due to rather low values of interfacial energy, ~, and small difference in the density of phases, emulsions are usually more stable towards sedimentation than foams. Sedimentation in emulsions may also be accompanied by the aggregation of emulsion droplets, referred to as flocculation. Flocculation leads to an increase in the effective size of settling aggregates, and as a result, leads to a higher sedimentation velocity. In dilute finely dispersed emulsions in which electrostatic stabilization is of primary importance, the major laws governing flocculation are close to those of coagulation ofhydrosols, and are given by the DLVO theory (see Chapter VII, 4). In such systems flocculation may be reversible. Flocculation is typical in inverse emulsions in which the long-range electrostatic repulsion is small due to a rather low charge on the emulsion droplets [41 ]. However, even if the droplets in inverse emulsion are charged, the presence of charge does not guarantee stability towards flocculation: the content of electrolyte and dielectric constant of dispersion medium are low, and hence the thickness of ionic atmosphere may be comparable to the distance between droplets. It is worth reminding one here that the position of potential barrier between interacting particles is determined by the balance between van der Waals attraction and electrostatic repulsion and corresponds to the distance of approximately twice the thickness of ionic atmosphere (See Chapter VII, 4). For this reason in sufficiently concentrated inverse emulsions the droplets are from the very beginning separated by distances at which the
619 potential energy barrier has been already overcome. Inverse emulsions may be stable towards flocculation in the presence of lyophilic structural-mechanical barrier that ensures a rather low energy of interaction between the droplets. in such cases an electrostatic repulsion may contribute to a weakening of attraction forces between particles. Stabilization of inverse emulsions against flocculation is of particular importance in the preparation of water-in-gasoline emulsions containing up to 30% of water. The introduction of emulsified water into gasoline accounts for a more complete combustion and higher octane number of fuels, and makes engine exhausts less harmful to the environment. C o a 1e s c e n c e is especially typical in concentrated emulsions. In such systems coalescence mainly determines the lifetime of emulsions prior to phase separation. In finely dispersed emulsions, both dilute and concentrated, the average size of drops may noticeably increase due to Ostwald ripening. At the same level of dispersion Ostwald ripening of emulsion droplets is a slower process than mass transfer of bubbles in foams [60]. This is due to a rather low interfacial energy, and consequently, low difference in chemical potentials of substance in droplets of different size, as well as due to a lower mutual solubility of liquids as compared to the solubility of gases in liquids. The methods of emulsion breaking (de-emulsification) are of importance in various areas of industry [39,61], especially in oil recovery" in crude petroleum the content of highly saline water may be as high as 50 - 60%. Oil-soluble surfactants present in petroleum (asphaltenes, porphyrines, etc) and those introduced during tertiary recovery form highly developed adsorption layers at the water surface, and thus create structural-mechanical
620 barrier, which ensures high stability of crude oil emulsions. At the same time the contact between emulsified water and parts of oil recovery and oil transport machinery should be possibly avoided, as the salts and hydrogen sulfide present in this water cause fast corrosion of equipment. Various methods of emulsion breaking include the use of effective surface active deemulsifying agents that displace stabilizing surfactants from water surface, chemical binding of the stabilizer, changes in the pH and electrolyte content of the medium, and the use of electric field, ultrasound and heat. Isolated emulsion films, and especially films of inverse emulsions, are important subjects of various studies [31]. Hydrocarbon films formed in aqueous medium and stabilized by surfactants constitute the simplest and at the same time representative model of biological membranes formed by mixtures of natural water- and oil- soluble surfactants, i.e. of proteins and lipids. Figure VIII-13 shows a common scheme of" membrane structure [62,63].
,.';;
Fig. VIII-13. Schematicrepresentation of membrane structure (After Singer, S.J., Nicolson, G.L., [62])
621 The studies of elememary films formed in inverse emulsions and stabilized by different symhetic and natural surfactants revealed that the membrane electric conductivity experiences a sharp increase upon the addition of some biologically active surfactants. For instance, membrane conductivity may increase by five orders of magnitude when trace amounts ofvalinomycin antibiotic are introduced imo the outer aqueous medium of lipid membrane. At the same time the membrane becomes permeable to potassium and hydrogen ions but impermeable to sodium ions. A sharp decrease in electric resistance of synthetic membranes is observed when proteins and enzymes with suitable substrates are introduced into them. By studying the properties of such membranes one may model important biological processes, e.g. the transfer of neural impulses. A step forward in the modeling of cellular membranes was made when synthetic methods leading to the formation of liposomes and vesicles were discovered and developed [46,64-67]. Liposomes and vesicles are peculiar colloidal particles representing continuous closed bilayer and multilayer membranes which separate inner and outer volumes of liquid phase. These systems allow one to simulate exchange processes between cells and outer medium. Liposomes are used in biomedical research aimed at increasing effectiveness of drug delivery. In such studies the active components of drugs are incorporated into liposomes which deliver them to the organs that require treatment. Liposomes and vesicles may be prepared by ultrasonic treatment of lipid suspensions, by replacing the solvent, removing the surfactants from solubilized lipids in mixed micelles by dialysis, or even by shaking an aqueous
622 phase in a flask with a lipid deposited on the walls. Size and structure ofliposomes strogly depend on the conditions under which they were synthesized.
One usually classifies liposomes as
multilamellar (Fig. VIII-14, a), with membranes consisting of several lipid bilayers, and monolalamellar that contain membranes consisting of only one lipid bilayer (Fig. VIII-14, b, c). Multilamellar liposomes are usually several microns in diameter. Monolamellar liposomes are typically subdivided into
microvesicles with diameters between 25 and 100 nm, and macrovesicles with diameters between 0.2 and 2 gm. 17
f
a
b Fig. VIII-14. The structure of liposomes and the scheme of incorporotion of molecules into liposomes: 1 - polar (ions), H- diphilic, III- non-polar; a - multilamellar liposome; b macrovesicle; c - microvesicle; d- a fragment of a shell of multilamellar liposome; e - a fragment of vesicle membrane Liposomes formed in complex aqueous solutions may include some of the solution compnents. Water soluble surface-inactive substances (inorganic electrolytes) may become passively incorporated into inner volume
623 of liposomes and into interlamellar spacing in multilamellar liposomes, as shown in Fig. VIII- 14, a - d, position I. The degree of incorporation of these components is determined by the volume of dispersion medium (aqueous solution) immobilized by liposomes. Diphilic surfactant molecules, including proteins, in addition to their incorporation into the inner volume, may also become incorporated into bilayers (Fig. VIII-14, d, e, position//), which results in their accumulation in liposomes. Non-polar molecules may accumulate inside bilayers (Fig. VIII-14, d, e, position III ). The ability of liposomes to retain an incorporated substance during their transport in the organisms depends on the permeability and stability of liposome membranes. The latter are well permeable for water, which favors osmotic processes upon lowering the electrolyte concentration in an outer solution. These processes may sometimes lead to the osmotic shock, which results in a membrane rupture due to absorption of excessive amount of water by a liposome. The permeability of membranes depends on the phase state of a lipid in bilayer. The membranes formed with low melting point lipids containing double bonds in hydrocarbon chains, usually have higher permeability. This feature is utilized by microorganisms for maintaining the level of membrane permeability needed for normal life cycle: when the temperature decreases, microorganisms increase the content of unsaturated lipids in their membranes, while at the elevated temperatures microorganisms increase the content of saturated lipids. One of the most important subjects in the studies ofliposomes, related to the core issues of theoretical biology, is the interaction between liposomes and cellular membranes. This interactions include the adhesion of liposomes
624 to cellular surface, interactions between cellular and liposome membranes, the mechanisms of substance transfer from liposomes to the cells and cellular structures. Such studies are directed towards understanding how one can control the interactions of liposomes with cells of various types, as well as intercellar interactions.
VIII.4. Suspensions and Sols
Disperse systems with solid dispersed phase and liquid continuous phase are referred to as sols in the case when solid particles are of colloidal dimensions, and as suspensions, in the case of coarser particles and instability towards sedimentation. Highly concentrated suspensions are referred to as
pastes. Sols represent classic objects for studies in colloid and surface science, while suspensions are the objects of various areas in chemical industry, such as manufacturing of fertilizers, catalysts, paints, ceramics, etc. Manufacturing of materials with desired properties in
many cases includes stages of
formation of colloidal particles (by dispersion or condensation), and their subsequent coagulation in liquid dispersion medium. Coagulation and precipitation of suspended matter is the basic stage in the process of water purification. This is not only related to hazardous household suspensions and industrial waste byproducts, but also to specially prepared sols of metal hydroxides that are introduced into water for entraping surfactants and heavy metal ions. The methods that are used to control these processes are based on general laws that govern the formation and degradation of colloidal systems in combination with the studies of specific properties of such systems, in
625 particular of the ability to form spacial structures with characteristic mechanical properties (Chapter IX). Many geological processes, including the formation of fertile soil layer on the surface of Earth's crust, are based on formation and coagulation of sols. The nature of stability of disperse systems with solid dispersed phase and liquid continuous phase against coagulation is determined by phase composition, particle size and particle concentration. The stability of hydrosols at low electrolyte concentrations is usually related to the electrostatic component of disjoining pressure (Chapter VII), arising from the overlapping diffuse parts of electrical double layers. At the same time stability of sols towards aggregation (Chapter VII, 1) may be of thermodynamic nature. The sols are stable if the depth of potential energy minimum is smaller than the gain in the free energy due to participation of particles in the thermal motion. The most effective means to protect the system, and especially the concentrated one, from coagulation is to introduce micelle-forming lowmolecular weight surfactants, or the high- molecular weight substances referred to as protective colloids [68]. These substances create structuralmechanical barrier at the surface of colloidal particles, which completely prevents coagulation and direct contact between the particles that may lead to irreversible changes in properties of disperse systems. Structuralmechanical barrier plays especially important role in the stabilization of inverse systems - suspensions and sols of polar substances in non-polar media, in which electrostatic repulsion is usually insignificant. A complete prevention of particle adhesion due to the formation of protective surfactant layers may
626 occur not only in dilute suspensions but also in concentrated pastes. In the latter case the surfactant acts as aplasticizer, ensuring a high mobility of the system (Chapter IX). Selection of surfactants suitable for stabilization of different suspensions and sols is similar to the choosing proper surfactants for stabilization of direct and inverse emulsions, i.e., one should use surfactants of 3 rd and
4 th groups
with high HLB numbers for stabilization of suspensions
and sols in polar media, and with low HLB numbers for stabilization in nonpolar media [46,47]. Coagulation of dilute sols due to insufficient stabilization (or due to introduced electrolytes, in the case of electrostatic stabilization of the system) usually results in formation of separate aggregates if the system originally consisted of isometric particles. As a result, the system loses its sedimentation stability and more or less loose p r e c i p i t at e forms. In systems consisting of highly anisometric particles, as well as in concentrated systems, coagulation may lead to the formation of s p a c i a l n e t w o r k s of dispersed particles. Without losing its sedimentation stability, the system may undergo a transition from free disperse (sol) to structured one (gel). Surfactants may not only stabilize system against coagulation, but may have an opposite effect, i.e. cause destabilization in cases when the surfactant adsorption proceeds against the polarity equalization rule (Chapter III,2), e.g., during chemisorption of surfactants from aqueous medium on a hydrophilic surface. For example, small additives of cationic surfactants cause coagulation of aqueous dispersions of clays and other silicates due to hydrophobization at F< 1-'max. Further increase in surfactant concentration results in the formation of a second (hydrophilizing) adsorption layer and leads to an increased
627 stability of sols. In suspensions coagulation may take place during sedimentation of dispersed particles as well as in already formed precipitate. Coagulation during sedimentation results in deviations in the shape of sedimentation curve from that typically observed in stable systems: if coagulation leads to the increase in aggregate size and sedimentation rate, the sedimentation curves may contain an inflection point, see Fig. VIII- 15. The change in the shape of sedimentation curve upon stabilization of sedimentationally unstable system by addition of a surfactant, may allow one to study coagulation in such systems. Stabilization
P
0
t
Fig. VIII-15. Change in shape of sedimentation curves caused by coagulation
Another result of coagulation taking place during suspension sedimentation is the change in the volume of precipitate. The formation of loose particle aggregates due to coagulation leads to an increase in the volume of precipitate, in comparison with stable systems, in which particles move freely with respect to each other and are capable of forming close packing. The addition small quantities (often less than 0.01%) of some polymeric surfactants to suspensions and sols leads to flocculation, which in its appearance resembles coagulation. Polymeric surfactants capable of causing flocculation are referred to asflocculants. Typical flocculants used to flocculate aqueous dispersions are polyacrilamides, polyethyleneimines, etc.
628 According to LaMer, flocculation, in contrast to coagulation, leads to the formation of more friable aggregates (flocks) in which particles are located at considerable distances from each others and are bound together by polymer molecules. At low concentrations a macromolecule of surface active polymer containing a large number of polymer segments in its chain may extend sufficiently far into the bulk of solution and bind to several particles at a time, tying them into a single loose flock (this bonding is especially strong in the case ofchemisorption). This type offlocculation, in which polymer molecules form bridges between the particles, is referred to as the bridgingflocculation. When present at high concentration, polymeric surfactant, due to its high adsorption, may form a dense lyophilizing adsorption layer at the particle surface. Under these conditions the same polymer acts as a stabilizer of colloid dispersion, stabilizing the latter by means of structural-mechanical barrier (Chapter VII). Flocculation and subsequent stabilization ofsols can also be caused by polymers that do not adsorb on the particle surface. In this case the mechanism of polymer action is different from the one described above and is related to the state of conformation of polymer molecules and change in the free energy of the system upon the transfer of polymer coil from gap between the particles into the solution bulk (a so-called
depletion flocculation) [69,70].
Flocculation has numerous industrial applications" in water treatment (for accelerating particle settling), in papermaking (for enhancing retention of fillers and fibers and increasing rate of dewatering), in agriculture (for controlling structural-mechanical properties of soils, increasing the resistance
629 of soils to wind erosion), in geological engineering (for fixation of grounds), and in many other areas.
VIII.5. Coagulation of Hydrophobic Sols by Electrolytes
The tendency of hydrophobic sols to coagulate upon the addition of small amounts of electrolytes was notices a long time ago and became a subject for numerous experimental and theoretical studies. It was established that the ability of different electrolytes to coagulate colloidal systems is primarily governed by the magnitude of charge on ions bearing the charge of sigh opposite to that of colloidal particles, i.e., the ions of the same charge (by sign) as the counter-ions. The coagulating action increases with an increase in the charge of coagulating ion. As the electrolyte concentration increases, coagulation becomes noticeable only above certain critical concentration, referred to as the critical coagulation concentration, or C.C.C. The ratios between critical coagulation concentrations of mono-, di-, and trivalent counterions are 1:0.016"0.0015. Consequently, the inverse values, referred as the coagulation ability, form series 1:60:700. This empirical relationship between the valence of counter-ion and the C.C.C. is known as the Schulze -
Hardy rule, and was theoretically explained by the DLVO theory (Chapter VII, 4), in agreement with which the concentration of electrolyte that is needed to completely coagulate sol of highly charged particles is proportional to the charge of coagulating ion in the negative sixth power. Consequently, in agreement with the DLVO theory coagulation abilities of mono-, di-, and trivalent ions are related as 16. 26. 36 = 1 964 9729, respectively. It was
630 observed that the value of ~ potential in the vicinity of the C.C.C. becomes lowered to ~30 mV (within the limits between 25 and 50 mV). Detailed studies of coagulation by different electrolytes with the same charge of coagulating ion revealed that these electrolytes form series similar to lyotropic ones, based on the participation in ion exchange and on the influence of electrolytes on the electrokinetic potential (Chapter III). When the electrolyte concentration increases further above the C.C.C., the coagulation rate first increases (Fig. VIII-16, region I), which corresponds to the slow coagulation (Chapter VII, 6) region, and then becomes independent of electrolyte concentration in the region of rapid, or fast
coagulation (Fig. VIII-16, region II). According to Smoluchowski's theory such dependence of coagulation rate on electrolyte concentration must be related to an increase in the number of effective collisions between the particles upon destabilization of the system. The experimental investigation of coagulation kinetics indicated that at low values of ~-potential one often observed a linear relationship between inverse concentration of aggregates,
1/n~, and time (Fig. VIII-17), which agrees with Smoluchowski's theory. It is 1 nz dn Z
H
clt
i
0
C.C.C.
_1
c
Fig. VIII-16. Coagulation rate, dn~/dt, as a function of electrolyte concentration
0
t
Fig. VIII-17. Fast coagulation kinetics, according to Smoluchowski
631 worth pointing out here that at low electrolyte concentrations, far from the region of rapid coagulation, the deviation between experimentally determined and theoretically predicted coagulation rate exists. Kihira, Ryde and Matij evid explained such a discrepancy by the segregation (i.e. non-uniform distribution over the particle surface) of particle surface charge. [71 ]. This is illustrated in Fig. VIII-18 which shows the experimentally measured and theoretically computed stability ratios as a function of ionic strength for polystyrene latex. The theoretical stability ratios were computed for different degrees of surface charge segregation. Figure VIII-18 shows that by assuming proper amount of surface charge segregation one can bring theoretical and experimental stability ratios into closer agreement with each other. Another explanation for such a deviation between experimentally measured and theoretically calculated coagulation rates was offered by G.A. Martynov who suggested that there may be an equilibrium between coagulation and peptization (see ref. [72]). i
I
,
I
,,,,
]
,
"
:
o
l
-3
,
-2
I
-I I(~(I/mol dm -a)
I
0
Fig. VIII-18. Effect of the surface charge segregation on the stability of polystyrene latex. Experimentally measured (symbols and dashed line) and theoretically computed (solid lines) stability ratios as a function of ionic strength. Different degrees of surface charge segregation wereassumed in the evaluation of theoretical stability ratio: a - 0 (totally smoothed out surface charges); b - 10%; c - 20%; d- 30%; e - 40%. (From ref. [71 ] with permission)
632 In the case of electrolytes that are capable of causing the reversal of surface charge of dispersed particles, coagulation occurs only at electrolyte concentrations corresponding to rather small values of the ~-potential. As electrolyte concentration becomes higher, the ~-potential decreases; coagulation starts as soon as the ~-potential drops below the critical value. This corresponds to thefirst critical coagulation concentration, C.C.C. - I, (c~ point in the curves shown in Fig. VIII-19). Further increase in electrolyte
+~cr i ,
c
-~cr !
dnz
i
....
dt
C1
r
e3
C
Fig. VIII-19. The electrokinetic (~) potential and coagulation rate, dnz/dt, of particles as a function of electrolyte concentration
concentration results in a decrease in the ~-potential to zero, and then to the reversal of surface charge due to change in the sign of % or q0j potentials, depending on the nature of electrolyte (Chapter III). The absolute value of starts to increase again, and when it reaches the critical value, peptization of a sol may start (Fig. VIII-19, point c2). If the amount of electrolyte necessary to recharge the surface is added quickly, the coagulation does not take place at all. This results in the appearance of the second stability zone of a sol. Further increase in electrolyte concentration leads to a compression of diffuse
633 double layers, w h i c h in turn results in a decrease of electrokinetic potential and causes coagulation. This corresponds to the second critical coagulation
concentration, C.C.C. - II, (Fig. VIII-19, point c3) and second zone o f c o a g u l a t i o n with a well established region o f fast coagulation. The C.C.C.-II is significantly higher than the value predicted by S c h u l z e - H a r d y rule. This p h e n o m e n a are referred to as the "irregular series" because the p r i m a r y stability region w a s not noticed during p r i m a r y studies.
When sols are coagulated by a mixture of electrolytes, the latter in some cases act independently, i.e., their effects are additive. For instance, if (C.C.C.)l and (C.C.C.)2 are critical coagulation concentrations of a given sol by two different electrolytes, the critical coagulation concentration of this sol by a mixture of these two electrolytes is given by (C.C.C)m~x = (1-%)(C.C.C.)I + q~2(C.C.C.)2, where % is the volume fraction of the second electrolyte in a mixture. This type of behavior is usually observed when both electrolytes contain coagulating ions of the same charge. In other cases the electrolytes may show an antagonistic or synergistic behavior, i.e. in their mixtures they either retard or promote individual coagulation abilities. Apparently, in the case of antagonistic behavior one has to add more electrolyte than is required by the additivity rule, while in the case of synergism the amount of mixed electrolyte that causes coagulation is lower than that predicted by the additivity rule. According to Yu. M. Glazman, antagonistic and synergistic behavior of electrolytes in the case of weakly charged sols may be explained by a competitive adsorption of ions, while for strongly charged interfaces it results from electrostatic interactions of diffuse ionic atmospheres. One must also take into account the possibility of chemical interaction between the components of electrolytes. Such an interaction may lead to the formation of insoluble compounds or complexes that are unable to cause coagulation. In a number of cases the sol becomes accustomed to the added electrolyte: the amount of electrolyte that is needed for coagulation is higher in the case when electrolyte is added slowly, as compared to the case when it is added rapidly. An opposite phenomena, the
634 so-called negative accustoming, has also been observed, i.e., lesser amount of electrolyte was needed to cause coagulation in the case of slow addition. Negative accustoming can be explained in terms of the DLVO theory. Slow addition of electrolyte causes gradual decrease in the height of potential barrier with a simultaneous increase in the number of effective collisions between particles. Under these conditions each subsequent portion of electrolyte is added to a d iffe r e n t, less stable sol, as compared to the initial one, and hence smaller amount of electrolyte is needed to cause coagulation. Coagulation, as well as the opposite phenomenon of peptization, are sensitive to thermal and mechanical actions, and to ultrasound. Aging also plays an important role: spontaneous coagulation, not assisted by electrolytes, is often observed when sols are aged for extended periods of time. Peptization usually takes place in rather fresh coagulates, while during prolonged aging the precipitates experience irreversible changes due to coalescence of particle by Ostwald ripening (see Chapter VII, 7) [72].
It has been already pointed out that the energy of interaction between dispersed particles depends on the particle size.
As a result,
for large
particles, and especially for anisometric ones, oriented in a certain way with respect to each other, the presence of a secondary minimum may be of importance. For such particles this secondary minimum may be sufficiently deep in comparison with kT. In some cases these systems may experience a peculiar "colloid phase transition" from a free disperse system (at low concentrations of dispersed phase) to crystal-like periodic structures consisting of colloidal particles in equilibrium with the dilute sol consisting of single particles. Such periodic structures are observed in some biological systems, e.g. in tobacco mosaic virus, in V205 sols and in latexes. Aggregation processes in systems that contain more than one suspended component have some interesting features. In such systems, in
635 addition to the aggregation of similar particles,
heterocoagulation, i.e. mutual
aggregation of dissimilar particles may also take place [73]. In these processes electrostatic and molecular components ofdisj oining pressure sometimes may "trade places". Indeed, as discussed in Chapter VII.2, the molecular component of disjoining pressure may be positive (A*<0) for non-symmetric films (in the present case the liquid gaps separating dissimilar particles). As a result, the dispersion medium may be sucked in between the particles by molecular forces. This phenomenon resembles the appearance of zero twosided angle at the zone of contact between liquid phase and grain boundary, when the surface energies of phases are interrelated by the Gibbs-Smith condition (Chapter III). Oppositely, if the surfaces of dissimilar particles bear charges of opposite signs, the electrostatic component of disjoining pressure is negative" particles bearing charges of opposite signs are attracted to each other; the higher the effective charges (the more developed the diffuse layers of counterions), the stronger the attraction. Theoretical analysis carried out by Derjaguin indicated that heterocoagulation should primarily take place at low electrolyte concentrations, as at high electrolyte content dissimilar particles may be resistant to aggregation, while homocoagulation will still take place. These results were experimentally confirmed by Yu.M. Chernoberezhskiy [74] in experiments with mixed sols consisting of colloidal gold and fen'ic hydroxide particles. Heterocoagulation in the systems containing dissimilar particles is of importance in the formation of soils and in water clarification processes.
636
VIII.6. Detergency. Microencapsulation Let us now give examples illustrating the use of surfactants for the purpose of stabilizing various disperse systems of complex nature, namely let us discuss detergency[75] and microencapsulation[76] (preparation of protective coating at the surface of various substances). The removal of solid and liquid, low and high molecular weight dirt is of importance in both household and industrial applications. The latter include decontamination of various surfaces prior to applying protective coatings, removal of oil and grease from the body and working parts of various machinery, etc. Synthetic surfactants that are broadly used in these applications are usually present as components of formulations, referred to as
synthetic detergents. Dirt removal process is rather complex: dirt is usually a mixture of different polar and non-polar solid and liquid components; the cleaning of fabrics is further complicated by mechanical entrapment of dirt between the fibers. The theory of detergency, the development of which has not yet been totally completed, helps one to properly prepare detergent formulations, choose application conditions and develop environmentally safe washing processes. The action of surfactants in detergents involves all of the mechanisms that were discussed in this book. The improvement of wetting of the treated surface by water is especially important in washing of fabrics, in which capillary forces may significantly retard the penetrating ability of detergent solutions. If one is attempting to remove wet oil contaminants, the p r e f e r e n t i a l wetting plays an important role: it aids in displacing the dirt
637 from contaminated surface by water. The stage of separating solid and liquid contaminants from the soiled surface, which is the major stage of the cleaning process, is related to the d i s p e r s i n g a c t i o n of surfactants. This process is aided by mechanical treatment, which is always involved in washing. An important condition for effective cleaning is the prevention of redeposition of removed dirt. This is achieved by
strong stabilization of removed
contaminants and by l y o p h i l i z a t i o n (hydrophilization) of the treated surface by surfactant solutions. Surfactants used in detergent formulations should be highly surface active; since the molecular solubility of these substances is rather limited, the sufficient amounts of surfactants in solutions can be only accumulated in the m i cell ar form. Effective stabilization of oily liquid contaminants may be related to their effective s o l u b i l i z a t i o n inside the micelles. The described mechanisms of detergency are put into practice by using synthetic micelle-forming surfactants, among which the mixtures of anionic and nonionic surfactants (particularly alkylsulfates and oxyethylenated alcohols) make 10 to 40 % of the total detergent formulation. Cationic surfactants (alkylamines) that are also included into synthetic detergent formulations may contribute up to 5% of the total amount of formulation. These substances reveal biocidal action and control micelle formation by forming mixed micelles. Detergent formulations may also contain sodium polyphosphate, which helps to stabilize dispersed dirt by increasing the surface potential due to adsorption ofpolyvalent anion. This substance also binds bivalent cations, and hence decreases water hardness. The use of polyphosphates is however
638 limited. There is an evidence that sodium polyphosphate promotes the growth of blue-green algae that overgrow natural water reservoirs. Among other components of synthetic detergents one can name sodium silicate, sulfate and carbonate, bentonite clays (used as individual detergent during the earlier days in history) and zeolites. Sodium silicate and sodium carbonate are added for the purpose of controlling the pH of detergents which influences the detergency performance of anionic surfactants. These substances also affect the surface properties of fabric fibers, in particular their ability to swell. An optimal pH range in cleaning of wool fabrics is between 7 and 8, for washing cotton fabrics it is between 9 and 10, while in the industrial cleaning it is 11 or higher. Lyophilization of the treated surfaces and of removed particles is achieved
by
the
addition
of
various
polymers,
such
as,
e.g.,
carboxymethylcellulose, which is introduced into synthetic detergent formulations in the amount of a few percent. Detergents also often contain enzymes that are capable of cleaving proteins present in soiled areas. It became possible for one to use enzymes only after the methods of their encapsulation were developed; encapsulation of enzymes prevented their degradation by other components present in synthetic detergents. Detergents that are intended for use in laundry machines contain increased amounts ofnonionic surfactants that prevent excessive formation of foam that may interfere with the normal working cycle of machines. The action of nonionic surfactants is based on their decreased tendency towards foaming, especially at elevated temperatures, at which they lose ability to form micelles due to dehydration of polar groups (Chapter VI, 3).
639 The same principle is utilized in the use of nonionic surfactants in the so-called thermoregulated detergents that are used to clean engines and tanks from oil, carbon, and grease deposits. The treatment of greasy parts with organic solvents containing up to 10% of mixed nonionic surfactants with oxyethylene chains of different lengths and subsequent washing with dilute aqueous solutions of the same surfactants allows one to achieve spontaneous grease removal (without extra mechanical action) and complete cleaning of surfaces. Increasing the temperature above the c lo ud point ofthe dispersion of nonionic surfactant (Chapter VI.3) leads to the phase separation in the formed complex disperse system [46]. As a result, one obtains water that is sufficiently clean to be reduced, and also recovers the surfactant. Consequently, one may carry out the closed cleaning cycle without discharging hazardous substances into the environment (nonionic surfactants, oils, fuel residues, etc.). Commercial detergents are typically sold as powders, liquid formulations or as pastes. Pastes are encountered more rearly because it is difficult for one to control their stability during prolonged storage and transportation, especially when pastes get exposed to severe hot and cold temperatures. The fast growing area of technology is microencapsulation of liquids and powders. During microencapsulation one creates protective coatings at the surface of substances. These coatings prevent an encapsulated substance from coming into direct contact with the surrounding medium. Such coatings are composed of polymer films that in some sense resemble cell membranes. The major routes of microencapsulation involve the adsorption of film-forming
640 polymers or the formation at the particle surface a thin film of a new liquid phase (coacervation). These films undergo further treatment (by tanning agents, changing pH, temperature, etc.), until they acquire solid-like properties. Various natural and synthetic substances are used for the preparation of films. Among the latter proteins (gelatin, albumin), polysaccharides,
cellulose
derivatives,
polyvinyl
alcohol
(PVA),
polyvinylacetate are especially common. Microencapsulation significantly enhances the areas of utilization of various products and improves properties of the latter. Characteristic properties ofmicroencapsulated liquid fuel are the high ignition temperatures and low explosion hazards. Briquettes of such "solidified" fuel may be transported without special packaging; in case if ignition such fuel may be quenched by water. Microencapsulation is used in the preparation of solid rocket propellants, which contain encapsulated oxidizing and reducing agents, a direct mixing of which prior to the moment of actual use is impossible due to their high activity. In pharmaceutical industry encapsulation is used to stabilize unstable drugs, to ensure drug delivery to proper spots in the organism, and to monitor the rate of drug release, duration of action, and the maximum level of concentration in the body [77]. Microencapsulation is also of importance in agricultural applications. Encapsulation of fertilizers ensures their slow release into soil and more uniform feeding to plants. Microencapsulation allows one to coat the surface of seeds with various important substances, such as growth agents, pesticides, and fertilizers.
641 The use of fodder additives and concentrates that contain aminoacids, proteins, fats, mineral salts, vitamins and antibiotics requires sufficient stability, controlled dosing and release, all that can be achieved by using microencapsulated components.
VIII.7. Systems with Solid Dispersion Medium
Let us now briefly describe disperse systems in which gaseous, liquid or solid inclusions are either distributed within the solid phase, or form networks of channels in a continuous solid phase. In the latter case the subdivision of system into the dispersed phase and dispersion medium is rather symbolic. Such bicontinual systems are quite common in nature and have broad industrial application. The examples of natural systems include pumice stones, tufa, dry and watery grounds, all multimineral rocks which, as a rule, contain several phases (that are often finely dispersed or even amorphous) and also gaseous and liquid inclusions. Numerous materials that are widely used in technology also belong to the class of systems with solid dispersion medium. The latter include various alloys, solidified polymer foams (polyurethane foam, styrofoam), ceramics, various catalysts and adsorbents, building materials, and others. Tissues of living organisms, and especially bones, are also representatives of the systems with solid dispersion media. Bone tissue is formed with ultrafine crystals of calcium hydrogen phosphate (apatite) with nearly theoretical strength that armor peculiar fibril collagen spiral structures "threaded" with different pitch in different directions. Many disperse systems with solid continuous phase are the common subjects for studies in such areas of science as material science, physics of materials, physics of metals and others. This is related to the existing great variety of such systems. Obviously, their properties (among which mechanical ones are of primary importance) are significantly different from those of systems with liquid dispersion medium. At the same time, the investigation of processes leading to the formation of such systems and their interactions with ambient media constitute direct subjects of colloid science.
642 The formation of systems with solid backbones is often the result of aggregation processes taking place in suspensions and sols which lead to the development of spacial networks and final conversion of disperse systems into materials with valuable properties (Chapter IX, 2). In some cases, e.g. during solidification of metal alloys, the processes of structuring accompany formation of new phases. The systems with solid dispersion medium also form upon the solidificationof a continuous phase in foams, emulsions, suspensions and sols. Due to high viscosity of continuous phase, collective recrystallization,related to the diffusional transport of inclusions through dispersion medium, is the principal and usually the only mechanism by which the degree of dispersion in these systems may be changed. Because of small diffusion coefficients of components, changes in the degree of dispersion are extremely slow and become noticeable only at sufficiently high temperatures. The extension of general laws of colloid science to systems with solid dispersion medium leads to the development of physical-chemical mechanics as an independent branch of science. The principles of physical-chemical mechanics are described in chapter that follows.
References ~
,
,
0
,
6. 0
Vold, R.D., Vold, M.J., Colloid and Interface Chemistry, AddisonWesley Publishing Inc., ReaDING, ma, 1983 Hidy, G.M., and Brock, J.R. (Editors), Topics in Current Aerosol Research, vols. 1-3, in "International Reviews in Aerosol Physics and Chemistry", Pergamon Press, New York, 1970-1972 Colbeck, I. (Editor), Physical and Chemical Properties of Aerosols, Blackie Academic and Professional Publ., London, 1998 Friedlander, S.K., Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2 nd ed., Oxford University Press, NY, 2000 Kerker, M., Adv. Colloid Interface Sci., 5 (1975) 105 Matijevid, E., and Partch, R. E., in "Surfactant Science Series", vol. 92, T. Sugimoto (Editor), Marcel Dekker, New York, 2000 Kodas, T.T., Hampden-Smith, M.J., Aerosol Processing of Materials, VCH-Wiley, New York, 1999
643 ~
9. 10. 11. 12. 13. 14.
15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
Hendricks, C.D., Babil, S., J. Phys E, 5 (1972) 905 Visca, M., Matijevid, E., J. Colloid Interface Sci., 68 (1979) 308 Sinclair, D, and LaMer, V.K., Chem Rev 44 (1949) 245 Nicolaon, D., Cooke, D.D., Kerker, M., Matijevid, E., J. Colloid Interface Sci., 34 (1970) 534 Nicolaon, D., Cooke, D.D., Davis, E.J., Kerker, M., Matijevid, E., J. Colloid Interface Sci., 35 (1971) 490 Hsu, W.P., Wang, G., Matijevid, E., Colloids Surfaces, A61 (1991) 255 Exerowa, D., Kruglyakov, P.M., Foam and Foam Films. Theory, Experiment, Application, in "Studies in Interface Science", vol.5, D. M6bius, R. Miller (Editors), Elsevier, Amsterdam, 1998 Tikhomirov, V.K., Foams. Theory and Practice of Their Generation and Breaking, Khimiya, Moscow, 1983 (in Russian) Bikerman, J.J., Foams, Springer-Verlag, New York, 1973 Schwartz, A.M., Perry, J.M., and Berch, I., Surface Active Agents and Detergents, Interscience Publishers, New York, 1958 Schultze, A.M., Physicochemical Elementary Processes in Flotation, Elsevier, Amsterdam, 1984 Domingo, X., Fiquet, L., and Meijer, H., Tenside, 29 (1992) 16 Akers, R.J., Foams, Academic Press, London, 1975 Mysels, K J. Phys. Chem., 68 (1964) 3441 Mysels, K.J., and Jones, M.N., Disc. Faraday Soc., 42 (1966) 42 Derjaguin, B.V., Titijevskaya, Kolloidn. Zh., 15 (1953) 416 Perrin, J., Ann. Phys., 10 (1918) 160 Sonntag, H., Netzel, J., and Clare, H., Kolloid-Z. Polym.,, B211 (1966) 121 Scheludko, A., Exerowa, D., Comm. Dept. Chem. Bulg. Acad. Sci., 7 (1959) 123 Scheludko, A., Adv. Colloid Interface Sci., 1 (1967) 391 Exerowa, D., Zacharieva, M., Cohen, R., and Platikanov, D., Colloid Polymer Sci., 257 (1979) 1089 Exerowa, D., Kaschiev, D., and Platikanov, D., Adv. Colloid Interface Sci., 40 (1992) 201 Khristov, Khr. I., Exerowa, D.R., and Kruglyakov, P.M., Colloid Polymer Sci., 261 (1983) 265 Kruglyakov, P.M., Rovin, Yu. G., Physical Chemistry of Black Hydrocarbon Films, Nauka, Moscow, 1978
644 32. 33. 34. 35. 36. 37. 38.
39. 40. 41. 42. 43. 44. 45. 46.
47.
48.
Ekserowa, D., Nikolov, A., Zakharieva, M., J. Colloid Interface Sci., 81 (1981) 419 de Vries, I.A., Proc. 3 rd Intern. Congr. Detergents, vol.2, Cologne, 1960, page 566 Pertsov, A.V., Chernin, V.N., Chistyakov, B.E., Shchukin, E.D., Dokl. Akad. Nauk SSSR, 238 (6) (1978) 1395 Garrett, P.R. (Editor), Defoaming: Theory and Industrial Applications, in "Surfactant Science Series", vol. 45, Dekker, New York, 1993. Rosen, M.J., Surfactants and Interfacial Phenomena, 2nd ed., Wiley, New York, 1989 Garti, N., and Aserin, A., in "Micelles, Microemulsionsand Monolayers", D.O. Shah (Editor), Dekker, New York, 1998 Friberg, S.E., in "Emulsions - A Fundamental and Practical Approach", J. Sj6blom (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 363, Kluwer, Dordrecht, 1992 Sj6blom, J. (Editor), Emulsions and Emulsion Stability, in "Surfactant Science Series", vol. 61, Dekker, New York, 1995 Breen, P..J., Wasan, D.T., Kim, Y-H., Nikolov, A.D., in "Surfactant Science Series", vol. 61, Dekker, New York, 1995 Kumar, K., Nikolov, A.D., Wasan, D.T., Ind. Eng. Chem. Res., 40 (2001) 3009 Zhukhovitsky, A.A., Grigoryan, V.A., Mikhalik, K., Dokl. Akad. Nauk SSSR, 155 (1964) 392 Zhukhovitsky, A.A., Grigoryan, V.A., Mikhalik, K., Zh. Fiz. Khim., 39 (1965) 1179 Shinoda, K., and Friberg, S., Emulsions and Solubilization, Wiley, New York, 1986. Shchukin, E.D., Amelina, E.A., Parfenova, A.M., Colloids Surf., A 176 (2001) 35 JOnsson, B., Lindman, B., Holmberg, K., and Kronberg, B., Surfactants and Polymers in Aqueous Solution, Wiley, Chichester, 1998 Kruglyakov, P.M., Hydrophilic - Lipophilic Balance" Physicochemical Aspects and Applications, in "Studies in Interface Science", vol.9, D. MObius, R. Miller (Editors), Elsevier, Amsterdam, 2000 Davies, J.T., Proc. 2 nd Int. Congr. Surface Activity, 1 (1957) 426
645 49. 50. 51. 52. 53.
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56. 57.
58.
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Izmailova, V.N., Yampolskaya, G.P., J. Mendeleev Chem. Soc., 24 (1989) 81 Izmailova, V.N., Yampolskaya, G.P., Summ, B.D.,Surface Phenomena in Protein Systems, Khimia, Moscow, 1988 (in Russian) Izmailova, V.N., Rehbinder, P.A., Structure Formation in Protein Systems, Nauka, Moscow, 1974 (in Russian) Izmailova, V.N., Yampolskaya, G.P.,in"Studies in Interface Science", vol.7, D. M6bius, R. Miller (Editors), Elsevier, Amsterdam, 1998 Rehbinder, P.A., in "Selected Works by P.A. Rehbinder", vol.1, Surface Phenomena in Disperse Systems. Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Wenstr6m, E.K., Rehbinder, P.A., Zh. Phys. Khim., 2 (1931) 754 Asua, J.M. (Editor), Polymeric Dispersions" Principles and Applications, NATO ASI Series E: Applied Sciences, vol.335, Kluwer, Dordrecht, 1997 Dickinson, E., Introduction to Food Colloids, Oxford University Press, New york, 1992 Larsson, K., in "Emulsions - A Fundamental and Practical Approach", J. Sj6blom (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 363, Kluwer, Dordrecht, 1992 Junginger, H.E., in "Emulsions - A Fundamental and Practical Approach", J. Sj6blom (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 363, Kluwer, Dordrecht, 1992 Wasan, D.T., in "Emulsions - A Fundamental and Practical Approach", J. Sj6blom (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 363, Kluwer, Dordrecht, 1992 Taylor, P., Adv. Colloid Interface Sci., 75 (1998) 107 Lissant, K., Demulsification, Industrial Applications, in "Surfactant Science Series", vol. 13, dekker, NY 1983 Singer, S.J., Nicolson, G.L., Science, 175 (1972) 720 Ebbing, D.D., General Chemistry, 3~ded., M.S. Wrighton (Consulting Editor)Houghton Mifflin Co., Boston, 1990 Lasic, D.D., Biochem. J., 256 (1988) 1 Simonnet, J., Cosmetics Toiletries, 109 (1994) 45 Huang, L., J. Liposome Research, 4 (1994) 397
646 67.
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Lindman, B., Kremer, F., Ninham, B.W., Colloid Science of Lipids: New Paradigms for Self-Assembly in Science, Springer-Verlag, New York, 1998 Harkins, W.D., J. Amer. Chem. Soc., 69 (1947) 1428 Napper, D.H., Polymeric Stabilization of Colloidal Dispersions, Academic Press, London, 1983 Evans, D.F., Wennerstr6m, H., The Colloidal Domain: Where Physics, Chemistry, and Biology Meet, 2nded., Wiley-VCH, New York, 1999 Kihira, H., Ryde, N., and Matijevid, E., J. Chem. Soc. Faraday Trans., 88 (1992) 2379 Shchukin, E.D., Amelina, E.A., Yaminsky, V.V., J. Colloid Interface Sci., 90 (1982) 137 Kihira, H., Matijevid, E., Adv. Colloid Interface Sci., 42 (1992) 1 Chernoberzhskiy, Yu. M., Golikova, E.V., Kolloidn. Zh., 36 (1974) 115 Cutler, W.G., and Davis, R.C., Detergency Theory and Test Methods, parts I-III, in "Surfactant Science Series", vol.5, Dekker, 1981 Benita, S. (Editor), Microencupsulation, Dekker, New York, 1996 Donbrow, M. (Editor), Microcapsuls and Nanoparticles in Medicine and Pharmacy, CRC Press, 1992
List of Symbols
Roman symbols A* B~ C c C.C.C D d E F g HLB
Hm
complex Hamaker constant HLB group number capacitance concentration critical coagulation concentration diffusion coefficient pore diameter strength of electric field force acceleration of gravity hydrophile-lipophile balance height of foam column
647 h Kn K k mM NA /7
film thickness Knudsen number foam number Boltzmann constant mass of a molecule Avogadro's number number of particles per unit volume pressure P capillary pressure P~ q effective charge R universal gas constant F particle radius r average particle size curvature radius of Gibbs-Plateau border Fcurv T absolute temperature t time average life time of a single foam cell tl time of aerosol degradation due to diffusion /dif time of aerosol degradation due to sedimentation tsed foam life time tf V volume Vm molar volume particle velocity average velocity of gas molecule VM We, work of particle charging in aerosol x, y, z Cartesian coordinates
Greek symbols F F max
q Ar AM ~o H He1
adsorption maximum adsorption electrokinetic potential dynamic viscosity mean free path of a particle free path of a particle electric conductivity of a medium disjoining pressure electrostatic component of disjoining pressure
648 I-Imo 1 7C
9s g go
%
molecular component of disjoining pressure 3.14159... surface charge density specific surface free energy, surface tension dielectric constant electrical constant surface potential
649 IX. PRINCIPLES OF PHYSICAL-CHEMICAL M E C H A N I C S
Aggregation processes that take place in disperse systems (Chapters VII, VIII) lead to the separation of these systems into macroscopic phases or to the development ofspacial networks, i.e. to the transition from free disperse systems to structured disperse ones. In structured disperse system the cohesive forces in the contacts are sufficiently high, so these systems are resistant to both the thermal motion and external action. At the same time radical changes in disperse system occur" it acquires new structural and mechanical (rheological) properties that characterize
ability of a system to resist
deformation and fracture. The system acquires mechanical strength, the most important feature of all solids and materials that determines their role in nature and technology. The laws (with the main focus on the role of interfacial phenomena) that govern structure formation in disperse systems, regulate mechanical properties of structured systems and of materials based on them, are the subject of a separate branch of colloid science referred to as the
physical-chemical mechanics [ 1-7]. The origin of physical-chemical mechanics dates back to 1930- 40's. In 50's it was established as a separate discipline, primarily in the works by Soviet scientists and in particular in those by P.A. Rehbinder. One can give numerous examples of objects and materials that are the subjects of investigation and application in physical-chemical mechanics. Among the latter one may name such natural objects as rocks, minerals, living tissues, and various disperse systems that one encounters in technology (pastes, powders, suspensions, etc.). This large variety of objects is stipulated by the universal
650 nature of dispersed state and the universal role of mechanical properties. There may be cases when mechanical strength is of outmost importance, such as when one deals with materials, construction, or strength of grounds, or when, oppositely, one needs to overcome system's resistance towards deformation and fracture, as, for instance, in mixing, molding, milling or mechanical treatment. Among disperse systems the fine disperse ones are especially important. Ensuring fine degree of dispersion as well as maximum uniformity of microheterogeneous structure, is the basis for increasing mechanical strength and durability of materials. At the same time, various heterogeneous processes, mass and heat transfer, most effectively occur in fine disperse systems with highly developed interfaces. Molding, however, is most difficult to take place particularly in fine disperse systems due to the presence of a large number of contacts between particles. The adhesion between the latter represents the action of physico-chemical surface interactions. One may gain the most effective control over the properties of systems by combining both mechanical action (including treatment by vibration) and physical-chemical methods of controlling the molecular interactions at interfaces by using suitable media and surfactants. By examining how mechanical properties of disperse systems and materials are linked to their structure and to phenomena taking place at interfaces, one can find new routes leading to a control over mechanical properties of solids and materials, which constitutes the objective of physicalchemical mechanics [2]. The concepts of rheology are used in this chapter for the description
651 of mechanical properties of various liquid- and solid-like bodies and materials. We shall also discuss the formation, nature and main characteristics of contacts between particles in structured systems. These two conceptual approaches will allow us to analyze characteristic mechanical properties of real disperse systems and identify the ways to control them [8]. Special attention will be devoted to Rehbinder's effect, which is the result of the adsorption effect of medium on the mechanical properties of solids [2, 8-13].
IX.1. Description of Mechanical Properties of Solids and Liquids
Investigation of mechanical properties of solids and liquids reveals generalities in laws that describe mechanical behavior of objects of different nature. One usually outlines several simplest types of mechanical behavior, a combination of which allows one to approximately describe more complex mechanical properties of real objects. The science that formulates rules and laws of generalized description of mechanical behavior of solid-like and liquid-like systems is referred to as the theology (originates from Greek pecoc~, the flow, and )~o3,o~, the study). The main approach used in rheology is the description of mechanical properties with the help of particular idealized models, the behavior of which can be described by small number of parameters. One usually restricts himself to a consideration of a simple uniform shear and low strain rates, constituting the quasi-stationary regime [14,15]. In a given physical object, let us select a cube with an edge of unit length. Let a tangential force, F, be applied to the opposite faces of this cube.
652 This force creates shear stress, -c, numerically equal to F (Fig. IX.I). T
T Fig. IX-1. Shear strain, Y, caused by the shear stress,
The applied stress results in the shear strain of the cube, i.e. the top face becomes shifted with respect to the bottom one by distance 7. This displacement is numerically equal to the tangent of a tilt angle of the side face, i.e. it is equal to the relative shear strain, y, and at small strains tany ~y. The relationship between shear stress, ~, and shear strain, 7, and the rates of change in these quantities with time, d~/dt=~, dy/dt@, represent
m e c hani c al
be h avi o r, which is the main subject in rheology. One usually begins the description of mechanical behavior with three elementary models, namely elastic, viscous, and plastic behavior. I. In the case of el ast i c b e h a v i o r stress and strain are proportional to each other, i.e. there is a linear dependence between 3' and ~, described by the Hooke law, namely T-G
7.
653 In the above equation the modulus of elasticity, G, is the shear ~ modulus, which has the units of N m 2, the same as the units of the shear stress, -c. Graphically Hooke's law can be represented by a straight line shown in Fig. IX-2. This line passes through the origin and its inverse slope is equal to the modulus G.
3, Fig. IX-2. Elastic deformation
An important feature of idealized elastic behavior is its complete mechanical and thermodynamic reversibility: as the load is taken off, the object restores its initial shape, and there is no dissipation of energy taking place upon applying and removing the load. The energy stored in a unit volume of elastically strained object is given by Y J~elast --
f'c(~)d 7 - G7 2
0
2
2
_- "c 2G
0x.1)
The elastic behavior can be modeled by a spring with stiffness given by the ratio of the applied force to the elongation caused by this force, F/AI,
In rheology of condensed systems the shear modulus, G, is usually used as a characteristic of elasticity. In mechanics of continuous (isotropic) media it was shown that the modulus of shear for solid-like objects equals approximately 2/5 of the Young modulus, E 1
654 numerically equal to the modulus of elasticity of a given object (Fig. IX-3). G
Fig. IX-3. The elastic behavior modeled by a spring The elastic behavior upon applied shear stress is primarily typical in the case of solids. The nature of elasticity is in the reversibility of small deformations of interatomic (or intermolecular) bonds. In the limit of small deformations the potential energy curve is approximated by a quadratic parabola, which corresponds to a linear z(~,) dependence. Elasticity modulus of solids depends on the type of interactions. For molecular crystals it is ~ 10 9 N m -2, while for metals and covalent crystals it is ~ 101~N m -2 or higher. The value of elasticity modulus is only weakly dependent (or nearly independent) on temperature. At the same time, elasticity may also have completely different, i.e. entropic, nature. For instance, the applied stress causes segments of macromolecules in polymers, or lamellar particles present in clay suspensions to co-orient in a way that leads to a decrease in entropy. In this case the tendency of an object to restore its original configuration is related to thermal motion, which distorts the acquired co-orientation. Under these conditions elasticity modulus (the entropic elasticity modulus) is small and to a significant extent depends on temperature. Elasticity modulus is expressed in N m -2, which is equivalent to J m -3. This means that in agreement with eq. (IX.l) one can formally view the elasticity modulus as twice the elastic energy stored in the unit volume during
655 unit deformation (if only so large deformation could have been possible). One should emphasize that according to eq. (IX. 1) at a given stress, ~, the lower the modulus G, the higher the elastic energy density stored by an object. In real systems elastic deformation of solids takes place only at stresses below some critical value, ~ , above which either a fracture (in the case of brittle objects for which the limiting value of elasticity corresponds to strength), or residual deformation (the objects reveal plasticity) are observed. II. V i s c o us
b e h a v i o r (viscous flow) is characterized
by
proportionality between the shear stress and strain rate, i.e. by linear dependence between x and the rate of shear, ~ =d?/dt, given by Newton' s law:
dr
where rl is the viscosity, expressed in Pa s or in N m -2 s (1 Pa s = 10 pois). Newton's equation can be represented graphically in a ~,- t coordinates as a straight line passing through the origin (Fig. IX-4). Inverse slope of this line yields the viscosity, rl. This idealized viscous behavior is completely irreversible, both mechanically and thermodynamically, meaning that the original shape of the object is not restored after one stops to apply stress. 7
/ 0 Fig. IX-4. Viscous flow
656 Viscous flow is accompanied by energy dissipation, which results in a conversion of all the work into heat. Energy dissipation rate, i.e. the power (energy per unit time) dissipated per unit volume, is given by IJ'rd - "I;'y - 1"15'2 .
Such quadratic dependence between dissipated power and the shear rate is characteristic of the viscous friction. A cylinder with a loose piston filled with some viscous medium (Fig. IX-5) may serve as a model of viscous behavior. In this model it is assumed that the ratio of applied force to the speed of piston motion, F / (dl/dt), is numerically equal to viscosity of described fluid, q, /
q
I7
Fig. IX-5. Viscous behavior modeled by a cylinder with loose piston
The nature of viscous flow is related to self-diffusion, which is the mass transfer occurring when atoms and molecules sequentially exchange their positions with respect to each other as a result of thermal motion. Applied stress lowers the potential barrier for such a migration in one direction, and at the same time increases the barrier for the motion into the opposite direction. As a result, macroscopic deformation gradually takes place. Viscous flow is thus a thermally activated process, and the viscosity, rl, is a characteristic exponential function of temperature. The range ofq values for real systems is very broad: for liquids with low viscosity, such as water and melted metals,
657 1"1 is around
1 0 -3
Pa s, while highly viscous Newtonian fluids may have
viscosities that are thousand or even million times greater than this value. Viscosities of structured systems can be greater than those of low viscous liquids by factors of billions. The probability of thermally activated acts (diffusion) grows with time even at a significant height of potential barrier. For this reason solid objects may reveal a liquid-like behavior, such as that encountered in geological processes, and viscosities may reach 10 ~5- 1020 Pa s or even higher values. III. In contrast to the two cases described above, p l a s t i c i t y (plastic flow) reflects a non-linear behavior, i.e. proportionality between applied stress and deformation is no longer present. For i d e a l i z e d plastic objects, in which one can ignore any elastic deformation, the deformation does not occur at stresses lower than the critical shear stress (the yield stress, or the yield
point), ~*, i.e. 3' = 0 and ~,= 0. When ~ = ~* the deformation starts to occur at a given rate, i.e., plastic flow, which does not require further noticeable increase in shear stress, begins (Fig. IX-6). 3'
-- I7" 17
1;
Fig. IX-6. Plastic flow. Similar
to
viscous
flow,
plastic
flow
is
irreversible
both
thermodynamically and mechanically. The energy dissipation rate in the case
658 of plastic flow is, however, given by the first power of the strain rate, namely
Wd -T*~ t" Such a dependence is characteristic of the
dry friction, i.e. it agrees with
Coulomb's friction law, Ff~ = fffN" Consequently, the model of plastic behavior of a material or disperse system may be represented by two surfaces (two plates) with a mutual friction coefficient, s pressed against each other with normal force, FN, causing the tangential force, Ff~, to be equal to the critical shear stress of material (Fig. IX-7).
1
T
Fig. IX-7. The model of plastic behavior The nature of plasticity is rupture and rearrangements of interatomic bonds which in crystalline objects involve peculiar mobile linear defects, referred to as dislocations. Temperature dependence of plasticity may significantly differ from that of Newtonian fluids. Under certain conditions (including the thermal ones) various molecular and ionic crystals, such as NaC1, AgC1, naphthalene, etc., reveal a behavior close to the plastic one. The values of z* typically fall into the range between 105 and 109 N m 2. At the same time, plastic behavior is typical for various disperse structures, namely powders and pastes, including dry snow and sand. In this case the mechanism of plastic flow is a combination of acts involving the establishment and rupture of contacts between dispersed particles. Plastic object, in contrast to a liquid, maintains the acquired shape after removal of the stress. It is worth
659 pointing out here that the plasticity (Greek ~k0to'coo) of wet clay gave rise to pottery which is the most ancient manufacturing process. This concludes our discussion of three elementary mechanical behaviors and corresponding rheological models. By combining these oneparameter models, one can come up with more complex models that describe rheological properties of various systems. Each combination is usually viewed within a framework of some particular deformation regime, peculiar to a given combination in which qualitatively new properties of a given model, not present within the constituent elements appear. Let us address below some typical combinations of the elementary one-parameter rheological models. I. The M a x w e l l
model
involves a sequential combination of
elasticity and viscosity (Fig. IX-8). According to Newton's Third Law, a sequential combination of such elements means that identical forces (shear stresses, ~) act on both constituents of this model, and the strains corresponding to elastic and viscous elements, 3'G and ~n, respectively, are additive, i.e. their sum yields the net strain, ,{ : t -~+
3'-3'G+3'n
G
-~ 0
1"1 '
Consequently, individual strain rates are added as well to yield the net strain rate: ~) - ) G + ") n"
(IX.2.)
A possible regime characteristic to specific mechanical behavior of such model involves
fast (instantaneous) deformation to the net strain of %
660 followed by maintaining further strain at the same level, i.e. 3' - 70 - const. G 1; i
Fig. IX-8. The Maxwell model
At the initial moment of time, t - 0, the viscous strain is zero, so the total strain and all of the work consumed are solely due to the elastic element. Consequently, the initial stress is given by ~0 - GT0. This stress results in the strain of viscous element. Since the net strain is constant, the elastic strain decreases, consequently, resulting in a decrease in stress. At 7=const, one can write eq. (IX.2) as 1 dT
~~+-G dt
-
O.
1"1
Integration of this linear equation using the initial condition of
z(t-
0 ) - ~0 =
GT0 yields
"c - Z o e x p ( - t/tr ). The quantity t~- rl/G with dimension of time is referred to as the relaxation
time. This quantity graphically corresponds to a point at which the line, tangent to the ~(t) curve at t - 0 point intersects with the x axis (Fig. IX-9). Such a gradual decrease of stress (stress relaxation) is typical for the considered elastoviscous system. In such a system the dissipation of energy originally stored in the elastic element takes place in the viscous element, and as a result, the behavior of system is, in principle, both thermodynamically and mechanically irreversible.
661
T~ \ \\
\ L,
t~ Fig. IX-9. Stress relaxation At stress action times longer than t~the described system has properties similar to those of liquids. As an example of such behavior one can name a flow of glaciers and other processes of rock deformation. Under the conditions when the stress is applied during times shorter than t~the systems behaves as an elastic solid. 2. The K e 1v i n m o d e 1 represents a parallel combination of two linear elements, i.e., of elasticity and viscosity (Fig. IX-10). In this case both strains are the same, while the shear stresses are additive, i.e. ~= % +
~n" An
interesting strain regime in this model is the one in which constant shear stress is applied, i.e. ~ - % - const. G
i Fig. IX- 10. The Kelvin model
662 In contrast to Maxwell's model, in Kelvin's model the presence of a viscous element does not allow for the elastic strain to appear instantly. As a result, the net strain gradually develops in time with a rate given by _dY__~zq _-z0-ZG dt
11
q
_-z~ 11
Integration of this equation yields the strain as a function of time:
' -- -d-O(1-- exp(which corresponds to a gradually slowing increase of strain up, to the limit of 7max= %/G, determined by the elasticity modulus ofHooke's element (Fig. IX11). Such process is referred to as the elastic aftereffect and can be found in solid-like systems that reveal an elastic behavior. Elastic aftereffect is mechanically reversible; the removal of applied stress results in gradual decrease of strain to zero due to the energy stored in the elastic element. The object thus restores its original shape. At the same time, in contrast to the case of a truly elastic body, the deformation of an object that follows Kelvin's model is thermodynamically irreversible due to the dissipation of energy in the tr ~max
/
"----- -'t
~,.~-~ ~ . . . . .
/ 0
Fig. IX-11. Elastic aftereffect.
t
663 viscous element. An example that corresponds to Kelvin's model is that of damping mechanical vibrations in rubber. 3. Let us now introduce a non-linear element. The model that describes i n t e r n a l r e s i d u a l s t r e s s e s involves a parallel combination of elastic element and dry friction (Fig. IX- 12). If the applied stress, ~, exceeds the yield T--T
stress (~ > x* ), the object acquires strain ~' -
, which results in the G
accumulation of energy by elastic element. If also ~ < 2~*, then due to the action of the dry friction element, the object maintains residual ("frozen") stress after the applied stress has been removed. This residual stress is equal to ~ - ,~* , and has the sign opposite to that of initial stress. Apparently, the absolute value of residual stress can not exceed ~*. G
Fig. IX-12. The model of internal residual stresses
4. The B i n g h a m m o d e 1 involves a parallel combination of a viscous Newtonian element and plasticity element, as shown in Fig. IX- 13. This model is widely used in description of colloidal structures, such as aqueous suspensions of clay minerals. rib
Fig. IX-13. The Bingham model
664 Since elements in Bingham' s model are parallel to each other, their strains are identical, and the stresses are additive. The shear stress in Coulomb's element can not exceed the critical value, ~*. Consequently, the strain rate generated by viscous element, should be proportional to the difference between the acting stress and the critical shear stress, namely" "~--'C ~
~
o
riB When ~ < ~*, deformation does not occur (Fig. IX-14). Since the parameter ofBingham's model, riB, defines the derivative, d~/d~,- riB, this constant value is referred to as the differential viscosity, in contrast to a variable effective
viscosity, "c/4[- qef('Y )" 7
q0 =
rib
~qo 0
r*
r
Fig. IX-14. The viscoplastic behavior
In order to describe rheological behavior of real systems, in particular under broadly varying conditions (time, stress), one often uses more complex combinations that include above described simplest rheological models. For instance, in order to adequately describe the system, one may need several relaxation times (not just one), or even a broad spectrum of them. Rheological models thus become more complex, and mathematical description of such models becomes cumbersome as well.
665 One of the approaches used to simplify the solution of such problems is based on the use of the so-called electro-mechanical analogies, it. This approach involves modeling of rheological properties with the help of electric circuits and is based on mathematically identical form of laws that describe the transport of electric current and stresses in solid and liquid objects. For instance, the expression for energy stored by a spring,
G72/2, is identical
to the expression for energy of charged capacitor, q2/2C;the expression for energy dissipation in viscous element, fly 2, is identical to the expression for heat release at ohmic resistance, given by R/2. This allows one to describe and model relaxation of mechanical stresses in Maxwell's model as the voltage decrease that occurs upon discharging a capacitor over the resistor in a chain with the time constant equal to t = RC = rl/G. At the same time it is often impossible to describe successfully real systems even by complex models consisting of elements with constant parameters G, q, r* that remain unchanged in the course of deformation. In such cases one needs to introduce models with variable parameters, which, for instance, include elements of non-linear elasticity, G = G(y), non-linear viscosity, 11___ 11(~'),and variable yield stress, i.e., work hardening, z*
__
~*( 3' ).
IX.2. Structure Formation in Disperse Systems
Structure formation that takes place in disperse systems is the result of spontaneous thermodynamically favorable processes of particle aggregation that lead to a decrease in free energy of the system. These processes include coagulation of dispersed phase or condensation of substance in the zones of direct particle contact. The development of spacial networks (disperse structures) of different kinds defines the ability of disperse system to be converted into a material with particular mechanical properties. Such system becomes qualitatively different from its initial, unstructured state. Material strength, Pc, expressed in N m -2, is an important mechanical characteristic, which determines the ability of a material to withstand external
666 stresses without becoming fractured. Let us address disperse structures of g l o b ul ar type, for many of which contributions into Pc value come from a combination of adhesive forces that act between particles at points of their contacts, i.e., from the strength of individual contacts between particles, p~ (expressed in the units of force, N), and the number of such contacts per unit area of a fracture surface, Z, expressed in m -2. Within the limits of additivity approximation, one can write that Pc ~ 7~Pl. The values ofp~ and Z can be both theoretically evaluated and experimentally assessed [ 16-19].
The value of 7~is defined by geometry of the system, primarily by the size, r, and the packing density of particles. The latter is characterized by porosity of the structure, H, which may be defined as the ratio of pore volume, Vp,to the total volume of porous structure, V, i.e., H = Vp/V. One can evaluate ?( = )~(r, H) dependence from particle dispersity and sample
porosity data by employing particular models of disperse structures. In the simplest case of porous monodisperse structures consisting of spherical particles that form crossing chains containing, on average, g particles from node to node (Fig. IX-15), one can describe this dependence for porosities, 1-I >_48%, using the following expressions [20,21]"
Z-
1 ( 2 r ) 2 ~2
;
II - 1 -
(3~-2). 6~ 3
For structures that are not too porous, within the first approximation one may write that
667 1
Z~
(2r)2 '
which allows one to obtain rough estimates for possible values of Z in real systems. For particles with a diameter 2r = 1O0 btm, Z ~ 103 - 104 contacts per 1 c m 2", for particles with 2r ~ 1 gm
X;~ 107- 10 8 cm 2", if 2r ~ 10 nm, ~ ~ 10 ~
to 10 ~2contacts per 1 c m 2. If one accounts for polydispersity and/or anisometric shape of particles, these values will change accordingly.
2r
n = 1.5 Fig. IX-15. A model of globular disperse structure This geometry is pre-defined by a combination ofphysico-chemical processes of the particle formation during dispersion or condensation. The physicochemical and chemical factors are represented to even greater extent and in greater variety in the strength of individual contacts, p~, characteristic that describes adhesive forces between individual particles. This description corresponds to the case of disperse structures of globular type in which the strength originates from a continuous skeleton that forms due to adhesion of individual particles upon the conversion of free disperse system into structured disperse system. There are, however, other types of structures, such as, e.g., cellular structures (in solidified foams and emulsions), in which the skeleton consists of continuous films of solid-like dispersion medium. Such structures, typical for some polymeric systems, may
668 form during the formation of a new phase in a mixture of polymers by condensation. Separate approach is also needed for the description of structures consisting of anisometric particles, fibers. At the same time, in addition to porous structures, one also encounters various compact microheterogeneous systems, which include ceramics, composite materials and natural "construction materials", such as wood, animal bones, etc. Depending on the nature of forces responsible for the adhesion of particles to each other, one can classify the contacts as coagulation and phase [ 16-22]. In the case of coagulation contacts interactions between particles are limited to a simple contact between them, either via a gap filled with the dispersion medium (Fig. IX-16, a), or directly (Fig. IX-16, b). Such contacts may form in cases when the DLVO potential barrier in the system is either surmounted or totally absent. Thus, coagulation contacts correspond to particles in the primary potential energy minimum (see Chapter VIII, 5). Disperse systems containing such contacts may be characterized by weak strength and by mechanical reversibility, which reflects their ability to restore an original structure upon mechanical destruction
(thixotropy).
~
--
I i I I ! I I 1 ,
|
|
w
u
!
|
I
a b c Fig. IX-16. Different contacts between particles: a and b - coagulation contacts; c - phase contact The strength of a coagulation contact, i.e., adhesive force between particles, is determined primarily by molecular forces. For spherical particles, in agreement with eq. (VII. 13), this force is given by
669 A*?"
Pl-
IF(h0)l ~
12h g '
where A* is the Hamaker constant ( with dispersion medium accounted for); h0 is the equilibrium gap width between particles, and r is the curvature radius of particles at the point of their contact. The energy of particle adhesion in a contact in agreement with eq. (VII. 15) is given by
A*r blc
12h o For example, let the coagulation contact be formed by two spherical particles with radii r ~ 1 gin, or by non-spherical particles touching each other with curvatures of the same radii. For h0 between a few tenths of a nanometer and 1 nm, and A*
10 -19 J, one hasp~ ~ 10 .7 - 10 .8 N. The adhesion energy in
a contact, u c, in this case falls within a range between 10 -16 and 10 -~7 J, which significantly exceeds the value of kT (at T ~ 293 K, kT ~ 4x 10 -2j J). For colloidal particles with radii r ~ 10 nm under the same conditions one obtains p~ ~ 10 .9 - 10-~~ determined
and Uc ~ 10 -~9 J, which is on the border of what can be
experimentally
for an individual
contact.
Oppositely,
for
macroscopic molecularly smooth spherical particles with r ~ 1 mm, one finds that p~ ~ 10 .5
-
1 0 -4
N and u c
~
1 0 -14 J ,
which is within
reach by direct
experimental measurements. It is important for one to realize that all of the above numbers are given for l y o p h o b i c
systems in which the interfacial
tension, cy, is on the order of 10 mJ m 2 or higher, and the Hamaker constant, A* ~24~h2c~ (see Chapter III, 1), reaches the value o f ~ 1 0 19 J. In this case in the contact corresponding to a primary potential energy minimum the adhered
670 particles can not be separated by Brownian motion. At the same time, within the same approximation, at low values of interfacial tension, ey, such as tenths and hundredths of mJ m -2, i.e., when A% 10 2~ - 10 .22 J, for particles of r ~ 1 ~tm, one obtains p~ _<10 -~~N and u c ~10-~9 _ 10-20 j. These numbers indicate that in such l y o p h i l i c system adhered particles can be separated by the energy of Brownian motion, which would oppose structure formation. Under the conditions of a complete displacement of dispersion medium from the gap between particles, i.e., upon rupture of adsorptionsolvation layer, or in a vacuum, a direct point-like contact between the particles may be established (Fig. IX-16, b). Such contact may be formed by one or several atomic cells. In this case, along with the van der Waals forces, the short-range (valent) forces acting over the area of contact may also play some role in particle adhesion. The contribution of such forces into the strength of contact can be estimated as p~ ~ .A/'e2/(b24~%), where Jg" is the number of valent bonds, e is the elementary charge, e0 is the electric constant, and b is typical interatomic distance (see Chapter I, 2). In this case for several valent bonds, A r~ 1 to 10, one finds thatpl ~ 10 .8 N, i.e. in lyophobic systems for particles with r ~ 1 gm the contribution of short-range forces into the strength of contact may be of the same order (or less) as the contribution of van der Waals forces. The given estimates of cohesive forces (CF) in contacts are confirmed by direct measurements. It was pointed out in Chapter VIII that the studies of long range forces in dilute (free disperse) systems require the measurements of force as a
671 function of separation distance between the surfaces. Such measurements can be carried out by various techniques involving
stiff (hard) dynamometric
devices (e.g. AFM, see Chapter VIII). Oppositely, in concentrated (structured) disperse systems the main characteristic of particle interactions, determining the rheology of the system is the c o he s iv e for c e between the particles brought into a direct contact. [22-26]. The measurement of such force does not require fixing the distance, and the measuring device can be soft (pliable) and thus highly sensitive towards measured force. The corresponding techniques and instruments, suitable for direct measurements of cohesive forces (CF) between individual macroscopic particles of any nature placed into any media, were designed and developed by Shchukin and co-workers [ 17-19,27,28]. The key element in this technique is the magnetoelectric galvanometer used as a precision dynamometer (Fig. IX- 17). One of the macroscopic particles, e.g., a glass bead, (a in Fig. IX- 17) is attached to the arm connected to the core of the galvanometer, while the
J ~2R I
Fig. IX-17. Schematic drawing of a magnetoelectric dynamometer; a, b - samples; F compression force; p~ - contact rupture force [ 16,17]
672 other one (b) is mounted on the manipulator, movable in any direction. The purpose of a manipulator is to bring particle a into contact with particle b. The application of direct current to the core of the galvanometer brings the particles together and compresses them with the force, F. Reversing the direction of the current takes the particles apart again, i.e. ruptures the contact. By measuring the amount of current that one needs to apply in order to rupture the contact, and using the calibration curve establishing the relationship between the applied current and the force, one can obtain the cohesive force, i.e. the strength of a contact, p~. The force-current dependence obtained by direct calibration is linear over a broad range of values. For force measurements in a liquid medium, the L-shaped holder is used to mount particle a (Fig. IX-18). The horizontal part of the holder with the particle is immersed into liquid medium, while the vertical part is mounted to the extension of rotating axle of the galvanometer. With such a setup the meniscus at air-solution interface (around the holder) does not create any additional momentum and thus does not influence the measurements.
Fig. IX-18. Schematic representation of a device for measurement of contact strength in liquid media: a, b - samples; 1 - magnetoelectric dynamometer; 2, 3 - holders; 4 - manipulator, 5 - container with a liquid [ 16,17]
673 The experiments with molecularly-smooth macroscopic particles, such as 1-3 mm melted glass or quartz beads allow one to determine the free energy of interaction (cohesion) in equilibrium immediate contact, A 5rf (h0) =p~/~r, with high precision and reproducibility. This in turn yields the estimate for the Hamaker constant, A*, from eqs. (IX. 13) and (IX. 14). If the zone of particle contact does not contain any traces of dispersion medium (as e.g. in the case of completely hydrophobic surfaces of methylated glass in the air), 89A g f yields the estimate of the surface free energy of the solid phase (see Chapter VIII, l). In the mentioned case of methylated glass in the air, cy- 89 ~-~-r-
=l/2p~/~r = 22 mJ/m 2, which agrees with the surface tension of a hydrocarbon. This rejects the once-common opinion about the impossibility of direct measurements of the surface energy of solids. Similar data for different very lyophilic and lyophobic systems are given in Table IX. 1.
TABLE IX. 1. Free energy of interaction (cohesion), 89 ~ , in contacts between moleculary smooth hydrophilic and hydrophobic spherical particles with radii, r of 1-2 mm in different polar and non-polar media ("extreme cases" of lyophilicity and lyophobicity) [ 17, 19,22] Particles
'/2 A J r , m j/m2 Air
Heptane
Water
glass
( >40 )
25
<0.01
methylated glass
22
<0.01
40
fluorinated glass
28
5
50
Figures IX-19 and IX-20 illustrate the isotherms of interaction free energy between methylated surfaces in solutions of different alcohols (i.e. in media with a "continuously changing" polarity) and in an aqueous solution of
674 sodium dodecylsulfate, respectively. In these examples of the reversible behavior, i.e. when dispersion m e d i u m is displaced from the contact zone vzAOff, mJ/m: (5, mJ/m'40
3o \
•
\ ,It~~~x
20
10 ..II ~ \
~
I
1
0.2 0.4 0.6 ~
1
1.0 ~0
Fig. IX- 19. The isotherms of free energy of cohesion between methylated particles in aqueous solutions of methanol (1), ethanol (2), 1-propanol (3), n-butanol (4); q~is the alcohol volume fraction; 2' is the interfacial free energy, '~12, isotherm for solid paraffin - ethanol solution interface, estimated from contact angle measurements as 189 g r - o L cos0 [ 17] V2A ~grr, mJ/m 2 ~AA ~r _/r~, mJ/m 2 40 o I ,~ 2 30 1 x
20 \
x x
\
1
1
1
1
2
3
c x 10 3, mol/l
Fig. IX-20. The isotherm of free energy of cohesion between methylated particles in aqueous sodium dodecylsulfate (SDS) solution of concentration c (1) and the interfacial energy, o~2 isotherm for methylated surface - SDS solution interface (2) estimated as 89 G-f -~rs (here rcs is the two-dimensional pressure obtained from the corresponding adsorption isotherm) [ 17]
675 upon the application of even negligibly small force, and returns back upon the rupture of contacts. This corresponds to the "non-specific" adsorption of molecules that face hydrophobic surface with their hydrocarbon chains. In the opposite case of adsorption from non-polar medium, e.g. adsorption ofoctadecylamine on glass in heptane, the adsorption layer reveals a certain finite strength, i.e. has properties of structural-mechanical barrier (Chapter VII, 5). The described technique and devices are also used for the investigation of formation and properties of phase contacts discusses below. If now one goes back to the estimation of the strength of coagulation structure formed by more or less densely packed particles with r-~ 1 ~m, for lyophobic system withA* ~ l 0 -19 J, one obtains Pc ~ ~ ~
(1/2r)2A*r/2ho 2
104 N m 2. For a suspension or powder this value of Pc has the meaning of a critical shear stress, ~* (see Chapter IX, 1). For a coarser disperse system with particles of r ~ 100 ~tm, the strength Pc ~ 102N m -2, which is characteristic of highly mobile systems, such as sand in the sand-glass. Oppositely, for fine disperse systems consisting of particles with r = 10 nm the value of Pc is around 106 N m -2 or higher, which is characteristic of a significant resistance of a system to molding. At the same time, liquid medium of a similar nature as well as the adsorption of surfactants may lower the interfacial energy, ~, and complex Hamaker constant, A*, by 2 - 3 or more orders of magnitude. In such lyophilized system the adhesion forces and energy are lowered by several orders of magnitude [17]. Under these conditions a system with low concentration of dispersed phase remains stable towards aggregation (see
676 Chapter VII, 5), while in highly concentrated system, in which the particles are mechanically brought into contact, lyophilization results in a substantial decrease in resistance towards strain, ~*, i.e. in plastification of the system(see Chapter VIII,4). The authors have also applied the method of molecular dynamics (see Chapter III,4) for the investigation of peculiarities in the formation and rupture of coagulation contacts at the atomic-molecular level [29]. It was established that at the nano-level under the conditions of a complete lyophilicity (macroscopically, at extremely low interfacial tension at solidliquid interface) the particle did not always readily separate from the substarate, because its separation required that a particular gap was to be formed before the dispersion medium could penetrate underneath the particle and fill this gap. This means that molecular attractive forces had to be mostly overcome before the work of wetting could be performed, which required either the work of external forces or waiting for a long period of time for a suitable fluctuation. Periodic oscillations of force, experimentally observed by J. Israelachvili [30], were also present at several molecular distances. In the case of phase c o n t a c t s
[17,18], particle attachment is
stipulated by short-range cohesive forces that act over an area with linear dimensions substantially exceeding those of elementary cell, i.e., the contact occurs over at least 102 - 103 interatomic bonds. In this case the surface of contact is similar to the region of a grain boundary in polycrystalline material, and the transition from a bulk volume of one particle into that of another particle takes place continuously within the same phase (Fig. IX-16, c), which explains the term "phase contact". The minimum value of strength of such
677 contacts can be estimated as pl ~ 102
e2/(b24~o)
~
10-TN.
More precise estimation for particular materials may be obtained, if one accounts for peculiar features of chemical bonding. Since the phase contact with an area s c ~ (102 - 103)b 2 ~ 10 ~6 m 2 can be considered defect-flee, its theoretical strength is the same as that of an ideal solid (see Chapter I, 2). By using this approach one finds that the minimum values ofp~ ~P~d Sc are about 10 .8 N for low-strength objects with low melting points, ~ 10 .7 N for ionic crystals and metals with an intermediate strength, and ~ 10 .6 N and higher for high-strength refractory materials. As the s c becomes more developed, the strength of phase contact increases to even higher values, reaching 10 .4 - 10 .3 N. In the limiting case of a continuous polycrystalline material (e.g., metal) one obtains the value of strength of cohesion at grain boundaries. Turning to the estimation of the strength of structure with phase contact, one finds that depending on the degree of dispersion, and, consequently, on a number of contacts per unit area and on the strength of an individual contact, i.e., depending on the chemical nature of particles and the entire combination of conditions of structure formation, the values o f P c ~ 7,P~ fall into a range between
10 4
and 108 N/m or broader. In contrast to
coagulation contacts, the rupture of phase contacts is irreversible. To some extent one can view the formation of phase contacts as a result of partial coalescence of solid particles due increased area of direct contact between them, resulting in a transition from a point-like contact to cohesion over an area significantly larger than atomic dimensions. In some
678 cases such transition may occur gradually, e.g. by diffusional transfer of substance into the contact zone, as in sintering. Experimental studies, however, confirmed that in most cases such transition takes place jumpwise. This is exactly the case if formation of a phase contact is associated with a need to overcome the potential energy barrier, determined by the work of formation of a stable under given conditions "nuclei contact", which is the primary bridge that connects the particles. The formation and further growth of such primary contact may result from mutual plastic deformation of particles at contact points due to the action of mechanical stresses that exceed the yield stress of a material. According to A. Polak [31 ], the formation of nuclei contact may also take place in a contact zone between newly formed crystals upon the formation of a new phase in metastable solutions. The coalescence of these crystals results in a formation of finely dispersed polycrystalline structures. According to studies carried out by E. A. Amelina and co-workers, one can experimentally observe the transition from coagulation contacts to phase ones and determine the energy and geometric parameters (work of formation and the size of critical nucleus contact) of such transition by a direct measurement of forces between individual particles in a contact [ 17,21 ]. Let us briefly describe the principles behind such measurements. Two crystals are allowed to touch each other inside a solution, supersaturated with respect to them. The crystals are then kept under the conditions leading to the formation of contact between them.
In these experiments the degree of
solution supersaturation, contact time, the force with which one crystal is pressed against the other, type and amounts of added surfactants, etc. are
679 broadly varied. After some time, one applies the force that separates the crystals and measures the strength of a contact, p~. Experiments of this kind usually yield broadly scattered data which correspond to the formation of microcontacts between different regions of surfaces of real particles, heterogeneous in energy and geometry. The results of measurements of adhesive forces between two CaSO4x2H20 crystals in supersaturated CaSO 4 solutions are shown in Fig. IX-21. These results are
presented as histograms, in which x axis shows the logarithm of contact strength, while the data on the y axis correspond to a fraction of contacts, p, the strength of which falls within a given range; a =
C/Co is
a degree of
supersaturation of solution, where c and Coare the solution concentration and the solubility of calcium sulfate dihydrate, respectively; t is the time of contact between crystals. Figure IX-21 shows that there are two types of contacts between crystals" those with the strength p~ ~<10-7N (coagulation contacts), and those with p~>~10-6 N (phase contacts). An increase in degree of supersaturation (Fig. IX-21, a) and time of contact (Fig. IX-21, b) does not result in a gradual increase in contact strength and further transition of coagulation contacts into phase ones. Had this been the case, one would have observed a smooth shift in a position of the maximum on a histogram. An increase in contact strength occurs jumpwise: a second maximum, separated from the first one by several orders of magnitude of p~, appears. This maximum corresponds to the formation of novel phase contacts, the fraction and average strength of which increase as the degree of supersaturation and duration of contact increase. A gradual shift of a position of this maximum to the right is caused by the growth of primary crystallization
680 bridge, which has fluctuationally formed in a gap between the two particles. Analogously one can investigate phase contacts that establish during the formation and bridging of particles of a new amorphous phase (either organic or inorganic) from a metastable solution, during mutual plastic deformation of particles, sintering, etc. ~t=l 1.0- .
a=3
I
~.
~=1.2
o.8-
I
I
I
0.4
I
I
0 _ ~_.._, -8
-7
,
-8 -7
t =100 s
1.0~ t = 1 0 s
0.4 0.2 0
I
', I
~ -3
t =100 s
Q
0.0
,
-8 -7 -6 -5 -4
!
-6 -5 -4
!
M']--, I
i
~
log pI(N)
t =1000 s
& I
-8 -7 -6 -5 -4
I
I
-87 -6 -5 -4 -3 -8 -7 -6 -5 -4 ~=1.8 "- l o g p l ( N )
Fig. IX-21. The histograms showing the distribution of contact strength between two crystals of calcium sulfate dihydrate in supersaturated calcium sulfate solutions upon varying a - the degree of solution supersaturation, and b - duration of contact, t [16,17]
Since contacts that originate between the particles are the main carriers of strength in disperse structures, a detailed investigation of mechanisms and laws of their formation under different physical-chemical conditions serves as a scientific base for the development of effective methods of controlling mechanical properties of disperse structures and materials. Depending on what type of interparticle contacts prevail, one can
681 classify disperse structures as belonging to either one of the two large groups: coagulation structures and the structures with phase contacts. C o a g u 1at i o n s t r u c t ur e s form when disperse system undergoes aggregation; if there is a sufficient amount of dispersed phase, armoring of the entire volume of disperse system takes place. The amount of dispersed phase that is needed to solidify liquid dispersion medium may sometimes be very small, especially if the system contains very fine and anisotropic particles" this may be only 6% by weight in the case of bentonite clay lamella suspension, and even less for a suspension of thread-like particles of V205. Along with a rather weak strength, the reversibility towards mechanical action, i.e., the ability to spontaneously restore the original configuration after mechanical fracture (in a mobile dispersion medium), are characteristic of coagulation structures. Their latter property is referred to as the thixotropy. Coagulation structures are formed by pigments and fillers of paints, varnishes and polymers. Spacial networks that form during coagulation of clay suspensions by electrolytes represent a characteristic example of thixotropic systems. Due to their ability to undergo structuring in aqueous media, finely dispersed bentonite and montmorillonite clays are widely used as main components of drilling liquids (see Chapter IX, 3) [32]. The formation of disperse s t r u c t u r e s with phase contacts takes place under a great variety of physical-chemical conditions, for instance during sintering and pressing of powders. Disperse systems with phase contacts that form in the course of condensation of a new phase from metastable solutions or melts, are commonly referred to as condensation. If the particles that form
682 such structures have a well defined crystalline structure, one refers to such structures as crystallization-condensation, or simply crystallization, in contrast to newly formed condensation structures composed of amorphous matter. The formation of polycrystalline metals during casting, and formation of natural minerals and rocks are based on the development of crystallization structures. The role crystallization structuring plays in the process of formation of artificial stone from hardening cement or concrete was described in the works by E.E. Segalova, V.B. Ratinov, and A.F. Polak [31 ]. Structures of this type also form during caking of granular, especially hygroscopic materials, i.e. during recrystallization that is followed by the growth of contacts between particles under the conditions of varying humidity. This brings problems into many technological applications, such as filling containers from bins, drug dosing, transport and application of powder fertilizers, oil transport at lowered temperatures (due to the crystallization of paraffins). In some cases, such as in the synthesis of materials, one is interested in creating optimal conditions that favor the development of crystallization structures, while in the others one should create conditions that retard the formation of such structures as much as possible. A relatively simple but at the same time "classical" example illustrating the formation of crystallization disperse structure is hardening of gypsum semihydrate upon reacting with water as shown below [16]: C a S O 4 . 1~2 H 2 0 + 1 1~2 H 2 0
-
C a S O 4 . 2H20
In a broad temperature range gypsum dihydrate, CaSO 4 •
is
thermodynamically more stable than semihydrate. At 20~ the solubility of
683 calcium sulfate dihydrate is approximately 2 grams per liter, while that of C a S O 4 x 1/2 H 2 0
is 6 to 8 g/l, depending on the type of calcium sulfate
modification. Because of this the liquid phase in sufficiently concentrated aqueous suspension ofCaSO 4 x 1/2H20 is saturated with respect to semihydrate but supersaturated with respect to dihydrate. Under these conditions a new colloidal disperse phase forms. This new phase consists of crystals of C a S O 4 x2H20 which together with the particles of initial mineral binder, i.e. C a S O 4 x 89 H20 ' initially form coagulation structure. A decrease in solution supersaturation due to new phase formation is balanced by the dissolution of new portions ofsemihydrate, and hence the crystals of gypsum dihydrate form and grow under continuous supersaturation conditions. The existence of supersaturation and its duration depend on the ratio between the rates at which the dissolution of semihydrate and crystallization of dihydrate take place. Sufficiently high supersaturation allows for the formation of nuclei of crystallization contacts between crystals of gypsum dihydrate at points of their contact. A rapid increase in the number of primary crystallization bridges and their further growth result in qualitative structural changes. Initially formed plastic thixotropic reversible coagulation structure turns into a strong, rigid, and brittle structure that undergoes irreversible fracture upon crushing. The formation of new phase contacts and the expansion of their area results in further increase of strength of the formed structure. As hydration of gypsum semihydrate progresses further, the degree of solution supersaturation in the system decreases, and, consequently, the probability of formation of phase contacts becomes lower. For this reason, at later stages of the process hydration no longer results in the formation of new contacts, but rather leads
684 to growth of crystals and to an increase in the strength of previously formed contacts. If up to this point one destructs the forming structure, the crystals will not bridge, and, as a result, one would end up with a reversible lowstrength coagulation structure. The above described experimental method for the observation of elementary acts of intergrowth of crystallites allowed one to investigate the physico-chemical nature of these processes. A typical feature of crystallization disperse structures is the development of internal stresses during their formation. These stresses result from the pressure generated during directional growth of crystals that are linked with each other into a spacial network. According to the studies by S.I. Kontorovich and L.M. Rybakova these internal stresses, evaluated from broadening of lines in the X-ray diffractograms may be as high as 107 N m -2 or higher [ 16]. If internal stresses that develop within the forming structure reach the strength of the structure, crystallization that takes place during the hydration of primary binder leads to a fracture of structure along the most weak regions. This destructive action of internal stresses may reveal itself in the lowering of structure strength occurring as the hydration proceeds. If internal stresses do not exceed the structure strength, no definite fracture takes place upon stress relaxation. In this case residual internal stresses remain present within the material in a form of elastic deformation and excessive energy related to it. During further use of such materials, especially in the media containing adsorption active substances (see Chapter IX,4), the action of these residual stresses results in a decrease of material's strength and durability.
Thus, lowering of internal stresses is crucial for improving
operational characteristics of disperse materials. As was shown by P.A.
685 Rehbinder, N.V. Mikhailov, and N.B. Ur' ev, this can be reached by preventing too rapid particle bridging at early stages of hydration by applying optimal combination of v i b r a t i o n a l action and using surfactant additives [2]. Internal residual stresses may also be generated during pressing of powders and in some other processes of structure formation. The formation of crystallization disperse structures in aqueous suspensions of other monomineral binders takes place in a similar way. For example, upon hydrational hardening of magnesium oxide, fine Mg(OH)2 structure forms. This process is used for the preparation of catalyst granules with high durability. The same physico-chemical phenomena are laying in the basis of hydrational hardening of calcium aluminosilicate cements (as well as of cements of other compositions) in the manufacturing of construction materials. As the examples of non-crystalline condensation disperse structures, one can name silicates and aluminosilicates (silica and aluminosilica gels, both hydrated and dehydrated). Silica gels form in the course of chemical reaction between sodium silicate and acid, namely [ 16,33]: Na2SiO 3 + 2HC1 + H 2 0 ---) 2NaC1 + Si(OH)4 (HO)3 = S i - OH + HO - S i--- (OH)3
q- " " - - }
--) (HO)3- S i [ - O S i ( O H ) 2 - ] n O - Si ---(OH)3 + ( n - 1)H20 Aluminum salts may also be included into these reactions. The primary polysilicic acid sol (or alumosilicic sol) coagulates and forms gel, the primary coagulation structure. In the presence of a supersaturated dispersion medium
gel particles bridge forming
phase
686 contacts, and the development of condensation structure takes place. The synthesis of many catalysts, carriers, and sorbents, such as the catalysts used in petroleum cracking is based on these processes. The formation of condensation structures in solidifying ethylsilicates is used in the preparation of casting molds for precise casting. Formation of condensation structures is the reason for gelation of solutions of various natural and synthetic polymers. Gelation may be accompanied by conformational changes of macromolecules, which occur in the case of gelling of gelatin and other biopolymers, or in the course of chemical reactions. For instance, according to Vlodavets, partial acetalization ofpolyvinyl alcohol with formaldehyde in acidic medium under the conditions of supersaturation yields fibers of polyvinyl formals which further undergo coalescence and form a network with properties similar to those of leather (and artificial leather substitute). Described examples of disperse structures and of materials based on them give one an opportunity to understand the universal role that structured disperse systems play in various applications. One of the main objectives of modern colloidal science is to provide one with scientific grounds for controlling the properties of disperse structures, among which the mechanical properties are of primary importance. Depending on a particular application, the task may either be to increase the structure strength or, on the contrary, to lower it. The dependence of structure strength, Pc ,on the value of Z and contact strength, p~, (Pc ~ 7,P~) gives the following guidelines for controlling mechanical properties: 1) changing the number of contacts by varying particle size (degree of dispersion) and packing density; 2) changing the strength of
687 individual contacts by varying the conditions under which such contacts form and develop. This allows one to work within a broad range of strengths, say from Pc ~ 104 N m -2 for coarse disperse structures with coagulation contacts to Pc ~ 107 - 108 N m 2 characteristic of fine disperse structures with phase contacts. Consequently, fine degree of dispersion is primarily responsible for high strength of materials, both porous and bulk. In the latter case fine degree of dispersion means that there are no large heterogeneities (defects). Dispersion methods usually do not allow one to obtain particles with r < 1 gm. Comminution (see Chapter IV) to lower sizes becomes much easier in surfactant solutions. Systems with very fine degree of dispersion with particle size down to 10-8 m form by condensation methods when particles of new phase are formed under metastable conditions. For materials to have high strength, the particles should be packed as densely as possible, and large number of strong phase contacts between them should be established. However, the process of molding becomes more complex especially in finely dispersed systems, where relatively weak coagulation contacts create a rather significant net internal strength. This often reveals itself during molding of powders and concentrated pastes. The use of higher pressures, as in powder pressing, introduces other complications, such as the appearance of internal stresses that oppose optimal formation of phase contacts and weaken the material during further use. Consequently, during the stages of preparation and molding one has to overcome high viscoplastic resistance of disperse system by liquefying it, i.e. by lowering fiefand ~*parameters (see Chapter IX, 3). The simplest way by which one can liquify the system, that is, by using the excess
688 of dispersion medium, is in most cases either disadvantageous or unsuitable at all. For instance, the excess of water in cement leads to a concrete that is not frost resistant. Water that is not bound into crystal hydrate and stays in the pores, upon freezing undergoes expansion and bursts the material. In order to reach a highest possible packing density, allowing for the formation of a maximum number of contacts in the structure, while avoiding high internal stresses, one commonly applies treatment by vibration. At the same time, in order to make particle attachment weaker (as in the preparation of dry and wet molds of catalytic and ceramic composites) one uses various surfactant additives, which through their adsorption at the particle surface lower the strength of coagulation contacts, and at certain stages prevent formation of phase contacts. In order to gain control over the structure formation during mineral binder hardening, one, along with surfactants, also introduces electrolytes into the system. This allows one to directionally change supersaturation, the conditions of crystallization, and the coalescence of newly formed hydrate structures, thus carrying out solidification under the optimal set of conditions. In any textile industry fabric fibers are coated with surfactant adsorption layers which prevent strong fiber adhesion and protect against damage during yarn spinning and weaving of fabrics. One also faces similar challenges in food, pulp and paper, and other industries [34]. The investigation of physico-chemical means of controlling the structure, and mechanical (rheological) properties of disperse systems and materials at various stages of their synthesis, molding, treatment and exploitation with a primary focus on a combined effect of mechanical action
689 and interfacial phenomena constitutes the subject of physical-chemical mechanics.
IX.3. Rheological Properties of Disperse Systems In this section we will use particular examples to illustrate the above discussed rheological behavior and structure formation in disperse systems. In real disperse systems broad spectra of particle sizes and interaction parameters stipulate great variety ofrheological properties, utilized in various applications. At the same time disperse systems are the main carriers of mechanical properties of various objects found in nature. A broad variety of rheological properties of disperse systems is reflected, for instance, in a large set of possible values of three main parameters: the shear modulus, G (or the Young modulus, E), the viscosity, q, and the critical shear stress, z* (the yield stress) [21 ]. In continuous systems consisting of solid phases, the parameter G is the modulus of elasticity of rigid body. Its values may fall in the range between 109 and 10 ~ N m -2. The elasticity modulus of common liquids under the conditions of uniform (hydrostatic) compression is also of the same order of magnitude. However, due to low viscosity, the shear elasticity of liquids may only be observed by rapid tests in which the time of stress action is close to the relaxation period. For this reason at typical times of mechanical action liquids with low r I values behave as viscous media. The elasticity modulus in systems with solid and liquid phases is determined by conditions of interaction between dispersed particles. In porous,
690 globular-type disperse structures with phase contacts the elasticity modulus of the system is determined by the elasticity modulus of the solid phase substance, and the number and area of contacts between particles, regardless whether the other phase is liquid or gaseous. The values of elasticity modulus of porous crystallization structures may, for instance, fall into the range between 10 s and 10~~ m -2. Such structures are often brittle, i.e. they reveal a tendency towards an irreversible fracture without a substantial prior residual strain. The fracture occurs at such critical stress at which plastic flow is not yet possible. The elasticity of coagulation colloidal structures with solid and liquid phases may have different nature, especially at relatively low solid volume content, high degree of dispersion, and, most important, in the case of anisometric particles. The examples of such structures include hydrogels of vanadium pentoxide and structured aqueous colloidal suspensions of bentonite clays. It was shown that in these systems the shear elasticity may be due to a lesser or greater degree of particle co-orientation (decrease in entropy) that appears under stress [35]. When stress is removed, the rotational Brownian motion restores chaotic orientation of particles and hence restores an initial shape of the body. Similarly to entropic nature of the elasticity of gas (pressure) and osmotic pressure, the shear elasticity is also stipulated by the change in entropy, which in the present case is the configurational entropy due to co-orientation of particles. Here the elasticity modulus,
Gel is on the order
of nkT, where n is the number of particles per unit volume participating in the Brownian motion (i.e. the number of kinetically independent units). For instance, in dilute suspension of finely dispersed clay for n ~ (3 - 5) x 1023
691 particles per
m 3 one
obtains
Gel ~
103 N m 2.
Elastic deformation also reveals itself in foams and concentrated emulsions. The shear stress in this case is determined by an increase in the interfacial area due to the deformation of the system. Mechanical properties of solidified foams and other solid-like cellular structures are governed by the degree of dispersion, type of backbone structure and a combination of mechanical characteristics of dispersed phase and dispersion medium. The viscosity of dilute fr e e d i s p e r s e systems is mainly determined by the viscosity of dispersion medium which may vary within several orders of magnitude, depending on the nature of the medium. For instance, viscosity of gases is on the order of 10.5 Pa s, for a large number of liquid-like objects it is between 10 .2 and 10 ~~Pa s, and for glasses and solids the viscosity is 1015 - 1020 Pa s or higher. It was shown by Einstein that viscosity undergoes an increase when dispersed particles are introduced into dispersion medium. The reason for an increase in viscosity is the energy dissipation upon rotation of particles in the shear field of the dispersion medium. At low concentration of dispersed phase, i.e. when the particles do not interact with each other, the increase in viscosity, q, is proportional to the volume fraction of dispersed particles, qo" q-qo 130 where rl0 is the viscosity of pure dispersion medium. For spherical particles the coefficient k = 2.5. Consequently, in the absence of particle interactions, the system containing isometric particles behaves as a N e w t o n i a n f l u i d with
692 a viscosity somewhat higher than that of a dispersion medium. If dispersed particles are anisometric (ellipsoidal, rodlike or platelike) or are able to undergo deformation (drops, macromolecules), different tendencies may appear in the flow of dispersion medium, depending on the nature and size of particles [36]. Shear stresses along with the rotational motion of particles caused by them tend also to deform the particles and orient them in the flow in a certain way. The orienting action is opposed by the rotational diffusion of particles. As a result, the degree of orientation is dependent, to a very significant extent, on a strain rate (Fig, IX-22). At low flow rates the orientation of particles may be totally random, while at high flow rates they may become co-oriented. Such co-orientation can be registered by optical methods (see Chapter V). This leads to the dependence of viscosity on the flow rate (or shear stress). 0 - - o - - 0 z-- ~ - - ~ - - 0 - o ~
_-- - -
-~_~_--~_
--~--- c ~ - - 0 - -
~-
~
c~__-
Fig. IX-22. The orientation of anisometric particles in a flow
In this case it is no longer sufficient for one to use the constant viscosity of Newtonian fluid ("Newtonian viscosity"), 11 = ~ / ~ - d~/d~, as a single characteristic of the system. The so-called effective viscosity, fief, along with the differential viscosity, d~/d~ (which in this case is also the function of j~), are thus introduced. At low strain rate (low shear stress) the effective viscosity is maximal. When the applied stress is increased further, the effective viscosity decreases to some minimal value and then remains unchanged with
693 further increase of stress. In free disperse systems this minimum value corresponds to particles, completely co-oriented in a flow (Fig. IX-23).
//
A
/// // ~~'~
J..111 ........ T
Fig. IX-23. Rheological curve of a free disperse system containing anisometric particles: cot q01 = qmax ; cot q)2--qmin
In other cases, including those involving considerable deformation of macromolecules (conformational changes) in the flow, an inverse phenomenon constituting an increase in viscosity with increasing flow rate may also take place. This phenomenon can not be described in terms of the simplest rheological models with constant parameters. Systems in which viscosity is dependent on the strain rate are referred to as ano m al o u s, o r no nN e w t o n i a n f l u i d s . In sufficiently dilute
systems, i.e. in the absence of
interactions between particles, changes in viscosity due to the orientation and deformation of dispersed particles are usually rather small. In structured disperse systems with coagulation contacts between particles the viscosity undergoes a much more abrupt change. In this case one may outline a broad spectrum of different states that fall in between the two extreme cases: fully structured systems and systems with totally destroyed structure. Depending on the magnitude of applied shear stress (flow rate), the rheological properties of structured disperse systems may change from those
694 characteristic of solid-like objects to those typical for Newtonian fluids. According to Rehbinder, this variety in rheological behavior of a real disperse system with coagulation structure may be described by a full rheological curve. The example of such curve, showing the ~,- ~, (~) dependence for a suspension of finely dispersed bentonite, is given in Fig. IX-24. Within this curve one can identify four characteristic regions, marked as I, II, III, and IV (Fig. IX-24).
,v/
iI///[
,'m/ I
Ii
i
i II / Ii//
II I
II
I 0
T~h~
Fig. IX-24. Full rheological curve of structured disperse system; cot q)ll---1]Schw;cot q~,~=rib At lowest shear stresses the behavior of bentonite clays may be the same as that of a solid-like system with high viscosity, which is consistent with the Kelvin model and corresponds to region I. The investigation of relaxation properties of coagulation structures forming in these moderately concentrated dispersions of bentonite clays revealed the existence of an elastic aftereffect at low shear stresses. This aftereffect is related to mutual coorientation ofanisometric particles that are capable of taking part in rotational Brownian motion without any rupture of contacts. Consequently, the nature of elastic aftereffect is entropic. In such systems high viscosities are related
695 to the flow of dispersion medium from shrinking cells via narrow gaps into the neighboring cells, and to the sliding of particles against each other. As the shear stress reaches some value, XSch~,, the region of slow viscoplastic flow, known as Schwedov's region (Fig. IX-24, region II ), is observed in the system with almost undestroyed structure. In this region the shear strain is caused by fluctuational process of fracture and subsequent restoration of coagulation contacts. Due to the action of external pressures this process becomes directed in a certain way. Such mechanism of creep may be described analogously to the mechanisms of fluid flow, the description of which was developed by Ya.B. Frenkel and G. Eiring. As a result of Brownian motion, the particles associated into a single coagulation structure undergo oscillations with respect to their position in contacts. Due to thermal fluctuations, some of these contacts become ruptured, but at the same time other contacts between particles establish at different spots. On average, the number of contacts in the structure remains constant in time and is close to the maximal number. In the absence of applied shear stress the rupture and re-establishment of contacts occur with equal probability in all directions within any cross-section of the system. Under the applied external stress field, the process of contact rupture and formation develops in a certain preferred direction, and one observes a slow macroscopic creep. The creep exists within some range of 9 values, at which the number of breaking and forming contacts is nearly constant and rather small. This region (region II in Fig. IX-24), qualitatively similar to the region III, can be described in terms of a viscoplastic flow model with low critical shear stress, ~Schw,and very high differential viscosity, rlSchw,i.e."
696 T - - "I; S c h w
=
11 S c h w ~t ,
where ]]Schwis the inverse slope, cot qDii, of the curve in region II (Fig. IX-24). Consequently, the values of a variable (effective) viscosity,
Z Tie f = ~
1
=
rlSch w
Y
1--
(* /) 1; S c h w
T
are also very high within this region. Generally, regions I and II, where shear stresses are small, can be characterized by rather low strains, 3', which are typically on the order of fractions of a percent. As some value of ~B is reached, the equilibrium between rupture and formation of contacts is shifted towards contact rupture; the higher the value of ~, the stronger the shift. This corresponds to a situation of intensively collapsing structure. This region of a viscoplastic flow (region III in Fig. IX24) can be described using Bingham's model with considerably different values of parameters, such as relatively high shear stress yield, XB, and rather low differential Bingham viscosity, riB"
T--T
B - - "I~B' ~ .
Bingham's critical shear stress, ~B, corresponding to the beginning of an intense fracture of the structure, may be regarded as the strength of the system (shear strength, to be exact).
697 A shift in the equilibrium towards the rupture of contacts leads to a decrease of the effective viscosity, sometimes by several orders of magnitude"
qef
=-
= qB
After a complete structure disintegration, the disperse system behaves as a Newtonian fluid under the laminar flow conditions (Fig. IX-24, region IV). The viscosity rlminof such relatively concentrated system is higher than that of dispersion medium by the amount exceeding the one predicted by Einstein's equation. In this case the concentration of dispersed phase is sufficiently high, and the particles interact with each other. Further increase in the applied shear stress results in deviations from Newton's equation due to the appearance of turbulence. Early transition of the flow to turbulent behavior sometimes results in the absence of region IV (Fig. IX-24). The full rheological curve of such thixotropic system may be presented as a graph showing the effective viscosity, llef- "[/'y~as a function of the shear stress, ~, (Fig. IX-25). In this Figure rlmincorresponds to viscosity of the system with completely disintegrated structure. For the above mentioned bentonite q
I [[ II
I III J I V
I
11rain
---[
-
I
t
i \
w t
I t
I
1
1
t \1 \t
.I
. I
Fig. IX-25. The full rheological curve presented in rl - $ coordinates
698 suspensions effective viscosity varies over several orders of magnitude, e.g. from 106 to ~ 10.2 Pa s. The rheological characteristics of structured disperse systems may significantly vary under applied vibration. Vibration favors rupture of contacts between particles, and hence leads to system liquefying at lower shear stresses. As a result of this, the ~z) curve is shifted to the left (Fig. IX-26). The vibrational action is commonly used for controlling rheological properties of various disperse systems, such as concentrated suspensions, pastes and powders.
I IZ
I ~'I
0
I
17
Fig. IX-26. The influence of vibrational action on rheological properties of structured disperse system
The thixotropic rheological behavior of structured disperse system to a significant extent depends on the direction towards which the equilibrium between rupture and re-establishment of interparticle contacts is shifted. Because of the finite rate of contact formation due to Brownian motion, a certain amount of time is needed
for the equilibrium to establish.
Consequently, some time is required for a spontaneous thixotropic restoration of structure destroyed by mechanical action. Due to a complete structure disintegration taking place in region IV (Fig. IX-24), the strength, i.e. the
699 critical shear stress, ~*, abruptly decreases (to zero in the extreme case), and the system acquires distinct liquid-like properties. At rest, the system with time regains its strength, i.e. restores solid-like behavior (Fig. IX-27). The strength (critical shear stress, ~*) of a completely restored structure does not depend on a number of structure disintegration cycles. The time required for a complete thixotropic restoration of a primarily disintegrated structure is referred to as the period of thixotropy, tT.
/ i/
0
1
I/
tT
/ 7
.
.
.
.
.
.
t
Fig. IX-27. The critical shear stress, z*, as a function of time, t, required for a thixotropic structure restoration
One can also study the system under the conditions when dynamic equilibrium between contact rupture and restoration has not yet been established. In this case system's dynamic properties will be dependent on the stage at which the measurements are performed, namely, whether a complete or only partial disintegration of structure had occurred; whether these measurements are carried out immediately after fracture or after some period of time.
Usually, when one deals with thixotropic systems, instead of equilibrium (true) values of rheological characteristics some reference (imaginary) values are measured, such as those recorded after a particular period of time had elapsed since a complete disintegration ofthixotropically reversible structure. In some cases, in order to reach an equilibrium between
700 the rupture and restoration of contacts, one needs to perform straining of system at constant rate for a very long time,whichmay not be always achieved in the laboratoryexperiment and, consequently, in practice.
The analysis of full rheological curve illustrates how the complex mechanical behavior can be subdivided into several regions, and how within each of these regions it can be represented by a simple model that utilizes only one or two constant parameters. For this reason, such phenomena as Schwedov's creep and Bingham's viscoplastic flow, whose molecular mechanisms are so different, can be described by substantially different parameters within otherwise the same model. Such subdivision of complex behavior into a finite number of simpler constituents with particular quantitative characteristics illustrates the universal role of macrorheology. At the same time, detailed description of a mechanism involved in each of these elementary stages requires the use of molecular-kinetic concepts and may be characterized as a
microrheologicalapproach.
Let us now turn to some examples illustrating the role which thixotropy of disperse systems plays in nature and technology. Because of their thixotropic properties, suspensions of bentonite clays are used as drilling liquids in oil recovery. When borer is working, such solutions behave as typical liquids. A stream of boring mud that is pumped into the borehole brings coarsely dispersed particles of bored rock to the ground surface. In case when the operation of borer is interrupted, there is a danger of quick settling of bored rock which would lead to jamming of the borer, and consequently, will result in a serious accident. Due to their thixotropic properties, clay suspensions retain rock particles within their restored coagulation network
701 structure, and hence prevent their settling. When the borer resumes its operation, the coagulation structure of clay particles is easily disintegrated and the system acquires liquid-like properties again. At the same time, it is necessary for one to take into account the thixotropic nature of irrigated grounds, especially of watery clay ones. It is of extreme importance in the development of technical protocols for building and road construction. Thixotropic properties of structures of pigments present in oil-based paints are responsible for important rheological properties of these commonly used materials. Stirring of paint results in disintegration of coagulation structure formed by pigment particles. This ensures high mobility of paint, and hence allows one to apply it in thin layers. Quick restoration of coagulation structure prevents draining of paint that is applied on a vertical surface. The formation of structure has to be accounted for when one introduces fillers into rubber and other polymeric materials. If, on the one hand, one desires to reach high strength and rigidity of material (at the expense of its elasticity), one should use as much filler as possible, so that dense particle packing can be achieved. One should prevent the formation of a loose spacial particle network, i.e. it is necessary to weaken mutual particle adhesion while maintaining strong adhesion between particles and the matrix. Since particles in the filling material are typically of polar nature, while the matrix is either non-polar or weakly polar, one can reach the desired effect by introducing surfactants whose adsorption or chemisorption results in the highest possible "oiling" (hydrophobization) of particle surface, i.e. in the highest lyophilization of the system (see Chapter III). For particles of quartz and aluminosilicates and of those of other acidic minerals the described effect
702 can be achieved with sufficient amounts of cationic surfactants. On the other hand, if one desires to maintain a considerable elasticity, a moderate filling leading to the formation of a more or less loose network of filler particles is necessary. In this case fine tuning of adhesive forces between filler particles is required. Indeed, at high degree of system lyophilization (excessively weak adhesion between particles), the filler undergoes sedimentation, which produces non-uniformity in material. However, if at the same time one maintains excessive lyophobicity (strong particle adhesion), the material will be non-uniform as well, due to filler aggregation (lumping). Studies by A.B. Taubman [1,2] indicated that optimal conditions for structuring may be achieved when adsorption coverage reaches approximately half of the monolayer. This example clearly illustrates a universal role that surfactants play in fine control of adhesion between dispersed particles, and, as a result, their role in governing structural and rheological (mechanical) properties of disperse systems and materials.
IX.4. Physico-Chemical Phenomena in Processes of Deformation and Fracture of Solids. The Rehbinder Effect
Let us now turn our attention to the investigation of a role that physicochemical interactions between bulk solids and the medium play in the processes of deformation and fracture. In this section we will be discussing various effects of plastic flow alleviation and decrease in the strength of solid materials. These effects, discovered by Rehbinder, occur due to a reversible physico-chemical action of the medium, related to lowering of the specific
703 surface free energy of solids, and, as a result, to the decrease in the work required for the formation of new surfaces during deformation and fracture processes [8]. These phenomena, commonly known as the Rehbinder effect, are observed upon a c o mb i ne d act i o n of the medium and particular mechanical stresses, when lowering of the interfacial energy by itself does not result in the development of a new surface, but only aids the external forces 2 When one talks about reversibility of the Rehbinder effect, the presence of a thermodynamically stable interface between mutually saturated solid phase and the liquid, as well as complete disappearance of these effects upon the removal of the medium (e.g. by evaporation) are implied. These features emphasize principal difference between the Rehbinder effect and corrosive action of the medium. At the same time, one has to realize that it is not possible to draw here a distinct border line. The term "disintegration" covers a broad range of processes from idealized cases of purely mechanical breaking to destruction by corrosion or dissolution. The Rehbinder effect, i.e. the lowering of strength due to adsorption and chemisorption, stress-caused corrosion, and corrosion fatigue, occupies some intermediate place between these extremes. All these phenomena represent a certain degree of combination between the mechanical work performed by external forces and chemical (physico-chemical) interaction with the medium. Depending on the nature of a solid and of dispersion medium, as well as on the conditions under which the influence of the latter on the former
Spontaneous dispersion becomes possible in the absence of external mechanical action (see Chapter IV) only under the conditions of a very strong lowering of the surface energy down to values close to critical, given by the condition c~ _
704 occurs, the described effects of reversible physico-chemical (adsorption) action of dispersion medium may reveal themselves in various forms and to different extent. A decrease in strength (down to the conditions close to spontaneous dispersion), or alleviated plastic deformation of solid (adsorption plasticizing) may occur. The possibility, type, and intensity of processes of adsorption-caused influence of dispersion medium on mechanical properties of the solid are determined by a number of factors that can be subdivided into the following three groups. I. Chemical nature ofthe medium and the solid, i.e. the nature of forces acting between molecules (atoms) of both phases and in particular at the interface. II. Real structure of solid, determined by a number and type of defects, including the size of grains, availability and sizes of primary microscopic cracks, etc. III. The conditions of deformation and fracture of solids, including type and magnitude of externally applied mechanical action, phase state and quantity of the medium, duration of contact between the dispersion medium and the object, temperature. Let us turn to a detailed examination of factors belonging to Group I., and just briefly mention the role that the structure of a solid and conditions of deformation play. The factors listed in Group I are those through which the Rehbinder effect, i.e. the sharp decrease in strength due to adsorption, reveals in a most drastic way. We will also discuss possible applications of these effects, as well as means to prevent the harm that they may cause [9].
705 IX.4.1. The Role of Chemical Nature of the Solid and the Medium in the Adsorption-Caused Decrease of Material Strength
Let us regard the interfacial energy as the main parameter that characterizes interactions between the solid and the medium, and is determined by their chemical composition. The following simplified considerations allow one to obtain a relationship between the strength and the surface energy for a solid with a defect in a form of a microcrack. Let us examine a plate of a unit thickness to which tensile stress, p, (in N m 2) is applied 3, as shown in Fig IX-28.
,I
H
Fig. IX-28. To the evaluation of the critical size of a crack According to Hooke's law, the elastic deformation of solid leads to the accumulation of elastic energy in it. The density of this energy is given by
p2 Welast- 2E' 3 Earlier in this chapter, when we discussed mechanical properties of disperse systems that revealed viscoplasic flow, we focused on the applied s h e a r s t r e s s and the values of G, r I, ~* related to it. The strength in such systems was identical to the critical shear stress. In the case of mechanical behavior of compact and primarily elasto-brittle solids it is more appropriate for one to use the uniaxial tension, thus replacing shear stress, ~, with tensile stress, p; the shear modulus, G, with the Young modulus, E, and using the rupture resistance, P,, as the strength characteristic (instead of~*)
706 where E is the Young modulus. Let a through crack (a notch) of length l to be formed in such an object. This results in a decrease of elastic deformation within the portion of the volume of this object, and consequently in the decrease of elastic energy density, Welas t. Within a certain degree of approximation, one may consider that such stress relaxation occurs in a region with size l2 (Fig. IX-28), i.e. decrease in the amount of the elastic energy stored within this object is proportional to the square of the crack length, namely
212
A '~elast
P 2E
At the same time, crack widening results in the increase in surface energy due to the formation of a new interface with area proportional to twice the crack length. Consequently, the system's free energy as a function of the crack size can be written as follows"
p212 A , ~ - ~ 2cyl
2E
i.e., when the crack ("nucleus of fracture" in a body) forms, the free energy, A ,~-, passes through a maximum, as is always the case in nucleation (Fig. IX29). The maximum of free energy corresponds to a critical crack size,
/cr
eyE p2 "
(IX.3)
Cracks with sizes greater than /cr are unstable and undergo spontaneous growth. This leads to the development of macroscopic cracks and eventual damage of the solid. One would expect that cracks with sizes smaller than/cr
707 would tend to shrink, i.e. to heal. However, due to slow diffusion, adsorption of admixtures (of oxygen in the case of metals), irreversible changes in the shape of crack walls due to plastic deformation, etc., in real solids such healing may take place only under exceptional conditions, such as during the cleavage of mica in vacuum. A,g-
I
Fig. IX-29. Free energy excess as a function of the crack length Equation (IX.3) can also be written as
(IX.4) In agreement with eq. (IX.4), which was for the first time derived by A. Griffits and is known as the Grifjqts equation, the real strength, P0, of a solid (elasto-brittle) object with a crack of size l is proportional to the square root of the surface energy, and inversely proportional to the square root of the crack length [37]. Taking into account the expression for the theoretical strength of an ideal solid (see Chapter I), one can write that
Pid
b
! cyg E ~ ,[ V b
and the Griffits equation can also be written as
708
Pid One can thus see that the ratio between real and ideal strengths of solid is determined by the ratio between the size of molecules (interatomic distance), b, and the size of a defect. This scheme of the loss of crack stability due to the action of external tensile stresses is valid only in the case of an ideally brittle fracture. Further, while discussing the role that deformation conditions and the structure of solid play in the Rehbinder effect, we will extend these considerations to objects in which fracture is accompanied by significant plastic deformation. We will also discuss the nature of primary cracks, as well as conditions under which they appear. One can use the Griffits equation for comparing the decrease in the surface energy, A~, with the decrease in strength, AP, in solids of different nature affected by the adsorption-active media. It was pointed out by Rehbinder and co-workers, that the highest decrease in the strength of a solid should take place upon its contact with medium of a similar nature, i.e. the one in which molecular interactions are similar to those in a solid undergoing deformation. Let us now turn to the discussion of some typical examples that illustrate a connection between the decrease in the surface energy and decrease in the strength of different solid materials in the presence of adsorption-active media [10-13]. Ionic crystals. Fine porous materials with well developed interfaces are convenient objects for investigation of a connection between the decrease in the interfacial energy and strength reduction. One example of such material
709 is fine disperse structure of magnesium hydroxide forming in the course of hydration hardening of magnesium oxide (see Chapter IX, 2). High specific surface area of such structure (as high as 102 m2/g) allows one to perform direct measurements of vapor adsorption by determining an increase in weight of samples (the adsorption of water vapor results in an increase in sample weight by ~1% per adsorbed monolayer!). Once the adsorption as a function of water vapor pressure, F(Pn20), is known, one can estimate the decrease in the surface energy, Ao, by using the Gibbs equation, written as pt
-Acy - o"o -cr - R T / F d l n p H 2 o . 0 Since materials of this kind undergo brittle fracture, the use of Griffits's equation (IX.4) leads to the following relationship between the strength of dry samples, P0, and the strength of samples that adsorbed moisture, PA" P02 - PA2
Ac~
P0
{50
This relationship is indeed confirmed in Fig. IX-30, which shows that the experimental data points plotted in proper coordinates indeed fall onto a straight line passing through the origin. This graph yields quite reasonable value for %, i.e. {So~ 300 mJ m -2. Special experiments involving the use of NMR indicated that no aqueous phase, capable of dissolving the contacts between particles, was present. Consequently, a decrease in strength was
710 caused solely by water adsorption layer. These studies represent a quantitative evidence supporting the adsorption nature of strength lowering effects.
!
0.5
0
100
200
_ Ac, m__NN m
Fig. IX-30. A comparison between strength lowering and a decrease in the surface energy of finely dispersed porous structure of magnesium hydroxide due to water vapor adsorption [9] The decrease in the surface energy of a solid occurs not only upon the adsorption of vapor, but also (to the same or greater extent) during capillary condensation with a continuous transition to a direct contact between the solid phase and the bulk volume of liquid. Consequently, the effects of strength decrease due to the contact of solids with a liquid phase are also included in the generalized concept of adsorption-induced strength lowering. Polar liquids (primarily, water and aqueous solutions or melts of salts) have nature similar to that of ionic crystals and in such crystals can cause strong strength lowering. The importance of similarity in the nature of ionic crystal and the medium for strength decrease is emphasized in Fig. IX-31, which shows the results of studies of the influence of different media on the strength of samples of polycrystalline potassium chloride. The use of liquid with intermediate polarity (dioxane) allowed one to study systems with a broad continuous range of polarities: from totally non-polar (heptane) to totally polar (water), and to measure the strength as a function of
711 concentration, i.e. to obtain the strength isotherm, P(c). In these experiments one dealt with brittle fracture of samples, thus allowing the application of the Griffits equation, namely
PA ~ O"1/2 and
dcy dc
Using the Gibbs equation (IX.4) one can convert the strength isotherm, PA(C) into the adsorption isotherm, F(c), namely 2PAC CY0 dPA F
m
m
RTPo2
dc
This expression allows one to obtain the estimate for the value of limiting adsorption (see Chapter II), and the area per molecule, SM, in the adsorption layer at the newly formed interface. Such estimates yield reasonable values for the order of magnitude of s M, which confirms that the approach relating the adsorption to the reduction of strength of solids in contact with liquid medium is correct. pAxlO
.6 N
m~
t
.
.
[
.
.
.
20 40 60 80 100 20 40 60 80 100 wt % Heptane Dioxane Water
Fig. IX-31. Influence of the medium on the strength of potassium chloride polycrystals
712 It was shown by N.V. Pertsov [3,38] that the presence of melts of oxides and silicates leads to a strength decrease of many minerals, which typically are the ionic compounds. These studies allow one to understand the nature of some geological phenomena. M o l e c u l a r crystals give one broad opportunities to study the influence
of liquid medium composition on the lowering of strength of solids. For instance, for non-polar substances, such as solid hydrocarbons, the highest decrease in strength is caused by the action of non-polar liquid media. In these systems an increase in polarity of a liquid medium results in an increase in interfacial energy and lesser decrease in strength. The studies with aqueous solutions of typical surfactants (fatty alcohols and fatty acids) indicated that the decrease in strength agreed with the Traube rule (see Chapter II,2 ): identical decrease in the strength of naphthalene samples was observed when solution concentration of each subsequent member of homologous series was lower than that of a preceding one by a factor of 3 to 3.5, as shown in Fig. IX-32.
p,,/eo I ,(I
0.8 0.6 I
0
0.2
,
I
i
0.4
0.6
..............................
0.8
1.0
c, mol/dm3 Fig. IX-32. The isotherms of adsorption-related strength decrease of naphthalene monocrystals immersed into aqueous solutions of various surfactants: 1 - CzHsCOOH; 2 C3HvCOOH; 3 - CnH9COOH;4 - CsHllCOOH [9]
713 A decrease in material strength is also observed when polymeric materials are immersed into the media with corresponding polarities. In the case of non-polar polymers, such as polyethylene, an increase in medium polarity causes a decrease in the degree of strength reduction, while in the case of polar materials, such as polymethylmetacrylate, the decrease in strength becomes greater as medium polarity increases. Metals and some covalent crystals. Liquid metals represent quite active media with respect to these materials [3,39]. A characteristic example is the effect of a thin layer of mercury on the strength of single crystal of zinc (Fig. IX-33). Pure single crystals can be stretched into thin bands. As the deformation increases, the load that one needs to apply to the sample in order to ensure a continued deformation increases as well. Such an increase in the plastic flow stress upon the increase in deformation is related to increase in the density of defects in a crystal, and is referred to as the mechanic strengthening, or work hardening, see Chapter IX, 1. The crystals undergo rupture only when applied loads reach high values, up to several kilograms per mm 2 (107 N m2), and the elongation reaches several times the length of original crystal. The microlayer of mercury placed at the surface of a single crystal dramatically changes its behavior: the specimen undergoes rupture when the degree of deformation is only about 10%. Such rupture results in a brittle cleavage along the cleavage plane of the crystal, i.e. the basis plane (0001), and the bursting strength reaches only hundreds of grams per mm 2 (10 6 N
mZ).
714 P kg/mm z 6' _
fractur~
4-
I I
3-
Zn
I i
2-
I
1-
I t
0.2
Znre ,,,, ,
0.1
0
I
I
,
,
I I
t I
20 40" 100 200 300 400 500 600 e, %
Fig. IX-33. Stretching load, P, as a function of the degree of deformation, e (%), of zinc single crystal" clean and in the presence of liquid mercury
The degree to which the strength of solid metal is lowered upon its contact with other liquid metal with a lower melting point is determined (to a large extent) by the energy of mixing of system's components, u0 (see Chapter III, 1). One can expect to observe strong decrease in the interfacial energy at low, on the order ofkT, positive values of the energy of mixing, u0. This is reflected in simple phase diagrams of binary systems with eutectic (Fig. IX-34). According to N.V. Pertsov and P.A. Rehbinder, strong effects of the adsorption-induced strength decrease are especially typical in systems T,~
432
29.5Ga
20
40
60
80
% Zn
Fig. IX-34. Phase diagram of Zn-Ga system
715 characterized by these simple phase diagrams with eutectic [3,38]. This observation may be used in predicting possible adsorption-related effects and choosing adsorption-active media for various solid objects. In a complete agreement with this rule, one finds the effects of large strength decrease in systems such as Zn-Hg, Zn-Ga, Cd-Ga, A1-Ga, Cu-Bi, Fe-Zn, Ti-Cd, Ge-Au, and others.
IX.4.2. The Role of External Conditions and the Structure of Solid in the Effects of Adsorption Action on Mechanical Properties of Solids
During our discussion of the conditions necessary for fracture of a real solid, we mentioned an important structural parameter that determines the strength of a solid object, namely the critical size of primary crack, lc~,present in the Griffits equation. In some cases, and in particular in those involving fracture of brittle objects (glass), such primary microcracks (patricularly the surface ones) may be present in the object prior to the application of load. The existence of these cracks may be related to surface defects, such as scratches of the surface. In porous materials these obvious defects are the pores themselves. In agreement with the Griffits equation, the largest pores are the ones that play the most important role in fracture. In the case when obvious cavities are not present, in heterogeneous material one can still find some weakened boundaries between particles of different phases (especially if brittle fillers are present) and other types of micro- and macro-inhomogeneities. The size of these inhomogeneities determine the effective value of a linear parameter, l, in the denominator of the Griffits equation. In objects that may undergo plastic flow (i.e. metals) these dangerous
716 defects may form even at the stage of initial plastic deformation (in the single crystal of zinc that we discussed above these defects are formed during crystal elongation by 10%, prior to brittle rupture of specimen covered with a layer of mercury). The plastic deformation in crystalline objects is related to the appearance and movement of specific linear structure defects, referred to as the
dislocations(see Chapter IV, 4) [37]. Within the slip plane, dislocation
separates the portion of a crystal in which the position of atoms was shifted by one interatomic distance from the portion of crystal in which such displacement has not yet occurred (Fig. IX-35). The movement of dislocation ii
_ i ii
/ i IL
[-
!i, ll ..i.,~ I I
_ki 7~ IJ I1
.... i
_
L__,__
__
i
_
1
i 1
/
Fig. IX-35. Displacement of crystal due to dislocation movement through the entire crystal results in the shift of slip plane position by one interatomic distance. Dislocation movement may be retarded by different defects in crystal lattice" foreign atoms, inclusions, other dislocations, boundaries between blocks of single crystals, twin boundaries, grain boundaries in a polycrystal, etc. Such retardation in the movement of dislocations may result in their accumulation at certain spots, i.e. in the formation of significant deformation inhomogeneities, which is followed by local concentration of stresses, resulting in the formation of primary microcracks. The size ofmicrocracks, and, consequently, the strength of object are determined by features of the structure, including the size of grains. For
717 this reason the strength of polycristalline materials in many cases is inversely proportional to the square root of the grain size, d, i.e. in this case l~ ~ d in the Griffits expression. We can thus formulate a general law which is confirmed by numerous experimental observations: strength of a real heterogeneous material object is the higher, the finer the degree of dispersion of its structure, and the lower the probability of presence of large inhomogeneities. The creation of structure with fine degree of dispersion is the main principle of increasing the strength of various materials, such as ceramics, tool materials, construction metals and alloys. Commonly used methods of doping, dispersion hardening, quenching of alloys, and work-hardening are all aimed at achieving fine degree of dispersion of the structure. Bone tissue of live organisms is an excellent example of finely dispersed composite material of high strength. One should emphasize that the adsorption-active medium itself does not create any defects within an object; it may only facilitate their development. For this reason some thread-like defect-free single crystals (whiskers) may be insensitive to the action of the medium. Another feature of the influence of structure of a solid on the intensity of adsorption-induced effects is related to the excessive free energy of defects. The example of such energy excess is the energy of grain boundaries in polycrystalline objects (see Chapter I, 2). This energy, related to the defects in the structure and stored within the object, leads to a thermodynamically more favorable development of primary cracks along such defects upon a contact of solid with adsorption-active medium. Under normal conditions the fracture of polycrystalline object takes place mainly across the body of grains, while in
718 the presence of active melts cracks develop primarily along the grain boundaries. The extreme case of such alleviated propagation of cracks along grain boundaries constitutes the thermodynamically favorable development of liquid interlayers along grain boundaries when the Gibbs-Smith conditions (see Chapter III) are maintained. If Gibbs-Smith conditions are maintained over a significant portion of grain boundaries, the liquid phase may spontaneously penetrate along the grain boundaries without any external mechanical action. Phenomena of this type were observed in such systems as polycrystalline zinc - liquid gallium, sodium chloride (or other alkaline halides) - salt solutions. Liquid penetration may sometimes occur with a substantial rate (~ 1 cm per day) and result in the formation of peculiar, low strength disperse systems in which particles are separated by thin (tens of nm) interlayers of dispersion medium. One more feature of structural defects that determines their role in the adsorption-induced strength decrease is that in most cases the penetration of liquid phase specifically along the defects facilitates the delivery of adsorption-active medium into a pre-fracture zone, and thus allows the medium to influence the development of cracks. In this sense the role of structural defects is closely related to the role of conditions under which deformation and fracture takes place. With respect to discussed case, these are the conditions under which penetration of active medium into the zone of crack formation and development takes place. The growth of macroscopic cracks is often determined by the kinetics of fluid penetration into crack tips and by the laws that govern viscous fluid flow inside the crack. It is obvious that solidification of liquid phase should
719 completely prevent the effects of adsorption-related strength decrease. Higher temperature may result in a weaker Rehbinder effect as well. This occurs due to the facilitation of a plastic flow at elevated temperatures. Thermal
fluctuations
microheterogeneities.
result As
in
the
a result,
relaxation
at
elevated
of
deformational
temperatures
local
concentrations of stresses are too low to initiate the formation of primary microcracks. An increase in temperature thus often leads to a transition from brittle fracture in the presence of adsorption-active medium to plastic deformation. The decrease in the rate of deformation of a solid has an analogous effect: slow deformation also results in an increased probability of the thermally activated relaxation of locally concentrated deformations and stresses. Under the conditions when transition to plastic flow takes place, the development of a crack in solid is accompanied by substantial deformation of the latter. The relationship between the body strength and the critical size of primary crack, lc~,can in this case be described by the equation similar to the Griffits expression, namely [37]:
Pcr "~
I
*E.
G" /cr
In this expression the effective surface energy, cy*,represents a specific (per unit area of a newly formed interface) work of fracture. This quantity, along with the true surface energy, c~, includes the work of plastic deformation per unit area of a surface of crack, i.e. the energy of lattice distortions and damages that occur during the development of a crack. The value of ~* may
720 exceed the true value of a surface energy of solid by several orders of magnitude. At the same time, numerous experiments indicated that the value of ~* itself is very sensitive to changes in the surface energy of a solid, and experiences a sharp decrease when solid object is placed in contact with strongly adsorption-active medium. Among other external factors that influence the adsorption-induced decrease of strength in solids one should mention the nature of applied stresses. As a rule, a decrease in strength is observed under the action of hard stressed states, among which the tensile components prevail. The amount of adsorption-active medium and the means by which it is delivered into the system may also play an important role. Deformation of a solid object in an adsorption-active medium under the conditions when no cracks develop and no fracture takes place, allows one to highlight another type of Rehbinder effect, namely the adsorption
plasticizing of a solid [ 10]. In this effect the adsorption-active medium, while lowering the surface energy, also facilitates the development of new surfaces, which always occurs during deformation of solids. If some constant load is applied to the solid object, the presence of active medium increases the rate of plastic deformation, de/dt (Fig. IX-36, a). At constant deformation rate the deformation resistance decreases, i.e. the yield stress, P*, becomes lower (Fig. IX-36, b). According to Shchukin [9], the mechanism of adsorption plasticizing is based on facilitation of dislocation movement. It was experimentally established that upon deformation of crystals (e.g., of naphthalene or sodium
721 chloride) placed into the active with respect to them medium, the distances of dislocation displacement increase. 0.2% palmitic acid in vaseline oil
P
Air
Po" p*
Air
A
0.2% oleic acid in vaseline oil
t
a
b
Fig. IX-36. The influence of adsorption-active medium on the mechanical properties of lead single crystal" a - the increase in the deformation rate,/:, at a constant load, P ; b - the lowering of the yield stress, P*, at constant deformation rate, ~ [3]
The method of molecular dynamics (MD) provides a remarkable opportunity for the observation of various mechanisms of processes taking place on a micro-(nano-) level, and for the evaluation of the probability of such processes by repeating experiments dozens of times. Figure IX-37 shows the MD simulation of the deformation and fracture of a two-dimensional crystal. Plastic deformation and formation of a dislocation (AB) at elevated temperature (upper part) and the formation of a brittle crack at low temperature (lower part) are shown in Fig. IX-37, a, while simultaneous processes of crack nucleation influenced by the presence of foreign atoms, and their propagation to the tip of the crack, taking place at elevated temperature, are illustrated in both lower and upper, portions of Fig. IX-37, b [40,41 ].
722
L~o~~o~~o 000
0
0
o8ooOoo,-,'o_fi.~
o " ood oOo
G O UoO~,f_..~,p..,~ O0 0 c r~,.---,
~o'-df 5
A
O00 O ~0~0~
00, -- C ~0~0~ C
~ o x o x o oXoXo,~ c X o x o x o oxoxo~, c 5o4oxo~oxoxo~. c n o~,oxoxoxoxo~ c
FoO
(7~o~
0o
oo
5o5o" "~%OoVO~c 5 Fig. IX-37. The molecular dynamic simulation of deformation and fracture of twodimensional crystal: a - plastic deformation and formation of a dislocation at elevated temperature (upper portion) and development of brittle crack at low temperature (lower portion); b - simultaneous processes of crack nucleation and foreign atom propagation at elevated temperature (both upper and lower portions)
The surface energy may also be lowered by surface polarization (the electrocapillary effect). In this case a decrease in the surface tension with increasing surface potential, q~, is described by the Lippmann equation (see Chapter III,3): d(y
dq~ - "
RS~
where psiS the surface charge density. Experimental studies [42] indicated that stretching of metal single crystals under the conditions of electrical polarization in electrolyte solutions, i.e. when the potential deviated from the zero charge potential, %, was indeed followed by an increase in the rate of deformation, k [7]. This is shown in Fig. IX-38 for the single crystal of lead.
723 8 , 10 "4 S"1
3.0 2.0
_
q~= -0.8
1.0
~//qo=q~o=-0.70V ~
/
[
2
1
I
4
6
,
p x 10 6, N / m =
Fig. IX-38. The effect of electrical polarization of the surface on the deformation rate of lead single crystal [42] It is worth pointing out here that if material that is subject to deformation is soluble in the liquid into which it has been immersed, one may observe the so-called
Ioffe effect.
This effect is, for instance, revealed when
brittle crystals of sodium chloride undergo plastic deformation in a pool of water that is not saturated with salt and dissolves the surface. In this case plasticity occurs not due to a decrease in resistance to plastic flow, as in the case of adsorption plasticizing, but rather due to an increase in the strength of crystals because of the dissolution of surface layer containing structural defects.
IX.4.3. The Application of Rehbinder's Effect
The ability of some media to facilitate fracture of" solids and to alter degree of dispersion (see Chapter IV, 5), for a long time has been involved in the technological processes such as grinding of ore prior to its flotational enrichment, milling of a cement, etc. Rehbinder pointed out that fine dispersion can not be achieved by solely mechanical means: the development of highly developed surfaces with enormous areas requires the involvement
724 of physico-chemical factors in controlling the phenomena taking place on the formed surfaces. The role of adsorption-related strength decrease is not only in facilitating disintegration of solids, but also in preventing aggregation due to alleviation or rupture of coagulation contacts between the particles. At the same time adsorption-active components are widely used as components in the formulations of cooling-lubricating fluids (CLF) that are employed to aid various mecahnical treatment (drilling, grinding, cutting, milling, polishing, buffing, etc.). All these types of mechanical treatment involve the dispersion of treated material [43]. The use of small amounts of low-melting point surface active metals in the treatment of quenched steel may serve as an illustration of capabilities of utilization of strong effects of adsorption-related strength decrease. For example, binding polymers of grinding wheels along with diamond powder may be also supplied with a powder of low melting point metal [44,45]. During grinding, microscopic quantities of this active metal melt at high temperatures in contact zone caused by friction. This results in a decrease of strength of treated materials, e.g. of hard alloys (sintered powder composites of tungsten and titanium carbides with cobalt). Strong decrease in the resistance of the treated material allows one to increase the speed of treatment, to improve the quality of treated surface and to extend the lifetime of grinding wheels. The surface effects in the dispersion processes are not limited to the decrease of surface energy and strength due to purely physical adsorption of molecules. In the practical use of CLFs various chemisorption and mechanochemical phenomena also play an important role. These phenomena are related to the destruction of molecules of organic substances due to a combined action
725 of mechanical stresses, high temperatures in the working zones, and interaction of molecules with newly formed solid surface characterized by an increased chemical activity. A broad area of application of p l a s t i c i z i n g a c t i o n of the medium is the use of surface active substances in the treatment of metals by pressure (dragging, rolling, impact extrusion). Plastification of surface layers of the treated metal due to the surface active components incorporated into greases and lubricants significantly reduces loads that one needs to apply in order to carry out these processes. It also improves the quality of the surface and decreases the level of internal stresses in the near-surface layers of a metal [46]. As in the case of metal treatment, simultaneous universal screening action of adsorption layers is revealed in these processes as well. Due to their resistance to displacement from the zone of contact, in particular when chemisorption takes place, adsorption layers
prevent adhesion (i.e. the
formation of phase contacts) between the surfaces of a tool and of the treated material. The role of lubricants is crucial in the performance of various joints and parts of machinery that undergo friction. In this case active components play a dual role: at initial stages they enhance wearing of parts in friction, resulting in a better mutual fit between these parts, while at later stages they protect the surfaces from premature wear. The investigation of a role that surface-active media play in the processes of friction and wear of mechanisms is a separate branch of physical-chemical mechanics, referred to as
tribology
[47]. The use of various catalysts and adsorbents represents a characteristic
726 example of adsorption-related strength decrease. The adsorption and chemisorption at the surface of solid phase and, consequently, the decrease in the surface energy and strength are always involved in the performance of catalysts and adsorbents [48]. Here one has to deal with the reciprocal influence of solid surface and surrounding medium: the contact with solid phase facilitates rupture and rearrangement of interatomic bonds in the adsorbed molecules. These processes of adsorption and rearrangement of the adsorbate molecules, in turn, result in weakened bonds within the near-surface layers of a catalyst. The presence of internal residual stresses originated at different stages of catalyst manufacturing may be sufficient for the increased wear of catalyst. The fracture is enhanced by the pressure exerted by the upper layer of catalyst granules, and is especially strong under intensive fluid regime. The effective means that one can use to prevent fracture of adsorbents and catalysts is creating between particles in a granule an optimum condensation (crystallization) structure with strong phase contacts. The laws linking the effects of strength reduction of hard metals in the presence of metal melts with the nature of interatomic interactions and types of phase diagrams, allow one to predict the possibility of a catastrophic decrease in the strength of constructions that may occur upon melting of antifriction alloys and anti-corrosion coatings, and during welding, soldering or brazing. One can thus effectively search the ways of protection against these dangerous phenomena. At the same time, one also needs to protect metal constructions in contact with a liquid metal from the selective influence of melt on the grain boundaries. This can be achieved by alloying hard metal with a component that by itself does not cause a decrease in strength, but by
727 concentrating at grain boundaries may prevent the penetration of active component. Some interesting results were obtained upon the application of concepts of physical-chemical mechanics to the analysis of processes that take place in the Earth's crust. This direction of studies, referred to as the physicalchemical geomechanics, investigates the processes of formation and disintegration of rocks and minerals. In these processes a combined action of mechanical stresses and adsorption-active liquid media, such as magmatic melts, high temperature fluids, and hydrothermal solutions, becomes pronounced. Experimental studies indicated that oxide and silicate melts are capable of decreasing the strength of minerals by several times. This allows one to compare the action ofmagmatic melts with similar composition in crust break-ups. In our concluding remarks we can emphasize that depending on the nature of interactions between the components that constitute the medium and the solid, as well as on a combination of external conditions, one may observe the effects of various types and intensity. These include the facilitation of plastic flow of solids, or, alternatively, brittle fracture due to the action of lowered stresses; mechanochemical phenomena in the zone of contact; mechanically activated corrosion (the stress corrosion); the processes that are close to the spontaneous dispersion (the so-called quasi-spontaneous dispersion), and the true spontaneous dispersion, leading to the formation of thermodynamically stable lyophilic system. A great variety of types of interactions that exist between the stressed solids and the medium in contact with it requires careful and thorough examination of conditions under which
728 such processes take place, as well as the laws governing them. Such detailed analysis can help one to utilize or, on the contrary, to prevent the consequences of the Rehbinder effect.
References
e
e
.
.
6. 7. 8.
0
10. 11. 12. 13.
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)Colloid Chemistry, Nauka, Moscow, 1978 (in Russian) Lichtman, V.I., Shchukin, E.D., Rehbinder, P.A., Physical-Chemical Mechanics of Metals. Moscow, Acad. Nauk SSSR, 1962 (in Russian); English translation: Jerusalem, Israel Program for Scientific Translations, 1964. Shchukin, E.D. (Editor), Advances in Colloid Chemistry and PhysicalChemical Mechanics, Nauka, Moscow, 1992 (in Russian). Shchukin, E.D., Colloid J. 61 (1999) 545 Shchukin, E.D., Colloids Surfaces, A149 (1999) 529 Shchukin, E.D., Proc. Acad. Sci. USSR, Chem Sci., 10 (1990) 2424 Lichtman, V.I., Rehbinder, P.A., Karpenko, G.V, Influence of SurfaceActive Media on the Deformation Processes in Metals, Acad. Sci. USSR Publ., Moscow,1954 (in Russian) Rehbinder, P.A., Shchukin, E.D., in "Progress in Surface Science", vol. 3, S.G.Davison (Editor), Pergamon Press, Oxford 1972 Latanision, R.M., Fourie, J.F. (Editors), Surface Effects in Crystal Plasticity, Leiden, Noordhoff, 1977 Westwood, A.R.C., Stoloff, N.S. (Editors), Environment-Sensitive Mechanical Behavior, Gordon and Breach, New York, 1966 Westwood, A.R.C., Ahern, J.S., Mills, J.J., Colloids Surfaces 2 (1981)1 Shchukin, E.D., in "Surface Effects in Crystal Plasticity", R.M. Latanision, J.F.Fourie (Editors), Leiden, Noordhoff, 1977
729 14. 15. 16.
17. 18.
19. 20. 21.
22. 23. 24. 25. 26. 27. 28.
29.
30.
31. 32.
Barnes, H.A., Hutton, J.F., Walters, K., An Introduction to Rheology, Elsevier, Amsterdam, 1989 Macosco, C., Rheology: Principles, Measurements, and Applications, Wiley-VCH, New York, 1994 Shchukin, E.D., Amelina, E.A., Kontorovich, S.I., in "Materials Science of Concrete III", Skalny, J. (Editor), American Ceramic Society, Westerville, OH, 1992 Shchukin, E.D., Amelina, E.A., Adv. Colloid Interface Sci. 11 (1979) 235 Shchukin, E.D., J. Colloid Interface Sci., 2001, in the press Shchukin, E.D., in "Fine Particles Science and Technology", E.Pelizzetti (Editor), Kluwer, Dordrecht, 1996 Shchukin, E.D., Kinetika i Kataliz, 6 (1965) 4 Shchukin, E.D., Pertsov, N.V., Osipov, O.I., Zlochevskaya, R.I. (Editors), Physical-Chemical Mechanics of Natural Disperse Systems, Moscow University Publ., Moscow, 1985 (in Russian) Yaminskii, V.V., Pchelin, V.A., Amelina, E.A., Shchukin, E.D., Coagulation contacts in Disperse Systems, Khimiya, Moscow, 1982 Tomlinson, G., Phil. Mag., 6 (1928) 695 Bradley, R.S., Phil. Mag., 13 (1932) 853 Malkina, A.D., and Derjaguin, B.V., Colloid J., 12 (1950) 431 Benitez, R., MacRitchie, F., J. Colloid Interface Sci., 40 (1972) 310 Shchukin, E.D., Yusupov, R.K., Amelina, E.A., and Rehbinder, P.A., Collloid J., 31 (1969) 913 Shchukin, E.D., Colloids surfaces, A149 (1999) 529 Shchukin, E.D., Yushchenko, V.S., Abstracts of the 5th International Conference on Surface and Colloid Science and 59 th ACS Colloid and Surface Science Symposium, Clarkson University, Potsdam, NY, June 1985 Israelachvili, J., Abstracts of the 5 th International Conference on Surface and Colloid Science and 59th ACS Colloid and Surface Science Symposium, Clarkson University, Potsdam, NY, June 1985 Polak, A..F., Solidification of Mineral Binders, Promstroyizdat, Moscow, 1966 (in Russian) van Oss, C.J., Giese, R.F., Colloid and Surface Phenomena in Clays and Related Materials, Dekker, New York, 2001
730 33.
34.
35. 36. 37. 38. 39. 40. 41. 42.
43. 44. 45. 46. 47. 48.
Iler, R.K., The Chemistry of Silica: solubility, Polymerization, colloid and Surface Properties, and Biochemistry, Wiley-Interscience, New York, 1979 Amelina, E.A., Shchukin, E.D., Parfenova, A.M., Pelekh, V.V., Vidensky, I.V., Bessonov, A.I., Aranovich, G., Donohue, M., Colloids Surfaces, A167 (2000) 215 Shchukin, E.D., Rehbinder, P.A., Kolloidn. Zh., 33 (1971) 450 Ur'ev, N.B., Potanin, A.A., Fluidity of Suspensions and Powders, Khimiya, Moscow, 1992 (in Russian) Cottrell, A.H., The mechanical Properties of Matter, Wiley, New York, 1964 Pertsov, N.V., Summ, B.D., The Rehbinder Effect, Nauka, Moscow, 1966 Rostoker, W., McCaughey, Y.M., Markus, H., Embrittlement by Liquid Metals, Reinhold, New York, 1960. Yushchenko, V.C., Grivtsov, A.G., Shchukin, E.D., Dokl. Akad. Nauk SSSR, 215 (1974) 148 Yushchenko, V.C., Grivtsov, A.G., Shchukin, E.D., Dokl. Akad. Nauk SSSR, 219 (1974) 162 Shchukin, E.D., Kochanova, L.A., Savenko, V.I., in "Modem Aspects of Electrochemistry", no.24, R.E.White, B.E.Conway, J.O'M.Bockris (Editors), 1993 Shchukin, E.D., Physical-Chemical Principles of the Intensification of Solids Treatment. Bull. Acad. Sci. USSR, 11 (1973) 30 Pertsov, N.V., Shchukin, E.D., Goryunov, Yu. V., Danilova, F.B., Almazy, 5 (1970) 42 Shchukin, E.D., Polukarova, Z.M., Yushchenko, V. S., Brukhanova, L.S., Rehbinder, P.A., Dokl. Akad. Nauk SSSR, 205 (1970) 86. Veiler, S. Ya., Lihtman, V.I., The Action of Lubricants in Pressure Metal Treatment, Moscow, 1960 (in Russian) Kragel' skiy, I.V., Friction and Wear, Moscow, 1968 (in Russian) Shchukin, E.D. Margolis, L.Ya., Kontorovich, S.I., Polukarova, Z.M.,. Russian Chem. Rev., 65 (1996) 881
731
List of Symbols
Roman symbols A* b C c E e F Ag AJ f
complex Hamaker constant interatomic distance electrical capacitance concentration Young modulus elementary charge force free energy free energy of film
A ~elast elastic energy
ffr G h0 I k l
/cr /7 P P0 PA Pc Pid p* P Pl PH20 q R R r M
friction coefficient elasticity modulus equilibrium gap thickness electric current Boltzmann constant size of stress relaxation region critical crack size number of valent bonds average number of particles stretching load real strength of a solid object strength of solid that adsorbed moisture ? p 312 rupture resistance, material strength theoretical strength of ideal solid yield stress tensile stress strength of individual contacts between particles water vapor pressure electric charge universal gas constant Ohmic resistance radius of particles area of phase contact area per molecule
732 T absolute temperature t time tr relaxation time tT period of thixotropy u0 energy of mixing Uc energy of particle adhesion V volume x, y, z Cartesian coordinates W energy dissipation rate melas t elastic energy density
Greek symbols F 7 7o 7n q rib fief rlSchw
II 71; 71;s
Ps O o* T "1;* TB "17 Schw '1~G
Z ~o
q9 qO
adsorption shear strain elastic shear strain viscous shear strain viscosity differential viscosity effective viscosity Schwedov's viscosity porosity 3.14159... two-dimensional pressure surface charge density interfacial energy, surface tension effective surface energy shear stress critical shear stress Bingam's critical shear stress Schwedov's critical shear stress elastic shear stress viscous shear stress number of particle contacts per unit area degree of deformation electrical constant volume fraction of dispersed particles (section IX.3) surface potential (section IX.4)
733
Subject Index activators 254 activity coefficient 73 adiabatic compressibility 413 adhesive force 702 admicelles 184 adsorbents 641 adsorption-related strength decrease(Rehbinder' s effect) 714 conditions 715 - effect of temperature 719 - in nature and technology 723 - the role of structure 715 the role of chemical composition of phases 715 - types 705, 719 adsorption limiting 97 at interface between condensed phases 176 at air-liquid interface 84 at oil-water interface 177 of surfactants on metal oxides 183 - in two-phase system 66 at grain boundaries at solid-gas interface negative 76,188 removal of toxic substances by 189 - self-consistent field lattice theory (SCFA) of 186 - energy (work of) 190 adsorption activity 99,105 adsorption isotherm 80 - four region 183 adsorption kinetics 56, 247 adsorption layers - area per molecule in 89,109 attraction and repulsions between molecules in 120 deformation of 124 -
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dilute 84 mechanical properties of 122, 123, 557 of insoluble surfactants 80 stability 108 solid 124 structure of 116, 125, 182 rheological properties of 124, 558 - thickness of 69 adsorption plasticizing 704, 720 adsorption potentials of ions 201 aeroflots 254 aerosols 584 degradation (entrapment) 585, 593 properties 587 - explosiveness 592 generation 589, 595 - in meterology 592, 593 in preparation of monodispersed partices 594 settling of particles in 588 - stability 592 aggregation number 472 alkylarylsulfonates 134, 136, 137, 152 -determination of 152, 154 Atomic force Microscopy (AFM), see microscopy autocorrelation function 443 average particle size 422 Avogadro's number -determination of 331,335, 341,342 alkylsulfates 134, 138 alloys 641 anionites 215 Antonow' s Rule 174, 175, 241 attenuation of sound waves 409 -
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Bakker equation 9 Bancroft rule 612 barometric Laplace equation 334
734 bicontinuous system xvi, 494 binary systems 166 binders 684 Bingham's rheological models 663,696 critical shear stress 696 - viscoplastic flow 700 biodegradeability 134 kinetics of 137 biopolymers 501 birefrigence 407 boiling 280 Boltzmann distribution 201,273 Born repulsion 20, 530, 549 Bougert-Beer-Lambert Law 400 Brewster angle 129 brittle fracture 727 Brownian motion 328, 337, 437, 510, 694 - theory of 338 -
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chromatography gas 159 gas-liquid 159 high performance liquid 158, 190 supercritical fluid 159 clouds (as disperse systems) 589 cloud point 483,639 clusters 312 coacervation 499, 640 coagulation 507 - ability 629 and surface charge segregation 631 antagonistic behaviour of electrolytes in 633 by charge neutralization 554 - computer modeling of 564 - fast (rapid) 561,630 --Smoluchowski theory 561,630 kinetics of 561 - of hydrophobic sols 629 - orthokinetic 565 perikinetic 565 - slow 562, 630 - synergistic behavior of electrolytes in 633 time of 562 coagulation-peptization equilibrium 631 coalescence 507 in emulsions 619 coefficient attenuation 410 extinction 153 - in Helmholtz-Smoluchowski equation 362 - of roughness 235 partition 178, 180 - phenomenological cross 363 - sedimentation 432 cohesive force 535,670 - measurement of 533,672 -in liquid medium 672 -
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capillary pressure, see pressure capillary waves 53 capillary wave method 53 capillary - attractive forces 38 condensation 43 effects of the second kind 518 - forces 37 - liquid flow through 373 - permeability 325 - resistance 325 rise 36 - superconductivity 382 cationites 215 catalysis 30 cavitation 280 cell model 420, 452 centrifuge 431 CF method, instrument 670, 671 chemical potential 42, 71, 84, 190, 262, 267,271,283 -
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735
co-ions 352 collective recrystallization 642 collectors 253 collision efficiency 562 "colloidal" crystals 565 colloidal particle scattering 533 colloidal solutions 327 colloidal systems, destabilization by electrolyte 345 colloid vibration potential (CVP) 323, 418,419,454 comminution 314, 724 condensation 681 construction materials 668 contact angle 35,225 - advancing 237 - apparent 237 - hysteresis of 246 - receding 237 contacts coagulation 668 - phase 668, 676, 677 - rupture of 676, 672, 694 molecular dynamics simulation of 676 - strength of 535,670, 672, 679, 680, 686 transition between coagulation and phase 678 CONTIN algorithm 448 controlled colloid formation 307 controlled double-jet precipitation 309 cooling- lubricating fluids 724 coordination number 14, 171,507, 511, 531 coprecipitation from homogeneous solutions 308 correlation delay time 443 corrosion 30, 703 - mechanically activated 727 Cotton-Mouton effect 407 Cottrell's smoke precipitator 594 Coulomb's dry friction law 658 -
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Coulter Counter 425 counter-ions 194, 352 crack 706 critical size of 705 - primary 719 creaming 617 critical - coagulation concentration 552, 629 - - first 632 - - second 633 emulsions, see emulsions micellization concentration 184, 475 - nucleus 274 - lifetime of 291 - shear stress 657 - specific surface free energy 268 state 469 surface (interfacial) tension 610 temperature (point) 13, 167, 465, 469 - - lower 168 - - upper 168 crushing 314 cryoscopy 346 crystal cleavage method 58 crystallization 280 - from melt 282 crystals - covalent 713 - defects in 300 equilibrium 43 ionic 21,708 liquid, see liquid crystals molecular 712 rupture of 713 Cunningham's equation 586 Curie-Wulff expression 43 -
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Debye effect 418 Debye-Huckel parameter 206 decay constant 445 deconvolution (inversion) 449
736 deemulsification, see emulsion degradation defoamers 607 depolarization of light 406 depressants 254 detectors flame ionization 158 mass-spectroscopic 158 detergency 636 detergents 133, 638 - commercial 639 - thermoregulated 639 diafragms 322, 376 - transfer processes in 373 ion selective 381 dialysis 378 dichroizm 408 dielectric spectroscopy 371 differential capacitance 156 differential sedimentation 565 diffraction pattern 438 diffusion 297, 329 kinetics of 348 - non-steady-state 332, 347 - steady-state 330 diffusion coefficient 298, 330 - measurement by PCS 443 - mutual 565 rotational 341 Einstein expression for 331 translational 298, 445 diffusional flux 330 diffusion-sedimentation equilibrium, see sedimentation-diffusion equilibrium diffusiophoresis 323,372 discontinuity surface 4, 66, 176 disinfectants 139 disjoining pressure 516 ionic-electrostatic component of 518, 543,602 -
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- isotherms 602 - molecular component of 518,522, 602 viscous component of 541 dislocations edge 29, 716 movement of 716 - screw 300, 716 disorientation 29 dispersants 132 disperse systems coarse xiv dilute xiv, 691 - fine xiv - formation by condenstation 300 - free disperse xiv, 321,325,485, 509, 691 - - transfer processes in 361 - lyophilic xix, 260, 268, 461 - - of polymers 498 - lyophobic xix 260, 268, 461,669 molecular interactions in 521 - pseudolyophilic 513 - role of entropic factor in stabilization of 266 stability -- aggregation 507 - - sedimentation 507 structured 321, 514 - - transfer processes in 373 - thermodynamics of 261 - with open porosity 322 dispersion analysis 421 dividing surface 5, 65 DLVO theory 543,668 Donnan equilibrium 378 Dorn effect, see sedimentation potential double refraction, see birefrigence drop-weight method -dynamic 55 -static 51 -
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737 Ducleaux-Traube rule 86, 98, 105 Dukhin number 420 Dupr6's experiment 2 dusts 584, 592 dynamic light scattering 442 dynamometric devices hard (stiff) 533, 671 magnetoelectric galvanometer 671 - soft (pliable) 535,671 -
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ebullioscopy 346 ECAH theory of acoustics 411 Eilers-Korff criterion 555 Einstein equation 331 theory of rotational diffusion 341 elastic aftereffect 662 elastic behavior 652, 654 elastic deformation 691 elastic energy density 706 electrical double layer (EDL) compression of 550 - dense part 197 - - charge in 199 - - Stern-Graham model of 198 - diffuse part 197, 201 - - charge in 204 - - effective thickness of 208 dynamic polarization of 418 flat 212 - Helmholtz model of 352 - influence of electrolytes 209 influence of pH 388 - of strongly charged surface 208 - of weakly charged surface 207 electrically conducting particles - electrophoresis of 366 light scattering from 405 electric charge, discrete nature of 588 electroacoustic electrode 420 electroacoustic phenomena 417 electrocapillary phenomena 220 electrocapillary curves 222 -
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effect of surfactant adsorption on 155,224 electrokinetic losses 417 electrokinetic phenomena 322, 349 electrokinetic potential 186, 357 electromechanical analogies 665 electroosmosis 322, 350, 421 electroosmotic flux 375 electroosmotic rise 350 - height of 376 electrophoresis 322, 351 - of conducting particles 366 - of non-conducting particles 366 electrophoretic deposition 365 electrophoretic mobility 370 dynamic 419 -phase angle 420 electrophoretic relaxation effect 369 electrophoretic velocity 358, 368 electrosonic amplitude, 421 electrospraying 316 ellipsometry 130 emulsification 609, 610 emulsifiers 609, 611 emulsion films 514 emulsion polymerization 492 emulsions concentrated 608 critical 343, 461,468 - particle size analysis in 471 degradation of (breaking) 609, 617, 619 dilute 608 direct 608 inverse 608 inversion 471,608 life time of 469 - lyophobic 609 preparation of 609 sedimentation in 618 spontaneous formation 610 stabilization of 560, 610, 618 - by microemulsion 610 - - by powders 616 -
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738 emulsion stability 469, 609 energy - characteristics of condensed phases 12 of cohesion 15 - - in condensed phases 526 - - between particles during coagulation 508,526 dispersion component of 23 - of film - excessive 516 - - free 516 interaction 508, 522, 531 internal excess of 10 of mixing 171 - ion-electrostatic component of 546 - Born, see Bom repulsion of sublimation (evaporation) 14 entrapment cross-section 570 entropy, excess of 10, 13 168 entropy, increase during surfactant adsorption 96 dispersion - of aggregates 179, 512 - - of macrophase 266 - micellization 479 equilibrium - crystal, see crystals sedimentation - diffusion, see sedimentation-diffusion equilibrium - metastable 274 equimolecular surface 7, 68 ethylene oxide 141 extinction 401 -
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fiber adhesion 688 Fick's laws 30, 332, 347 films - condensed 118 gaseous 112 - emulsion 620, 621 of globular proteins 110 - Langmuir-Blodget 92, 534 liquid condensed 114 liquid expanded 113 preparation of 640 solid 114 thin, see thin films - X-type 92 - Y-type 92 - Z-type 92 film tension 515 - excessive 516 full 517 film thickness 515 - primary minimum 549 secondary minimum 548 film thinning 518, 539, 542 filtration 232, 323,373 flocculants 627 flocculation bridging 628 depletion 628 of emulsions 618 - of suspensions 627 Flory-Huggins interaction parameter 185 flotation 251 flow ultramicroscope 437 fluctuations 243,337, 466 - of concentration 342 -light scattering by 43 8 - in light scattering 442 - Einstein-Smoluchowski theory of 340 - thermal 509, 719 fluorescence 394 foam cell 597 foam column height 598,605 foaming 607 -
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falling film aerosol generator 595 fatty amines 139 acids 142, 144 - synthetic 144 - alcohols, primary and secondary 140, 483 -higher 144 -
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739 foam number 596, 604, 605 foams - breaking of 600, 606, 607 generation of 600 hydrostatic equilibrium in 604 - solid 599, 641,667 stability o f 606 foam films 514, 599, 600 thickness of 600 foam lifetime 603 fogs 584 forced hydrolysis 308 forces of inertia 570 fractals 563 Free energy 464 density 4 - - excess 172 component of 27 - excess 2 non-dispersion component of 27 -of cohesion 673 - -isotherms of 674 -of film 517 Freundlich adsorption isotherm 103 friction coefficient 324, 329, 363,433 Frumkin-Freundlich -Guggenheim isotherm 185 Fuchs expression 569 -
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gangue 250 Gedroiz complex 217 gel 514 gelation 686 gel electophoresis 365 Gibbs adsorption equation(isotherm) 72,74, 180, 709, 711 Gibbs effect 536 Gibbs free energy Gibbs-Plateau channel (border) 518, 597, 601,604 capillary pressure in 518 reflection of light from 521 -
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Gibbs-Smith conditions 240, 326 Gibbs-Freundlich-Ostwald equation 42 Girifalco-Good-Fowkes equation 174 gold sols 302, 311,405,436 Gouy-Chapman layer 197, 201 gravimetry 147 gravity-chemical potential 334 Griffits equation 707, 719 grain boundaries - active 326 adsorption at 191 - grove formation at 239 high-angle 29 liquid penetration along 240, 718 - network of 327 segregation of admixtures at 192 - specific surface energy of 29 - two sided-angle of 240 grinding 314, 724 gypsum hardening 682 -
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Hamaker and De Boer macroscopic theory 24 Hamaker constant 25, 173,522, 559, 669, 673 - complex 173 Hartley-Rehbinder micelles 483 heat capacity excess 12 heat of evaporation 14 Helmholtz-Smoluchowski equation 355 hemimicelles 184 Henry region 85, 99 Henry's equation 368 Henry's law 178, 184 heterocoagulation 635 heterogeneous nucleation, see nucleation heteropolyacids 147 homogeneous nucleation, see nucleation Hooke's law 653,705 HPLC, see chromatography Huygens-Fresnel principle 391 Hyamine 1622 150 hydration of polar groups 179
740 Hydrophile-lipophile balance (HLB) 473, 612, 626 calculation of 613 - group numbers 613 numbers 613 hydrophilization 245,626, 637 hydrophobization 245,626 -
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ion pairs 153 ion-surface adsorption interaction 199 ion vibration potential 417 irreversible thermodynamics 322 isoelectric point 184, 385,388, 389 - from electroacoustic measurements 419 of metal oxides 184 of silver iodide 388 - shift of with pH 389, 390 isoionic point 389 isothermal gas transfer 603 isothermal mass transfer 507, 571 - application of 577 - Lifshitz-Slezov theory of 573 -
image analysis 45, 54, 130 indicatrixes of scattered light 396, 400, 403,404 indifferent electrolytes 214 influence of medium on strength 711,712 interaction - dipole - dipole 22, 28 - dispersion (London) 23, 172 - hydrophobic 96 - intermolecular 170, 526 non-dispersion 27, 172 - particle-particle 454 - permanent dipole/induced dipole 23 interfacial tension 168 - at oil-microemulsion interface 497 at water-microemulsion interface 497 - at water-oil interface 497 ion atmosphere thickness 206, 544 interlamellar spacing 623 internal pressure, see pressure ion double layer internal stresses 689 - residual 663 intrinsic losses 416 Ioffe effect 723 ion exchange - capacity 215 - conditional 216 - in clay minerals 216 in soils 216 resins 219 - wastewater purification by 220 ionic crystals 21 ionic lattice 22 -
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Kelvin - expressions, see capillary wave method - law, see Thomson law - rheological model 661,694 Kerr effect 407 Kirpichev-Kick rule 315 Knudsen number 585 Krafft point 480 Langmuir adsorption equation (isotherm) 102, 185, 199 - derivation of 102 balance 91,178 Langmuir-Blodgett technique 92 Langmuir-Blodgett films, see films Laplace law 32 lauric acid 121 Lennard-Jones interaction potential 22 Lifshitz theory of attraction 27, 525,527 light-absorbing particles, scattering from 405 linear tension 228 lipids 142, 143,622 -
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741 liposomes 620 - monolamellar 622 - multilamellar 622 Lippmann equation 221,590 liquid crystals 485,493 liquid metal embrittlement 713 Lorenz-Lorentz relationship 395 luminescence 394 luminous flux 394, 404 lyophilization 637, 638 lyophobicity 702 lyotropic series 384 macrorheology 700 macrovesicles 623 Mandel'shtam waves 465,550 Marangoni-Gibbs effect 538, 542, 601 material strength 665,675 maximum pressure method static 48 dynamic 56 Maxwell effect 407 Maxwell rheological model 659, 665 McBain method 83 micelles 484 mechanical behavior 652 membrane electrodes 157 membranes - biological 620 -electrical conductivity of 324 - semipermeable 347 - transfer processes in 373 - transport of electrolyte through 379 mercury porosimetry 37 metastable state 290 metastable equilibrium, see equilibrium method of cumulants 447 micelles 480 inverse 486, 496 - of hydrophobic sols 361 - o f surfactants 184, 472,475,476, 481 - types of 483 -
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micellar catalysis 492 micellar solutions 488,496, 497 micellization - enthalpy of 479 in non-aqueous media 486 - driving force of 479 - thermodynamics of 476 Michaelis equation 389 microelectrophoresis 365 microemulsions 493 microencapsulation 639 microheterogeneous systems 668 microprofilogram 235 microrheological approach 700 microscopy - atomic force (AFM) 501,533,534 - Brewster angle (BAM) 129 - fluorescence 129 - interfacial force 533 - scanning electron 425,534 - total internal reflection 533 - transmission electron 425 microvesicles 623 Mie theory 403,405 mills 314, 609 modifiers of the first kind 311 modifiers of the second kind 305 molecular dynamics simulation (MD) of formation and rupture of contacts 676 - of Rehbinder effect 721 of wetting and spreading 243 monodisperse sulfur sols 302,403 monodisperse systems 304 - applications of 305 - preparation of 305 --in aerosols 594 - La Mer's theory of formation 306 moving boundary method 365 -
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nanochemistry 312 nanoparticles 310, 311 -in microemulsions 497
742 nefelometry 436 Nernst equation 196 Neuman equation 238 "neutral drop" method 239 neutron reflection 125 Newton's law of viscous flow 354, 357, 655,697 Newtonian fluids 655,658, 691 Nikolsky equation 214 noise 448 non-DLVO stability factors 555 non-indifferent electrolytes 214 non-Newtonian fluids 693 non-wetting 228 du Notiy method 50 nucleation burst of 306, 310 - energy barrier in 294 heterogeneous 284 - homogeneous 273, 591 - - thermodynamics of 277 kinetics of 289 - on rough surface 289 - zone 309 nucleus critical, see critical nucleus - frequency of formation 289, 291, 293 height of 286 two-dimensional 291 Obreimow's method, see crystal cleavage method oiling 701 oil recovery 249, 497, 609 oleophilicity 231,232 oleophobicity 231,232 Onsager relationships 323,364, 375 opalescence 391,394, 406 optical density 401 oscillating bubble method 54 oscillating dipole 391 oscillating drop method 54 oscillating jet method 52 -
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osmosis 346 - reverse 377 osmotic shock 623 Ostwald ripening 507, 571,619, 634 - application of 576 decrease in rate o f 576 Ostwald's rule 304 overoiling 246 overvoltage 366 -
paints 250 paper electrophoresis 365 parismony 450 particle aggregation particle size distribution differential function 422 from sedimentation 430 - in metastable system 290 integral function 422 particles anisometric 341,407, 668 --orientation of in flow 692 - anisotropic 407 bridging 685 - flux of 325 - growth of 295 - - diffusionally controlled 296, 306, 310 - kinetically controlled 298 monodispersed 308 - network of 626, 641 of mixed internal composition 309 - strongly charged 553 surface charge segregation 631 - weakly charged 553 partition coefficient 178, 180 pastes 624, 639 peptization 510, 512, 556, 634 percolation theory 326 periodic oscillations of force 676 Petryanov filter 594 phospholipids 142 phosphorescence 394 -
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743 Photon correlation spectroscopy 443 photophoresis 372 pigments 250 plane of shear 359 plastic behavior 658 plasticity (plastic flow) 657 plasticizers 132, 626 plasticizing action 725 pluoronics 141, 614 Pockels-Langmuir technique 81 point of zero charge 222 Poiseuille equation 373 Poisson distribution function 344 - equation 202 Poisson-Boltzmann equation 203,544, 556 - boundary conditions for 204 polarity equalization rule 176, 182, 189 polarizability 391,392, 395,522 anisotropic 406 polarography 154 polyelectrolyte 380 polydispersity index 447 polyoxyethylene 141,147 polyoxypropylene 141, 147 polypeptides 142 polysilicic acid sol 685 porosity 326 porosimetry, see mercury porosimetry potential determining ions 194, 352, 387 potential energy minimum - primary 529, 556, 669 - secondary 548 potentiometry 153 powders 315, 584, 616 precipitation from homogeneous solutions 307 pressure -capillary 32, 518, 540, 602 condensation 118 disjoining, see disjoining pressure - hydrostatic 10 -
internal (molecular) 17 - osmotic 344, 346, 381,546 expansion of 440 - two-dimensional 81, 107, 120, 226 Privman's aggregation model 310 processes - crossed (cross-) 321 direct 321 protective colloids 132, 557, 625 proteins 380, 501 - adsorption of 143 conformation of 111 - electrophoresis of 362, 365 - in preparation of films 640 isoelectric point of 388 primary structure of 142 solutions 345 surface activity of 142 pumice 641 pyridinium salts 139 -
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quadrupole generation 403 quaternary ammonium salts 139 radiotracer technique 128 radius of curvature, principal 34, 518 Raman scattering 394 random walk 337, 338 Rayleigh scattering 390, 395 reflectance 125 regularization 450 regularized non-negative least-squares technique, see CONTIN algorithm Rehbinder effect 703 - at charged surfaces 220 in dispersion processes 315 Rehbinder-Shchukin criterion 464 relaxation time 660 Reynolds equaton 540, 601 rheological curve 693 rheology 651 Rittinger rule 315 -
744 scaling law 327 scattering multiple 406 - of polarized light 397, 399 of unpolarized light 398, 399 scattering length density 125 scattering losses 415 scattering vector 446 Schulze-Hardy rule 629 Schwedov rheological model 695 Schwedov's creep 700 screening of charge 209 secondary ore deposits 216 second virial coefficient 441 sedimentation 323,329 - analysis 426 - -in centrifuge 431,433 - coefficient, see coefficient - flux 329 sedimentation-diffusion equilibrium 333 sedimentation potential 351,364 sedimetry 148 seeding 295 sessile drop 45 sessile and pendant drop method 45 shear modulus 653,689 shear strain 652 shear stress 652 651,689, 699 - generation by electric field 354 Sheludko cell 601 sieve analysis 424 silica 685 silica gel 685 sintering 577 sitalles 295 size selection 310 slipping plane, see plane of shear smogs 584 smokes 584 soaps 136 soils 217, 218 -
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solubility - colloidal 269 molar 95 solubilization 483,487, 637 direct 488 - enthalpy o f 490 inverse 488 sols 624 - accustoming to electrolytes 633 - second stability zone 632 Spans 142 specific heat of melting 283 spectroscopy acoustic 452 - Auger electron 438 - infrared/near-infrared 151 - low energy electron diffraction 438 - secondary ion mass (SIMS) 43 8 second harmonic generation 130 - sum frequency generation 130 ultraviolet visible (UV) 151 spectroturbidimetric titration 405 spinning drop method 46 spontaneous dispersion 268 spreading 228 kinetics of 241 stability ratio 562, 569, 631 Stefan rule 16 steric factor (stabilization) 558 Stem-Helmholtz layer 187 streaming current 375 streaming potential 421 strength - of ideal crystal 19 - of contacts, see contacts stress relaxation 660 structural losses 411, 417 structural-mechanical barrier 542, 556 Rehbinder's demonstration of 617 stabilization of emulsions by 560, 616,619 superequivalent adsorption of ions 386 -
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745 supramolecular chemistry 313 surface - activity 75, 76 - charge density 200, 367 concentration 69 - conductivity 375 - films, see films - hydrophilic 36, 231 - hydrophobic inactivity 75 - recharging 385 - strongly charged 208 - weakly charged 207 "surface azeotrope" 187 surface free energy 2 excessive xvi, 2 of grain boundary 29 of solids 56 - determination of surface force apparatus 533 surface tension 2, 32, 179 at water-oil interface 181 determination of - - dynamic methods 52 - semi-static methods 48 - static methods 44 - dispersion component of 231 - isotherm 75, 78, 80, 100 - lowering 181 - non-dispersion component of 231 surfactants - amphyphilic 139 analysis of 148 - -by chromatography 158 -electrochemical 154 -gravimetric 147 157 - -spectroscopic 151 - - volumetric 148 anionic 135 action of 139 biologically active 142 cationic 138 -
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dispersing action of 637 fluorinated 134, 606 - modification of solid phase with 247 non-ionic 139 Rehbinder's classification of 131 silicon-organic 133 in textile industry 248 surfactant-selective electrodes 157 suspension effect 364 suspensions 624 - coagulation of 627 - sedimentation in 627 syneresis 600 Smoluchowski equation for rapid coagulation 561 for electrokinetic potential 355 Stokes law 367, 585 Stokes-Einstein equation, see Einstein equation streaming current 375 streaming potential 351,376 Structures - globular 666, 667 - cellular 667 coagulation 681 - -elasticity of 690 crystallization 682, 684 - periodic 634 Svedberg-Oden equation 430 Szyszkowski adsorption equation (isotherm) 97, 474 -
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tangential electric field 420 tensammetry 156 tensile stress 705 theory of fractals 563 thermal depth 413,453 thermal losses 413 thermocapillary phenomena 323 thermodynamic potential 357 thermoosmosis 323 thermophoresis 323
746 thin films - common black 550 - energy of 516 - hydrocarbon 620 - interferometry of 520 - Newton black 550, 603 - two-sided symmetric 515,525 wetting 516 thixotropic restoration 698 thixotropy 668, 681 - period of 699 Thomson (Kelvin) law 41,272, 572, 590 titration 147 - two-phase 148 tribology 725 turbidimetry 436 turbidity 395,400, 403 two-dimensional ideal gas law 88 two-dimensional pressure, see pressure Tweens 142 Tyndall effect 391 Tyndall spectra 403 -
ultracentrifuge 336 ultradisperse systems 311 ultrafiltration 377 ultramicroscopy 365,436 uncompensated oscillating dipole 392 van't Hoff law 344 vesicles 621 vibrational action 685,698 viscoplastic resistance 687 viscosity - differential 664 - dynamic 324, 329 - effective 664, 692 Einstein's law 691 viscous behavior (flow) 655 energy dissipation rate 656 viscous depth 411,453 viscous friction 656 viscous losses 411 -
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volume charge density 358 Vonnegut expression 46 Water null reflecting 126 softening 218 surface activity of 77 wavelength of light 390 of neutron beam 125 - of sound wave 411 - of surface ripples 53 wavenumber 409 "wedge" theory 611 wetting -control by surfactants 244 critical surface tension of 234 - hysteresis of 236 - improvement by surfactants 636 inversion point 245 kinetics of 241 line of 225 - perimeter of 225 - selective (preferential) 132, 230, 246, 636 - specific heat of 233 whiskers 717 Winzor equilibria 495 work hardening 665, 713 work of - adhesion 170, 229 - charging 587 - cohesion 15, 27, 229 comminution 315 - critical nucleus formation 274, 276, 279,289 critical two-dimensional nucleus formation 299 crushing 315 - dispersion 262 fracture 719 grinding 315 heterogeneous nucleation 287 -
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747 homogeneous nucleation 287 - particle formation 264, 273 - spreading 229 - transfer of C H 2 group 87, 95 -
xanthates 254 Xelan 139, 140 X-ray analysis 46 - -of disperse systems 437 diffraction 310, 438 low-angle scattering 438 reflection of 128 -
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yield stress (yield point) 657, 665,689 Young's -elasticity modulus 58, 653,689, 706 equation 226, 288 -
zeolites 218, 312 zero creep method 57 zero creep point 57 zeta potential, see electrokinetic potential zeta potential probe 420 Zimm plots 442 Zsigmondi method 305
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