Particles at Fluid Interfaces and Membranes (Studies in Interface Science)

STUDIES

IN I N T E R F A C E

SCIENCE

Particles at Fluid Interfaces and Membranes Attachment of Colloid Particles and Proteins to Interfaces and Formation of Two-Dimensional Arrays

STUDIES

IN I N T E R F A C E

SERIES D. M 6 b i u s

Vol. I Dynamics of Adsorption at Liquid Interfaces

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

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

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

SCIENCE

EDITORS and R. M i l l e r

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-Lipophile Balance of Surfactants and Solid Particles

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

Attachment of Colloid Particles and Proteins to Inteocaces and Formation of Two-Dimensional Arrays by P.A. Kralchevsky and K. Nagayama

Particles at Fluids Interfaces and Membranes Attachment of Colloid Particles and Proteins to Interfaces and Formation of Two-Dimensional Arrays

PETER A. KRALCHEVSKY

Laboratory of Chemical Physics and Engineering Faculty of Chemistry, University of Sofia, Sofia, Bulgaria KU N IAKI N A G A Y A M A

Laboratory of Ultrastructure Research National Institute for Physiological Sciences, Okazaki, Japan

200I

ELSEVIER Amsterdam - London-

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9 2001 Elsevier Science B.V.

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First edition 2001 L i b r a r y o f C o n g r e s s C a t a l o g i n g in P u b l i c a t i o n D a t a A c a t a l o g r e c o r d f r o m t h e L i b r a r y o f C o n g r e s s h a s b e e n a p p l i e d for.

ISBN: 0 444 50234 3 O

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PREFACE

In the small world of micrometer to nanometer scale many natural and industrial processes include attachment of colloid particles (solid spheres, liquid droplets, gas bubbles or protein macromolecules) to fluid interfaces and their confinement in liquid films. This may lead to the appearance of lateral interactions between particles at interfaces, or between inclusions in phospholipid membranes, followed eventually by the formation of two-dimensional ordered arrays. The present book is devoted to the description of such processes, their consecutive stages, and to the investigation of the underlying physico-chemical mechanisms. For each specific theme the physical background is first given, that is the available experimental facts and their interpretation in terms of relatively simple theoretical models are presented. Further, the interested reader may find a more detailed theoretical description and review of the related literature. The first six chapters give a concise but informative introduction to the basic knowledge in surface and colloid science, which includes both traditional concepts and some recent results. Chapters 1 and 2 are devoted to the basic theory of capillarity, kinetics of surfactant adsorption, shapes of axisymmetric fluid interfaces, contact angles and line tension. Chapters 3 and 4 present a generalization of the theory of capillarity to the case, in which the variation of the interfacial (membrane) curvature contributes to the total energy of the system. Phenomenological and molecular approaches to the description of the interfacial bending moment, the curvature elastic moduli and the spontaneous curvature are presented. The generalized Laplace equation, which accounts for the latter effects, is derived and applied to determine the configurations of free and adherent biological cells; a convenient computational procedure is proposed. Chapters 5 and 6 are focused on the role of thin liquid films and hydrodynamic factors in the attachment of solid and fluid particles to an interface. The particles stick or rebound depending on whether repulsive or attractive surface forces prevail in the liquid film. Surface forces of various physical nature are presented and their relative importance is discussed. In addition, we consider the hydrodynamic interactions of a colloidal particle with an interface (or another particle), which are due to flows in the surrounding viscous liquid. Factors and mechanisms for detachment of oil drops from a solid surface are discussed in relation to washing. Chapters 7 to 10 are devoted to the theoretical foundation of various kinds of capillary forces. When two particles are attached to the same interface (membrane), capillary interactions, mediated by the interface or membrane, may appear between them. Two major kinds of capillary interactions are described: (i) capillary immersion force related to the surface wettability and the particle confinement into a liquid film (Chapter 7), (ii) capillary flotation force originating from interfacial deformations due to particle weight (Chapter 8). Special attention is paid to the theory of capillary immersion forces between particles entrapped in spherical liquid films (Chapter 9). A generalization of the theory of immersion forces allows

vi one to describe membrane-mediated interactions between protein inclusions into a lipid bilayer (Chapter 10). Chapter l l is devoted to the theory of the capillary bridges and the capillary-bridge forces, whose importance has been recognized in phenomena like consolidation of granules and soils, wetting of powders, capillary condensation, long-range hydrophobic attraction, bridging in the atomic-force-microscope measurements, etc. The treatment is similar for liquid-in-gas and gasin-liquid bridges. The nucleation of capillary bridges, which occurs when the distance between two surfaces is smaller than a certain limiting value, is also considered. Chapter 12 considers solid particles, which have an irregular wetting perimeter upon attachment to a fluid interface. The undulated contact line induces interfacial deformations, which are theoretically found to engender a special lateral capillary force between the particles. Expressions for the dilatational and shear elastic moduli of such particulate adsorption monolayers are derived. Chapter 13 describes how lateral capillary forces, facilitated by convective flows and some specific and non-specific interactions, can lead to the aggregation and ordering of various particles at fluid interfaces or in thin liquid films. Recent results on fabricating twodimensional (2D) arrays from micrometer and sub-micrometer latex particles, as well as 2D crystals from proteins and protein complexes are reviewed. Special attention is paid to the methods for producing ordered 2D arrays in relation to their physical mechanisms and driving forces. A review and discussion is given about the various applications of particulate 2D arrays in optics, optoelectronics, nano-lithography, microcontact printing, catalytic films and solar cells, as well as the use of protein 2D crystals for immunosensors and isoporous ultrafiltration membranes, etc. Chapter 14 presents applied aspects of the particle-surface interaction in antifoaming and deJoaming. Three different mechanisms of antifoaming action are described: spreading mechanism, bridging-dewetting and bridging-stretching mechanism. All of them involve as a necessary step the entering of an antifoam particle at the air-water interface, which is equivalent to rupture of the asymmetric particle-water-air film. Consequently, the stability of the latter liquid film is a key factor for control of ~baminess.

The audience of the book is determined by the circle of readers who are interested in systems, processes and phenomena related to attachment, interactions and ordering of particles at interfaces and lipid membranes. Examples for such systems, processes and phenomena are: formation of 2D ordered arrays of particulates and proteins with various applications: from optics and microelectronics to molecular biology and cell morphology; antifoaming and defoaming action of solid particles and/or oil drops in house-hold and personal-care detergency, as well as in separation processes; stabilization of emulsions by solid particles with application in food and petroleum industries; interactions between particulates in paint films; micro-manipulation of biological cells in liquid films, etc. Consequently, the book could be a useful reading for university and industrial scientists, lecturers, graduate and post-graduate students in chemical physics, surface and colloid science, biophysics, protein engineering and cell biology.

vii

Prehistory. An essential portion of this book, Chapters 7-10 and 13, summarizes results and research developments stemming from the Nagayama Protein Array Project (October 1990 - September 1995), which was a part of the program "Exploratory Research for Advanced Technology" (ERATO) of the Japanese Research and Development Corporation (presently Japan Science and Technology Corporation). The major goal of this project was formulated as follows: Based on the molecular assembly of proteins, to fabricate macroscopic structures (2D protein arrays), which could be useful in human practice. The Laboratory of Thermodynamics and Physicochemical Hydrodynamics (presently lab. of Chemical Physics and Engineering) from the University of Sofia, Bulgaria, was involved in this project with the task to investigate the mechanism of 2D structuring in comparative experiments with colloid particles and protein macromolecules. These joint studies revealed the role of the capillary immersion forces and convective fluxes of evaporating solvent in the 2D ordering. In the course of this project it became clear that the knowledge of surface and colloid science was a useful background for the studies on 2D crystallization of proteins. For that reason, in 1992 one of the authors of this book (K. Nagayama) invited the other author (P. Kralchevsky) to come to Tsukuba and to deliver a course of lectures for the project team-members entitled: "Interfacial Phenomena and Dispersions: toward Understanding of Protein and Colloid Arrays". In fact, this course gave a preliminary selection and systematization of the material included in the introductory chapters of this book (Chapters 1 to 6). Later, after the end of the project, the authors came to the idea to prepare a book, which is to summarize and present the accumulated results, together with the underlying physicochemical background. In the course of work, the scope of the book was broadened to a wider audience, and the material was updated with more recent results. The major part of the book was written during an 8-month stay of P. Kralchevsky in the laboratory of K. Nagayama in the National Institute for Physiological Sciences in Okazaki, Japan (September 1998 - April 1999). The present book resulted from a further upgrade, polishing and updating of the text. Acknowledgments. The authors are indebted to the Editor of this series, Dr. Habil. Reinhard Miller, and to Prof. Ivan B. Ivanov for their moral support and encouragement of the work on the book, as well as to Profs. Krassimir Danov and Nikolai Denkov for their expert reading and discussion of Chapters 1 and 14, respectively. We are also much indebted to our associates, Dr. Radostin Danev and Ms. Mariana Paraskova, for their invaluable help in preparing the numerous figures. Last but not least, we would like to acknowledge the important scientific contributions of our colleagues, team-members of the Nagayama Protein Array project, whose co-authored studies have served as a basis for a considerable part of this book. Their names are as follows. From Japan: Drs. Hideyuki Yoshimura, Shigeru Endo, Junichi Higo, Tetsuya Miwa, Eiki Adachi and Mariko Yamaki; From Bulgaria: Drs. Nikolai Denkov, Orlin Velev, Ceco Dushkin, Anthony Dimitrov, Theodor Gurkov and Vesselin Paunov. October 2000

Peter A. Kralchevsky and Kuniaki Nagayama

viii

Photograph of the process of 2D array formation from latex particles, 1.7 gm in diameter, under the action of the capillary immersion force and an evaporation-driven convective flux of water (see Chapter 13); the tracks of particles moving toward the ordered phase are seen [from N.D. Denkov, O.D. Velev, P.A. Kralchevsky, I.B. Ivanov, H. Yoshimura, K. Nagayama, Langmuir 8 (1992) 3183].

ix

CONTENTS

Preface CHAPTER 1. PLANAR FLUID INTERFACES

1

1.1. 1.1.1. 1.1.2. 1.1.3. 1.1.4.

Mechanical properties of fluid interfaces The Bakker equation for surface tension Interfacial bending moment and surface of tension Electrically charged interfaces Work of interfacial dilatation

2 2 6 8 11

1.2. 1.2.1. 1.2.2. 1.2.3. 1.2.4. 1.2.5.

Thermodynamical properties of planar fluid interfaces The Gibbs adsorption equation Equimolecular dividing surface Thermodynamics of adsorption of nonionic surfactants Theory of the electric double layer Thermodynamics of adsorption of ionic surfactants

12 12 14 15 20 25

1.3. 1.3.1. 1.3.2. 1.3.3. 1.3.4. 1.3.5. 1.4. 1.5.

Kinetics of surfactant adsorption Adsorption under diffusion control Adsorption under electro-diffusion control Adsorption under barrier control Adsorption from micellar surfactant solutions Adsorption from solutions of proteins Summary References

37 38 41 48 53 55 56 58

CHAPTER 2. INTERFACES OF MODERATE CURVATURE: THEORY OF CAPILLARITY

64

2.1. The Laplace equation of capillarity 2.1.1. Laplace equation for spherical interface 2.1.2. General form of Laplace equation

65 65 66

2.2. 2.2.1. 2.2.2. 2.2.3.

Axisymmetric fluid interfaces Meniscus meeting the axis of revolution Meniscus leveling off at infinity Meniscus confined between two cylinders

71 72 75 77

2.3. 2.3.1. 2.3.2. 2.3.3. 2.3.4. 2.4. 2.5.

Force balance at a three-phase-contact line Equation of Young Triangle of Neumann The effect of line tension Hysteresis of contact angle and line tension Summary References

8O 8O 85 87 92 98 99

CHAPTER 3. SURFACE BENDING MOMENT AND CURVATURE ELASTIC MODULI

105

3.1. 3.1.1. 3.1.2. 3.1.3.

Basic thermodynamic equations for curved interfaces Introduction Mechanical work of interracial deformation Fundamental thermodynamic equation of a curved interface

106 106 106 109

3.2.

Thermodynamics of spherical interfaces

112

3.2.1. Dependence of the bending moment on the choice of dividing surface 3.2.2. Equimolecular dividing surface and Tolman length 3.2.3. Micromechanical approach

3.3. 3.3.1. 3.3.2. 3.4. 3.5.

Relations with the molecular theory and the experiment Contributions due to various kinds of interactions Bending moment effects on the interaction between drops in emulsions Summary References

112 115 117

123 123 129 132 133

CHAPTER 4. GENERAL CURVED INTERFACES AND BIOMEMBRANES

137

4.1. 4.2. 4.2.1. 4.2.2. 4.2.3. 4.2.4.

Theoretical approaches for description of curved interfaces Mechanical approach to arbitrarily curved interfaces Analogy with mechanics of three-dimensional continua Basic equations from geometry and kinematics of a curved interface Tensors of the surface stresses and moments Surface balances of the linear and angular momentum

138 140 140 142 145 147

4.3. 4.3.1. 4.3.2. 4.3.3. 4.3.4.

Connection between the mechanical and thermodynamical approaches Generalized Laplace equation derived by minimization of the free energy Work of deformation: thermodynamical and mechanical expressions Versions of the generalized Laplace equation Interfacial rheological constitutive relations

151 151 154 157 158

4.4. Axisymmetric shapes of biological cells 4.4.1. The generalized Laplace equation in parametric form 4.4.2. Boundary conditions and shape computation

162 162 164

4.5. 4.5.1. 4.5.2. 4.6. 4.7.

168 168 174 178 179

Micromechanical expressions for the surface properties Surface tensions, moments and curvature elastic moduli Tensors of the surface stresses and moments Summary References

CHAPTER 5. LIQUID FILMS AND INTERACTIONS BETWEEN PARTICLE AND SURFACE

183

5.1. 5.1.1. 5.1.2. 5.1.3. 5.1.4.

184 184 186 191 197

Mechanical balances and thermodynamic relationships Introduction Disjoining pressure and transversal tension Thermodynamics of thin liquid films Derjaguin approximation for films of uneven thickness

5.2. Interactions in thin liquid films 5.2.1. Overview of the types of surface forces 5.2.2. Van der Waals surface forces 5.2.3. Long-range hydrophobic surface force 5.2.4. Electrostatic surface force 5.~2.5. Repulsive hydration force 5.2.6. Ion-correlation surface force 5.2.7. Oscillatory structural and depletion forces 5.2.8. Steric interaction due to adsorbed molecular chains 5.2.9. Undulation and protrusion forces 5.2.10. Forces due to deformation of liquid drops

201 201 203 211 212 216 220 224 231 235 237

xi

5.3. 5.4.

Summary References

240 241

CHAPTER 6. PARTICLES AT INTERFACES: DEFORMATIONS AND HYDRODYNAMIC INTERACTIONS

248

6.1. Deformation of fluid particles approaching an interface 6.1.1. Thermodynamic aspects of particle deformation 6.1.2. Dependence of the film area on the size of the drop/bubble

249 249 254

6.2. 6.2.1. 6.2.2. 6.2.3. 6.2.4. 6.2.5. 6.2.6. 6.2.7.

Hydrodynamic interactions Taylor regime of particle approach Inversion thickness for fluid particles Reynolds regime of particle approach Transition from Taylor to Reynolds regime Fluid particles of completely mobile surfaces (no surfactant) Fluid particles with partially mobile surfaces (surfactant in continuous phase) Critical thickness of a liquid film

258 259 260 261 261 263 264 265

6.3. 6.3.1. 6.3.2. 6.3.3. 6.4. 3.5.

Detachment of oil drops from a solid surface Detachment of drops exposed to shear flow Detachment of oil drops protruding from pores Physicochemical factors influencing the detachment of oil drops Summary References

268 268 276 280 282 284

CHAPTER 7. LATERAL CAPILLARY FORCES BETWEEN PARTIALLY IMMERSED BODIES 287

7.1. 7.1.1. 7.1.2. 7.1.3. 7.1.4. 7.1.5.

Physical origin of the lateral capillary forces Types of capillary forces and related studies Linearized Laplace equation for slightly deformed liquid interfaces and films Immersion force: theoretical expression in superposition approximation Measurement of lateral immersion forces Energy and force approaches to the lateral capillary interactions

288 288 294 296 299 303

7.2. 7.2.1. 7.2.2. 7.2.3.

Shape of the capillary meniscus around two axisymmetric bodies Solution of the linearized Laplace equation in bipolar coordinates Mean capillary elevation of the particle contact line Expressions for the shape of the contact line

3O8 308 312 314

7.3. 7.3.1. 7.3.2. 7.3.3. 7.3.4.

Energy approach to the lateral capillary interactions Capillary immersion force between two vertical cylinders Capillary immersion force between two spherical particles Capillary immersion force between spherical particle and vertical cylinder Capillary interactions at fixed elevation of the contact line

316 316 321 327 328

7.4. 7.4.1. 7.4.2. 7.4.3. 7.5. 7.6.

Force approach to the lateral capillary interactions Capillary immersion force between two cylinders or two spheres Asymptotic expression for the capillary force between two particles Capillary immersion force between spherical particle and wall Summary References

334 334 341 343 345 347

xii CHAPTER 8. LATERAL CAPILLARY FORCES BETWEEN FLOATING PARTICLES

351

8.1. 8.1.1. 8.1.2. 8.1.3. 8.1.4. 8.1.5.

Interaction between two floating particles Flotation force: theoretical expression in superposition approximation "Capillary charge" of floating particles Comparison between the lateral flotation and immersion forces More accurate calculation of the capillary interaction energy Numerical results and discussion

352 352 354 356 358 361

8.2. 8.2.1. 8.2.2. 8.2.3. 8.2.4. 8.2.5. 8.2.6. 8.2.7. 8.3. 8.4.

Particle-wall interaction: capillary image forces Attractive and repulsive capillary image forces The case of inclined meniscus at the wall Elevation of the contact line on the surface of the floating particle Energy of capillary interaction Application of the force approach to quantify the particle-wall interaction Numerical predictions of the theory and discussion Experimental measurements with floating particles Summary References

367 367 369 374 376 379 382 386 392 394

CHAPTER 9. CAPILLARY FORCES BETWEEN PARTICLES BOUND TO A SPHERICAL INTERFACE

396

9.2. 9.2.1. 9.2.2. 9.2.3. 9.2.4.

Origin of the "capillary charge" in the case of spherical interface Interfacial shape around inclusions in a spherical film Linearization of Laplace equation for small deviations from spherical shape "Capillary charge" and reference pressure Introduction of spherical bipolar coordinates Procedure of calculations and numerical results

397 401 401 404 406 409

9.3. 9.3.1. 9.3.2. 9.3.3. 9.3.4. 9.4. 9.5.

Calculation of the lateral capillary force Boundary condition of fixed contact line Boundary condition of fixed contact angle Calculation procedure for capillary force between spherical particles Numerical results for the force and energy of capillary interaction Summary References

412 413 414 417 420 422 424

9.1.

CHAPTER 10. MECHANICS OF LIPID MEMBRANES AND INTERACTION BETWEEN INCLUSIONS

426

10.1. 10.2. 10.2.1. 10.2.2. 10.2.3.

Deformations in a lipid membrane due to the presence of inclusions "Sandwich" model of a lipid bilayer Definition of the model; stress balances in a lipid bilayer at equilibrium Stretching mode of deformation and stretching elastic modulus Bending mode of deformation and curvature elastic moduli

427 430 430 435 438

10.3. 10.3.1. 10.3.2. 10.3.3. 10.3.4. 10.3.5.

Description of membrane deformations caused by inclusions Squeezing (peristaltic) mode of deformation: rheological model Deformations in the hydrocarbon-chain region Deformation of the bilayer surfaces The generalized Laplace equation for the bilayer surfaces Solution of the equations describing the deformation

444 444 446 447 450 452

xiii

10.4. Lateral interaction between two identical inclusions 10.4.1. Direct calculation of the force 10.4.2. The energy approach

454 454 457

10.5. 10.6. 10.7.

460 463 465

Numerical results for membrane proteins Summary References

CHAPTER 11. CAPILLARY BRIDGES AND CAPILLARY BRIDGE FORCES

469

11.1. 11.2. ll.2.1. 11.2.2.

Role of the capillary bridges in various processes and phenomena Definition and magnitude of the capillary bridge force Definition Capillary bridge in toroid (circle) approximation

470 472 472 474

11.3. 11.3.1. 11.3.2. 11.3.3. 11.3.4.

Geometrical and physical properties of capillary bridges Types of capillary bridges and expressions for their shape Relations between the geometrical parameters Symmetric nodoid-shaped bridge with neck Geometrical and physical limits for the length of a capillary bridge

477 477 480 483 486

11.4. 11.4.1. 11.4.2. 11.5. 11.6.

Nucleation of capillary bridges Thermodynamic basis Critical nucleus and equilibrium bridge Summary References

492 492 496 498 499

CHAPTER 12. CAPILLARY FORCES BETWEEN PARTICLES OF IRREGULAR CONTACT LINE

503

12.1. Surface corrugations and interaction between two particles 12.1.1. Interfacial deformation due to irregular contact line 12.1.2. Energy and force of capillary interaction

505 505 508

12.2. 12.2.1. 12.2.2. 12.3. 12.4.

512 513 514 515 516

Elastic properties of particulate adsorption monolayers Surface dilatational elasticity Surface shear elasticity Summary References

CHAPTER 13. TWO-DIMENSIONAL CRYSTALLIZATION OF PARTICULATES AND PROTEINS

517

13.1. 13.1.1. 13.1.2. 13.1.3. 13.1.4. 13.1.5.

518 518 522 524 527 529

Methods for obtaining 2D arrays from microscopic particles Formation of particle 2D arrays in evaporating liquid films Particle ordering due to a Kirkwood-Alder type phase transition Self-assembly of particles floating on a liquid interface Formation of particle 2D arrays in electric, magnetic and optical fields 2D arrays obtained by adsorption and/or Langmuir-Blodgett method

13.2. 2D crystallization of proteins on the surface of mercury 13.2.1. The mercury trough method 13.2.2. Experimental procedure and results

530 530 532

xiv 13.3. Dynamics of 2D crystallization in evaporating liquid films 13.3.1. Mechanism of two-dimensional crystallization 13.3.2. Kinetics of two-dimensional crystallization in convective regime

535 535 542

13.4. Liquid substrates for 2D array formation 13.4.1. Fluorinated oil as a substrate for two-dimensional crystallization 13.4.2. Mercury as a substrate for two-dimensional crystallization

55O 550 554

13.5.

Size separation of colloidal particles during 2D crystallization

556

13.6. Methods for obtaining large 2D-crystalline coatings 13.6.1. Withdrawal of a plate from suspension 13.6.2. Deposition of ordered coatings with a "brush"

561 561 564

13.7. 2D crystallization of particles in free foam films 13.7.1. Arrays from micrometer-sized particles in foam films 13.7.2. Arrays from sub-micrometer particles studied by electron cryomicroscopy

566 566 568

13.8. 13.8.1. 13.8.2. 13.8.3. 13.9. 13.10.

Application of 2D arrays from colloid particles and proteins Application of colloid 2D arrays in optics and optoelectronics Nano-lithography, microcontact printing, nanostructured surfaces Protein 2D arrays in applications

Summary References

CHAPTER 14. EFFECT OF OIL DROPS AND PARTICULATES ON THE STABILITY OF FOAMS 14.1. 14.1.1. 14.1.2. 14.1.3.

Foam-breaking action of microscopic particles

14.2. 14.2.1. 14.2.2. 14.2.3. 14.2.4. 14.2.5.

Mechanisms of foam-breaking action of oil drops and particles

14.3. 14.3.1. 14.3.2. 14.3.3. 14.4. 14.5.

Control of foam stability; Antifoaming vs. defoaming Studies with separate foam films Hydrodynamics of drainage of foam films Scheme of the consecutive stages Entering, spreading and bridging coefficients Spreading mechanism Bridging-dewetting mechanism Bridging-stretching mechanism

Stability of asymmetric films: the key for control of foaminess Thermodynamic and kinetic stabilizing factors Mechanisms of film rupture Overcoming the barrier to drop entry

Summary and conclusions References

572 572 573 577 58O 582

591 592 592 594 600 602 602 606 611 613 615 617 617 620 623 626 628

Appendix 1A: Equivalence of the two forms of the Gibbs adsorption equation Appendix 8A: Derivation of equation (8.31) Appendix 10A: Connections between two models of lipid membranes

633

Index Notation

641 651

635 636

CHAPTER 1 PLANAR FLUID INTERFACES

An interface or membrane is one of the main "actors" in the process of particle-interface and particle-particle interaction at a fluid phase boundary. The latter process is influenced by mechanical properties, such as the interfacial (membrane) tension and the surface (Gibbs) elasticity. For interfaces and membranes of low tension and high curvature the interracial bending moment and the curvature elastic moduli can also become important. As a rule, there are surfactant adsorption layers at fluid interfaces and very frequently the interfaces bear some electric charge. For these reasons in the present chapter we pay a special attention to surfactant adsorption and to electrically charged interfaces. Our

purpose

is to

introduce

the

basic

quantities

and

relationships

in

mechanics,

thermodynamics and kinetics of fluid interfaces and surfactant adsorption, which will be further currently used throughout the book. Definitions of surface tension, interfacial bending moment, adsorptions of the species, surface of tension and equimolecular dividing surface, surface elasticity and adsorption relaxation time are given. The most important equations relating these quantities are derived, their physical meaning is interpreted, and appropriate references are provided. In addition to known facts and concepts, the chapter presents also some recent results on thermodynamics and kinetics of adsorption of ionic surfactants. Four tables summarize theoretical expressions, which are related to various adsorption isotherms and types of electrolyte in the solution. We hope this introductory chapter will be useful for both researchers and students, who approach for a first time the field of interracial science, as well as for experts and lecturers who could find here a somewhat different viewpoint and new information about the factors and processes in this field and their interconnection.

2

Chapter 1

1.1.

MECHANICAL PROPERTIES OF PLANAR FLUID INTERFACES

1.1.1.

THE BAKKER EQUATION FOR SURFACE TENSION

The balance of the linear momentum in fluid dynamics relates the local acceleration in the fluid to the divergence of the pressure tensor, P, see e.g. Ref. [1 ]: dv

,o~=-V.P dt

(1.1)

Here ,o is the mass density of the fluid, v is velocity and t is time; in fact the pressure tensor P equals the stress tensor T with the opposite sign: P = - T . In a fluid at rest v - 0 and Eq. (1.1) reduces to V.P=O

(1.2)

which expresses a necessary condition for hydrostatic equilibrium. In the bulk of a liquid the pressure tensor is isotropic, P=PsU

(1.3)

as stated by the known Pascal law (U is the spatial unit tensor; P~ is a scalar pressure). Indeed, all directions in the bulk of a uniform liquid phase are equivalent. The latter is not valid in a vicinity of the surface of the fluid phase, where the normal to the interface determines a special direction. In other words, in a vicinity of the interface the force acting across unit area is not the same in all directions. Correspondingly, in this region the pressure tensor can be expressed in the form [2,3]: P = Pr (exex +eyey)+PNeze: Here ex, ey and e~ are the unit vectors along the Cartesian coordinate

(1.4) axes, with ez being

oriented normally to the interface; PN and Pr are, respectively, the normal and the tangential components of the pressure tensor. Due to the symmetry of the system PN and Pr can depend on z, but they should be independent of x and v. Thus a substitution of Eq. (1.4) into Eq. (1.3) yields one non-trivial equation:

9PN = 0 9z

(1.5)

Planar Fluid hTterfaces

3

In other words, the condition for hydrostatic equilibrium, Eq. (1.3), implies that PN must be constant along the normal to the interface; therefore, PN is to be equal to the bulk isotropic pressure, PN = P8 = const. Let us take a vertical strip of unit width, which is oriented normally to the interface, see Fig. 1.1. The ends of the stripe, at z = a and z = b, are supposed to be located in the bulk of phases 1 and 2, respectively. The real force exerted to the strip is b FT(real) - I e r (z)dz

(1.6)

a

On the other hand, following Gibbs [4] one can construct an idealized

system consisting of

two uniform phases, which preserve their bulk properties up to a mathematical dividing surface modeling the transition zone between the two phases (Fig. 1.1). The pressure everywhere in the idealized system is equal to the bulk isotropic pressure, P8 =PN. In addition, a surface tension cy

[Real System,]

z=b PT...PN

Phase 2

[Idealized System[

ez~l~ey

Phase 2

0".~

transition zone

Z=Z 0

\ [,,~ividing

surface!

PT. PN Phase 1

Phase 1 z=a

Fig. 1.1. Sketch of a vertical strip, which is normal to the boundary between phases 1 and 2.

4

Chapter I

is ascribed to the dividing surface in the idealized system. Thus the force exerted to the strip in the idealized system (Fig. 1.1) is b

F T idealized)

=

f

PN de - Cr

(1.7)

a

The role of cy is to make up for the differences between the real and the idealized system. Setting

-r/7(idealized)-----7"v(re~ from Eqs. (1.6) and (1.7) one obtains the Bakker [5] equation for the

surface tension" +oo

r - I (PN -

Pr )&

(1.8)

--oo

Since the boundaries of integration z = a and z = b are located in the bulk of phases 1 and 2, where the pressure is isotropic ( P T - PN), we have set the boundaries in Eq. (1.8) equal to +oo. Equation (1.8) means that the real system with a planar interface can be considered as if it were composed of two homogeneous phases separated by a planar membrane of zero thickness with

26 24 22 20

~

0

18 e..)

;a~ "r

14

M

10

%

Liquid, phase I

Gas, phase II

8 4 2 0

.......

- -.

-2 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

z-zv , Angstroms Fig. 1.2. Anisotropy of the pressure tensor, AP, plotted vs. the distance to the equimoleqular dividing surface, .:-Z.v, for interface liquid argon-gas at 84.3 K; Curves 1 and 2 are calculated by the theories in Refs. [8] and [10].

Planar Fluid Interfaces

5

tension 6 given by Eq. (1.8). The latter equation gives a hydrostatic definition of surface tension. Note that this definition does not depend on the exact location of the dividing surface. The quantity AP - PN -

Pr

(1.9)

expresses the anisotropy of the pressure tensor. The function AaD(z) can be obtained theoretically by means of the methods of the statistical mechanics [6-9]. As an illustration in Figure 1.2 we present data for AP vs. Z-Zv for the interface liquid argon-gas at temperature T = 84.3 K; Zv is the position of the so called "equimolecular" dividing surface (see Section 1.2.2 below for definition). The empty and full points in Fig. 1.2 are calculated

by means of the

theories from Refs. [8] and [10], respectively. As seen in Fig. 1.2, the width of the transition

450 .................. ~. . . . . . . . . . . . . . . , ...... 400 ............................................................................. (1) ........................................................ i...........................

300 ~ .~

....i ! i

~00 i.....i-............i............i........... ........................i.....

~' ....i....................................................

0I ~ 150 100 i.....i...........i.............i............i............!............ j i , ../ , 1. , ! ? " "~

~o

~

~ .....................

~...........

i::

i k . "........ . . .{............. . . . }............{...........i...........i I

..............i........................

~.~-~o .ii i i i i i

v

iiii

-100

150 2OO -250

-14 -12 -10

-8

-6 z-z

-4 v

,

-2 0 2 Angstroms

4

6

8

10

1

--.o-(1) .......

(2)

.....

(3)

Fig. 1.3. Anisotropy of the pressure tensor, zXP, plotted vs. the distance to the equimoleqular dividing surface, =-Zv, calculated by the theory in Ref. [10] for the phase boundaries n-decane-gas (curve 1), gas-water (curve 2) and n-decane-water (curve 3).

6

Chapter 1

zone between the liquid and gas phases (in which AP :/: 0) is of the order of 10 A. On the other hand, the maximum value of the anisotropy AP(z) is about 2 x 108 dyn/cm, i.e. about 200 atmospheres, which is an impressive value. The area below the full line in Fig. 1.2 gives the surface tension of argon at that temperature, a = 13.45 mN/m, in accordance with Eq. (1.8). Curves 1, 2 and 3 in Fig. 1.3 present AP(z) calculated in Ref. [10] for the interfaces ndecane/gas, gas/water and n-decane/water, respectively. One see that kP(z) typically exhibits a single maximum for a liquid-gas

interface, whereas AP(z) exhibits a loop (maximum and

minimum) for a liquid-liquid interface. For all curves in Fig. 1.3 the width of the interfacial transition zone is of the order of 10 A.

1.1.2

INTERFACIALBENDING MOMENTAND SURFACE OF TENSION

To make the idealized system in Fig. 1.1 hydrostatically equivalent to the real system we have to impose also a requirement for equivalence with respect to the acting force moments (in addition to the analogous requirement for the acting forces, see above). The moment exerted on the strip in the real system (Fig. 1.1) is

M

(real)

b j PT (Z) zdz

(1. lO)

O

Likewise, the moment exerted on the stripe in the idealized system is [11]: b

M(idea~zed)-- f Pu zdz - OZo + 7l B 0

(1 11)

a

Here z = z0 is the position of the dividing surface and B0 is an interfacial bending moment (couple of forces), which is to be attributed to the dividing surface in order to make the idealized system equivalent to the real one with respect to the force moments. Setting M~ide'li~ed~=M t'~''l' fiom Eqs. (1.8), (1.10) and (1.11) one obtains an expression for the

interfacial bending moment:

P la n a r Flu id In te ifa ce s

7

~c,o

Bo

- 2 ~ (PN - iT ) (~.o - z)dz

(1.12)

-co

As in Eq. (1.8) we have extended the boundaries of integration to +~,. From the viewpoint of mechanics p o s i t i v e Bo represents a force moment (a couple of forces), which tends to bend the dividing surface around the phase, for which ex is an outer normal (in Fig. 1.1 this is phase 1). The comparison of eqs (1.8) and (1.12) shows that unlike o, the interfacial bending moment Bo depends on the choice of position of the dividing surface z0. The latter can be defined by imposing some additional physical condition; in such a way the "equimolecular" dividing surface is defined (see Section 1.2.2 below). If once the position of the dividing surface is determined, then the interfacial bending moment B0 becomes a physically well defined quantity. For example, the values of the bending moment, corresponding to the equimolecular dividing surface, for curves No. 1, 2 and 3 in Fig. 1.3 are, respectively [10]: B0 = 2.2, 2.3 and 5.2x 10-11N. One possible way to define the position, z0, of the dividing surface is to set the bending moment to be identically zero:

I

Bo zo:z,

=0

(1.13)

Combining eqs (1.8), (1.12) and (1.13) one obtains [2] +co

~.s - - -

-IT

(7

~1.14)

-co

Equation (1.14) defines the so called surface o f tension . It has been first introduced by Gibbs [4], and it is currently used in the conventional theory of capillarity (see Chapter 2 below). At the surface of tension the interface is characterized by a single dynamic parameter, the interfacial tension cy; this considerably simplifies the mathematical treatment of capillary problems. However, the physical situation becomes more complicated when the interfacial tension is low such is the case of some emulsion and microemulsion systems, lipid bilayers and biomembranes. In the latter case, the surface o f tension can be located far from the actual transition region between the two phases and its usage becomes physically meaningless.

8

Chapter I

Indeed, for o--90 Eq. (1.14) yields z, -9oo. Therefore, a mechanical description of an interface of low surface tension needs the usage of (at least) two dynamic quantities: interfacial (surface) tension and bending moment. In fact, B0 is related to the so called spontaneous curvature of the interface. In Chapter 3 we will come to this point again.

]. ].3. ELECTRICALLYCHARGEDINTERFACES As a rule, the boundaries between two phases (and the biomembranes, as well) bear some electric charge. Often it is due to the dissociation of surface ionizable groups or to adsorption of charged amphiphilic molecules (surfactants). It should be noted that even the boundaries water-air and water-oil (oil here means any liquid hydrocarbon immiscible with water) are electrically charged in the absence of any surfactant, see e.g. refs. [12] and [13]. If the surface of an aqueous phase is charged, it repels the coions, i.e. the ions of the same charge, but it attracts the counterions, which are the ions of the opposite charge, see Fig. 1.4. Thus a nonuniform distribution of the ionic species in the vicinity of the charged interface appears, which is known as electric double laver (EDL), see e.g. Ref. [14]. The conventional model of the EDL stems from the works of Gouy [15], Chapman [16] and Stern [ 17]. The EDL is considered to consist of two parts: (I) interfacial (adsorption) layer and (II) diffuse layer. The interfacial (adsorption) layer includes charges, which are immobilized (adsorbed) at the phase boundary; this includes also adsorbed (bound) counterions, which form the so called Stern layer, see Fig. 1.4. The diffuse layer consists of free ions in the aqueous phase, which are involved in Brownian motion in the electrical field created by the charged interface. The boundary, which separates the adsorption from the diffuse layer, is usually called the Gouv plane. The conventional theory of the electric double layer is briefly presented in Section 1.2.4 below. For our purposes here it is sufficient to take into account that the electric potential varies across the EDL: Iff = ~z). The thickness of the diffuse EDL could be of the order of hundred (and even thousand) nm, i.e. it is much greater than the thickness of the interracial transition zone (cf. Figs. 1.2 and 1.3). This fact requires a special approach to the theoretical description of the

9

Planar Fluid Interfaces

charged interfaces, which can be based on the expression for the M a x w e l l electric stress tensor

[181 Non - aqueous ^ , , . i f ' ~ phase ~J" V N.I~-'J,,~

Aqueous phase

!

/D

@

Coions

@

Counterions

@

,/V%

Diffuse layer

4

Surfactant adsorption

| =-

Stern layer

of adsorbed counterions

layer

L c-"

._o ,,.., ell)

Counterions

Coo

c--

S

.o_ r o

Coions

0

r

Z

Fig. 1.4. Sketch of the electric double layer in a vicinity of an adsorption monolayer of ionic surfactant. (a) The diffuse layer contains free ions involved in Brownian motion, while the Stern layer consists of adsorbed (bound) counterions. (b) Near the charged surface there is an accumulation of counterions and a depletion of coions, whose bulk concentrations are both equal to c~. E

~

E

Pk - (P, + - - E - ) ~;k - - - E E k (i, k - 1,2,3) 8n 4n" '

(1.~5)

Here ~;k is the K r o n e c k e r symbol (the unit matrix), 8 is the dielectric permittivity of the m e d i u m ( usually water), E, is the i-th c o m p o n e n t of the electric field,

I0

x = ,,vl1 y = x? and

z

= -1-3 are Cartesian crmrciiriales, ariti P., i s ari isnti-opic pressui-r, u!hich

ciin vary across thc EDL due to the osmotic effect of the dissolved ionic species. Ar; already mentioned. ill the

UBSC

of plaiic iiitcrfacc

WE

havc ly =

W,x;)>and rhcn Eq. (1.15)

reduces to the following two expreasioris: (1.17)

c P,.= P,,t = P,,r= P , t--

( 4 J1‘

(1.18)

-

8a d?

Eqs. (1.17) 2nd ( I . IS) can be applied to dcscribc thc. prc.ssurc tc.tisoi-within h c difl’usc par1 uf the electric double laycr.

Now! let us Ir)cate the plane := 0 in the Chuy plane separating the diffuse (at z > 0 ) froin t.hc ndstrrpliori layer. Then by means o l the H:tkker equation ( I -8) one can r e p r t x n l h e s u r t x e tcnsion 0 as I suiii uf corilributions from the adsorption and diffuse layers:

whci-c

(1.20)

Substiluting Eqs. (I.17) and ( 1 , 18) inlo thc ahovc cquiition fur

00.

m c obt.ains

a gerirral

expression for thc cunrrihutiori of the r l j / r i $ e layer, to the interfacial tcnsion [ 19,201: 11.21)

Planar Fluid Interfaces

11

interfacial tension o'. Explicit expressions for O'd, obtained by means of the double layer theory for various types of electrolytes, can be found in Table 1.3 below.

1.1.4.

WORK OF INTERFACIAL DILATATION

Let us consider an imaginary rectangular box containing portions of phases 1 and 2, and of the interface between them. As before, we will assume that the interface is parallel to the coordinate plane xy, and the sides of the rectangular box are also parallel to the respective coordinate planes. Moving the sides of the box one can create a small change of the volume of the box, tSV, with a corresponding small change of the interfacial area, 6A. The work tSW carried out by the external forces to create this deformation can be calculated by means of a known equation of fluid mechanics [1 ]: 6W - - f ( P " v

t~D)dV

(1.22)

Here 5D is the strain tensor (tensor of deformation) and ":" denotes double scalar product of two tensors (dyadics): (AB) : (CD) = (A- D ) ( B - C )

(1.23)

Since we consider displacements of the sides of our rectangular box along the normals to the respective sides, the strain tensor has diagonal form in the Cartesian basis [21,22]: ~D - exex

~(dx) ~(dy) ~(dz) dx + e y e Y dy + e z e z dz

(1.24)

Here 6(dx) denotes the extension of a linear element dx of the continuous medium in the course of deformation. Equation (1.24) shows that the eigenvalues of the strain tensor are the relative extensions of linear elements along the three axes of the Cartesian coordinate system. Substituting Eqs. (1.4) and (1.24) into Eq. (1.22) one can derive [22]:

-

+ dx

dy

+ ~ dz

dx4vdz+

(PN -- P;

6(dx) dx

6(dy) dy

"

The increments of the elementary volume and area in the process of deformation are

12

Chapter 1

6 ( d V ) = dydz 6(dx) + dxdz 6 ( @ ) + dxdy 6(dz),

6(dA) = dy 6(dx) + dx 6(dy)

(1.26)

Combining Eqs. (1.8), (1.25) and (1.26) one finally obtains (1.27)

a W = -PN 0~" + CY6A

Here --PNO~7 expresses the work of changing the volume and c~6A is the work o f interfacial dilatation. Equation (1.27) gives a connection between the mechanics and thermodynamics of

the fluid interfaces.

1.2.

T H E R M O D Y N A M I C A L P R O P E R T I E S OF PLANAR FLUID I N T E R F A C E S

1.2.1.

THE GraBS ADSORPTION EQUATION

Let us consider the same system as in section 1.1.4 above. The Gibbs fundamental equation, combining the first and the second law of thermodynamics, is [2,4] d U - T d S - PNdV +erdA + ~_~lttidN~ ,

(1.28)

i

where T is the temperature; U and S are the internal energy and entropy of the system, respectively; J./i and Ni are the chemical potential and the number of molecules of the i-th component (species); the summation in Eq. (1.28) is carried out over all components in the system. Equation (1.28) states that the internal energy of the system can vary because of the transfer of heat (TdS) and/or matter ( ~ j2idN i ), and/or due to the mechanical work, 6W, carried i

out by external forces, see Eq. (1.27). Following Gibbs [4], we construct an idealized system consisting of two bulk phases, which are uniform up to a mathematical dividing surface modeling the boundary between the two phases. Since the dividing surface has a zero thickness, the volumes of the two phases in the idealized system are additive: V = V~+

V ~2~

(1.29)

Planar Fluid Interfaces

13

We assume that the bulk densities of entropy, s (k), internal energy, u (k), and number of )

molecules, n~k , are known for the two neighboring phases (k = 1,2). Then the entropy, internal energy and number of molecules for phase "k" of the idealized system are: S (k)= s(k)V (k~"

U (k)= u(k)V (k)"

N5 k)= n~k)V (k)

( k - 1,2)

(1.30)

Each of the two uniform bulk phases has its own fundamental equation [2,4]" a u " ' - T a s '~' - P . a v " ' + ~ l a , aN) ~' i

(1.31)

dU(2' - rdS'2) - V~ dV'2' + Z It, dN~ 2) i

It is presumed that we deal with a state of thermodynamic equilibrium, and hence the temperature T and the chemical potentials Pi are uniform throughout the system [23]" in addition, PN = Pg = const., see Eq. (1.5) above. Next, we sum up the two equations (1.31) and subtract the result from Eq. (1.28); thus we obtain" dU (') - T dS C') + cy dA + ~__~p, dN i(') ,

(1.32)

i

where U (s) --~ U - U

(l) - U

(2) ,

S (s) -~

S -

S (1) - S (2) ,

N(')i

= N i _ Ni- (1) _ N ~ 2 )

(1.33)

are, respectively, surface excesses of internal energy, entropy and number of molecules of the i-th species; these excesses are considered as being attributed to the dividing surface. Equation (1.32) can be interpreted as the fundamental equation

of the interface [4,24]. Since the

interface is uniform, then dl~U ~, dS (') and dNi' ~can be considered as amounts of the respective extensive thermodynamic parameters corresponding to a small portion, dA, of the interface; then Eq. (1.32) can be integrated to yield [2,4]: U (s) - T S Is) + e r A + Z I I i N ~ st ,

(1.34)

i

Finally, we differentiate Eq. (1.34) and compare the result with Eq. (1.32); thus we arrive at the Gibbs [4] adsorption equation:

Chapter 1

14

da

St"~ - -~dV A

(1.35)

- EFidldi i

where F, = N~" = A

(n, ( ~ ) - nr 1) )d~, + -~

(H i ( ~ ) -

H i-('~) ]/d7

(1.36)

zo

is the adsorption of the i-th species at the interface; ni(z)

is the actual concentration of

component "i" as a function of the distance to the interface, z, cf. Eq. (1.33); z0 denotes the position of the dividing surface. Figure 1.5a shows qualitatively the dependence ni(z) for a nona m p h i p h i l i c component, i.e. a component, which does not exhibit a tendency to accumulate at

the interface; if phase l is an aqueous solution, then the water can serve as an example for a non-amphiphilic

component.

On the other hand, Figure

1.5b shows qualitatively the

dependence ni(z) for an amphiphilic component (surfactant), which accumulates (adsorbs) at the interface, see the maximum of hi(Z) in Fig. 1.5b.

1.2.2.

EQUIMOLECULAR DIVIDING SURFACE

As discussed in section 1.1.2 above, the definition of the dividing surface is a matter of choice. In other words, one has the freedom to impose one physical condition in order to determine the position of the dividing surface. This can be the condition the adsorption of the i-th component to be equal to zero [4]:

F;1:0:~v- 0

(equimolecular dividing surface)

(1.37)

The surface thus defined is called equimolecular dividing surface with respect to component "i". In order to have F ; - 0 the sum of the integrals in Eq. (1.36) must be equal to zero. This

means that the positive and negative areas, which are comprised between the continuous and dashed lines in Fig. 1.5a,b and denoted by (+) and (-), must be equal.

15

P l a n a r Fluid Interfaces

ni

n~'

(a)

"i

!

~

,(z)

(b)

_,

Zv

Z

0

Zv

Z

Fig. 1.5. Illustrative dependence of the density ni of the i-th component on the distance z to the interface for (a) non-amphiphilic component and (b) amphiphilic component; Zv denotes the position of the equimoleqular dividing surface; n,(~) and nit2~are the values of n,. in the bulk of phases 1 and 2. As seen in Fig. 1.5a, if component "i"

is non-amphiphilic (say the water as a solvent in an

aqueous solution), the equimolecular dividing surface, z = Zv, is really situated in the transition zone between the two phases. In contrast, if component "i"

is an amphiphilic one, then the

equimolecular dividing surface, z = Zv, is located far from the actual interracial transition zone (Fig. 1.5b). Therefore, to achieve a physically adequate description of the system, the equimolecular dividing surface is usually introduced with respect to the solvent; it should never be introduced with respect to an amphiphilic component (surfactant).

1.2.3.

T H E R M O D Y N A M I C S OF A D S O R P T I O N OF N O N I O N I C S U R F A C T A N T S

A molecule of a nonionic surfactant (like all amphiphilic molecules) consists of a hydrophilic and a hydrophobic moiety. The hydrophilic moiety (the "headgroup") can be a water soluble polymer, like poly-oxi-ethylene, or some polysaccharide [251; it can be also a dipolar headgroup, like those of many phospholipids. The hydrophobic moiety (the "tail") usually consists of one or two hydrocarbon chain(s). The adsorption of such a molecule at a fluid interface is accompanied with a gain of free energy, because the hydrophilic part of an adsorbed molecule is exposed to the aqueous phase, whereas its hydrophobic part contacts with the nonaqueous (hydrophobic) phase. Let us consider the boundary between an aqueous solution of a nonionic surfactant and a hydrophobic phase, air or oil. We choose the dividing surface to be the equimolecular dividing

16

Chapter 1

surface with respect to water, that is Fw - 0. Then the Gibbs adsorption equation (1.35) reduces to do" - - F 1 d/.t 1

(T = const.)

(1.38)

where the subscript "1" denotes the nonionic surfactant. Since the bulk surfactant concentration is usually relatively low, one can use the expression for the chemical potential of a solute in an ideal solution [23]" It 1 - ll~ ~ + k T In c I

(1.39)

where Cl is the concentration of the nonionic surfactant and /t{ ~ is a standard chemical potential, which is independent of Cl, and k is the Boltzmann constant. Combining Eqs. (1.38) and (1.39) one obtains dcy - - k T F 1d In c I

(1.40)

The surfactant adsorption isotherms, expressing the connection between Fj and c~ are usually obtained by means of some molecular model of the adsorption. The most popular is the Langmuir [26] adsorption isotherm, F1

Kc 1

~ = ~ F~ 1+ K c 1 which stems from a lattice model of localized adsorption of n o n - i n t e r a c t i n g

(1.41) molecules [27].

In Eq. (1.41) F~ is the maximum possible value of the adsorption (Fj-->F= for c~--+oo). On the other hand, for c~---)0 one has FI -- Kc~; the adsorption parameter K characterizes the surface activity of the surfactant: the greater K the higher the surface activity. Table 1.1 contains the 6 most popular surfactant adsorption isotherms, those of Henry, Freundlich, Langmuir, Volmer [28], Frumkin [29], and van der Waals [27]. For Cl--+0 all other isotherms (except that of Freundlich) reduce to the Henry isotherm. The physical difference between the Langmuir and Volmer isotherms is that the former corresponds to a physical model of localized adsorption, whereas the l a t t e r - to non-localized adsorption. The Fmmkin and van der Walls isotherms generalize, respectively, the Langmuir and Volmer isotherms for the case, when there is interaction between the adsorbed molecules; fl is the parameter,

Planar Fluid Interfaces

17

Table 1.1. The most popular surfactant adsorption isotherms and the respective surface tension isotherms. 9 Surfactant adsorption isotherms

(for nonionic surfactants" at. ,, ---- C 1 )

r~

Henry Kal, ' =

Freundlich

i

Kai, ' -

r,

11/m

Langmuir Kal. ` =

Volmer

~ exp/ ~ /

Kal, =

Frumkin

Ka I = "

van der Waals

F1 -r,

F

exp kT

~a,= F1 exp/~ ~

F ' -F~

2~ /

F ' -F~

kT

9 Surface tension isotherm

o r - o r o - k T J +or d

(for nonionic surfactants: o-d - O) Henry

j-~

Freundlich

j_~ m

Langmuir J--F

In 1 - ~ - f

Volmer F Frumkin

- Fl

J--rlnl-~

E ] 13Fl2 ~r

van der Waals j

m

F -F 1

kT

18

Chapter 1

which accounts for the interaction. In the case of van der Waals interaction ]3 can be expressed in the form [30,31]:

u(r) 1 - exp ----~--

[3--zckT ro

where

u(r) is

~--rc f u(r)rdr ti)

the interaction energy between two adsorbed molecules and r0 is the distance

between the centers of the molecules at close contact. The comparison between theory and experiment shows that the interaction parameter ]3 is important for air-water interfaces, whereas for oil-water interfaces one can set ,B = 0 [32,33]. The latter fact, and the finding that ]3 > 0 for air-water interfaces, leads to the conclusion that fl

takes into account the van der Waals

attraction between the hydrocarbon tails of the adsorbed surfactant molecules across air (such attraction is missing when the hydrophobic phase is oil). What concerns the parameter K in Table 1.1, it is related to the standard free energy of adsorption, A f - / / I ~ -/~1, . (0~, which is the energy gain for bringing a molecule from the bulk of the water phase to a diluted adsorption layer [34,35]:

~/

K - ~1 exp

F

kT

(1.42)

Here 8~ is a parameter, characterizing the thickness of the adsorption layer, which can be set (approximately) equal to the length of the amphiphilic molecule. Let us consider the integral

J-

c! dc, C , ~C1-

i 0

dlnC'dFl r, dr 1

(1.43)

The derivative d In c~/dFl can be calculated for each adsorption isotherm in Table 1.1, and then the integration in Eq. (1.43) can be carried out analytically. The expressions for J, obtained in this way, are also listed in Table 1.1. The integration of the Gibbs adsorption isotherm, Eq. (1.40), along with Eq. (1.43) yields o =G o

-kTJ,

(1.44)

which in view of the expressions for J in Table 1.1 presents the surfactant adsorption isotherm, or the two-dimensional (surface) equation of state.

Planar Fluid Interfaces

19

Table 1.2. Expressions for the Gibbs elasticity of adsorption monolayers (valid for both nonionic and ionic surfactants), which correspond to the various types of isotherms in Table 1.1. Type of surface tension isotherm

Gibbs elasticity Ec

Henry

E G =kTF 1

Freundlich

EG

=kT F1 m

Langmuir E G - kTF 1

F~ - F1

Volmer

F2 e c - kTr~ (v= - Vl

Frumkin

)2

Ec -kTF~(F~F~-F12/~F, )kT

van der Waals EG - kTF1

F2 (F~ - F 1)2

2flF 1 kT

An important thermodynamic parameter of a surfactant adsorption monolayer is its Gibbs (surface) elasticity: (1.45)

Expressions for Ec,, corresponding to various adsorption isotherms, are shown in Table 1.2. As an example, let us consider the expression for Ec;, corresponding to the Langmuir isotherm" combining results from Tables 1.1 and 1.2 one obtains Ec - F

kTKc 1

(for Langmuir isotherm)

(1.45a)

One sees that for Langmuirian adsorption the Gibbs elasticity grows linearly with the surfactant concentration c~. Since the concentration of the monomeric surfactant cannot exceed the critical micellization concentration, CI~ CCMC, then from Eq. (1.45a) one obtains

20

Chapter 1

EG ~ (EG)max --

~kT

KCcMc

(for Langmuir isotherm)

(1.45b)

Hence one could expect higher elasticity Ec for surfactants with higher CCMC;this conclusion is consonant with the experimental results [36]. The Gibbs elasticity characterizes the lateral fluidity of the surfactant adsorption monolayer. For high values of the Gibbs elasticity the adsorption monolayer at a fluid interface behaves as tangentially immobile. Then, if a particle approaches such an interface, the hydrodynamic flow pattern, and the hydrodynamic interaction as well, is approximately the same as if the particle were approaching a solid surface. For lower values of the Gibbs elasticity the so called "Marangoni effect" appears, which can considerably affect the approach of a particle to a fluid interface. These aspects of the hydrodynamic interactions between particles and interfaces are considered in Chapter 6 below. The thermodynamics of adsorption of ionic surfactants (see Section 1.2.5 below) is more complicated because of the presence of long-range electrostatic interactions in the system. As an introduction, in the next section we briefly present the theory of the electric double layer.

1.2.4.

THEORY OF THE ELECTRIC DOUBLE LAYER. B o l t z m a n n equation a n d activity coefficients. When ions are present in the solution, the

(electro)chemical potential of the ionic species can be expressed in the form [23] ]1 i - ].1~O) + k T In a i + Z i e l l t

(1.46)

which is more general than Eq. (1.39) above; here e is the elementary electric charge, gt is the electric potential, Zi is the valency of the ionic component "i", and a/is its activity. When an electric double layer is formed in a vicinity a charged interface, see Fig. 1.4, the electric potential and the activities of the ionic species become dependent on the distance z from the interface: ~t = ~z), ai = ai(z). On the other hand, at equilibrium the electrochemical potential, p,, is uniform throughout the whole solution, including the electric double layer (otherwise diffusion fluxes would appear) [23]. In the bulk of solution (z.--->~) the electric potential tends to a constant value, which is usually set equal to zero; then one can write

( 1.47)

( I .48) Setting equal the expression for p , at z - w and that for pi at some finite z. and using Eqs. ( I .46) and ( I .47), one obtains [ 2 3 ] :

where ui- denotes the value of the activity of ion "i" in the bulk of solution. Equation (1.49) shows that the activity obeys a Boltzmann type distribution across the electric double Iaycr

(EDL). If the activity in the bulk, u,-, is known, then Eq. (1.49) dekrrnines the activity a,(zj in each point of the EDL. Thc studics on adsorption of ionic surfactants [32.33,20] show that a good agreement between theory and experiment can be achieved using the following expression for

(l,-

:

( 1S O )

where c,~.,is the bulk concentration of the respective ion, arid the activily coefl'icienl y? is to be calculated from the known semicmpirical formula [37]

logy, = -

AIZ+Z I J I I+Brl,&

+bl

(1.51)

which originates from the Debye-Huckel theory; I denote? the ionic strcngth of thc solution:

whcre the surnmation is carried out o w r all ionic spccics in thc solution. When the solution onntairis a mix1tir.r of scvei-;~l trlecti-oly~es.[hen .kq. ( I .S 1 ) defines

for each separaie

eleclrulytc, with Z , and Z being thc valcnces of the cations and anions fur this clcctrolyte, but with I hcinp ltic fntirl ionic strcngth o f thc solution, accounting Ihr all dissolved ionic species

[37]. The log in Eq. ( 1 .S 1 ) is decimal. tl, is the diainetcr of thc ion. A, H , a n d b are parameters,

Chapter 1

22

whose values can be found in the book by Robinson and Stokes [37]. For example, if the ionic strength I is given in moles per liter (M), then for solutions of NaC1 at 25~

the parameters

values are A = 0.5115 M -1/2 Bdi = 1.316 M -1/2 and b = 0.055 M -l Integration

of Poisson-Boltzmann

equation.

The Poisson equation relating the

distribution of the electric potential ~ z ) and electric charge density, pe(z), across the diffuse double layer can be presented in the form [14] d21// 4Jr ----5- = - ~ P e dz e

,

(1.53)

Let us choose component 1 to be a coion, that is an ion having electric charge of the same sign as the interface. It is convenient to introduce the variables ~(Z)

-

Zle~(z)

.-.

k------~'

_

Z~ -_ Z k

Pe

Z1

Pe - Zl e ,

For symmetric electrolytes ~ and

/~e

(k = 1,2 .... N)

(1.54)

thus defined are always positive irrespective of whether

the interface is positively or negatively charged. Combining Eqs. (1.49), (1.53) and (1.54) one obtains d 2ci:)

2

=

2

1 ~r fie -

dz

1

2

N

Kc Z ziai ~ exp(-zi~) i=l

(1.55)

where 2 81~Z1e2 tr C ekT

(1.56)

As usual, the z-axis is directed along the normal to the interface, the latter corresponding to z = 0. To obtain Eq. (1.55) we have expressed the bulk charge density in terms of effective concentrations, i.e. activities, pe(Z)= ~_~Zieai(z), rather than in terms of the net concentrai

tions, P e ( Z ) - Y _ ~ Z i e c , ( z ) . For not-too-high ionic strengths there is no significant quantitative i

difference between these two expressions for Pe (z), but the former one considerably simplifies the mathematical derivations; moreover, the former expression has been combined with

23

Planar Fluid Interfaces

Eq. (1.49), which is rigorous in terms of activities (rather than in terms of concentrations). Integrating Eq. (1.55) one can derive dO / 2 N -~z - tr Z a;= [exp(-z,O)- 1]

(1.57)

i=1

where the boundary conditions O[z_~= =0 and (dO/dz)~._,= =0 have been used, cf. Eqs. (1.47), (1.48) and (1.54). Note that Eq. (1.57) is a nonlinear ordinary differential equation of the first order, which determines the variation of the electric potential O(z) across the EDL. In general, Eq. (1.57) has no analytical solution, but it can be solved relatively easily by numerical integration. Analytical solution can be obtained in the case of symmetric electrolyte, see Eq. (1.65) below. Further, let Ps be the surface electric charge density, i.e. the electric charge per unit area of the interface. Since the solution, as a whole, is electroneutral, the following relationship holds [ 14]:

cx~ (1.58)

Ps - - ~ p ~ ( z ) & 0

Substituting Pe(Z) from Eq. (1.55) into Eq. (1.58) and integrating the second derivative, d 2 9 / dz 2, one derives

/d

/2

_

~--o - ~ ~c/5,,

P"-Z~e

The combination of Eqs. (1.57) and (1.59) yields a connection between the surface charge density, Ps, and the surface potential, Os = O(z=0), which is known as the Gouy equation [15,38]: 112 Ps ---

tCc

aioo

i=1

s

,

s

--

kT

(1.60)

Note that because of the choice component 1 to be a coion, the sign of Os and /5, is always positive and that is the reason why in Eq. (1.60) we have taken sign "+" before the square root.

Cl1crpter I

24

To obtain an expression for calculating the diffuse layer contribution to the surface tension, o,/, we first corribirie Eqs. (1.2 I ) and ( I S4):

(1.61) A substitution of Eq. ( 1.57) into Eq. (1.6 I ) yields

( I .62)

Expressions for flf,obtained by tneans

or Eq. ( I 5 2 ) for solutions of surfactant and various

e.lectrolytes, can he found in Table 1.3 below, as well as in Ref. [20]. Atiulylicul expressiorz.s ,Jir 21:21 elactrolyte. Analytical cxprcssion for O(z) can bc

obtained in t.hc siiiiplcr case- when the solution contains only symmetric, Z1:Z,electi-olyte, that is Z. = -.ZI (Z, = 0 for i > 2 ) . In this case Eq. ( 1.57) can be rcpi-escntcd in the form

(2, :z, electi-0lyt.c)

( I .63)

where

c N

K

I

;K (

( I .64)

1-1

is known a s the Debye screerlirig parai~ieter.The integration of Eq. ( I .63) yields ;in analylical expression for the variation of thc electric potential @ ( z ) across the EDL 1141:

(I)( z ) = 4arctanh tad1 Equation

(

2

1

>xp(-m)

(2l:Zl electrolyte)

( I .6S)

I .65) shows that the electric potcntial, crcatcd by the charged interface, decays

exponenlially i n the depth of solLilion, that is O(:) = exp(-o) for z+m.

The irlversc Debye

paraiiictcr, dl,trcprcscnts a decay length, which characterizes the thickness of thc EDL. The

Gouy equation ( I .60). giving the conriectiori bctwccrl sut-l'acc cliargc and surl'ucc potcntial, also sirriptifits l o r 2,:Zj eleclrolyte:

25

Planar Fluid Interfaces

( Z 1:Z 1 electrolyte)

(1.66)

where F~ and F2 are the adsorptions of the ionic components 1 and 2, respectively. For the same case the integration in Eq. (1.62) can be carried out analytically and the following simpler expression for the diffuse layer contribution to the surface tension can be derived [19,38,39]"

O'd = - ~

cosh

- 1

(Z1 :Zl electrolyte)

(1.67)

K"c

The above equations serve as a basis of the thermodynamics of adsorption of ionic surfactants.

1.2.5.

THERMODYNAMICS OF ADSORPTION OF IONIC SURFACTANTS

Basic equations. Combining Eqs. (1.46), (1.47) and (1.49) one obtains a known expression for the chemical potential" /.t;-/.t~ ~ + k T l n a i .

The substitution of the latter

expression into the Gibbs adsorption equation (1.35) yields [ 19,33,40,41 ]: N

dcr - -kT~.~ Fi d In a~=

(T = const)

(1.68)

i=1

Here with F i we denote the adsorption of the i-th component; F; represents a surface excess of component "i" with respect to the uniform bulk solution. For an ionic species this means that ,-,.,

F i is a total adsorption, which include contributions from both the adsorption layer (surfactant adsorption layer + adsorbed counterions in the Stern layer, see Fig. 1.4) and the diffuse layer. Let us define the quantities

Ai - f[a~(z)-a~=]dz ,

Fi - F i - A ~

(1.69)

0

Ai and Fi can be interpreted as contributions of the diffusion and adsorption layers, ,-,.,

respectively, into the total adsorption F;. Using the theory of the electric double layer and the definitions (1.69) one can prove (see Appendix 1A) that the Gibbs adsorption equation (1.68) can be presented into the following equivalent form [20]

26

Chapter

1

N

do- a - - k T Z Ei d In ais

(T = const)

(1.70)

i=1

where ~, = o- - o-d is the contribution of the adsorption layer into the surface tension, o-j is the contribution of the diffuse layer, defined by Eq. (1.21), and

ais

-

aioo e x p ( - Z i ~ s ) ,

Zi

Zi Z1

(1.71)

is the subsurface activity of the i-th ionic species. The comparison between Eqs. (1.68) and (1.70) shows that the Gibbs adsorption equation can be expressed either in terms of o-, F; and aim, or in terms of o-~, Fi and ais. In Appendix 1A it is proven that these two forms are

equivalent. To derive explicit adsorption and surface tension isotherms, below we specify the type of ionic surfactant and non-amphiphilic salt in the solution.

Surfactant and salt are 1:1 electrolytes. We consider a solution of an ionic surfactant, which is a symmetric 1"1 electrolyte, in the presence of additional common symmetric l'l electrolyte (salt). Here we assume that the counterions due to the surfactant and salt are identical. For example, this can be a solution of sodium dodecyl sulfate (SDS) in the presence of NaC1. We denote by Cl~, c2~ and c3= the bulk concentrations of the surface active ions, counterions, and coions, respectively. For the special system of SDS with NaC1 cl=, c2= and c3~ are the bulk concentration of the DS-, Na + and CI- ions, respectively. The requirement for the bulk solution to be electroneutral implies c2~ = cl= + c3~. The multiplication of the last equation by ~'+, which according to Eq. (1.51) is the same for all monovalent ions, yields a2~ = al~+ a3~

(1.72)

The adsorption of the coions of the non-amphiphilic salt is expected to be equal to zero, F3 = 0, because they are repelled by the similarly charged interface (however, A3 :~ 0: the integral in N

Eq. (1.69) gives a negative A3, see Fig. 1.4; hence F 3 - A 3 r 0). Then the Gibbs adsorption equation (1.70) can be presented in the form

do- a - - k T ( F l d l n a j s

+ F2dlnazs)

(1.73)

The differentials in the right-hand side of Eq. (1.73) are independent (one can vary independently the concentrations of surfactant and salt), and moreover, d o , , is an exact (total) differential. Then according to the Euler condition [23] the cross derivatives must be equal, viz. (1.74)

A surfactant adsorption isotherm,

r, = r,(a,~,a 2 , ) , and

r2= r2(u, , u 2 , ) ,are thrrmodynuinically ~

a counterion adsorption isotherm,

cornpatible if they satisfy Eq. (1.74). Integrating Eq.

( I .74) one obtains

r, =-

dJ

(1.75)

d In q c

where we have introduced the notation

( I .76)

To determine the integration constant in Eq. (1.75) we have used the condition that for u l \= 0 (no surfactant in the solution) we have

r,= 0 (no surfactant adsorption) and r,=0 (no binding

of counterions at the headgroups of adsorbed surfactant). The integral J in Eq. (1.76) can be taken analytically for all popular surface tension isotherms, see Table 1 . 1 . Differentiating Eq. (1.76) one obtains

rl = d J / d l n u , , \ . The substitution of the latter equation, together with Eq.

(1.75) into Eq. ( I .73), after integration yields

o(,

-

kTJ ,

( 1.77)

where 4) is the value of o for pure water. Combining Eqs. ( I . 19) and (1.77) one obtains the surface tension isotherm of the ionic surfactant:

where o,/ is given by Eq. (1.67) and expressions for J , corresponding to various adsorption isotherms, are available in Table 1 . 1 . Note that for each of the isotherms in Table 1 . 1 depends on the product K u , , , that is

r,

r,= r,( K U , , ~ Then ). Eq. (1.76) can be transformed to read

Chnpter 1

28

(1.79)

Differentiating Eq. ( I .79) one can bring Eq. (1.75) into the form [20]

r, = r,

dInK

(1 3 0 )

~

d In a 2 s

which holds for each of the surfactant adsorption isotherms i n Table 1 . 1 . Note that Eq. ( 1 2 0 ) is valid for a general form of the dependence K = K( a ? , ) , which expresses the dependence of the equilibrium constant of surfactant adsorption on the concentration of the salt in solution. Let us consider a linear dependence K = K( u z 5), that is K =KI +K 2 ~ 2 s

(1.81)

where K I and K2 are constants. The physical meaning of the linear dependence of K on a 2 , in Eq. (1.8 1 ) is discussed below, see Eqs. ( I . 1 18)-( 1.128) and the related text. The substitution of Eq. (1.8 1) into Eq. (1 3 0 ) yields [20] (1.82) Equation (1.82) is in fact a form of the Stern isotherm [ 17,381. One can verify that the Euler condition (1.74) is identically satisfied if

r?is substituted

from Eq. (1 3 2 ) and l-1 is expressed

by either of the adsorption isotherms in Table 1.1. In fact, Eq. (1.81) is the necessary and sufficient condition for thermodynamic compatibility of the Stern isotherm of counterinn adsorption, Eq. (1.82), with either of the surfuctant adsorption isotherms in Table I . I . In other words, a given isotherm from Table I . I , say the Langmuir isotherm, is thermodynamically compatible with the Stern isotherm, if only the adsorption parameters K, K, and K? in these isotherms are related by means of Eq. (1.81). The constants K I and K2 have a straightforward physical meaning. In view of Eqs. ( I .42) and (1.81)

(13 3 )

29

Planar Fluid Intelfaces

where A/.t(~ has the meaning of standard free energy of adsorption of surfactant from ideal dilute solution to ideal adsorption monolayer in the absence of dissolved non-amphiphilic salt; the thickness of the adsorption layer ~ is about 2 nm for SDS. Note that the Langmuir and Stern isotherms, Eqs. (1.41) and (1.82), have a similar form, which corresponds to a statistical model considering the interface as a lattice of equivalent, distinguishable, and independent adsorption sites, without interactions between bound molecules [27]. Consequently, an expression, which is analogous to Eq. (1.83), holds for the ratio K2/K~ [the latter is a counterpart of K in Eq. (1.41)]: - (0)] K---L= c~--L2exp A]./2

K1

Foo

(1.84)

kT

where (39 is the thickness of the Stern layer (c.a. the diameter of a hydrated counterion) and A

~2(0) has the meaning of standard free energy of adsorption (binding) of a counterion from an

ideal dilute solution into an ideal Stern layer. In summary, the parameters K~ and/(2 are related to the standard free energies of surfactant and counterion adsorption. The above equations form a full set for calculating the surface tension as a function of the bulk surfactant and salt concentrations (or activities), o " - o'(aloo,a2oo). There are 6 unknown variables: o, ~,,a~.,.,F~, a2., and F 2. These variables are to be determined from a set of 6 equations as follows. Equation (1.49) for i = 1,2 provides 2 equations. The remaining 4 equations are: Eqs. (1.66), (1.78), (1.82) and one surfactant adsorption isotherm from Table 1.1, say the Langmuir isotherm.

Comparison of theory and experiment. As illustration we consider an interpretation of experimental data by Tajima et al. [42,43] for the surface tension vs. surfactant concentrations at two concentrations of NaCI: c>o= 0 and c3~ = 0.115 M, see Fig. 1.6. The ionic surfactant used in these experiments is tritiated sodium dodecyl sulfate (TSDS), which is 1:1 electrolyte (the radioactivity of the tritium nuclei have been measured by Tajima et al. to determine directly the surfactant adsorption). Processing the set of data for the interracial tension O"

=O'(Cloo,C2oo) as a function of the bulk concentrations of surfactant ions, c~oo, and

counterions, c2oo, one can determine the surfactant adsorption, Fl(clo~,c2oo), the counterion

Chapter I

30

TSDS-water-air

/

40 A

9 9

E

z E

30

No salt 0.115MNaCI .

.

.

.

/

, ~

.

t,_

L

20

U L

1 0 -2

2

i

3

i

i

i

,1LI

0-1

2

i

3

i

i

r i.~,1

100

i

2

r

3

,i

~ i i ]1

101

TSDS concentration (mM)

Fig. 1.6. Surface pressure at air-water interface, o'0-o-, vs. the surfactant (TSDS) concentration, cl=, for two fixed NaCI concentrations: 0 and 0.115 M; the symbols are experimental data from Refs. [42] and [43]; the continuous lines represent the best fit by means of the theory from Ref. [20]. adsorption, F2 (cloo,c2=) , and the surface potential, ~t (c~oo,c2=) . To fit the data in Fig. 1.6 the Frumkin isotherm is used (see Table 1.1). The theoretical model contains four parameters, /3, F=, K~ and/(2, whose values are to be obtained from the best fit of the experimental data. The parameters values can be reliably determined if only the set of data for o'-o-(c1=,c2= ) contains experimental points for both high and low surfactant concentrations, and for both high and low salt concentrations; the data by Tajima et al. [42,43] satisfy the latter requirement. (If this requirement is not satisfied, the merit function exhibits a flat and shallow minimum, and therefore it is practically impossible to determine the best fit [20]). The value of F~, obtained in Ref. [20] from the best fit of the data in Fig. 1.6, corresponds to 1 / F ~ - 37.6 .~2. The respective value of K~ is 156 m3/mol, which in view of Eq. (1.83) gives a standard free energy of surfactant adsorption A/,t~~ = 12.8 kT per TDS- ion, that is 31.3 kJ/mol. The determined value of K2/K~ is 8.21x10 -4 m3/mol, which after substitution in Eq. (1.84) yields a standard free energy of counterion binding A~t~~ kJ/mol.

1.64 kT per Na + ion, that is 4.04

31

Planar Fluid Interfaces

TSDS-water-air 1.0 E

0.9

._o

0.8

o

0.7

-o

0.6

Q.

ltl

0.5

= o

0.4

1/) = (1)

E

Q

o

m

0.3

0.2

TDS- adsorption ] ~ Na+ adsorption ~

.........

/ It

I

0.1 0.0

10-5

i

2

3

~

'

i

i-,l

1 0-4

t

2

i

3

i

,

i1/

[

10-a

i

2 3

i

I

i

i

I

t

10-2

TSDS concentration (M) Fig. 1.7. Plots of the calculated adsorptions of surfactant F~/Foo (the full lines), and counterions F2/Foo (the dotted lines), vs. the surfactant (TSDS) concentration, c~oo. The lines correspond to the best fit of the data in Fig. 1.6 obtained in Ref. [20]. The value of the parameter fl is positive

(2flF~kT = +0.8), which indicates attraction between

the hydrocarbon tails of the adsorbed surfactant molecules. Figure 1.7 shows calculated curves for the adsorptions of surfactant, F 1 (the full lines), and counterions, F 2 (the dotted lines), vs. the TSDS concentration, cloo. These lines represent the variation of Fj and F 2 along the two experimental curves in Figure 1.6. One sees that both F~ and F 2 are markedly greater when NaC1 is present in the solution. The highest values of FI for the curves in Fig. 1.7 are 4.30 x 10 -6 mol/m 2 and 4.20 • 10 .6 mol/m 2 for the solutions with and without NaC1, respectively. The latter two values compare well with the saturation adsorptions measured by Tajima [42,43] for the same system by means of the radiotracer method, viz. F 1 = 4.33 x 10 -6 mol/m 2 and 3.19 x 10 -6 mol/m 2 for the solutions with and without NaC1. In Fig. 1.8 the occupancy of the Stern layer, 0 - F 2 / F 1 , concentration for the curves in Fig. 1.7. For the solution

is plotted vs. the surfactant

without NaCI F 2 /F~ rises from 0.15

Chapter 1

32

TSDS-water-air 0.8 0.7

,.-" CO t'O

1 5 M NaCI

0.6

salt

0.5

0.4

o t-

O.. O O

O

0.3 0.2 ,1

i

ii

10-2

~

2

3

i

i

LI

1 0-1

t

2

3

P

~

I

L

ql

1 00

1

2

3

I

I

II

1 01

TSDS concentration (mM)

Fig. 1.8. Calculated occupancy of the Stern layer by adsorbed counterions, F2/Fj, vs. the surfactant (TSDS) concentration, Cl=, for two fixed NaC1 concentrations: 0 and 0.115 M. The lines correspond to the best fit obtained in Ref. [20] for the data in Fig. 1.6. up to 0.74 and then exhibits a tendency to level off. As it could be expected, the occupancy F 2 /F~ is higher for the solution with N a C I even at TSDS concentration 10 .5 M the occupancy is about 0.40" for the higher surfactant concentrations 0 levels off at F 2 / F] = 0.74 (Fig. 1.8). The latter value is consonant with data of other authors [44-47], who have obtained values of F 2 /F~ up to 0.70 - 0.90 for various ionic surfactants; pronounced evidences for counterion binding have been obtained also in experiments with solutions containing surfactant micelles [48-53]. These results imply that the counterion adsorption (binding) should be always taken into account. The fit of the data in Fig. 1.6 gives also the values of the surface electric potential, g t . For the solutions with salt the model predicts surface potentials varying in the range I W, I= 55 - 95 mV within the experimental interval of surfactant concentrations, whereas for the solution without salt the calculated surface potential is higher: lip'., I= 150 - 180 mV (note that for TSDS I/t has a negative sign). Thus it turns out that measurements of surface tension, interpreted by means

Planar Fluid Interfaces

33

of an appropriate theoretical model, provide a method for determining the surface potential N.~ in a broad range of surfactant and salt concentrations. The results of this method could be compared with other, more direct, methods for surface potential measurement, such as the electrophoretic ~'-potential measurements [12,13,54,55], or Volta (AV)potential measurements, see e.g. Ref. [56].

Surfactant is 1:1 electrolyte, salt is Z3:Z4 electrolyte. In this case we will number the ionic components as follows: index "1" - surfactant ion, index "2" - counterion due to the surfactant, index " 3 " - c o i o n

due to the salt, and index " 4 " - c o u n t e r i o n due to the salt. As

before, we assume that the coions due to the salt do not adsorb at the interface:

F3 = 0. The

counterions due to the surfactant and salt are considered as separate components, which can exhibit a competitive adsorption in the Stern layer (see Fig. 1.4). The analogs of Eqs. (1.81) and (1.82) for the case under consideration are [20]:

K = K l + K2a2s + K4a4s

Fi

-- = F1

Kiais K1 + K2a2s + K4a4s

(1.85)

(i = 2 , 4 )

(1.86)

where KI, K2 and K4 are constants. All expressions for surfactant adsorption isotherms and surface tension isotherms given in Table 1.1 are valid also in the present case. Different are the forms of the Gouy equation and of the expression for o j , which depend on z3 and z4 in accordance with Eqs. (1.60) and (1.62). In particular, the integration in Eq. (1.62) can be carried out analytically for some types of electrolyte. Table 1.3 summarizes the expressions for the Gouy equation and o-(l, which have been derived in Ref. [20] for the cases, when the salt is 1:1, 2:1, 1:2 and 2:2 electrolyte. (Here 2:1 electrolyte means a salt of bivalent counterion and monovalent coion.) One may check that in the absence of salt (a4~ = 0) all expressions in Table 1.3 reduce either to Eq. (1.66) or to Eq. (1.67). More details can be found in Ref. [20].

Gibbs elasticity ,for ionic" surfactants. The definition of Gibbs (surface) elasticity is not well elucidated in the literature for the case of ionic surfactant adsorption monolayers. That is the reason why here we devote a special discussion to this issue.

Chapter I

34

Table 1.3. Special forms of the Gouy equation (1.60) and of the expression for

CQ

, Eq. (1.62), for

solutions of surfactant which is 1: 1 electrolyte, and salt which is 1: I , 2: I , 1.2 and 2:2 electrolyte

Type

Expressions

of salt

obtained from Eqs. (1.60) and ( I .62)

1:1

2: I

I :2

g,

= (1 - v z + V

I/?

Y )

q=

2:2

I

I

5)

Planar Fluid Interfaces

35

The physical concept of surface elasticity is the most transparent for monolayers of insoluble surfactants. The changes of cy and FI in the expression Ec, =-F~(0o/0F 91 correspond to variations in surface tension and adsorption during a real process of interracial dilatation. In the case of a soluble nonionic surfactant the detected increase of cy in a real process of interfacial dilatation can be a pure manifestation of surface elasticity only if the period of dilatation, At, is much shorter than the characteristic relaxation time of surface tension, At 0) the diffusion transport of surfactant tends to saturate the adsorption layer, and eventually to restore the equilibrium in the system. In other words, the interfacial expansion happens only at the initial moment, and after that the interface is quiescent and the dynamics in the system is due only to the diffusion of surfactant. The adsorption process is a consequence of two stages: the first one is the diffusion of surfactant from the bulk to the subsurface and the second stage is the transfer of surfactant molecules from the subsurface to the surface. When the first stage (the surfactant diffusion) is much slower than the second stage, and consequently determines the rate of adsorption, the process is termed adsorption under diffusion control; it is considered in the present section. The opposite case, when the second stage is slower than the first one, is called adsorption under

barrier (or kinetic) control and it is presented in Section 1.3.3. If an electric double layer is present, the electric field to some extent plays the role of a slant barrier; this intermediate case of adsorption under electro-diffusion control, is presented in Section 1.3.2. Here we consider a solution of a nonionic surfactant, whose concentration, c 1 =c~(z,t), depends on the position and time because of the diffusion process. As before, z denotes the distance to the interface, which is situated in the plane z = 0. The surfactant adsorption and the surface tension vary with time: F~ = F~(t), cr = o(t). The surfactant concentration obeys the equation of diffusion: c~ c c)t

t

= D, 692c~ c? Z

(z > 0, t > 0)

(1.90)

Planar Fluid Interfaces

39

where D~ is the diffusion coefficient of the surfactant molecules. The exchange of surfactant between the solution and its interface is described by the boundary condition d F~ = D1 o1 cl

dt

(z = 0, t > 0)

(1.91)

~z

which states that the rate of increase of the adsorption F~ is equal to the diffusion influx of surfactant per unit area of the interface. The three equations necessary to determine the three unknown functions, cl(z,t), Fl(t) and o'(t), are in fact Eqs. (1.90), (1.91) and one of the adsorption isotherms, F~ = F~(c~), given in Table 1.1. Except the Henry isotherm, all other isotherms in Table 1.1 give a nonlinear connection between F1 and c~. As a consequence, an analytical solution of the problem can be obtained only if the Henry isotherm can be used, or if the deviation from equilibrium is small and the adsorption isotherm can be linearized:

r, (,) - tie

e

--

]

(1.92)

Cls stands for the subsurface concentration; here and hereafter the subscript "e" means that the respective quantity refers to the equilibrium state. The set of three linear equations, Eqs. (1.90)-(1.92), have been solved by Sutherland [65]. The result, which describes the relaxation of a small initial interfacial dilatation, reads:

F~(t)-Fle =exp

O'(t)--l~e

rfc

(1.93)

where (1.94)

is the characteristic relaxation time of surface tension and adsorption, and 2

oo

2

erfc(x) - ----~exp(-x )dx

(1.95)

~//17 x

is the so called complementary error function [67, 68]. The asymptotics of the latter function for small and large values of the argument are [67, 68]:

2

A'+ o(.x')

erfc(.x) = I - --x

c- , ?

& + (I(

for x > I

(1 3 6 )

Combining Eqs. ( I ,931 and (1.96) one obt.ains the short-time and longtime asymptotics of the surface tcnsion rclaxation:

(f

>> z),

( I 38)

Equation (1.98) is oftcn used as a test t o vei-ify whether the adsorption process is under diffiision control: data for A q t ) = q t ) - 0, are plotted vs. I / &

and it is checked if- tho plot

complics wich a straight line. We recall that Eqs. (1.97) a n d ( I .9X') arc valid in the case of a snirrll initial pei-tiii-bal.ion;alternalive asymptotic expi-essions for the case of h - g e inilia1

perturbation hnvc bccn derived for nonionic surfactants by Hansen [h9l and for ionic surfactants by Danov ct al. [70].

Using thermodynamic trilnslormations one can relate the dci-ivative in Eq. (1.94) to the Gihhs elasticity Ec;; thus ELI.( 1.94) can be expressed i n an altcmativc form:

Substituting &; frorri Table 1.2 i n t o Eq. (1.99) one could obtain cxprcssions foi-

Z ,

corresponding to the various adsorption isothcrrns. In the special case of Langmuir adsorption isothcrrn onc can present Eq. ( 1.99) in [he Lor111

IT

I

( ~ r . " .-, )I '

: -

r),

(I

-I- Kc, I'

I), (I

( AT-,)

'

.

+ I> r ; , i = 1,2)

(1.105)

where the adsorption relaxation time ri is defined as follows [60,70]" ri -

g~2 + gG2 (,~)

gi,G, (~,) +

(i = 1,2)

(1.106)

P

where ~cis the Debye screening parameter, Eq. (1.64), and the following notation is used"

g = gl,g22 - gl2g2,,

g ji ~ t r

(i,j = 1,2) e

P - I + ~ "2 + ( g l , - g21)~"3 + ( g 2 2 - g ~ 2 ) / ~ ,

--h-

/

1 -7"/ D,

7/ +-D3

11'2 ,

... q-

1

7"/

~" - exp(-~,~,~ / 2)

1

+--+ D2

l-r/ D3

/.'2 ,

7/=

c~ c2~

44

il= 1

for small initial perturbation: Ar,(O) > T ~ )

for [urge initial perturbation ( I ,109b)

where the characteristic relaxation time is determined by the expression [70]

w = 2 tanh-

@\.'c~. Note also that

V2

keeps always smaller than "c~ and "c,~, that is the

adsorption of counterions relaxes always faster than does the adsorption of surfactant ions and the surface tension. Moreover, r 2 exhibits a non-monotonic behavior (Figure 1.9). The initial increase in "c2 with the rise of the TSDS concentration can be attributed to the fact that the strong increase of the occupancy of the Stern layer, F 2 / F I , with the rise of surfactant concentration (see Fig. 1.8, the curve without salt) leads to a decrease of the surface charge density and a proportional decrease of the driving force of counterion supply, V ~ . To demonstrate the effect of salt on the relaxation time of surface tension, in Fig. 1.10 r~ is plotted vs. Clo~ for a wider range of surfactant concentrations (than that in Fig. 1.9) and for 4 different salt concentrations denoted in the figure. A g a i n , one sees that "c~ varies with many orders of magnitude: from more than 100 s down to 10 -5 s. As seen in Fig. 1.10, the addition of salt (NaC1) accelerates the relaxation of the surface tension for the higher surfactant

47

Planar Fluid Intel~lces

TSDS-water-air

No salt

mM NaCI 50 mM NaCI

20

....... 1 02

~-~.~Qz~:~.~L~.

55.

.........

" *-.~

9

115 mM NaC!

. . . . . . . . . . . . . . . . .

1 01 (/) v

o~

E o

1 0o 1 0-1

..i-.,

to x to

1 0-2

n"

1 0-3

m

(1.1

,

\ ,

\

9

~ 9 9.

,, 9

1 0-4

,~

9

9 9.

,,

...

1 0-5

i I

0-a

i

2

i

i

i

3 4 5

i

i

i ~ J

1 0 -2

l

2

h

L

t

3 4 5

I

i

i

i I

1 0 -1

i

i

2

3 4 5

;

i

t I

100

TSDS concentration (mM)

Fig. 1.10. Relaxation time of surface tension, z',~, vs. surfactant (TSDS) concentration, cj=, calculated in Ref. [70] by means of Eq. (1.110) for four different NaCI concentrations using parameters values determined from the best fit of the data in Fig. 1.6; a large initial perturbation is assumed. concentrations, but decelerates it for the lower surfactant concentrations. This curious inversion of the tendency can be interpreted in the following way. The accelerating effect of salt at the higher surfactant concentrations can be attributed to the suppression of the electric double layer by the added salt. For the lower surfactant concentrations (in the region of Henry) the latter effect is dominated by another effect of the opposite direction. This is the increase of (o~F/c)c,~)c~

due to the added salt. Physically, the effect of (o~F/c?c,~)c~

can be explained

as follows [60]. At low surfactant concentrations the diffusion supply of surfactant is very slow and it controls the kinetics of adsorption. In the absence of salt the equilibrium surfactant adsorption monolayer is comparatively diluted, so the diffusion flux from the bulk is able to

quickly equilibrate the adsorption layer. The addition of salt at low surfactant concentrations strongly increases the equilibrium surfactant adsorption (see Fig. 1.7); consequently, much longer time is needed for the slow diffusion influx to equilibrate the interface (the left-hand side branches of the curves in Fig. 1. I0). More details can be found in Ref. [60,70].

48

Chapter 1

1.3.3.

ADSORPTION UNDER BARRIER CONTROL

The adsorption is under barrier (kinetic) control when the stage of surfactant transfer from the subsurface to the surface is much slower than the diffusion stage because of some kinetic barrier. The latter can be due to steric hindrance, spatial reorientation or conformational changes accompanying the adsorption of the molecules. The electrostatic (double-layer) interaction presents a special case, which is considered in the previous Section 1.3.2. First, we will restrict our considerations to the case of pure barrier control without double layer effects. In such a case the surfactant concentration is uniform throughout the solution, Cl = const., and the increase of the adsorption Fl(t) is solely determined by the "jumps" of the surfactant molecules over the adsorption barrier, separating the subsurface from the surface:

dVl

~ = dt rad s

and

rde s

(1.111)

Q - rads(C,,F~)--rdes(F1)

are the rates of surfactant adsorption and desorption. The concept of barrier-limited

adsorption originate from the works of Bond and Puls [101], and Doss [102]. Further this theoretical model has been developed in Refs. [103-110]. Table 1.4 summarizes the most popular expressions for the total rate of adsorption under barrier control, Q, see Refs. [108112]. The quantities Kads and Kdes in Table 1.4 are the rate constants of adsorption and desorption, respectively. Their ratio gives the equilibrium constant of adsorption K e = Kads / Kdes = lPooK ,

(1.112)

The expression Ke = F=K, is valid for the Henry, Langmuir, Frumkin, Volmer and van der Waals isotherms; likewise, for the Freundlich isotherm Ke --- F=KF; the parameters Foo, K and KF are the same as in Table 1.1. Setting Q = 0 (which corresponds to the equilibrium state of

the system) each expression in Table 1.4 reduces to the respective equilibrium adsorption isotherm given in Table 1.1, as it should be expected. In addition, for

/3

= 0 the Frumkin

expression for Q reduces to the Langmuir expression. For F~ 1 Eq. (1.117), along with Eq. (1.112), gives the expression for diffusion control, Eq. (1.94), for the Henry isotherm. Other results for the mixed diffusion-barrier problem can be found in Refs. [114-1181.

51

Planar Fluid Interfaces

The case of mixed barrier-electrodiffusion control also deserves some attention insofar as it can be important for the kinetics of adsorption of some ionic surfactants. We will consider the same system as in Section 1.3.2, that is a solution of an ionic surfactant M+S - with added non-amphiphilic salt M+C -. Here S- is the surfactant ion, M + is the counterion and C- is the coion due to the salt. First, let us consider Langmuir-type adsorption, i.e. let us consider the interface as a twodimensional lattice. Further, we will use the notation 00 for the fraction of the free sites in the lattice, 0~ for the fraction of the sites containing adsorbed surfactant ion S-, and 02 for the fraction of the sites containing the complex of adsorbed surfactant ion with a bound counterion. Obviously, one can write 00 + 01 + 02 = 1

(1.1 18)

The adsorptions of surfactant ions and counterions can be expressed in the form: F1/Foo = 01 + 02 ;

F2/F= = 02

(1.1 19)

Following Kalinin and Radke [1 19], let us consider the "reaction" of adsorption of S- ions: A0 + S - = AoS-

(1.120)

where A0 symbolizes an empty adsorption site. In accordance with the rules of the chemical kinetics one can express the rates of adsorption and desorption in the form: rl,ads = Kl,ads00 Cls ,

rl,des = Kl,des01

(1.1 21 )

where, as before, C~s is the subsurface concentration of surfactant; Kl,ads and Kl,des a r e constants. In view of Eqs. (1.1 18)-(1.1 19) one can write 00 = (F~ - FI)/F~ and 01 = (Fl - F2)/F~. Thus, with the help of Eq. (1.121) we obtain the adsorption flux of surfactant: Q! - rl,ads- rl,des = Kl,adsCls(F~,- F I ) / F ~ - Kl,des(Fl -- F2)/F~

(1.122)

Next, let us consider the reaction of counterion binding: AoS-+ M += AoSM

(1.123)

The rates of the straight and the reverse reactions are, respectively, r2,ads --= K2,ads01 C2s,

r2,des ----K2.des02

(1.124)

Chapter 1

52

where g2,ads and g2,des are the respective rate constants, and C2s is the subsurface concentration of counterions. Having in mind that 01 = (F1 - F2)/F= and 02 = F2/F=, with the help of Eq. (1.124) we deduce an expression for the adsorption flux of counterions" Q2 - r2,ads- r2,des = K2,ads C2s(Fl - F 2 ) / F o o - K2,des F2/Foo

(1.125)

Up to here, we have not used simplifying assumptions. If we can assume that the reaction of counterion binding is much faster than the surfactant adsorption, then we can set Q2 - 0, and Eq. (1.125) reduces to the Stern isotherm: F2 F1

-

Kstc2s Kstc2s

Kst -= K2,ads/K2,des

,

(1.126)

1+

Note that Eq. (1.126) is equivalent to Eq. (1.82) with/(st = K2/K~. Next, a substitution of F2 from Eq. (1.126) into Eq. (1.122) yields Q1 - rl,ads- rl,des = Kl,ads Cls(F~- F I ) / F ~ - Kl,des(1 +/(St CZs)-1F1/F~

(1.127)

Equation (1.127) shows that in the adsorption flux of surfactant is influenced by the subsurface concentration of counterions, Czs. If there is equilibrium between surface and subsurface, then we have to set Q1 - 0 in Eq. (1.127), and thus we obtain the Langmuir isotherm for an ionic surfactant:

Kcls = F1/(Foo-

F1),

K-= (Kl,ads/Kl,des)(1 + Kst Czs)

(1.128)

Note that the above expression for the adsorption parameter K is equivalent to Eq. (1.81), with K! = Kl,ads/Kl,des. This result demonstrates that the linear dependence of K on C2s can be deduced from the reactions of surfactant adsorption and counterion binding, Eqs. (1.120) and (1.123). In the case of

Frumkin-type adsorption isotherm an additional effect of interaction between the

adsorbed surfactant molecules is taken into account. Then, instead of Eq. (1.122), one can derive QI - r l , a d s - rl,des = Kl,ads C l s ( F ~ - F l ) / F ~ - fl,des(Fl - Fz)/F~o

(1.129)

where Fl,des depends on Fj, because an adsorbed surfactant molecule "feels" the presence of other adsorbed molecules at the interface. The latter dependence can be expressed as follows

53

Planar Fluid Interfaces

Fl,des -- Kl,desexp I - 2flF~ kT J

(~.130)

see Table 1.4 and Ref. [110].

1.3.4.

ADSORPTION FROM MICELLAR SURFACTANT SOLUTIONS

As known, beyond a given critical micellization concentration (CMC) surfactant aggregates (micelles) appear in the surfactant solutions. In general, the micelles exist in equilibrium with the surfactants monomers in the solution [50,51]. If the concentration of the monomers in the solution is suddenly decreased, the micelles release monomers until the equilibrium concentration, equal to CMC, is restored at the cost of disassembly of a part of the micelles. The relaxation time of this process is usually in the millisecond range. The dilatation of the surfactant adsorption layer leads to a transfer of monomers from the subsurface to the surface, which causes a transient decrease of the subsurface concentration of monomers. The latter is compensated by disintegration of a part of the micelles in the

IIIII 0

0

0

0

0

O

0

demicellization

SUBSURFACE LAYER

v

. . . .

fdi~fusi-'on . . . . . . . . I convection

~di~fiasion" "- " / convection

demicellization ,.-~ O ~ g

BULK

assembly

Fig. 1.11. In the neighborhood of an expanded adsorption monolayer the micelles release monomers to restore the equilibrium surfactant concentration at the surface and in the bulk. The concentration gradients give rise to diffusion of micelles and monomers.

54

Chapter 1

subsurface layer, see Fig. 1.11. This process is accompanied by a diffusion transport of surfactant monomers and micelles due to the appearance of concentration gradients. In general, the micelles serve as a powerful source of monomers which is able to quickly damp any interfacial disturbance. Therefore, the presence of surfactant micelles strongly accelerates the kinetics of adsorption. The theoretical model by Anianson et al. [120-123] describes the micelles as polydisperse aggregates, whose growth or decay happens by exchange of single monomers:

K7 Aj+-->Aj_ 1 +A 1

(/'=2 .....M)

x;

(1.131)

+

Here j denotes the aggregation number of the micelle; K j and K j are rate constants of micelle assembly and disassembly. The general theoretical description of the diffusion in such a solution of polydisperse aggregates taking part in chemical reactions of the type of Eq. (1.131 ) is a heavy task; nevertheless, it has been addressed in several works [ 124-127]. Approximate models, which however account for the major physical effects in the system, also have been developed [ 128-134]. The basis of these models is the experimental fact that the size distribution of the micelles has a well pronounced peak, so they can be described approximately as being monodisperse with a mean aggregation number, m, corresponding to this peak. Other simplification used is to consider small deviations from equilibrium. In this case any reaction mechanism of micelle disassembly gives a linear dependence of the reaction rate on the concentration, i.e. one deals with a reaction of "pseudo-first order". As an example, we give an expression for the relaxation of surface tension of a micellar solution at small initial deviation from equilibrium derived in Ref. [125]:

a(t)-a~

r, ( t ) - r, ~

1

a(o)-a

F, ( 0 ) - F,.~

g, - g2

where E ( g , v )

- g exp(g 2r)erfc(g~-),

(1.132)

Z -- t/Zd.,

g,.2 -- [ l + ~/l + 4 v , / V,,, ] / 2 ,

r, - (OF, / ?c, ; ~ / D, .

(1.133)

55

Planar Fluid Interfaces

"c,,, and z~l are the characteristic relaxation times of micellization and monomer diffusion, see Ref. [135]; K~l is rate constant of micelle decay; as earlier, the subscript "e" refers to the equilibrium state and m is the micelle aggregation number. In the absence of micelles one is to substitute za/z,,,--->O; then gl = 1, g2 = 0, and Eq. (1.132) reduces to Eq. (1.93), as it should be expected. One can estimate the characteristic time of relaxation in the presence of micelles by using the following combined expression: 4za

(1.134)

According to the latter expression zo ~ z,,, for 2~ >> rm and zo~'cj for 2"j 90 ~ hr becomes negative and the inner meniscus is situated below the level of the outer liquid far from the cylinders in Fig. 2.4. To obtain the equations for the shape of the lower interface, u(r), of the fluid particle in Fig. 2.3 let us fix the coordinate origin at the bottom of the particle. Combining Eqs. (2.5), (2.7), (2.24) and (2.26) one can obtain the Laplace equation in the form: dsinq9 sinq9 2 fic ~ + ~ = -+e z, dr r b -b-y-

tanq9 . . . .

dz dr

b-

20~ Pl0 -P20

(2.28)

Here b is the radius of curvature at the bottom of the bubble (drop) surface, where z = 0, see Fig 2.3; the parameter e takes values +1. For sessile type drops or bubbles the mass density of the

Interfaces of Moderate Curvature: Theory of Capillarity

73

fluid particle (Phase 1) is smaller than that of Phase 2, Pl < P2, and e = +1; for pendant type drops/bubbles Pl >/)2 and e = -1. The definition of the sign of tan(p in Eq. (2.28) leads to (p = n:

at

z = 0.

(2.29)

Equations (2.28) allow one to determine the meniscus profile in a parametric form, that is r = r((p) and z = z(q~). Let us consider three cases corresponding to different values of the capillary number tic.

(i) No gravity deformation: /3c = 0. Such is the case of small fluid particles (drops, bubbles) for which the gravitational deformation can be neglected. In this case the only solution of the Laplace equation, Eq. (2.28), is a spherical meniscus: r((p) = sinq~ b

9

z(q~) b

= 1 + cos(p.

(2.30)

If the boundaries of small fluid particles or biological cells have a shape, which is different from spherical, this is an indication about the presence of an effect of the interfacial (membrane) bending elasticity, see Chapter 3.

(ii) Small gravity deformation:/~c ~1, see Fig. 2.13b). From a macroscopic viewpoint the force balance in Fig. 2.13b can be preserved if only a line tension term, c~r = ~rc, is introduced, see Eq. (2.71) with o~= o~2. Indeed, the surface tensions o', ~y~,.

94

Chapter 2 (y G 0.2s ,,._

0.2s

/ ./////z ///,

(a)

/

,c / / / / / / / / ,

7~7/

'

///////

"~/

(b)

Fig. 2.13. Sessile liquid drop on a solid substrate. (a) Balance of the forces acting per unit length of the contact line, of radius r, ; o" is the surface tension of the liquid, o-is and o-2.,,are the tensions of the two solid-fluid interfaces, al is contact angle. (b) After liquid is added to the drop, hysteresis is observed: the contact angle rises to a2 at fixed r, ; the fact that the macroscopic force balance is preserved (the contact line remains immobile) can be attributed to the action of a line tension effect o-~. and o2s are the same in Figs. 2.13a and 2.13b, and the difference between the contact angles (a2 > oq) can be attributed to the action of a line tension. The interpretation of the contact-angle hysteresis as a line tension could be accepted, because, as already mentioned, the two phenomena have a similar physical origin: local microscopic deviations from the macroscopic Young-Laplace model in a narrow vicinity of the contact line. When the meniscus advance is accompanied by an increase of the contact radius re, a positive line tension must be included in the Young equation to preserve the force balance (Fig. 2.13b). In the opposite case, if the meniscus advance is accompanied by a decrease of rc, then a negative line tension must be included in the Neumann-Young equation to preserve the force balance. The shrinking bubbles, like that depicted in Fig. 2.3, correspond to the latter case and, really, negative line tensions have been measured with such bubbles [109,110, 100-102]; see also the discussion in Ref. [111]. Theoretical calculations, which do not take into account effects such as surface roughness or heterogeneity, or dynamic effects with adhesive thin films, usually predict very small values of the line tension from 10-~ to 10-~3 N, see Table 2.1. On the contrary, the experiments which deal with real solid surfaces, or which are carried out under dynamic conditions, as a rule give much higher values of tc (Table 2.1). The values of tc in a given experiment often have variable magnitude, and e v e n -

variable sign [70-73,100-102,109,110]. Moreover, the values of ~"

Interfaces of Moderate Curvature: Theory of CapillariO'

95

determined in different experimental and theoretical works vary with 8 orders of magnitude (Table 2.1). Table 2.1. Comparison of experimental and theoretical results for line tension to. Researchers

Theory /

System

Value(s) of line tension K (N)

Experiment Tarazona & Navascues [112]

Theory

Solid-liquid-vapor contact line

-2.6 to -8.2 x 10-11

Navascues & Mederos [113]

Experiment

Nucleation rate of water drops on Hg

-2.9 t o - 3 . 9 x 10-~~

de Feijter & Vrij [52]

Theory

Kolarov & Zorin [114]

! Foam films

= - 1 • 10-12

Experiment

Foam films

- 1.7 x 10- 1 0

Denkov et al. [115]

Theory

Emulsion films

-0.95 t o - 1 . 5 7 x 10-13

Torza and Mason [59]

Experiment

Emulsion films

Ivanov & coworkers [100-102, 109-111]

Experiment

Foam film at the top of shrinking bubbles

-1 x 10.7 to = 0

Wallace & Schtirch [116,117]

Experiment

Sessile drop on monolayer

+1 to +2.4 x 10-8

Neumann & coworkers [118-121]

Experiment

Sessile drops

+1 to +6 x 10- 6

Gu, Li & Cheng [122]

Experiment

Interface around a cone

= +1 x 10- 6

Nguyen et al. [123]

Experiment

Silanated glass spheres on water-air surface

+1.2 to +5.5 x 10-6

,

!

-0.6 t o - 5 . 8 x 10-s

|

There could be some objections against the formal treatment of the contact angle hysteresis as a line-tension effect. Firstly, some authors [124-126] interpret the hysteresis as an effect of static friction (overcoming of a barrier), which is physically different from the conventional molecular interpretation of line tension, see e.g. Ref. [112]. Secondly, a hysteresis of contact angle can be observed also with a straight contact line (r~.-->oo); if such hysteresis is interpreted as a line tension effect, one will obtain ;c~oo, but cr~:= to/re will remain finite.

96

Chapter 2

If o'~ in the Neumann "quadrangle" Eq. (2.73), is a manifestation of hysteresis, then o'~ is not expected to vary significantly with the size of the particles (solid spheres, drops, bubbles, lenses). On the other hand, rc can vary with many orders of magnitude. Consequently, if the line tension effect in some system is a manifestation of a contact angle hysteresis, then one could expect that the measured Itcl = Io%clrc will be larger for the larger particles (greater rc) and smaller for the smaller particles (smaller rc). Some of the reported experimental data (Table 2.1) actually exhibit such a tendency. For example, in the experiments of Neumann and coworkers [118-121] and Gu et al. [122] rc = 3 mm and one estimates an average value o-~ = 1 mN/m; in the experiments of Ivanov and coworkers [ 100-102,109,110] the mean value of the contact radius is rc-- 35 lain and one estimates Io,cl = 1.4 mN/m; in the experiment of Torza & Mason [59] rc = 15 gm in average and then Io-~1 = 2 raN/m; in the experiments of Navascues & Mederos [113] rc = 23 nm and one obtains Io~1 = 20 mN/m. One sees, that in contrast with Itr which varies with many orders depending on the experimental system, Io-~1 exhibits a relatively moderate variation. Then a question arises whether tr or o~ is a better material parameter characterizing the linear excess at the three-phase contact line. In the experiments with slowly diminishing bubbles from solutions of ionic surfactant [100102] it has been firmly established that the shrinking of the contact line is accompanied by a rise (hysteresis) of the contact angle, o~, and appearance of a significant negative line tension, K-. When the shrinking of the contact line was stopped (by control of pressure), both c~ and Itcl relaxed down to their equilibrium values, which for tc turned out to be zero in the framework of the experimental accuracy (_+1.5 x 10-s N). This effect was interpreted [100-102,111] as a "dynamic" line tension related to local deformations in the zone of the contact line, which are due to the action of attractive (adhesive) forces opposing the detachment of the film surfaces in the course of meniscus advance. Arguments in favor of such an interpretation are that a measurable line tension effect is missing in the case of (i) receding meniscus (expanding bubbles) [100-102] and (ii) shrinking bubbles from nonionic surfactant solution [108]. In the latter case the adhesive surface forces in the film are negligible. Finally, let us summarize the conclusions stemming from the analysis of the available experimental and theoretical results for the line tension:

Interfaces of Moderate Curvature: Theoo' of Capillarity

97

1) The line tension of three-phase-contact-lines can vary by many orders of magnitude depending on the specific system, configuration (contact-line radius) and process (static or dynamic conditions). The sign of line tension could also vary, even for similar systems [70-73]. In some cases this could be due to the fact, that the measured line tension is a manifestation of hysteresis of contact angle; in this case the variability of the magnitude and sign of the line tension is connected with the indefinite value of the contact angle. Hence, unlike the surface tension, the line tension, to, strongly depends on the geometry of the system and the occurrence of dynamic processes. This makes the theoretical prediction of line tension a very hard task and limits the importance and the applicability of the experimentally determined values of tc only to the given special system, configuration and process. 2) The line tension of three-phase-contact lines is usually a small correction (an effect of secondary importance) in the Young equation or Neumann triangle, and it could be neglected without a great loss of accuracy. 3) In contrast, the line tension of the boundary between two surface phases (see e.g. Fig. 2.1 lb) is an effect of primary importance, which determines the shape and the stability of the boundaries between domains (spots) in thin liquid films and Langmuir adsorption films.

2.4.

SUMMARY

The pressure exhibits a jump on the two sides of a curved interface or membrane of non-zero tension. This effect is quantitatively described by the Laplace equation, which expresses the force balance per unit area of a curved interface. In general, the Laplace equation is a second order nonlinear partial differential equation, Eq. (2.20), determining the shape of the interface. This equation, however, reduces to a much simpler ordinary differential equation for the practically important special case of axisymmetric interfaces and membranes, see Eqs. (2.22)(2.25). There are three types of axisymmetric menisci. (I) Meniscus meeting the axis of revolution: the shapes of sessile and pendant drops and some configurations of biological cells belong to this type (Section 2.2.1). (II) Meniscus decaying at infinity: it describes the shape of the fluid interface around a vertical cylinder, floating solid or fluid particle (including gas bubble and oil lens), as well as around a hole in a wetting film (Section 2.2.2). (Ill) Meniscus

98

Chapter 2

confined between two cylinders (Section 2.2.3): in the absence of gravitational deformation the shape of such a meniscus is described by the classical curves "nodoid" and "unduloid", which represent linear combinations of the two elliptic integrals of Legendre; such menisci are the capillary "bridges", the Plateau borders in foams, the shape of the free surface of a fluid particle or biological cell pressed between two plates. For all types of axisymmetric menisci the available analytical formulas are given, and numerical procedures are recommended if there is no appropriate analytical expression. In reality the fluid interfaces (except those of free drops and bubbles) are bounded by threephase contact lines. The values of the contact angles subtended between three intersecting phase boundaries are determined by the force balance at the contact line, which is termed Young equation in the case of solid particle, Eq. (2.64), and Neumann triangle in the case of fluid particle, Eq. (2.66). It is demonstrated that the force balance at the contact line (likewise the Laplace equation) can be derived by variation of the thermodynamic potential. Linear excess energy (line tension) can be ascribed to a contact line. The line tension can be interpreted as a force tangential to the contact line, which is completely similar to the tension of a stretched string of fiber from mechanical viewpoint. When the contact line is curved, the line tension gives a contribution, o~, in the Young and Neumann equations, see Figs. 2.10, 2.1 l a and Eqs. (2.72) and (2.73). The latter equations express force balances, which influence the equilibrium position of a particle at an interface. The rule how to calculate the net force exerted on such a particle is presented and illustrated, see Fig. 2.12. The accumulated experimental results for various systems show that the line tension a of threephase-contact line can vary by many orders of magnitude, and even by sign, depending on the specific system, configuration and process. In some cases the measured macroscopic line tension can be a manifestation of contact angle hysteresis; in such a case the variability of the magnitude and sign of the line tension is connected with the indefinite value of the contact angle. The line tension of three-phase-contact-lines (see Table 2.1) is usually dominated by the surface tensions of the adjacent interfaces, and therefore it is a small correction in the Young equation or Neumann triangle. In contrast, the line tension of the boundary between two surface phases (see Fig. 2.1 l b and Eq. 2.75) is an effect of primary importance, which determines the shape and the stability of the respective contact lines.

Interfaces of Moderate Curvature: Theory of Capillarity

2.5.

99

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1O0

Chapter 2

23. A. Scheludko, Proc. Koninkl. Nederl. Akad. Wet., B65 (1962) 87. 24. A. Scheludko, Adv. Colloid Interface Sci. 1 (1967) 391. 25. K.J. Mysels, J. Phys. Chem. 68 (1964) 3441. 26. A. Hadjiiski, R. Dimova, N.D. Denkov, I.B. Ivanov, R. Borwankar, Langmuir 12 (1996) 6665. 27. I.B. Ivanov, A. Hadjiiski, N.D. Denkov, T.D. Gurkov, P.A. Kralchevsky, S. Koyasu, Biophys. J. 75 (1998) 545. 28. J. Plateau, Mere. Acad. Roy. Soc. Belgique 33 (1861), sixth series and preceding papers. 29. J.W. Gibbs, The Scientific Papers of J.W. Gibbs, Vol. 1, Dover, New York, 1961. 30. A.S. Dimitrov, P.A. Kralchevsky, A.D. Nikolov, D.T. Wasan, Colloids Surf. 47 (1990) 299. 31. G. Bakker, Kapillarit~it und Oberfl~ichenspannung, in: Handbuch der Experimentalphysik, Band 6, Akademische Verlagsgesellschaft, Leipzig, 1928. 32. I.B. Ivanov, P.A. Kralchevsky, A.D. Nikolov, J. Colloid Interface Sci. 112 (1986) 97. 33. J. Gaydos, A.W. Neumann, Thermodynamics of Axisymmetric Capillary Systems, in: A.W. Neumann & J.K. Spelt (Eds.) Applied Surface Thermodynamics, Marcel Dekker, New York, 1996, p. 53. 34. S. Ljunggren, J.C. Eriksson, P.A. Kralchevsky, J. Colloid Interface Sci. 191 (1997) 424. 35. T. Young, Philos. Trans. Roy. Soc. London 95 (1805) 55. 36. A. Dupr6, Theorie Mecanique de la Chaleur, Paris, 1869, p. 368. 37. J.C. Eriksson, S. Ljunggren, Langmuir 11 (1995) 2325. 38. P.A. Kralchevsky, Langmuir 12 (1996) 5951. 39. R. Shuttleworth, Proc. Phys. Soc. (London) A63 (1950) 444. 40. C. Herring, in: W.E. Kingston (Ed.) The Physics of Powder Metallurgy, McGraw-Hill, New York, 1951. 41. G.C. Benson, K.S. Yun, in: E.A. Flood (Ed.) The Solid-Gas Interface, Vol. 1, Marcel Dekker, New York, 1967. 42. J.C. Eriksson, Surface Sci. 14 (1969) 221. 43. A.I. Rusanov, Kolloidn. Zh. 39 (1977) 711; J. Colloid Interface Sci. 63 (1978) 330. 44. F.E. Neumann, in: A. Wangerin (Ed.) Vorlesungen tiber die Theorie der Kapillaritfit, Teubner, Leipzig, 1894, p. 161.

Interfaces of Moderate Curvature: Theory of Capillarit3'

101

45. F.P. Buff, H. Saltsburg, J. Chem. Phys. 26 (1957) 23. 46. F.P. Buff, in: S. Fltigge (Ed.) Encyclopedia of Physics, Vol. 10, Springer, Berlin, 1960, 298. 47. N.F. Miller, J. Phys. Chem. 45 (1941) 1025. 48. P.R. Pujado, L.E. Scriven, J. Colloid Interface Sci. 40 (1972) 82. 49. P.A. Kralchevsky, K.D. Danov, I.B. Ivanov, Thin Liquid Film Physics, in: R.K. Prud'homme (Ed.) Foams: Theory, Measurements and Applications, Marcel Dekker, New York, 1995, Section 3.3. 50. J.K. Spelt, E.I. Vargha-Butler, Contact Angle and Liquid Surface Tension Measurements, in: A.W. Neumann & J.K. Spelt (Eds.) Applied Surface Thermodynamics, Marcel Dekker, New York, 1996, p. 379. 51. B.A. Pethica, J. Colloid Interface Sci. 62 (1977) 567. 52. J.A. de Feijter, A. Vrij, J. Electroanal. Chem. Interfacial Electrochem. 37 (1972) 9. 53. L. Boruvka, A.W. Neumann, J. Chem. Phys. 66 (1977) 5464. 54. V.S. Veselovsky, V. N. Pertsov, Z. Phys. Khim. 8 (1936) 245. 55. I.B. Ivanov, B.V. Toshev, B.P. Radoev, in: J.F. Padday (Ed.) Wetting, Spreading and Adhesion, Academic Press, New York, 1978, p. 37. 56. G.A. Martynov, I.B. Ivanov, B.V. Toshev, Kolloidn. Zh. 38 (1976) 474. 57. B.A. Pethica, Rep. Prog. Appl. Chem. 46 (1961) 14. 58. A.I. Rusanov, Phase Equilibria and Surface Phenomena, Khimia, Leningrad, 1967 (in Russian); Phasengleichgewichte und Grenzfl~ichenerscheinungen, Akademie Verlag, Berlin, 1978. 59. S. Torza, S.G. Mason, Kolloid-Z. Z. Polym. 246 (1971) 593. 60. I.B. Ivanov, P.A. Kralchevsky, in: I.B. Ivanov (Ed.) Thin Liquid Films, Marcel Dekker, New York, 1988, p. 91. 61. P.M. Kruglyakov, in: I.B. Ivanov (Ed.) Thin Liquid Films, Marcel Dekker, New York, 1988, p. 767. 62. D. Exerowa, P.M. Kruglyakov, Foam and Foam Films, Elsevier, Amsterdam, 1998. 63. A.D. Nikolov, D.T. Wasan, P.A. Kralchevsky, I.B. Ivanov, Ordered Structures in Thinning Micellar Foam and Latex Films, in: N. Ise and I. Sogami (Eds.) Ordering and Organisation in Ionic Solutions, World Scientific, Singapore, 1988, p. 302. 64. P.A. Kralchevsky, A.D. Nikolov, D.T. Wasan, I.B. Ivanov, Langmuir 6 (1990) 1180.

102

Chapter 2

65. H. M6hwald, Annu. Rev. Phys. Chem. 41 (1990) 441. 66. U. Retter, K. Siegler, D. Vollhardt, Langmuir 12 (1996) 3976. 67. M.J. Roberts, E.J. Teer, R.S. Duran, J. Phys. Chem. B 101 (1997) 699. 68. V.N. Paunov, P.A. Kralchevsky, N.D. Denkov, K. Nagayama, J. Colloid Interface Sci. 157 (1993) 100. 69. R. Aveyard, B.P. Binks, P.D.I. Fletcher, C.E. Rutherford, Colloids Surf. A 83 (1994) 89. 70. A.B. Ponter, A.P. Boyes, Canadian J. Chem. 50 (1972) 2419. 71. A.P. Boyes, A.B. Ponter, J. Chem. Eng. Japan 7 (1974) 314. 72. A.B. Ponter, M. Yekta-Fard, Colloid Polym. Sci. 263 (1985) 1. 73. M. Yekta-Fard, A.B. Ponter, J. Colloid Interface Sci. 126 (1988) 134. 74. D. Platikanov, M. Nedyalkov, V. Nasteva, J. Colloid Interface Sci. 75 (1980) 620. 75. J. Gaydos, A.W. Neumann, Line Tension in Multiphase Equilibrium Systems, in: A.W. Neumann & J.K. Spelt (Eds.) Applied Surface Thermodynamics, Marcel Dekker, New York, 1996, p. 169. 76. D. Li, Colloids Surf. A 116 (1996) 1. 77. J. Drelich, Colloids Surf. A 116 (1996) 43. 78. Y. Gu, D. Li, P. Cheng, Colloids Surf. A 122 (1997) 135. 79. A. Marmur, Colloids Surf. A 136 (1998) 81. 80. R.J. Good, M.N. Koo, J. Colloid Interface Sci. 71 (1979) 283. 81. N.K. Adam, The Physics and Chemistry of Surfaces, Oxford University Press, Oxford, 1941. 82. R. Shuttleworth, G.L.J. Bailey, Discuss. Faraday Soc. 3 (1948) 16. 83. R.E. Johnson Jr., R.H. Dettre, in: E. Matijevic (Ed.) Surface and Colloid Science, Vol. 2, Wiley, New York, 1969, p. 85. 84. A. Marmur, Adv. Colloid Interface Sci. 50 (1994) 121. 85. F.Z. Preisach, Z. Phys. 94 (1935) 277. 86. D.H. Everett, W.I. Whitton, Trans. Faraday Soc. 48 (1952) 749. 87. R.J. Good, J. Phys. Chem. 74 (1952) 5041~ 88. J.A. Enderby, Trans. Faraday Soc. 51 (1955) 835. 89. R.E. Johnson Jr., R.H. Dettre, J. Phys. Chem. 68 (1964) 1744.

Interfaces of Moderate Curvature: Theory of Capillarity

103

90.

T.D. Blake, J.M. Haynes, in: J.F. Danielli et al. (Eds.) Progress in Surface and Membrane Science, Vol. 6, Academic Press, 1973, p. 125.

91.

J.D. Eick, R.J. Good, A.W. Neumann, J. Colloid Interface Sci. 53 (1975) 235.

92.

A.W. Neumann, in: J.F. Padday (Ed.) Wetting, Spreading and Adhesion, Academic Press, New York, 1978, p. 3.

93.

R.J. Good, in: R.J. Good & R.R. Stromberg (Eds.) Surface and Colloid Science, Vol. 11, Plenum Press, 1979, p. 1.

94.

I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, New York, 1991.

95.

A. Marmur, J. Colloid Interface Sci. 168 (1994) 40.

96.

J. Drelich, J.D. Miller, R.J. Good, J. Colloid Interface Sci. 179 (1996) 37.

97.

G.A. Martynov, V.M. Starov, N.V. Churaev, Kolloidn. Zh. 39 (1977) 472.

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V.M. Starov, Adv. Colloid Interface Sci. 39 (1992) 147.

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K.D. Danov, P.A. Kralchevsky, I.B. Ivanov, Chapter 9 in: G. Broze (Ed.) Handbook of Detergents, Part A.: Properties, Surfactant Science Series, Vol. 82. Marcel Dekker, New York, 1999, Section V.C.

100. I.B. Ivanov, A.S. Dimitrov, A.D. Nikolov, N.D. Denkov, P.A. Kralchevsky, J. Colloid Interface Sci. 151 (1992) 446. 101. A.S. Dimitrov, A.D. Nikolov, P.A. Kralchevsky, I.B. Ivanov, J. Colloid Interface Sci. 151 (1992) 462. 102. I.B. Ivanov, P.A. Kralchevsky, A.S. Dimitrov, A.D. Nikolov, Adv. Colloid Interface Sci. 39 (1992) 77. 103. J. Mingins, A.D. Nikolov, Ann. Univ. Sofia Fac. Chem. 75 (1981) 3. 104. R. Aveyard, J.H. Clint, J. Chem. Soc. Faraday Trans. 93 (1997) 1397. 105. A. Dussaud, M. Vignes-Adler, Langmuir 13 (1997) 581. 106. P. Chert, S.S. Susnar, A. Amirfazli, C. Mak, A.W. Neumann, Langmuir 13 (1997) 3035. 107. P. Chen, S.S. Susnar, C. Mak, A. Amirfazli, A.W. Neumann, Colloids Surf. A 129 (1997) 45. 108. L.A. Lobo, A.D. Nikolov, A.S. Dimitrov, P.A. Kralchevsky D.T. Wasan, Langmuir 6 (1990) 995, Figs. 10 and 13. 109. P.A. Kralchevsky, A.D. Nikolov, I.B. Ivanov. J. Colloid Interface Sci. 112 (1986) 132.

104

Chapter 2

110. A.D. Nikolov, P.A. Kralchevsky, I.B. Ivanov. A New Method for Measuring Film and Line Tensions, in: K.L. Mittal & P. Bothorel (Eds.) Surfactants in Solution, Vol. 6, Plenum Press, New York, 1987, p. 1.537.

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119. D. Li, A.W. Neumann, Colloids Surf. 43 (1990) 19.5. 120. D. Duncan, D. Li, J. Gaydos, A.W. Neumann, J. Colloid Interface Sci. 169 (199.5) 2.56. 121. A. Amirfazli, D.Y. Kwok, J. Gaydos, A.W. Neumann, J. Colloid Interface Sci. 20.5 (1998) 1.

122. Y. Gu, D. Li, P. Cheng, J. Colloid Interface Sci. 180 (1996) 212. 123. A.V. Nguyen, H. Stechemesser, G. Zobel, H.J. Schulze, J. Colloid Interface Sci. 187 ( 1 997) 547. 124. R. Finn, M. Shinbrot, J. Math. Anal. Appl. 123 (1987) 1. 12.5. S.D. Iliev, J. Colloid Interface Sci. 194 (1997) 287. 126. S.D. Iliev, J. Colloid Interface Sci. 2 13 (1 999) 1.

105

CHAPTER 3 SURFACE BENDING MOMENT AND CURVATURE ELASTIC MODULI

This chapter is devoted to a generalization of the theory of capillarity to cases, in which variations of the interfacial (membrane) curvature give an essential contribution to the total energy of the system. An interface (or membrane) possesses 4 modes of deformation" dilatation, shearing, bending and torsion. The first couple of modes represent two-dimensional analogues of respective deformations in the bulk phases. The bending and torsion modes are related to variations in the two principle curvatures of the interface, that is to the presence of two additional degrees of freedom. From a thermodynamic viewpoint, the curvature effects can be accounted for as contributions of the work of interfacial bending and torsion to the total energy of the system; the respective coefficients are the interfacial (surface) bending and torsion moments, B and |

The most popular model of the interfacial curvature effects provides

an expression for the mechanical work of flexural deformation, which involves 3 parameters: bending and torsion elastic moduli, kc and k C, and spontaneous curvature, H0. Initially we consider the simpler case of spherical geometry. The dependence of the bending moment B on the choice of the dividing surface at fixed physical state of the system is investigated. The connection between the quantities bending moment, Tolman length and spontaneous curvature is demonstrated. Micromechanical expressions are derived, which allow one to calculate the surface tension and the bending moment if an expression for the pressure tensor is available. From the viewpoint of the microscopic theory, various intermolecular forces may contribute to the interfacial moments B, |

and to the curvature elastic moduli, kc and kc. Such are the van

der Waals forces, the steric and electrostatic interactions. The interfacial bending moment may give an essential contribution to the interaction between deformable droplets in emulsions. In general, the curvature effects are expected to be significant for interfaces of low tension and high curvature, including biomembranes.

106

Chapter 3

3.1.

BASIC THERMODYNAMIC EQUATIONS FOR CURVED INTERFACES

3.1.1.

INTRODUCTION

The curvature dependence of the interfacial tension was first investigated by Gibbs in his theory of capillarity [1 ]. The approach of Gibbs has been further developed by Tolman [2], who established that such curvature dependence appears for sufficiently small liquid drops or gas bubbles, whose radii are comparable with the so called Tolman length, ~o" The latter represents the distance between the surface of tension and the equimolecular dividing surface, see Chapter 1. Further development in the thermodynamics of curved interfaces was given in the works of Koenig [3] and Buff [4-6]. Kondo [7] investigated how the choice of dividing surface affects the surface thermodynamic parameters; see also Refs. [8-11 ]. An additional interest in the curvature effects has been provoked by the studies on microemulsions [12-20]. The biomembranes, lipid bilayers and vesicles represent another class of systems, for which the curvature effects play an essential role on the background of a low interfacial tension. The predominant number of works on lipid membranes is based on the mechanics of shells and plates, originating from the studies by Kirchhoff [21], Love [22], see also Refs. [23-25], and on the related theory of liquid crystals [26-28], rather than on the Gibbs thermodynamics. The mechanics of biomembranes is a complex and rich in phenomena field, whose importance is determined by the fact that such membranes are basic structural and physiological element of the cells of all living organisms. In particular, in Chapter 10 below we apply the mechanics of curved interfaces to describe theoretically the membrane-mediated interaction between proteins incorporated in a lipid bilayer.

3.1.2.

MECHANICAL WORK OF INTERFACIAL DEFORMATION

First we will make an overview of the most important equations in the thermodynamics of the curved interfaces. The work of deformation of an elementary parcel, AA, of a the boundary between two fluid phases, can be expressed in the form [29-31 ] (5%= 7 fi a + ~ fi fl + B 6 H + O fi D ,

&-

~AA)/AA

(3. l)

Surface Bending Moment and Curvature Elastic Moduli

107

Here 5Ws is the mechanical work of deformation per unit area of the phase boundary; 5o~ is the relative dilatation (increase of the area) of the surface element AA. If 5u~1 and ~tt22 are the two eigenvalues of the surface strain tensor, then 5 a and 513 can be expressed as follows: ~ a -- I~Ull -k- ~tt22 ,

(3.2)

~/~ = (~Ull-- I~b/22

see also Eq. (4.22) below. Consequently, 5o~ and 513 characterize the isotropic and the deviatoric part of the surface strain tensor. In particular, 513 characterizes the interfacial deformation of shear, see Fig. 3.1. Likewise, the surface curvature tensor has two eigenvalues, Cl and c2, representing the two principal curvatures; then 1 n - -~(c I nt- c 2 ),

1

O = ~(c,- ce)

(3.3)

are the mean and deviatoric curvature; the latter is a measure for the local deviation from the spherical shape.

Dilation

A//~~NkC~

D Bending

Shear

Reference State

B

D

C

A

A

B

D

Reference State

C

Torsion

Fig. 3.1. Modes of deformation of a surface element: dilatation, shear, bending and torsion.

Chapter 3

108

Equation (3.1), without the term ~Sfl, was first formulated in the classical work by Gibbs [1 ], and without the curvature terms - in the study by Evans & Skalak [25]. In particular, ? a a is the work of pure dilatation (aft = 0; &?l

= (~72 = 0 ) ;

~

is

the work of pure shearing (ao~ = 0;

(~1 = (~}C2 = 0), BaH is the work of pure bending (&l = &2; ao~ = aft = 0) and OaD is the work

of pure torsion (&l = -&2; ao~ = aft = 0), termed also "work of saddle-shape deformation", see Fig. 3.1. Correspondingly, B and | are called the interfacial bending and torsion moments [15]. Often in the literature the Gaussian curvature

K = c1c2 = H 2 _

(3.4)

D 2

is being chosen as an independent thermodynamic parameter, instead of the deviatoric curvature D; then Eq. (3.1) is transformed in the equivalent form

aw~ = ?~o~ + {afl + CIaH + C2~K

(3.5)

Equation (3.5), without the term ~ f l , is used in the works by Boruvka & Neumann [32] and Markin et al. [33]. A comparison between Eqs. (3.1) and (3.5) yields [31,34]: B = C 1+

2 C 2 H,

|

= - 2C 2D

(3.6)

Equation (3.1) is more convenient to use for spherical interfaces (D = 0), and Eq. (3.5) - for cylindrical interfaces (K = 0). Below we will follow the Gibbs approach, and will use H and D as thermodynamic variables; the latter have a simple geometric meaning (Fig. 3.1), and the respective moments B and | have the same physical dimension, in contrast with C 1 and C 2. In general, the surface moments B and | depend on the curvature. The latter dependence can be expressed in an explicit form by introducing some model of the interfacial flexural theology. The following rheological constitutive relation, introduced by Helfrich [23,24], is frequently used in literature

W f - 2kc(H-Ho)2+ k--cK

(3.7)

Surface Bending Moment and Curvature Elastic Moduli

109

Here wf is the work of flexural deformation per unit area of the interface" Ho, k c and k c are constant parameters of the rheological model" H0 is called the spontaneous curvature, k c and k c are the bending and torsion (Gaussian) surface elastic moduli. From Eq. (3.1) it follows

awf-

(3.8)

BaH + OaD

Combining Eqs. (3.4), (3.7) and (3.8) one derives [34,35]

B-

I 3H

6)

D - B~ + 2(2kc+k-c)H'

_( 0 for By > 0, and vice versa, & < 0 for Bv < 0.

3.2.3.

MICROMECHANICAL APPROACH

Mechanical definitions of surface tension and bending moment. The hydrostatic approach to the theoretical description of curved interfaces has been developed by Buff [5], Ono & Kondo [8] and Rusanov [9]. Owing to the spherical symmetry, the pressure tensor can be expressed in the form [8]

P = PN erer + Pr(eoeo + % % )

(3.43)

where (r, 0, q~) are polar coordinates with center in the center of spherical symmetry; e r, e 0 and e~0 are the unit vectors of the curvilinear local basis; PN and P:r represent the normal and tangential component of the tensor P with respect to the spherical interface. Let us consider a part of the system, which is confined between two concentric spheres of radii rl and r2, see Fig. 3.3. The total force acting on the shaded sectorial strip (Fig. 3.3) is [8] r2

(3.44)

dO I PT rdr rl

see Fig. 3.3 for the notation. The respective force moment is given by the expression r2

dO I PT r2 dr

(3.45)

rl

Following Gibbs [1] we define an idealized (model) system consisting of one spherical dividing surface of radius a and two bulk fluid phases, I and II, which are uniform and isotropic up to the very dividing surface. The pressure in the idealized system can be expressed in the form

~-={PI for eli for

ra

As noted in Chapter 1 (Figs. 1.1 - 1.3) the pressure tensor P is not isotropic in a vicinity of an interface. To compensate this difference between the real and the idealized system, the dividing

118

Chapte r 3

Ztd0

Z

\

X

(a)

O-

\

~

~

(b)

/

~

_IM

~

rl

I

/

Y~

v

Fig. 3.3. Sketch of the real and idealized systems, and of the sectorial strip (shaded) used to give a mechanical definition of the surface tension, o', and the bending moment, M.

surface is treated as a membrane with surface (membrane) tension o" and surface bending moment M, see Fig. 3.3b. Then the counterparts of Eqs. (3.44) and (3.45) for the idealized system are [311" r~ dO ~ -firdr - crdO

(13.47)

rl

r2

dO

f -fir 2dr - ~ a - d O

+ MadO

(3.48)

rl

To make the idealized system mechanically equivalent to the real one, we require that the force and the moment acting on the sectorial strip in the two systems (Fig. 3.3) to be equal. Thus setting equal the expressions in Eqs. (3.44) and (3.47) we obtain

oa - f (P--Pr)rdr rl

Likewise, from Eqs. (3.45) and (3.48) we derive

(3.49)

Surface Bending Moment and Curvature Elastic Moduli

0-a 2 -

M a - f (-fi-Pr )r2dr

119

(3.50)

I"1

In the above mechanical derivation we deliberately have used the notation o- and M for the

mechanical surface tension and moment. Indeed, it is not obligatory the latter to coincide with their

thermodynamic analogues, y and B, defined by Eq. (3.1). Relationship between the mechanical and thermodynamical surface tension. Under

conditions of hydrostatic equilibrium the divergence of the pressure tensor is zero, that is V.P = O. In the considered case of spherical symmetry the latter equation yields [39] d

~tr(r2 pN )= 2rPr

(3.51)

Integrating the latter equation we derive

IPrrdr=2(Pnr22-pIq2 )

(3.52)

rl

Substituting Eqs. (3.46) and (3.52) into (3.49) we obtain a version of the Laplace equation" 20-

a

(3.53)

= P~-P.

The comparison of Eqs. (3.53) and (3.3 l) yields B

? ' - 0-

(3.54)

2a

To find a unique relationship between the couple of mechanical parameters (0-, M) and the couple of thermodynamical parameters (y, B) we need a second relationship, in addition to Eq. (3.54). Such an equation can be obtained in the following way. Let us consider a purely lateral displacement of the conical surface depicted in Fig. 3.3a. The work of this displacement, carried out by the outer forces, is [8] p~

r2

dW - - f (Pr 2ffrsin Odr)rdO - - door Pr r2 dr rl

rI

(3.55)

120

Chapter 3

where 27r

d m - sin OdO f dq9 = 2Jr sin0 dO

(3.56)

0

is the increment of the spatial angle at the vertex of the cone corresponding to the considered infinitesimal displacement of the lateral surface. An alternative expression for dW is provided by thermodynamics [8]"

dW - - P I dVI - Pn dVn + ydA

(3.57)

where V~ and Vn represent the volumes of phases I and II, and A is the area of the spherical dividing surface. By means of geometrical considerations one obtains a

dVi - dco f dr r 2" q

r2

dVit - doo f d r r 2 a

dA=a2 dco

(3.58)

Setting equal the two expressions for dW, Eqs. (3.55) and (3.57), and using Eqs. (3.46) and (3.58), one deduces [8] r2

(3.59) rl

Finally, by comparing Eqs. (3.50) and (3.59) we obtain the sought for second equation connecting the mechanical and thermodynamical parameters" y=cr

M

(3.60)

Equations (3.54) and (3.60) imply the following relationship between B and M: B = 2M

(3.61)

Generalized versions of Eqs. (3.54) and (3.61) for an arbitrarily curved interface are derived below, see Eqs. (4.79) and (4.81). Equation (3.54) shows that for a curved interface there is a difference between the mechanical and thermodynamical surface tension. This difference is zero only if the dividing surface is defined as surface of tensions, for which B = 0 by definition, cf. Eq. (3.25). However, from a

Surface Bending Moment and Curvature Elastic Moduli

121

physical viewpoint the surface of tension not always provides an adequate description of the real phase boundary or membrane. To demonstrate the latter fact we will use the equation [31 ] 3

~-a v 1-

1/3

O-v - TBv / a v

(3.62)

O" v

which follows from Eqs. (3.54) and (3.42). For interfaces of low interfacial tension, Ov--+0, e.g. microemulsions or lipid membranes, Eq. (3.62) gives ~---~ ,,% that is the surface of tension is situated far away from the real boundary between the two phases; see also Ref. [ 17].

Micromechanical expressions for ~, 7 and B. The functions Pu(r) and Pr(r) provide a micromechanical description of the stresses acting in the transitional zone between the two neighboring phases [5]. Such a description takes an intermediate position between the

macroscopic description in terms of quantities like o, 7 and B, and the microscopic description in terms of the correlation functions of the statistical mechanics, see e.g. Refs. [39-42]. Convenient for applications are expressions which represent the macroscopic parameters as integrals of the function

AP(r) = PN (r)- PT (r),

(3.63)

AP(r) characterizes the anisotropy of the pressure tensor P in a vicinity of the phase boundary, see Eqs. (1.8) and (1.12), as well as Figs. 1.2 and 1.3. For a spherical interface Buff [5] has derived the expression

7,-

i

r2 AP (r)--Tdr,

r~

(3.64)

as

which is valid only for the surface of tension. Below we describe the derivation of other micromechanical expressions obtained in Ref. [31], which are valid for an arbitrary choice of the spherical dividing surface. Equation (3.51)can be represented in the form

dPN - 2AP dr

r

(3.65)

122

Chapter

3

The integration of Eq. (3.65), along with Eq. (3.53), yields r2

~r - S Ap a

(3.66)

r

rl

The latter equation specifies that the analogous expression, derived by Goodrich [43], refers to the mechanical surface tension, o, rather than to the thermodynamical one, y. Further, from Eqs. (3.46) and (3.59) we obtain ae

-

l a 3 (PI

3

-

PH ) -

r: -31 (plr13-pllr32)-fPTr2dr

(3.67)

!-1

On the other hand, the integration of Eq. (3.51 ) yields re

re

2fPTr2dr-Pi,

rzS-Piq3-~PNredr

rl

(3.68)

rl

With the help of Eqs. (3.53), (3.66) and (3.68) one can eliminate PI and Pu from Eq. (3.67)

[31].

i ,r)l ar -+U ldr

(3.69)

rl

In accordance with Eq. (3.24) we differentiate Eq. (3.69) to derive a micromechanical expression for the interfacial bending moment B [31 ]"

2 zXP(r) B _ ~rl

/a / --r

a

dr

(3.70)

The same expression for B can be obtained by substitution of the expressions for cr and ?', Eqs. (3.66) and (3.69), into Eq. (3.54). Moreover, the differentiation of Eq. (3.70), in accordance with Eq. (3.33), leads to Eq. (3.69). The latter facts demonstrate that the theory is selfconsistent. Equations (3.66), (3.69) and (3.70), which are valid for an arbitrary choice of the spherical dividing surface, have been used in Refs. [44, 45] to calculate the contribution of the van der Waals forces to the interfacial bending moment B.

Surface Bending Moment and Curvature Elastic Moduli

123

3.3.

R E L A T I O N S W I T H T H E M O L E C U L A R T H E O R Y AND T H E E X P E R I M E N T

3.3.1.

CONTRIBUTIONS DUE TO VARIOUS KINDS OF INTERACTIONS

A typical example for an electrically charged fluid interface is shown in Fig. 3.4: the surface charge is due to the presence of an adsorption layer of ionic surfactant. Upon bending of the interface (decrease of the radius a of the equimolecular dividing surface) the distance between the charges of the surface-active ions increases. This is energetically favorable owing to the presence of repulsive forces between ions of the same electric charge. As a result, a surface bending moment appears, which tends to bend the interface around the non-aqueous phase. In reality, not only the electrostatic interactions, but also other type of forces contribute to the interfacial bending and torsion moments; such are the van der Waals forces and the steric interactions between the hydrophilic headgroups and the hydrophobic tails of the surfactant molecules (Fig. 3.4). From Eq. (3.16) it follows

B

~ OH

s,,v~,a,fi,D

Insofar as the van der Waals, the electrostatic, and the steric interactions can be considered to be independent, they give additive contributions to the surface density of the internal energy u~. Then, from Eq. (3.71) it follows that these interactions give also additive contributions to the interfacial bending and torsion moments, B - B ~w + B ~l + B st ,

O --O ~w +O ~1 +O st

(3.72)

Here and hereafter the superscripts "vw", "el" and "st" denote terms related to the corresponding interactions. In view of Eqs. (3.9) and (3.72) B0, kc and k C can be expressed in the form Bo

-

l:~ VW

el

st

--0 +B0 + B 0 ,

kc-kc

vw

el

st

+kc +kc ,

--

--wv

--el

--st

k,.-k c +k C +k C

On the other hand, having in mind Eq. (3.10), one sees that the spontaneous curvature,

(3.73)

124

Chapter 3

B ov w + B(~t + B~t H0 = -

(3.74)

4(kc"w + k~el + k st )

is not additive with respect to contributions from the various interactions; instead, H0 represents a ratio of additive quantities. In Ref. [45] an expression for the van der Waals contribution, Bo w, to the bending moment of the boundary between two fluid phases has been derived:

B~

5

1

~5JrAn

(3.75)

AH _ ~2 (O, l l p 2 _ 20~12/91P2 A- Of22P 2)

(3.76)

Here 7'0 is the interfacial tension of the planar boundary between the two pure fluids (without surfactants) AH is the Hamaker constant, p~ and /92 are the number densities of the two neighboring phases, a;k are the constants in the van der Waals potential: ui~ = -ai~/r6;

the

subscripts "1" and "2" refer to the phase inside and outside the fluid particle, respectively. In general, Bo w tends to bend around the phase, which has a larger Hamaker constant [45]. Equation (3.75) has been derived by means of Eq. (3.70) and an appropriate model expression for the anisotropy of the pressure tensor, AP. For an oil-water interface Eq. (3.75) predicts Bow = 5 • 10-11N. Theoretical expressions for k~ w and k~.w are not available in the literature. The contribution of the steric interaction can be related to the size and shape of the tails and headgroups of the surfactant molecules [46-53]. The following expression was proposed [52] for such amphiphiles as the n-alkyl-poly(glycol-ethers), (CzH4)n(OCHzCH2)mOH:

Bg' = -

where ~ - ( n -

lr Zv 2b~ kT

4a M 4

gr Zv3 b k T k~ t = ~ ( 1 64a M 5

+ lZg')

(3.77)

m ) / ( n + m) characterizes the asymmetry of the amphiphile, v is the volume of

an amphiphile molecule, aM is the interfacial area per molecule, k is the Boltzmann constant, b is a molecular length-scale in the used self-consistent field model [52].

Surface Bending Moment and Curvature Elastic Moduli

125

o t 1

/

/ (a) Fig. 3.4.

(b)

Sketch of a "molecular condenser" of thickness d, which is formed (a) from adsorbed surfactant ions and their counterions and (b) from adsorbed zwitterionic surfactant. The dividing surface (of radius a) is chosen to be the boundary between the aqueous and the nonaqueous phase; l~ and 12 are the distances from the "charged" surfaces to the dividing surface.

Expressions for the electrostatic components of the bending moment, B~ t , and the curvature elastic moduli, k~,' and k~e' , have been also derived. For example, one can relate B~' , k~' and k~.~l to the surface Volta potential, AV, which is a directly measurable parameter [54]:

1 + --d--

e d (AV)2 /

k, '-

ce'

.

. . 24rc

.

(3.78)

II 112 / l + 3--d + 3 ~Z-

d

+3

(3.79)

/ d2

(3.80)

Here e is the dielectric constant, d is the distance between the positive and negative charges; the other notation is explained in Fig. 3.4. In Eqs. (3.78) - (3.80) AV must be substituted in CGSE-

Chapter 3

126

units, i.e. the value of AV in volts must be divided by 300. Note that AV expresses the change of the surface potential due to the presence of an adsorption monolayer. AV can be measured by means of the methods of the radio-active electrode or the vibrating electrode [55], which give the change in the electric potential across the interface. Equations ( 3 . 7 8 ) - (3.80) could be used when the model of the "molecular condenser" is applicable, viz.: (i) when there is an adsorption layer of zwitterions or dipoles, such as nonionic and zwitterionic surfactants or lipids, at the interface; (ii) when the electrolyte concentration is high enough and the counterions are located in a close vicinity of the charged interface to form a "molecular capacitor"; (iii) when the surface potential is low: then the Poisson-Boltzmann equation can be linearized and the diffuse layer behaves as a molecular capacitor of thickness equal to the Debye screening length [56]. For example, taking experimental value of the Volta potential for zwitterionic lipids [57], AV = 350 mV, and assuming e = 78.2, d - 5 ' ,

ll/d 0, then k~yt is

positive, whereas k~ ~ is negative and k-~Y~ = - k f z/2. It is interesting to note that the same relationship, k-~YI = - k [ 1/2, has been obtained by Ennis [51] in the framework of a quite different model taking into account the steric interactions. The surface charge density o,, i.e. the electric charge Q per unit area of the "plate" of the molecular condenser (Fig. 3.4), is simply related to AV:

Surface Bending M o m e n t and Curvature Elastic Moduli

127

Q eAV as . . . . A 4rd

(3.81)

Then a substitution of AV from (3.81) into Eqs. (3.78) - (3.80), in view of the identity d = 12- 1~, leads to Bo I - ~2zr o ' , 2 (12z - l()

(3.82)

E

ke t

- 4reOs2

k~fl _

9

1

-2kc

el

(3.83)

As mentioned in Chapter 1, see Fig. 1.4, the double electric layer consists of a Stern layer and a diffuse layer, composed, respectively, of bound and free counterions. Correspondingly, the bending moment and the curvature elastic moduli are composed of contributions from these two layers [31,58]" Bo I

1~Stn l~ dif = "-'o + ~o ,

kcel = kcStn + kcdif ,

--el -- Stn -kc = kc + kc dif

(3.84)

If the Stern layer is situated at a distance 12 from the dividing surface, then it can be proven [31 ] that n-'0 9 Stn , kcStn and ~?tn . can be expressed by analogues of Eqs. (3.82) and (3.83)

BStn 0

2re _2 (122

-

-

~O-s 6

l 2) -

(3.85)

-

kS],, _ 47c a ~2(l 3 - 13 ), 3e

k2stn --

1

k cStn

(3.86)

2

where, as before, 1~ is the distance between the surface charges and the dividing surface, see Fig. 3.4a. In the case of low surface electric potential, the Poisson-Boltzmann equation, describing the diffuse electric double layer (see Chapter 1) can be linearized. In such a case it turns out that the counterions can be treated as being situated at a distance 12 + ts-1 from the dividing surface, where U l is the Debye length, see Eq. (1.56) and (1.64)" the derived expressions for Bo l, k,el and k~ 1 in this case are [31]

Clrcrpier-3

128

BG'

771 =-af[(12+K-l)2-If]

(3.87)

&

(3.88) The latter equations look like Eqs. (3.82)-(3.83) in which

12

is formally replaced by

(12

+ K-').

In accordance with Eq. (3.84) the respective contributions of the diffuse part of the electric double layer can be obtained by subtraction of Eqs. (3.85)-(3.86) from their counterparts among Eqs. (3.87)-( 3 38):

2n B t f =--0'(21, &

K-' + K - ? )

(low surface potential)

(3.89)

(3.90)

In the case of moderate and high surface electric potentials the expressions related to the Stern layer- Eqs. (3.85)-(3.86) can be applied again, whereas Eqs. (3.89)-(3.90) are no longer valid. In this more complicated case expressions for B,d", k:1- and

FF have been derived by means

of a thermodynamic approach 1311, and independently in Ref. [59] by using a hydrostatic approach based on Eqs. (3.146), (4.149j and (4.150'1 - see below. The results are [3 1,581

K-

Here

1

(3.91)

is the bulk concentration of a Z : Z electrolyte; x and y are quantities related to the

C~J

surface potential as followx

(3.94)

Surface Bending Moment and Curvature Elastic Moduli

129

In the latter expression the signs "+" and " - " refer to an electric double layer, respectively, inside an aqueous drop and outside a non-aqueous drop (bubble). Setting 12 = 0, that is

neglecting the distance between the equimolecular dividing surface and the surface of location of the bound counterions, Eqs. (3.91)-(3.93) are reduced to the expressions derived by Lekkerkerker [59]. Numerical calculations based on Eqs. (3.84)-(3.86) and (3.91)-(3.93) show that Bo t is dominated by ~0Rst", i.e. by the contribution of the Stern layer, whereas k,el and k~.~t contain a considerable contribution from the diffuse layer, that is from k{ 1if and k-flU . The magnitude of B~l, k~land k-c~l is higher for lower electrolyte concentrations. For example, for co = 10-5 M one computes B~l-- 5 x 10-11 N, k~/~ 1 x 10-19 J and ~-e/=-3 x l0 -19 J [58].

3.3.2.

BENDING MOMENT EFFECTS ON THE INTERACTION BETWEEN DROPS IN EMULSIONS Interaction between deforming emulsion drops. The collisions between the drops in an

emulsion are accompanied with a flattening in the zone of contact between such two drops, see Fig 3.5. The conditions for the formation of a flat film between two similar droplets have been studied by Danov et. al. [60] and Denkov et al. [61]. A modeling of the shape of a deformed drop with portions of a sphere and a plane (Fig. 3.5) proved to be a very good approximation [62]. Despite the fact that area of the formed flat film is relatively small, its appearance leads to a strong enhancement (with dozens of kT) of the energy of interaction between the two drops.

I I

h' Fig. 3.5. Scheme of two emulsion drops deformed upon collision; the magnitude of the radius, r, of the formed flat film and its thickness, h, are exaggerated.

130

Chapter 3

We have in mind interactions due to the various components of the disjoining pressure: electrostatic, van der Waals, steric, depletion, oscillatory-structural, etc., see Chapter 5 for details. It has been taken into account [61,63] that if an initially spherical droplet deforms at fixed volume, its surface area increases, which gives rise to an effective interdroplet repulsion. Moreover, work of flexural (bending) deformation is conducted when the initially spherical interface in the zone of contact is converted into the planar surface of the film (Fig. 3.5). To estimate this work one can use Eq. (3.7); viz. by expanding Eq. (3.7) in series, keeping linear terms with respect to the curvature one obtains [63] WT ( H ) - 2re r 2 w f = 27r r 2 (2k CH 2 + B o l l + . . . ) ,

( r / a ) 2 0 for emulsions type "oil-in-water", whereas B 0 < 0 for emulsions type "water-in-oil". Consequently, in view of Eq. (3.96) the energy of interfacial

Surface Bending Moment and Curvature Elastic Moduli

131

bending deformation contributes to a repulsion between oil drops in water (AWr > 0), but to an attraction between water drops in oil (AW r < 0 ) [63]. For example, Koper et al. [64] have observed formation of doublets (with energy of bonding = l0 kT ) from aqueous microemulsion drops dispersed in oil; this phenomenon could be attributed, at least in part, to the effect of AWf < O. Interactions between drops in double emulsions. As mentioned above, the effect of the interfacial bending moment B0 can be important for emulsion systems of low interfacial tension. It has been demonstrated by Binks [65], that after intensive stirring in such systems one could observe a simultaneous formation of emulsion and microemulsion drops, see Fig. 3.6. Having in mind the above discussion, one expects that AWu > 0 helps for stabilization of the formed emulsion. Indeed, for AWT > 0 the bending moment opposes the flattening of the drops in the contact zone thus decreasing the probability for formation (and consecutive rupturing) of a thin liquid film between two colliding droplets; in other words, the bending moment counteracts the coalescence of emulsion drops. It is reasonable to assume that the condition AWl > 0 is fulfilled for the microemulsion drops, which do not increase their size with time, despite the intensive Brownian collisions between them. For the system depicted in Fig. 3.6a the microemulsion drops are in the continuous phase, i.e. the emulsion and microemulsion drops have the same sign of the curvature. Therefore, one could expect that

9 (a)

(b)

Fig. 3.6. Sketch of an emulsion drop coexisting with smaller microemulsion drops (a) in the continuous and (b) in the disperse phase. The running unit normal to the interface, n, is directed from phase 1 toward phase 2.

Chapter3

132

AWI > 0 for the emulsion drops, as well. On the contrary, for the system depicted in Fig. 3.6b, that is the microemulsion drops are in the

disperse phase, the emulsion and microemulsion

drops have the opposite sign of the curvature, which is determined by the direction of the unit normal n, see Eq. (3.21). Hence, if

A WI > 0 for the micro-emulsion drops, then A Wf < 0 for

the emulsion drops. Consequently, for the system in Fig. 3.6b the bending moment contributes to an

attraction between the emulsion drops and favors their coalescence.

According to Davies and Riedal [66] both types of emulsions, those from Fig. 3.6a,b and 3.6b, are formed upon stirring. That type of emulsion survives, for which the coalescence is slower. If the effect of the interfacial bending moment dominates the interactions between the emulsion droplets, one can expect that the emulsion in Fig. 3.6a will survive [63]. In fact, this is observed experimentally; viz. the emulsion which contains microemulsion drops in the continuous phase is more stable [65]. It is worthwhile noting that in emulsions the effect of the interfacial bending moment acts simultaneously with other type of interactions between the droplets, such as the surface forces of various origin: van der Waals interactions, electrostatic (double-layer), steric, depletion, oscillatory-structural, hydration and other forces. For that reason, the analysis of the stability of an emulsion needs a careful estimate of the relative contributions of various factors to the dropdrop interaction energy; see Ref. [67] for details. It should be also noted that the bending effects may influence the stability of an emulsion when the rupturing of thin emulsion films happens through nucleation of pores, see e.g. Ref. [68].

3.4.

SUMMARY

An interface (or membrane) possesses 4 modes of deformation: dilatation, shearing, bending and torsion (Fig. 3.1). The first couple of modes represent two-dimensional analogues of respective deformations in the bulk phases. The bending and torsion modes are related to variations in the two principle curvatures of the interface, that is to the presence of two additional degrees of freedom. From a

thermodynamic viewpoint, the curvature effects can be

accounted for as contributions of the work of interfacial bending and torsion to the total energy

Surface Bending Moment and Curvature Elastic Moduli

133

of the system; the respective coefficients are the interfacial (surface) bending and torsion moments, B and |

see Eq. (3.1). The most popular model of the interfacial curvature effects

provides an expression for the mechanical work of flexural deformation, Eq. (3.7), which involves 3 parameters: bending and torsion elastic moduli, kc and k C, and spontaneous curvature, H0. First we have considered the simpler case of spherical geometry. The dependence of the bending moment B on the choice of the dividing surface at fixed physical state of the system is investigated, see Eq. (3.32), (3.33) and Fig. 3.2. The connection between the quantities bending moment, Tolman length and spontaneous curvature has been demonstrated, see Eqs. (3.10) and (3.39). Micromechanical expressions, Eqs. (3.69) and (3.70), allow one to calculate the surface tension and the bending moment if an expression for the pressure tensor is available. From the viewpoint of the microscopic theory, various intermolecular forces may contribute to N

the interfacial moments B, |

and to the curvature elastic moduli, k~ and k c , see Eqs. (3.72)

and (3.73). Such are the van der Waals forces, Eq. (3.75), the steric interactions, Eq. (3.77) and the electrostatic interactions, Eqs. (3.78)-(3.80) and (3.82)-(3.93). The interfacial bending moment may give an essential contribution to the interaction between deformable droplets in emulsions, see Eq. (3.96). In general, the curvature effects are expected to be significant for interfaces of low tension and high curvature. An example are the biomembranes, which usually have low tension. The present chapter is an introduction to the next Chapter 4, in which the general theory of curved interfaces and biomembranes is considered.

3.5.

REFERENCES

1. J.W. Gibbs, The Scientific Papers of J.W. Gibbs, Vol. 1, Dover, New York, 1961. 2. R.C. Tolman, J. Chem. Phys. 17 (1949) 333. 3. F.O. Koenig, J. Chem. Phys. 18 (1950) 449. 4. F.P. Buff, J. Chem. Phys. 19 (1951) 1591. 5. F.P. Buff, J. Chem. Phys. 23 (1955) 419.

134

Chapter 3

6. F.P. Buff, The Theory of Capillarity, in: S. Fltigge (Ed.), Handbuch der Physik, Vol. X, Springer, Berlin, 1960. 7. S. Kondo, J. Chem. Phys. 25 (1956) 662. 8. S. Ono, S. Kondo, Molecular Theory of Surface Tension in Liquids, in: S. Fltigge (Ed.), Handbuch der Physik, vol. 10, Springer, Berlin, 1960, p. 134. 9. A.I. Rusanov, Phase Equilibria and Surface Phenomena, Khimia, Leningrad, 1967 (in Russian); Phasengleichgewichte und Grenzfl~ichenerscheinungen, Akademie Verlag, Berlin, 1978. 10. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1982. 11. J. Gaydos, Y. Rotenberg, L. Boruvka, P. Chen, A.W. Neumann, The Generalized Theory of Capillarity, in: A.W. Neumann & J.K. Spelt (Eds.) Applied Surface Thermodynamics, Marcel Dekker, New York, 1996, p. 1. 12. J.E. Bowcott, J.H. Schulman, Z. Elektrochem. 59 (1955) 283. 13. J.H. Schulman, J.B. Montagne, Ann. N.Y. Acad. Sci. 92 (1961) 366. 14. W. Stoeckenius, J.H. Schulman, L.M. Prince, Kolloid-Z. 169 (1960) 170. 15. C.L. Murphy, Thermodynamics of Low Tension and Highly Curved Interfaces, Ph.D. Thesis (1966), University of Minnesota, Dept. Chemical Engineering); University Microfilms, Ann Arbour, 1984. 16. M.L. Robbins, in: K.L. Mittal (Ed.) Micellization, Solubilization and Microemulsions, Vol. 2, Plenum Press, New York, 1977. 17. C.A. Miller, J. Dispersion Sci. Technol. 6 (1985) 159. 18. P.G. de Gennes, C. Taupin, J. Phys. Chem. 86 (1982) 2294. 19. J.Th.G. Overbeek, G.J. Verhoeckx, P.L. de Bruyn, H.N.W. Lekkerkerker, J. Colloid Interface Sci. 119 (1987) 422. 20. N.D. Denkov, P.A. Kralchevsky, I.B. Ivanov, C.S. Vassilieff, J. Colloid Interface Sci. 143 (1991) 157. 21. G. Kirchhoff, Crelles J. 40 (1850) 51. 22. A.E.H. Love, Phil. Trans. Roy. Soc. London A 179 (1888) 491. 23. W. Helfrich, Z. Naturforsch. 28c (1973) 693. 24. W. Helfrich, Z. Naturforsch. 29c (1974) 510. 25. E.A. Evans, R. Skalak, Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton, Florida, 1979.

Surface Bending Moment and Curvature Elastic Moduli

135

26. P.G. de Gennes, Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. 27. H.W. Huang, Biophys. J. 50 (1986) 1061. 28. A.G. Petrov, The Lyotropic State of Matter: Molecular Physics and Living Matter Physics, Gordon & Breach Sci. Publishers, Amsterdam, 1999. 29. I.B. Ivanov, P.A. Kralchevsky, in: I.B. Ivanov (Ed.) Thin Liquid Films, Marcel Dekker, New York, 1988, p. 91. 30. P.A. Kralchevsky. J. Colloid Interface Sci. 137 (1990) 217. 31. T.D. Gurkov, P.A. Kralchevsky, Colloids Surf. 47 (1990) 45. 32. L. Boruvka, A.W. Neumann, J. Chem. Phys. 66 (1977) 5464. 33. V.S. Markin, M.M. Kozlov, S.I. Leikin, J. Chem. Soc. Faraday Trans. 2, 84 (1988) 1149. 34. P.A. Kralchevsky, J.C. Eriksson, S. Ljunggren, Adv. Colloid Interface Sci. 48 (1994) 19. 35. S. Ljunggren, J.C. Eriksson, P.A. Kralchevsky, J. Colloid Interface Sci. 161 (1993) 133. 36. C.E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge University Press, Cambridge, 1939. 37. A.J. McConnell, Application of Tensor Analysis, Dover, New York, 1957. 38. P.A. Kralchevsky, T.D. Gurkov, I.B. Ivanov. Colloids Surf. 56 (1991) 149. 39. J.G. Kirkwood, F.P. Buff, J. Chem. Phys. 17 (1949) 338. 40. J.H. Irving, J.G. Kirkwood, J. Chem. Phys. 18 (1950) 817. 41. T.L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley, Reading, MA, 1962. 42. F.M. Kuni, A.I. Rusanov, in: The Modern Theory of Capillarity, A.I. Rusanov & F.C. Goodrich (Eds.), Akademie Verlag, Berlin, 1980. 43. F.C. Goodrich, in: Surface and Colloid Science, Vol. 1, E. Matijevic (Ed.), Wiley, New York, 1969; p. 1. 44. P.A. Kralchevsky, T.D. Gurkov, Colloids Surf. 56 (1991) 101. 45. T.D. Gurkov, P.A. Kralchevsky, I.B. Ivanov, Colloids Surf. 56 (1991) 119. 46. J.L. Vivoy, W.M. Gelbart, A. Ben Shaul, J. Chem. Phys. 87 (1987) 4114. 47. I. Szleifer, D. Kramer, A. Ben Shaul, D. Roux, W. M. Gelbart, Phys. Rev. Lett.60 (1988) 1966. 48. S.T. Milner, T.A. Witten, J. Phys. (Paris) 49 (1988) 1951.

136

Chapter 3

49. I. Szleifer, D. Kramer, A. Ben Shaul, W.M. Gelbart, S. Safran, J. Phys. Chem. 92 (1990) 6800. 50. Z.G. Wang, S. A. Safran, J. Chem. Phys. 94 (1991) 679. 51. J. Ennis, J. Chem. Phys. 97 (1992) 663. 52. N. Dan, P. Pincus, S.A. Safran, Langmuir 9 (1993) 2768. 53. P.A. Barneveld, J.M.H.M. Scheutjens, J. Lyklema, Langmuir 8 (1993) 3122; Langmuir 10 (1994) 1084. 54. P.A. Kralchevsky, T.D. Gurkov, K. Nagayama, J. Colloid Interface Sci. 180 (1996) 619. 55. A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, Wiley, New York, 1997. 56. J.Th.G. Overbeek, Electrochemistry of the Double Layer, in: Colloid Science, Vol. 1, H. R. Kruyt, Ed., Elsevier, Amsterdam, 1953. 57. S.A. Simon, T.J. McIntosh, A.D. Magid, J. Colloid Interface Sci. 126 (1988) 74. 58. P.A. Kralchevsky, Curved Interfaces and Capillary Forces between Particles, Thesis for Doctor of Science, Faculty of Chemistry, University of Sofia, Sofia, 2000. 59. H.N.W. Lekkerkerker, Physica A 167 (1990) 384. 60. K.D. Danov, N.D. Denkov, D.N. Petsev, I.B. Ivanov, R.P. Borwankar, Langmuir 9 (1993) 1731. 61. N.D. Denkov, D.N. Petsev, K.D. Danov, Phys. Rev. Lett. 71 (1993) 3326. 62. N.D. Denkov, D.N. Petsev, K.D. Danov, J. Colloid Interface Sci. 176 (1995) 189. 63. D.N. Petsev, N.D. Denkov, P.A. Kralchevsky, J. Colloid Interface Sci. 176 (1995) 201. 64. G.J.M. Koper, W.F.C. Sager, J. Smeets, D. Bedeaux, J. Phys. Chem. 99 (1995) 13291. 65. B.P. Binks, Langmuir 9 (1993) 25. 66. J.T. Davies, E. K. Rideal, Interfacial Phenomena, Academic Press, New York, 1963. 67. I.B. Ivanov, K.D. Danov, P.A. Kralchevsky, Colloids Surf. A 152 (1999) 161. 68. A. Kabalnov, H. Wennerstr6m, Langmuir 12 (1996) 276.

137

CHAPTER 4 GENERAL CURVED INTERFACES AND BIOMEMBRANES

Mechanically, the stresses and moments acting in an interface or biomembrane can be taken into account by assigning tensors of the surface stresses and moments to the phase boundary. The three equations determining the interfacial/membrane shape and deformation represent the three projections of the vectorial local balance of the linear momentum. Its normal projection has the meaning of a generalized Laplace equation, which contains a contribution from the interfacial moments. Alternatively, variational calculus can be applied to derive the equations governing the interfacial/membrane shape by minimization of a functional - "the thermodynamic approach". The correct minimization procedure is considered, which takes into account the work of surface shearing. Thus it turns out that the generalized Laplace equation can be derived following two alternative approaches: mechanical and thermodynamical. In fact, they are mutually complementary parts of the same formalism; they provide a useful tool to verify the selfconsistency of a given model. The connection between them has the form of relationships between the mechanical and thermodynamical surface tensions and moments. Different, but equivalent, forms of the generalized Laplace equation are considered and discussed. The general theoretical equations can give quantitative predictions only if theological constitutive relations are specified, which characterize a given interface (biomembrane) as an elastic, viscous or visco-elastic two-dimensional continuum. Thus the form of the generalized Laplace equation can be specified. Further, it is applied to determine the axisymmetric shapes of biological cells; a convenient computational procedure is proposed. Finally, micromechanical expressions are derived for calculating the surface tensions and moments, the bending and torsion elastic moduli, kc and k C, and the spontaneous curvature, H0, in terms of combinations from the components of the pressure tensor.

Chapter4

138

4.1.

THEORETICAL APPROACHES FOR DESCRIPTION OF CURVED INTERFACES

A natural mathematical description of arbitrarily curved interfaces is provided by the differential geometry based on the apparatus of the tensor analysis. Our main purpose in this chapter is to demonstrate the application of this apparatus for generalization of the relationships described in Section 3.2 for spherical interface. Firstly, the generalization of the theory includes presentation of the surface stresses and moments as tensorial quantities. Secondly, generalizations of the theoretical expressions, such as the Laplace equation and the micromechanical expressions (3.69)-(3.70), are considered. A special attention is paid to the connection between the two equivalent and complementary approaches: the thermodynamical and the mechanical one.

The thermodynamical approach to the theory of the curved interfaces, which is outlined in Sections 3.1 and 3.2 above, originates from the works of Gibbs [1] and has been further developed by Boruvka & Neumann [2]. In this approach a heterogeneous (multiphase) system is formally treated as a combination of bulk, surface and linear phases, each of them characterized by its own fundamental thermodynamic equation. The logical scheme of the Gibbs' approach consists of the following steps: (1) The extensive parameters (such as internal energy U, entropy S, number of molecules of the i-th component Ni, etc.) and their densities far from the phase boundaries are considered to be, in principle, known. (2) An imaginary idealized system is introduced, in which all phases (bulk, surface and linear) are uniform, the interfacial transition zones are replaced with sharp boundaries (geometrical surfaces and lines), and the excesses of the extensive parameters (in the idealized with respect to the real system) are ascribed to these boundaries; for example - see Figs 1.1 and 3.3. (3) The Gibbs fundamental equations are postulated for each bulk, surface and linear phase. Since the densities of the extensive parameters can vary along a curved interface, a Local formulation of the fundamental equations should be used, see Eq. (3.13). (4) The last step is to impose the conditions for equilibrium in the multiphase system. These are (i) absence of hydrodynamic fluxes (mechanical equilibrium) (ii) absence of diffusion fluxes

General Curved Interfaces and Biomembranes

139

(chemical equilibrium) and (iii) absence of heat transport (thermal equilibrium). As known [3], the conditions for thermal and chemical equilibrium imply uniformity of the temperature and the chemical potentials in the system. The conditions for mechanical equilibrium are multiform; examples are the Laplace and Young equations (Chapter 2). All conditions for equilibrium can be deduced by means of a variational principle, that is by minimization of the grand thermodynamic potential of the system, see Chapter 2 and Section 4.3.1.

The mechanical approach originates from the theory of elastic deformations of "plates" and "shells" developed by Kirchhoff [4] and Love [5]; a comprehensive review can be found in Ref. [6]. For the linear theory by Kirchhoff and Love it is typical that the stress depends linearly on the strain, and that the elastic energy is a quadratic function of the deformation. Similar form has Eq. (3.7), postulated by Helfrich [7,8], which expresses the work of flexural deformation as a quadratic function of the variations of the interfacial curvatures. Evans and Skalak [9] demonstrated that a relatively complex object, as a biomembrane, can be treated mechanically as a two-dimensional continuum, characterized by dilatational and shearing tensions, and elastic moduli of bending and torsion. The logical scheme of the mechanical approach consists of the following steps: (1) Strain and stress tensors, as well as tensors of the moments (torques), are defined for the bulk phases and for the boundaries between them. A phase with a (nonzero) tensor of moments is termed continuum of Cosserat [10]; the liquid crystals represent an example [11]. Here we will restrict our considerations to bulk fluid phases without moments; action of moments will be considered only at the interfaces. (2) Equations expressing the balances of mass, linear and angular momentum are postulated; they provide a set of differential equations and boundary conditions, which describe the dynamics of the processes in the system. In particular, the Laplace equation (2.17) can be deduced as a normal projection of the interfacial balance of the linear momentum; see Eq. (4.51) below and Refs. [ 12-14]. (3) The properties of a specific material continuum are taken into account by postulating appropriate theological constitutive relations, which define connections between stresses (or moments) and strains. In fact, the rheological constitutive relations represent mechanical

Chapter 4

140

models, say viscous fluid, elastic body, visco-elastic medium, etc. In Section 4.3.4 we demonstrate that the Helfrich equation (3.7) leads to a constitutive relation for the tensor of the interfacial moments. In summary, the thermodynamical and the mechanical approaches are based on different concepts and postulates, but they are applied to the theoretical description of the same subject: the processes in multiphase systems. Then obligatorily these two approaches have to be equivalent, or at least complementary. One of our main goals below is to demonstrate the connections between them. The combination of the two approaches provides a deeper understanding of the meaning of quantities and equations in the theory of curved interfaces (membranes) and provides a powerful apparatus for solving problems in this field. Below we first present the mechanical approach to the curved interfaces and membranes. Next we consider the connections between the thermodynamical and mechanical approaches. Further, we give a derivation of the generalized Laplace equation by minimization of the free energy of the system. A special form of this equation for axisymmetric interfaces is considered with application for determination of the shape of biological cells. Finally, some micromechanical expressions for the interfacial (membrane) properties are derived.

4.2.

M E C H A N I C A L APPROACH TO ARBITRARILY CURVED INTERFACES

4.2.1.

A N A L O G Y WITH MECHANICS OF THREE-DIMENSIONAL CONTINUA

Balance of the linear momentum. First, it is useful to recall the "philosophy" and basic equations of the mechanics of three-dimensional continua. Consider a material volume V, which is bounded by a closed surface S with running outer unit normal n. On the basis of the second Newton's law it is postulated (see e.g. Ref. 15) d dt f dV p v - ~ dsn . T + f dV p f v

S

(4.1)

v

Here t is time, v is the velocity field; T is the stress tensor; ds is a scalar surface element; p is the mass density; f is an acceleration due to body force (gravitational or centrifugal). Equation

General Curved Interfaces and Biomembranes

(4.1) expresses the

integral balance

141

of the linear momentum for the material volume V; indeed,

Eq. (4.1) states that the time-derivative of the linear momentum is equal to the sum of the surface and body forces exerted on the considered portion of the material continuum. Using the Gauss theorem and the fact that the volume V has been arbitrarily chosen, from Eq. (4.1) one

local balance

can deduce the

of the linear momentum [ 15]:

dv p ~ = V-T + pf

(4.2)

dt

In the derivation of Eq. (4.2) the following known hydrodynamic relationships have been used:

d !dVpvdt

@ dt

~-+

ldvId(pv) v L dt

+ pvV. vI

pV. v = 0

(4.3)

(4.4)

Equation (4.3) is a corollary from the known Euler formula, whereas Eq. (4.4) is the continuity equation expressing the local mass balance [15].

Rheological models.

The continuum mechanics can give quantitative predictions only if

a model expression for the stress tensor T is specified. As a rule, such an expression has the form of relationship between stress and strain, which is termed rheological constitutive relation (defining, say, an elastic or a viscous body, see below). The vectors of displacement, u, and velocity, v, are simply related: du v=~

(4.5)

dt

Further, the strain and rate-of-strain tensors are introduced: = [ g u + ( g u ) T]

kit --[Vv +(Vv) T]

(strain tensor)

(4.6)

(rate-of-strain tensor)

(4.7)

As usual, V denotes the gradient operator in space and "T" denotes conjugation. The tensors and ~ are related as follows: = ~ 8t Here 8t denotes an infinitesimal time interval.

(4.8)

142

Chapter 4

An elastic body is defined by means of the following constitutive relation [ 16] T = ~ Tr(O)U + 2/~ [ O - .~1Tr(O) U]

(Hooke's law)

(4.9)

where U is the spatial unit tensor; ,~ and/1 are the dilatational and shear bulk elastic moduli; as usual, "Tr" denotes trace of a tensor. Note that ,~ and/,t multiply, respectively, the isotropic and the deviatoric part of the strain tensor O. (The trace of the deviatoric part is equal to zero, i.e. no dilatation, only shearing deformation). Similar consideration of the isotropic part (accounting for the dilatation) and deviatoric part (accounting for the shear deformation) is applied also to viscous bodies and two-dimensional continua (interfaces, biomembranes), see below. The substitution of Eq. (4.9) into the balance of linear momentum, Eq. (4.2), along with Eq. (4.6), yields the basic equation in the mechanics of elastic bodies [I 6]:

dZu 1/t)VV u + f p T t ~ - ~ v ~ . + (x+~9 p

(Navier equation)

(4.10)

Likewise, a viscous body (fluid) is defined by means of the constitutive relation [ 17] T = - P U + ~'vTr(~) U + 2 r/v [~ - ~Tr(~) U]

(Newton's law)

(4.11 )

where P is pressure, ~'v and r/v are the dilatational and shear bulk viscosities; in fact r/v is the conventional viscosity of a liquid, whereas ~',, is related to the decay of the intensity of sound in a liquid. The substitution of Eq. (4.11) into the balance of linear momentum, Eq. (4.2), along with Eq (4.7), yields the basic equation of hydrodynamics [15, 17]: P dv d--t - _Vp+rlvVZv + (~'v +~r/v )VV-v + p f

(Navier-Stokes equation)

(4.12)

In Section 4.2.4 we will consider the two-dimensional analogues of Eqs. (4.1)-(4.12). Before that we need some relationships from differential geometry.

4.2.2.

BASIC EQUATIONS FROM GEOMETRY AND KINEMATICS OF A CURVED SURFACE

The formalism of differential geometry, which is briefly outlined and used in this chapter, is described in details in Refs. [12, 18-20]. Let (u I ,u 2) be curvilinear coordinates on the dividing surface between two phases, and let R(ul,uZ,t) be the running position-vector of a material

General Curved Interfaces and Biomembranes

143

point on the interface, which depends also on the time, t. We introduce the vectors of the surface local basis and the surface gradient operator:

3R

a~

= ~

3u ~ ,

V -a ~'~ s

c?

( / t = l 2)

9u"

'

(4.13)

Here and hereafter the Greek indices take values 1 and 2; summation over the repeated indices is assumed. The curvature tensor b is defined by Eq. (3.21); b is a symmetric surface tensor, whose eigenvalues are the principal curvatures c~ and c2. The surface unit tensor U, and curvature deviatoric tensor q are defined as follows

1

U, =a/~ a ~ ,

q =-~ (b-HU,)

(4.14)

where, as before, H and D denote the mean and deviatoric curvature, see Eq. (3.3). For every choice of the surface basis the eigenvalues of Us are both equal to 1; the tensor q has diagonal form in the basis of the principal curvatures and has eigenvalues 1 a n d - 1 . In particular, the covariant components of Us, a~v = a/~. a v

(4.15)

represent the components of the surface metric tensor; here "-" is the standard symbol for scalar product of two vectors. The covariant derivative of auv is identically zero, a;~u,v = 0, whereas the covariant derivative q,~,.v of the components of the tensor q is not zero, although its eigenvalues are constant at each point of the interface; in particular, the divergence of q is [ 13]:

q,~u,~ _ 1 (a,~uH ~

-D

-q~.uD '~ )

(4.16)

In view of Eq. (4.14), the curvature tensor can be expressed as a sum of an isotropic and a deviatoric part:

bzu = H a;w + Dqxu The velocity of a material point on the interface is defined as follows

(4.17)

144

Chapter 4

(4.18) II

, Ll

According to Eliassen [21], the interfacial rate-of-strain tensor, which describes the twodimensional dilatational and shear deformations, is defined by the expression:

du v _-10auv _-2~Vu,vl" 2 ~t 2

+ Vv _ 2buy v ('))' '~

(4.19)

v~ - a v. v 9

v (n) = n. v

(4.20)

where

are components of the velocity vector v. Despite the fact that we will finally derive some quasistatic relationships, it is convenient to work initially with the rates (the time derivatives) of some quantities. For example, if ~t is a small time interval and /-;/ is the time derivative of the mean curvature then ~H = H~t is the differential of H, which takes part in Eq. (3.1). We will restrict our considerations to processes, for which the rate-of-strain tensor has always diagonal form in the basis of the principal curvatures. For example, such is the case of an axisymmetric surface subjected to an axisymmetric deformation [9]. A more general case is considered in Ref. [22]. In such cases the quantities ~ and fl, defined by the relationships

r

av duv,

fi = q/~V duv,

(4.21)

express the local surface rates of dilatation and shear [23]. Then, the infinitesimal deformations of dilatation and shearing, corresponding to a small time increment, ~St, will be ~o~ = a ~ t;

~fl = fi ~ t

(4.22)

The latter two differentials also take part in Eq. (3.1). In view of Eq. (4.21) we can present duv as a sum of an isotropic and a deviatoric part 1

duv - 2 6~auv +2 ~ quv

(4.23)

145

General Curved Interfaces and Biomembranes

4.2.3.

TENSORS OF THE SURFACE STRESSES AND MOMENTS

Owing to the connections between stress and strain, an expression similar to Eq. (4.23) holds for the surface stress tensor [24]" (4.24)

0"uv - 0" auv + 77 q~v

(Each of Eqs. 4.23 and 4.24 represent the respective tensor as a sum of an isotropic and a deviatoric part.) Let 0"1 and 0"2 be the eigenvalues of the tensor 0"uv. The tensions 0"1 and 0"2 are directed along the lines of maximum and minimum curvature. From Eq. (4.24) it follows 1 0. -- --(0.1 -'1-0" 2 )"

1 77 -- 7 (0"1 -- 0"2)

2

(4.25)

n

----//,

l_ t3"11

M12

(a)

(b)

n

2 N21 u

,ii;

"- N22

N12

Nil

(c)

Fig. 4.1. Mechanical meaning of the components of (a) the surface stress tensor, O'uuand o'u(n)" (b) the tensor of surface moments Muv" (c) the tensor of surface moments Nuv. The relationship between Muv and Nuv is given by Eq. (4.29).

146

Chapter 4

In fact, the quantity 0- is the conventional mechanical surface tension, while r/is the mechanical shearing tension [9,24]. The physical meaning of the components 0-~v (/t,v = 1,2) in an arbitrary basis is illustrated in Fig. 4.1. Note that in general the surface stress tensor ~ is not purely tangential, but has also two normal components, o"~n~,/t = 1,2, see Refs. [6, 12]" ffq_ - a~av0- ~v + a/~no"/~(n)

(4.26)

In other words, the matrix of the tensor __ffis rectangular:

_ (0..11 0.12 0-1(n)] O" [0.21 0.22 0.2(n)

(4.27)

Let us consider also the tensor of the surface moments (torques), 1 M/a v - ~ ( M 1

1 + M z ) ajav +-~ (M 1 - M z )

(4.28)

qjav ,

which is defined at each point of the interface; M1 and M2 are the eigenvalues of the tensor Muv supposedly it has a diagonal form in the basis of the principal curvatures. In the mechanics of the curved interfaces the following tensor is often used [6, 12] (4.29)

N~v = M ,a, ev,~

where euv is the surface alternator [19]" el2-qt-a -, e 2 1 - - ~ a ,

ell = e 2 2 - - 0 "

a is the

determinant of the surface metric tensor auv, see Eq. (4.15) The mechanical meaning of the components of the tensors Muv and Nuv is illustrated in Fig. 4.1b,c. One sees that N~l and N22 are normal moments (they cause torsion of the surface element), while N~2 and N21 are tangential moments producing bending. In this aspect there is an analogy between the interpretations of Nuv and 0-uv (Fig. 4.1 a). Indeed, if v is the running unit normal to a curve in the surface (v is tangential to the surface), then the stress acting per unit length of that curve is t - v.~ and the moment acting per unit length is m - v.N. On the other hand, as seen from Eqs. (4.28) and (4.29), the tensor Nuv is not symmetric. For that reason the symmetric tensor Muv is often preferred in the mechanical description of surface moments [6, 9]. A necessary condition for mechanical equilibrium of an interface is

General Curved Interfaces and Biomembranes

buv Nuv - e aft ~

147 (4.30)

,

which expresses the normal resultant of the surface balance of the angular momentum; see also Eq. (4.44) below. Substituting O'nv from Eq. (4.24) and Nnv from Eqs. (4.28)-(4.29) into Eq. (4.30) one could directly verify that the latter condition for equilibrium (with respect to the acting moments) is satisfied by the above expressions for Ouv and Nu,,.

4.2.4.

SURFACE BALANCES OF THE LINEAR AND ANGULAR M O M E N T U M

Balance of the linear momentum. First, let us identify the surface (two-dimensional)

analogues of Eqs. (4.1)-(4.4). Following Podstrigach and Povstenko [12] we consider a material volume V, which contains a portion, A, from the boundary between phases I and II, together with the adjacent volumes, V~ and VII, from these phases, see Fig. 4.2. In analogy with Eq. (4.1) one can postulate the integral balance of the linear momentum for the considered part of the system [ 12]:

(4.31) Y=I,II Vy

A

=, I Sy

Vy

a

L

II

Fig. 4.2. Sketch of a material volume V, which contains a portion, A, of the boundary between phases I and II; V~, VII and $I, S~I are parts of the volume V and its surfaces, which are located on the opposite sides of the interface A.

In Eq. (4.31) the subscript "s" denotes properties related to the interface; VI, VII and SI, SII are the two parts of the considered volume and its surface separated by the dividing surface A; L is the contour which encircles A; v is an unit normal, which is simultaneously perpendicular to L

148

Chapter 4

and tangential to A (Fig. 4.2); F is the surface excess of mass per unit area of the interface; as before, n is running unit normal and ~ is the surface stress tensor, see Eq. (4.26) and Fig. 4.1. For the sake of simplicity we will assume that the normal component of the velocity is continuous across the dividing surface: (v I - v, )-n - (v n - v, )-n - 0

(4.32)

Then the surface analogues of Eqs. (4.3) and (4.4) have the form [21]:

d l dVFv =! dV [ d ( F V )s+ F v ~ V . v I dt a dt 9 , s

(4.33)

dF dt

(4.34)

FV.,. 9v s - 0

--+

The latter equation is valid if there is no mass exchange between the bulk phases and the interface. Next, we will transform Eq. (4.31) with the help of the integral theorem of Green, which in a general vectorial form reads [12, 25]"

IdAV, . T - ~ d l v . T - I d A A

L

2H n . T

(Green theorem)

(4.35)

A

Here T is an arbitrary vector or tensor; the meaning of v is the same as in Fig. 4.2. If T is a purely surface tensor, viz. T-aUaVT, v, or if T has a rectangular matrix like that in Eqs. (4.26)-(4.27), then n .T = 0 and the last integral in Eq. (4.35) is zero. In particular, the Green theorem (4.35), along with Eq. (4.26), yields

~dlv " ~ - I d A V s "~ L

(4.36)

A

With the help of Eqs. (4.32)-(4.34), (4.36) and the versions of Eq. (4.1) for the material volumes VI and VII, from Eq. (4.31) one deduces the local surface balance of linear momentum [121" F dv, _ V .,..or + Ff, + n- (T n - T~ )

dt

(4.37)

Here T~ and Tn are the subsurface values of the respective tensors. The last term in Eq. (4.37),

General Curved Interfaces and Biomembranes

149

which accounts for the interaction of the interface with the two neighboring bulk phases, has no counterpart in Eq. (4.2). Differential geometry [18-21] yields the following expression for the surface divergence of the tensor o" defined by Eq. (4.26):

V s -o- - (o-,vn -bvno-v(n))a n +(bnv(Tnv +o-~(n))n

(4.38)

Here bnv are components of the curvature tensor, see Eq. (4.17). In view of Eq. (4.38), the projections of Eq. (4.37) along the basis vectors a n and n have the form F i n - f f V v ~ - b v U f f v(n~ + V f ~

+(rll n)n -T, (n'u)

~t= 1,2

(4.39)

(normal balance)

(4.40)

Fi (n) -o--,~(n) +bnvt~Tnv +Ffs (n) +(Til n)(n) _Zi (n)(n))

where i n and i(n) are components of the vector of acceleration, dvs/dt. Equations (4.39) and (4.40) coincide with the first three basic equations in the theory of elastic shells by Kirchhoff and Love, see e.g. Refs. [6, 12]. B a l a n c e o f the a n g u l a r m o m e n t u m .

In mechanics the rotational motion is treated

similarly to the translational one. In particular, instead of velocity and force, angular velocity and force moment (torque) are considered. Three- and two-dimensional integral balances of the angular momentum, i.e. analogues of Eqs. (4.1) and (4.31), are postulated; see Refs. [6, 12] for details. From them the local form of the surface balance of the angular momentum can be deduced [ 12]" a F(&o/dt)

- V, 9N + F m , + n- [(~" _~)U, + ~ . ~]

(4.41)

The last equation is the analogue of Eq. (4.37) for rotational motion. Here o~ is a coefficient accounting for the interfacial moment of inertia; co is the vector of the angular velocity; m, is a counterpart of f, in Eq. (4.37)" N - a u a v N ~v + a u n N n(n)

is the tensor of the surface moments (Fig. 4.1c); _~;- anavenv

(4.42) is the surface alternator.

Comparing Eqs. (4.37) with (4.41) one sees that for rotational motion N plays the role of ff for translational motion. Using Eq. (4.38) with N instead of o , along with Eqs. (4.26) and (4.42),

150

Chapter 4

one can deduce equations, which represent the projections of the surface balance of the angular momentum, Eq. (4.41), along the basis vectors a~, and n [12]: a Fj ~ - N,Vvu - b ~v' N v (n ) + Fm sP + 8 v'lv av~" cr )~(n)

aFJ

(n) -

Here j~ and

NPp (n), +buy NUV + Fm(n)s -e~uv ~

j(n)

/1 = 1,2.

(4.43)

(normal balance)

(4.44)

are the respective projections of the angular acceleration vector & o / d t .

Equations (4.43) and (4.44) represent the second group of three basic equations in the theory of elastic shells by Kirchhoff and Love, see e.g. Refs. [6, 12]. S i m p l i f i c a t i o n o f the equations. For fluid interfaces and biomembranes one can simplify

the general surface balances of the linear and angular momentum, Eqs. (4.39)-(4.40) and (4.43)-(4.44), by using the following relevant assumptions: (i) Negligible contributions from the body forces (fs = 0) and couples (ms = 0). (ii) Quasistatic processes are considered ( i a = i (") = O, j a = j(n)= 0). (iii) The stress tensors in the bulk phases are isotropic: T I =-P~U,

T n =-PI, U

(4.45)

(iv) O'uv and Muv are symmetric surface tensors defined by Eqs. (4.24) and (4.28). (v) The transversal components of the tensor N are equal to zero (Na{n)= 0); in such a case, with the help of Eq. (4.29), we can transform Eq. (4.42): N-a

(4.46)

a a/~ N ~/~ - a a a/~ M y e

If the above assumptions are fulfilled, then Eq. (4.44), representing the tangential resultant of the surface balance of the angular momentum, is identically satisfied; see Eq. (4.30) and the related discussion. The remaining balance equations, (4.39), (4.40) and (4.43), acquire the form: v

vin)

o'~ + (q q~ ),v - b~v o"

,

2Her + 2Dq + o'~ (n) - Pn - PI

p = 1,2.

(4.47)

(normal balance of linear momentum)

(4.48)

General Curved Interfaces and Biomembranes

O"p(n) --

- M , v,L/V

151 /.t = 1,2.

(4.49)

To derive Eqs. (4.48) and (4.49) we have used Eqs. (4.17) and (4.29), respectively. Finally, s u b s t i t u t i n g o "p(n)

from Eq. (4.49) into Eqs. (4.47) and (4.48) we obtain

cr,u + (71 q v ) ,v - - b u y M vz.~. ]-/V

2Her + 2Drl - M,~ v - Pn - PI

/.t = 1,2. (generalized Laplace equation)

(4.50) (4.51)

Eqs. (4.50) and (4.51) have the meaning of projections of the interfacial stress balance in tangential and normal direction with respect to the dividing surface. In fact, Eq. (4.51) represents a generalized form of the Laplace equation of capillarity. Indeed, if the effects of the shearing tension and surface moments are negligible (r/= 0, MUV= 0), then Eq. (4.51) reduces to the conventional Laplace equation, Eq. (2.17). In addition, a version of the generalized Laplace equation can be derived by using the thermodynamic approach, that is by minimization of the free energy of the system, see Eq (4.71) below. Of course, the two versions of that equation must be equivalent. Thus we approach the problem about the equivalence of the mechanical and thermodynamical approaches, which is considered in the next section.

4.3.

C O N N E C T I O N B E T W E E N M E C H A N I C A L AND T H E R M O D Y N A M I C A L A P P R O A C H E S

4.3.1. GENERALIZED LAPLACE EQUATION DERIVED BY MINIMIZATION OF THE FREE ENERGY

In this section we will follow the pure thermodynamic approach, described in Sections 3.1 and 3.2, to derive the generalized Laplace equation. Our goal is to compare the result with Eq. (4.51) which has been obtained in the framework of the mechanical approach. With this end in view we consider the two-phase system depicted in Fig. 4.2. The grand thermodynamic potential of the system can be expressed in the form ~:2- f2 b + f2,

(4.52)

where ~)b and ~2~ account for the contributions from the bulk and the surface, respectively. For the bulk phases one has

152

Chapter 4

5 f~b = -P~ 5V~ - Pn 5Vn,

V = V~ + V u

(4.53)

The integral surface excess of the grand thermodynamic potential f~,, and its variation 8f2,, can be expressed in the form f2, - f (o, dA,

(4.54)

A

5 a s - 5 I o o , dA - I ( S cos +(.o, 5 o~) dA , A

(4.55)

A

where COs is the surface density of f~s, defined by Eq. (3.17); we have used the fact that 5a = ~dA)/dA, and the integration is carried out over the dividing surface between the two phases. A substutution of 5(o, from Eq. (3.18) into Eq. (4.55), along with the condition for constancy of the temperature and chemical potentials at equilibrium, yields 5 f~, - I 0 ' 5 a +~ 5 ~ + B S H + 0 5 D) dA

(4.56)

A

The combination of Eqs. (4.52), (4.53) and (4.56) yields [23, 26]: 5 ~2 - -(P~ - PH )6V~ - P~ O3/ + f O/"Sa +~ 5[3 + BSH + 0 5 D ) d A

(4.57)

A

Let us consider a small deformation of the dividing surface at fixed volume of the system, 5V = 0. The change in the shape of the interface is described by the variation of the positionvector of the points belonging to the dividing surface, 5 R =~'" a , + g t n ,

(4.58)

at fixed boundaties: ~'1 _ ~-- 2 _ Iff -- 0 ,

lff./t -- 0

over the contour L

(4.59)

Here ~'u = vUSt and gt = v(n)st are infinitesimal displacements; L is the contour encircling the surface A. With the help of Eqs. (4.19)-(4.22) we obtain [23, 26] 5o~ = auG~u.G - 2HN,

5 fl = qU~u.~ - 2 Dgt.

(4.60)

General Curved Interfaces and Biomembranes

153

Likewise, using the identities [ 12, 21 ] (n)v /2/- H,ov c~ + ( H 2 + D 2 ) v (n) + ~ a vv v,p

(4.61)

D - D.crv 'r + 2 H D v (n) + i q,V v(n).,v

(4.62)

we deduce ~H - H , ~ "~ + ( H 2 + D2)lff +~ a"vV,ov "

riD- D.a~ ~ + 2HDllt +~'4

q'.,v

(4.63)

The substitution of Eqs. (4.60) and (4.63) into Eq. (4.57), along with the condition for thermodynamic equilibrium, fig2 = 0, leads to [23, 26]: 0 = I { ( B H , , +|

~ + [PII - P I - 2 H T - 2 D ~

+ ( H 2 + D2)B + 2 H D |

A

A

(4.64) A

where we have used the auxiliary notations M "G _ 71 B a U ~ +71 | q.O

(4.65)

'?'"~ - 7 a v~ +~ qV~

(4.66)

f ~ is called the "thermodynamic surface stress tensor" [22]. Next, we transform the integrands of the last two integrals in Eq. (4.64) using the identities: ~, , ~ ~-,.o = ( 7'"~ ~-, ).o - ~. , o .o ~',

M .o- V,vo- - (M vo- gt,. ),~ - M "~

(4.67) V,.

(4.68)

M uo ,o-V,. - (M uo ,o- V),, - M uo ,,o- V

(4.69)

The first term in the right-hand side of Eqs. (4.67)-(4.69) represents a divergence of a surface vector. Integrating the latter, in accordance with the Green theorem, Eq. (4.35), we obtain an integral over the contour L, which is zero in view of Eq. (4.59). Thus the last two integrals in Eq. (4.64), which contain derivatives of ~', and I/L are reduced to integrals containing ~', and

154

Chapter 4

themselves. Finally, we set equal to zero the coefficients multiplying the independent variations ~'~ and I//in the transformed Eq. (4.64); thus we obtain the following two condition for mechanical equilibrium of the interface [26,23,22,13]"

y,; + (~ q (~),a - BH,; + | -2HT-2D~

,

/1 - 1,2;

+ ( H 2 + D 2 ) B + 2 H D O + I ( B a ~v +|

(4.70)

-PI-Pn

(4.71)

Equation (4.71) represents a thermodynamic version of the generalized Laplace equation. From a physical viewpoint, Eqs. (4.70)-(4.71) should be equivalent to Eqs. (4.50)-(4.51). This is proven below, where relationships between the thermodynamical parameters 7, ~, B, |

and the

mechanical parameters or, r/, M~, M2 are derived. For other equivalent forms of the generalized Laplace equation - see Section 4.3.3.

4.3.2.

W O R K OF D E F O R M A T I O N : T H E R M O D Y N A M I C A L A N D M E C H A N I C A L E X P R E S S I O N S

Relationships between the mechanical and thermodynamical surface tensions and moments. In the thermodynamic approach the work of surface deformation (per unit area) is expressed as follows"

5Ws-~'5o~ +~Sfl + BSH + 0 5 D ,

(4.72)

see Eq. (3.1). On the other hand, the mechanics provides the following expression for the work of surface deformation (per unit area and per unit time)" (~ W s

6t

= (s:(V~ v + Us x co) + N'(V~ o)),

(4.73)

see e.g. Eq. (4.26) in the book by Podstrigach and Povstenko [12]. As in Eqs. (1.22)-(1.23), here the symbol ":" denotes a double scalar product of two tensors. The vector co expresses the angular velocity of rotation of the running unit normal to the surface, n, which is caused by the change in the curvature of the interface [24]:

General Curved Interfaces and Biomembranes

m -nx~

155

dn

(4.74)

dt

Here "x" denotes vectorial product of two vectors. Hence, the vector m is perpendicular to the plane formed by the vectors n and dn/dt, which means that m is tangential to the surface. Equation (4.73) (multiplied by &) should be equivalent to the thermodynamic relationship, gq. (4.72). Since _c is a surface tensor ( n . ~ = 0), one can prove that ~ :(Us x m) - 0. Next, with the help of Eqs. (4.23) and (4.24), one derives [24]

,," (Vs v)- o ' , t . -oa+,t

(4.75)

Further, after some mathematical transformations described in Ref. [24], one can bring the term with the moments in Eq. (4.73) to the form N:(Vs m) = (Ml + m2) (/-I + 5I H a + 51 D/~) + (M1 - M2) (O + 7I D a +17 U/~)

(4.76)

Combining Eqs. (4.73), (4.75) and (4.76) one obtains (~Ws (~t

=o-a+rID +(M 1+M2)(H+IHa+IDfl) 2

1

(4.77) 1

+ ( M 1- M e ) ( b + - D a + - H / ~ ) 2 2 Further, taking into account that ~H - H~t, ~D - / ) ~ t and using Eq. (4.22), we get

' l +M2)H+2(M1-M2)D , ] ~0~+ (~ws- [cr+2(M + rl+2(M l

+M2)D+2(M1-M2)H (~+

(4.78)

+(M 1+Mz)~H +(M1-M2)~D which represents a corollary from Eq. (4.73). Comparing Eqs. (4.72) and (4.78) one can identify the coefficients multiplying the independent variations [24]" B - Ml+M2

(4.79)

|

(4.80)

l -M 2

Chapter 4

156

y-o'+

(4.81)

l OD 7 B H + -~

- 7 / + 7I B D + I ~ |

(4.82)

Discussion. Equations (4.79)-(4.80) show that the bending and torsion moments, B and

|

represent is|

and deviatoric scalar invariants of the tensor of the surface moments, M.

The substitution of Eqs. (4.79)-(4.80) into (4.28) yields 1 0 q ~v M ~v = 71 B a ~v + -~

(4.83)

In addition, Eqs. (4.81) and (4.82) express the connection between the mechanical surface tensions, o-, r/, and the thermodynamical surface tensions ~' and ~. For a spherical interface D = 0 and MI = M2 = M. Then Eqs. (4.79) and (4.81) are reduced to Eqs. (3.61) and (3.60), respectively, while Eqs. (4.80) and (4.82) yield | latter relationship holds for an is|

0, ~ - 7 " / - 0" the

deformation of a spherical surface.

The concepts for the surface tension as (i) excess force per unit length and (ii) excess surface energy per unit area are usually considered as being equivalent for a fluid phase boundary [2729]. Equation (4.81) shows that this is fulfilled only for a planar interface. In the general case, the difference between o" and ?' is due to the existence of surface moments. This difference could be important for interfaces and membranes of high curvature and low tension, such as microemulsions, biomembranes, etc. An interesting consequence from Eq. (4.82) is the existence of two possible definitions o f f l u i d interface [ 13, 14]. From a mechanical viewpoint we could require the two-dimensional stress tensor, o'v,,, to be is|

for a fluid interface (a two-dimensional analogue of the Pascal law).

Thus from Eq. (4.25) we obtain as the mechanical definition of a fluid interface in the form 7"/-0. The intriguing point is that for 7"/=0 Eq. (4.82) yields ~ = 7I BD + 7~|

and

consequently the mechanical work of shearing, ~3/3, is not zero if surface moments are present. On the other hand, from a thermodynamical viewpoint we may require the work of quasistatic shearing to be zero, that is ~ - 0 , for a f l u i d interface. However, it turns out that for such an

157

General Curved lntelfaces and Biomembranes

interface the surface stress tensor ouv is not isotropic: indeed, setting { - 0 in Eq. (4.82) we obtain q = - ( 7I B D + 7l o l l ) . In our opinion, it is impossible to determine which is the "true" definition of a fluid interface (7"/=0 or ~=0) by general considerations. Insofar as every theoretical description represents a model of a real object, in principle it is possible to establish experimentally whether the behavior of a given real interface agrees with the first or the second definition (q = 0 or ~ = 0).

4.3.3.

VERSIONS OF THE GENERALIZED LAPLACE EQUATION

First of all, using Eqs. (4.81)-(4.83), after some mathematical derivations one can transform Eqs. (4.70)-(4.71) into Eqs (4.50)-(4.51), and vice versa [14]. This is a manifestation of the equivalence between the mechanical and thermodynamical approaches, which are connected by Eqs. (4.81)-(4.83). In particular, Eqs. (4.51) and (4.71) represent two equivalent forms of the generalized Laplace equation. Another equivalent form of this equation is [14,22]: -2HT-2D~

+( H2 + D 2 ) B + 2 H D O + ( V s V s )

:M

(4.84)

= P~ - P n

The most compact form of the latter equation is obviously Eq. (4.51). In terms of the coefficients Cl and C2, see Eqs. (3.5) and (3.6), the generalized Laplace equation can be represented in another equivalent (but considerably longer) form [35]: 2 H 7 + 2 D ~ - C l (2H 2 _ K ) - 2 C 2 H K l v 2-y

s Cl - 2HV~2 C 2 + b ; Vs v s C 2

= /19II --

PI

(4.85)

Boruvka and Neumann [2] have derived an equation analogous to (4.85) without the shearing term 2D~. These authors have used a definition of surface tension, which is different from the conventional definition given by Gibbs; the latter fact has been noticed in Refs. [26] and [30]. In an earlier work by Melrose [31] an incomplete form of the generalized Laplace equation has been obtained, which contains only the first, third and fourth term in the left-hand side of Eq. (4.85). Another incomplete form of the generalized Laplace equation was published in Refs. [32-34]. Further specification of the form of the surface tangential and normal stress balances, Eqs. (4.50)-(4.51), can be achieved if appropriate rheological constitutive relations are available, as discussed in the next section. For other forms of the generalized Laplace equation - see Eqs.

Chapter 4

158

(4.99), (4.103), (4.107) and (4.110) below.

4.3.4.

INTERFACIAL RHEOLOGICAL CONSTITUTIVE RELATIONS

To solve whatever specific problem of the continuum mechanics, one needs explicit expressions for the tensors of stresses and moments. As already mentioned, such expressions typically have the form of relationships between stress and strain (or rate-of-strain), which characterize the theological behavior of the specific continuum: elastic, viscous, plastic, etc." see e.g. Refs. [6,20,35,36]. In fact, a constitutive relation represents a theoretical model of the respective continuum; its applicability for a given system is to be experimentally verified. Below in this section, following Ref. [14], we briefly consider constitutive relations, which are applicable to curved interfaces.

Surface stress tensor if_. Boussinesq [37] and Scriven [38] have introduced a constitutive relation which models a phase boundary as a two-dimensional viscous fluid:

rY,v -rYa,v + rlda,v d~ + 2r/s (d,v - 2 a , v d~ )

(4.86)

where d,v is the surface rate-of-strain tensor defined by Eq. (4.19), d~ is the trace of this tensor; r/d and r/s are the coefficients of surface dilatational and shear viscosity, cf. Eq. (4.11). The elastic (non-viscous) term in Eq. (4.86), o" a,v, is isotropic. Consequently, it is postulated that the shearing tension 7/ is zero, see Eq. (4.25), i.e. there is no shear elasticity. In other words, in the model by Boussinesq-Scriven the deviatoric part of the tensor O,v has entirely viscous origin. In Eq. (4.86) o" is to be identified with the mechanical surface tension, cf. Eq. (4.81). For emulsion phase boundaries of low interfacial tension the dependence of o" on the curvature should be taken into account. From Eqs. (3.39) and (4.81), in linear approximation with respect to the curvature, we obtain o" _ 70 + 7I Bo H + O ( H 2 )

(4.87)

where 7o is the tension of a flat interface. For example, for an emulsion from oil drops in water we have B0

=

10 -10

N and H =

105cm -~" then we obtain that the contribution from the

curvature effect to ry is 7 Boll -~ 0.5 mN/m. For such emulsions the latter value could be of the

General Curved Interfaces and Biomembranes

159

order of 7o, and even larger. Therefore, the curvature effect should be taken into account when solving the hydrodynamic problem about flocculation and coalescence of the droplets in some emulsions. Since the surface stress tensor ~ has also transversal components, cr ~(n), see Eq. (4.26), one needs also a constitutive relation for o~(n) ( y = 1,2). In analogy with Eq. (4.86) cr ~(n) can be expressed as a sum of a viscous and a non-viscous term [ 14]" 0"/~(n) -- O'ffvOn) + tw/'t(n) "" (0)

(4.88)

The viscous term can be expressed in agreement with the Newton's law for the viscous friction

[14]: O./~(n) (v)

- Z , v (n),/~

(4.89)

v (n) is defined by Eq. (4.20). As illustrated schematically in Fig. 4.3, equation (4.89) accounts for the lateral friction between the molecules in an interfacial adsorption layer; this effect could be essential for sufficiently dense adsorption layers, like those formed from proteins. Zs is a coefficient of surface transversal viscosity, which is expected to be of the order of r/s by magnitude. For quasistatic processes (v --)0 and cr/~,,c)n) --)0 ), the transversal components of _~ reduce to cr (~'(n) Then, in keeping with Eqs. (4.49), (4.83), (4.88) and (4 89), we can write [14]: 0) "

i-

I

lateral friction between ~ s u r f a c e molecules

I

Fig. 4.3. An illustration of the relative displacement of the neighboring adsorbed molecules (the squares) in a process of interfacial wave motion; u is the local deviation from planarity.

160

Chapter 4

O'S(n) t~'sV(n)'/a - 7 l(Ba~V + O q /Jv ),v

(4.90)

_

The surface rheological model, based on the constitutive relations (4.86) and (4.90), contains 3 coefficients of surface viscosity, viz. r/d, r/s and Z,- Moreover, the surface bending and torsion elastic moduli do also enter the theoretical expressions through B and O, see Eqs. (3.9) and (4.90). Tensor o f the surface m o m e n t s M. Equations (3.9) and (4.83) yield an expression for

the non-viscous part of M: M(0)~,v _ 89

+ Oq uv ) _ [ 8 9

uv

(4.91)

In keeping with Eqs. (4.17), (4.49), (4.90) and (4.91) the total tensor of the surface moments (including a viscous contribution)can be expressed in the form [14]: M ~ = [-~B o + 2(k,: + k c ) H ]a/~v - k~.b ~v - Zs v~176 a~v

(4.92)

The latter equation can be interpreted as a rheological constitutive relation stemming from the Helfrich formula, Eq. (3.7). With the help of the Codazzi equation, b ~'v'z = b vz'~ , see e.g. Ref. [ 19], we derive: b~,v - avzbpV, ~ = avzb vz,u - 2 H , ~

(4.93)

The combination of Eqs. (4.92) and (4.93) yields (4.94)

M uv,v - 2kc H ' u - Zs v~

In the derivation of Eq. (4.94) we have treated B0 as a constant. However, if the deformation is accompanied with a variation of the surface concentration F, then Eq. (4.94) should be written in the following more general form: M m' l ' F 4' + 2 k c H '~ - Z ' V (")'~ ; ,v _ 7Bo

B ~'

3B~ OF

(4.95)

General Curved Interfaces and Biomembranes

161

p The term with B0 has been taken into account by Dan et al. [39], as well as in Chapter l0 below, for describing the deformations in phospholipid bilayers caused by inclusions (like membrane proteins). In the simpler case of a quasistatic process (v (n~ - 0) and uniform surface concentration (F '~' = 0) Eq. (4.95) reduces to a quasistatic constitutive relation stemming from the Helfrich model: M uv ,V _ 2k~. H 'u

(4.96) m

It is worthwhile noting that the torsion (Gaussian) elasticity, k C, does not appear in Eqs. (4.94)-(4.96). Then, in view of Eq. (4.49) and (4.94), k c will not appear explicitly also in the tangential and normal balances of the linear momentum, Eqs. (4.39) and (4.40), which for small Reynolds numbers (inertial terms negligible) acquire the form

(T,v~ nt- bg (2kc s ' v -/~s v(n)'v ) - (TI(n)tt - ZIl n)p )

/1 = 1,2.

b~v 17/~v- (2kc n'pv -)(,sv(n)'~V )a~v - (TI(n)(n) -TII n)(n) )

(4.97)

(4.98)

o vv is to be substituted from Eq. (4.86). It should be noted that Eq. (4.98) is another form of the generalized Laplace equation. In vectorial notation and for quasistatic processes (v (n~ --> 0) Eq. (4.98) reads (4.99)

b ' f f - 2kcV,2H = n - ( T i - Tn).n

Application to capillary waves. As an example let us consider capillary waves on a flat

(in average) interface. It is usually assumed that the amplitude of the waves u (see Fig. 4.3) is sufficiently small, and consequently Eqs. (4.97) and (4.98) can be linearized:

O" V 2 b/

keg2 V 9

V ~, o ' + r / j V

s

V .vn +r/, s

2 6~b/

V

,vn 2

-- Ti(in)(n)

_

_

n.(T I - T n)-U,

where we have used the constitutive relation, Eq. (4.86), and the relationships

(4.100)

(4.1ol)

Chapter 4

162

2H = V ,u, 2

v (n) = ~3/,/ 'at

vii-aUvu

One sees that in linear approximation the dependent variables u and

(4.102)

VII

are separated: the

generalized Laplace equation, Eq. (4.100), contains the displacement u along the normal, whereas the two-dimensional Navier-Stokes equation (4.101) contains the tangential surface velocity, vii. In the linearized theory the curvature elasticities participate only trough kc in the b

normal stress balance, Eq. (4.100)" k C does not appear.

4.4.

AXISYMMETRICSHAPESOFBIOLOGICALCELLS

4.4.1.

THE GENERALIZEDLAPLACEEQUATIONIN PARAMETRICFORM

Equation (4.99) can be used to describe the shapes of biological membranes. For the sake of simplicity, let us assume that the phases on both sides of the membrane are fluid, i.e. Eq. (4.45) holds (the effect of citoskeleton neglected). Then substituting Eqs. (4.17), (4.24) and (4.45) into Eq. (4.99) one derives [14] 2Ho" + 2 D r / - 2kcV,2H = P n - PI

(4.103)

Further, let us consider the special case of axisymmetric membrane and let the z-axis be the axis of revolution. In the plane xy we introduce polar coordinates (r,q~); z = z(r) expresses the equation of the membrane shape. Then V,.2H can be presented in the form (see Ref. 37, Chapter XIV, Eq. 66):

VgH 1(| -

'

- -

r

+

z , -9 ) - l / 2 d I ( r 1 + z

t2 )-1/2_~F1

~rr

(4.104)

where & z' - - - - tan 0

dr

(4.105)

with 0 being the running slope angle. The two principal curvatures of an axisymmetric surface are ci = d(sinO)/dr and c2 = sinO/r. In view of Eq. (3.3), we have

GeneralCurvedInterfacesandBiomembranes dsin0 sin0 2 H = ~ + ~ ,

dr

163

dsin0 2D=--

r

sin0 --,

dr

(4.106)

r

Finally, with the help of Eqs. (4.104)-(4.106) we bring Eq. (4.103) into the form [14]

( d. s i n O . cr(dsinO + .s i n O )+7/ . dr

r

s i n O .) ~AP + r

dr

cos0 d

dr

{ r c o s 0 - d- I l ~ r dr

(rsin0)

1}

(4.107)

where AP = P u - PI. Equations (4.105) and (4.107) determine the generatrix of the membrane profile in a parametric form: r =

r(O), z = z(O). In the

special case, in which 7/= 0 and k c = 0

(no shearing tension and bending elasticity), Eq. (4.107) reduces to the common Laplace equation of capillarity, Eq. (2.24). The approach based on Eq. (4.107) is equivalent to the approach based on the expression for the free energy, insofar as the generalized Laplace equation can be derived by minimization of the free energy, see Section 4.3.1. The form of Eq. (4.107) calls for discussion. The possible shapes of biological and model membranes are usually determined by minimization of an appropriate expression for the free energy (or the grand thermodynamic potential) of the system, see e.g. Refs. [7,9,40-49]. For example, the integral bending elastic energy of a tension-free membrane is given by the expression [7] We - ~[2kc (H - H0) 2 +

kcK]dA

see Eq. (3.7). The above expression for

(4.108)

We contains

as parameters the spontaneous curvature

H0 and the Gaussian (torsion) elasticity k c , while the latter two parameters are missing in Eq. (4.107). As demonstrated in the previous section H0 and

k C must

not enter the generalized

Laplace equation, see Eq. (4.99); on the other hand, H0 and k C can enter the solution trough the boundary conditions [22]. For example, Deuling and Helfrich [43] described the myelin forms of an erythrocyte membrane assuming tension-free state of the membrane, that is cr = 7/= 0 and AP = 0; then they calculated the shape of the membrane as a solution of the equation ld ---

r dr

(r sin 0) = 2H0 = const.

(4.109)

It is obvious that for o" = 7/= 0 and AP = 0 every solution of Eq. (4.109) satisfies Eq. (4.107),

164

Chapter 4

and that the spontaneous curvature H0 appears as a constant of integration. In a more general case, e.g. swollen or adherent erythrocytes [50], one must not set 0" = 0 and AP = 0, since the membrane is expected to have some tension, though a very low one. To simplify the mathematical treatment, one could set r / = 0 in Eq. (4.107), i.e. one could neglect the effect of the shearing tension. Setting 71 = 0 means that the stresses in the membrane are assumed to be tangentially isotropic, that is the membrane behaves as a two-dimensional fluid. In fact, there are experimental indications that 77 0) and attractive (1-I < 0). A repulsive disjoining pressure may keep the two film surfaces at a given distance apart, thus creating a stable liquid film of uniform thickness, like that depicted in Fig. 5.1a. In contrast, attractive disjoining pressure destabilizes the liquid films. In the case of two solid surfaces interacting across a liquid H < 0 leads to adhesion of the two solids. If one of the film surfaces is fluid, the attractive disjoining pressure enhances the amplitude of the thermally excited fluctuation capillary waves, which grow until the film ruptures [5-9], see Section 6.2. In the case of a solid particle approaching a solid surface, the gap between the two surfaces can be treated as a liquid film of nonuniform thickness (Fig. 5.1b). Similar configuration may happen if the particle is fluid, but its surface tension is high enough, and/or its size is sufficiently small. If the interface is fluid, it undergoes some deformation produced by the interaction with the

185

Liquid Films and Interactions between Particle and Surface

approaching particle (Fig. 5.1c). When the liquid film ruptures, one says that the particle "enters" the fluid phase boundary. The occurrence of "entry" is important for the antifoaming action of small oil drops; this is considered in more details in Chapter 14 of this book. If a particle is entrapped within a liquid film (Fig. 5.1 d), two additional liquid films appear in the upper and lower part of the particle surface. Such a configuration is used in the film trapping technique (FTT), which allows one to measure the contact angles of Bm-sized particles [10], and to investigate the adhesive energy and physiological activation of biological cells [11,12]. (See also Fig. 5.6 below.) In this chapter we first derive and discuss basic mechanical balances and thermodynamical equations related to thin liquid films and equilibrium attachment of particles to interfaces (Section 5.1). Next, we consider separately various kinds of surface forces in thin liquid films (Section 5.2). In Chapter 6 we present an overview of the hydrodynamic interactions particle-interface and particle-particle. (Section 6.2). film of uneven thickness \

film ~,

, l h .solid

~, ,

I

,~

,

, so!id, iquid

~

llqmd

(a)

(b)

_ _ _ . f film 3

film

liquid ~

(c)

film I

(d)

Fig. 5.1. Various configurations particle-interface which are accompanied with the formation of a thin liquid film: (a) fluid particle (drop or bubble) at a solid interface" (b) solid particle at a solid surface; (c) solid or fluid particle at a fluid interface; (d) particle trapped in a liquid film.

186 5.1.2.

Chapter 5

DISJOINING PRESSURE AND TRANSVERSAL TENSION

Figure 5.2 shows a sketch of a fluid particle (drop or bubble) which is attached to a solid substrate. At equilibrium (no hydrodynamic flows) the pressure Pt in the bulk liquid phase is isotropic. The pressure inside the fluid particle, Pin, is higher than Pt because of the interfacial curvature (cf. Chapter 2): 20" R

= P/n - P~ - P c

(5.1)

where 0" is the fluid-liquid interfacial tension, Pc is the capillary pressure (the pressure jump across the curved interface), and R is the radius of curvature. The force balance p e r unit area of the upper film surface (Fig. 5.2) is given by the equation [ 13] Pin = P! + 1-I(h)

(5.2)

In other words, the increased pressure inside the fluid particle (Pin > Pl) is counterbalanced by the repulsive disjoining pressure I-I(h) acting in the liquid film. For a given I-I(h)-dependence, this balance of pressures determines the equilibrium thickness of the film. The comparison of Eqs. (5.1) and (5.2) shows that at equilibrium the disjoining pressure is equal to the capillary pressure: Fl(h) = P,:

(5.3)

Next, let us consider the force balance per unit length of the contact line, which encircles the plane-parallel film [ 14,15]: + _of + __x= 0

(5.4)

The vectors ~, _or and _xare shown in Fig. 5.2" 0" f is the tension of the upper film surface, which is different from the liquid-fluid interfacial tension 0" (as a rule 0" f < 0"), see Eq. (5.5) below. ~: is the so called transversal tension which is directed normally to the film surface. The transversal tension is a linear analogue of the disjoining pressure" r accounts for the excess interactions across the liquid film in the narrow transition zone between the uniform film and the bulk liquid phase. (Microscopically this transition zone can be treated as a film of uneven thickness and a micromechanical expression for ~"can be d e r i v e d - see Ref. 15.) Note that, in

Liquid Films and Interactions between Particle and Surface

187

4t liquid

. /

......

.... Ih "4,'

Fig. 5.2. Sketch of a fluid particle which is attached to a solid surface. A plane-parallel film of thickness h and radius rc is formed in the zone of attachment; Pin and Pt are the pressures in the inner fluid and in the outer liquid; II is disjoining pressure; o" and d are surface tensions of the outer fluid-liquid phase boundary and of the film surface; ~"is transversal tension. general, Eq. (5.4) may contain an additional line-tension term, cf. Eq. (2.73), which is usually very small and is neglected here; see Section 2.3.4 and Eq. (5.31) below. The horizontal and vertical projections of Eq. (5.4) have the form: o- y = o" c o s a

(5.5)

~'-asina

(5.6)

where o~ is the contact angle. Since c o s a < l, Eq. (5.5) shows that o" f

< O'.

In addition, Eq. (5.6)

states that the transversal tension ~"counterbalances the normal projection of the surface tension with respect to the film surface. To understand deeper the above force balances, we will use a thermodynamic relationship,

3a ~ Oh

=-H,

(wetting film)

(5.7)

which is derived in the next Section 5.1.3. The integration of the latter equation, along with the

188

Chapter 5

boundary condition lim o- f (h) - o-, yields h----),,~

cx~

o-f (h) - o- + II-I(h)dh

(5.8)

(wetting film)

h

In fact, the integral oo

I(h)--- fn(h; h

(5.9)

h

expresses the work (per unit area) performed against the surface forces to bring the two film surfaces from an infinite separation to a finite distance h; f(h) has the meaning of excess free energy per unit area of the thin liquid film. Comparing Eqs. (5.5) and (5.8) one obtains

1 i II(h)dh

cosa - 1 +--

- 1 -t- f~( h )

(3" h

(wetting film)

(5.10)

(f/o- 0. The exact balance of these two forces of opposite direction, expressed by Eq. (5.13), determines the state of equilibrium attachment of the particle to the interface. Note that the conclusions based on Eq. (5.13) are valid not only for particle-wall attachment, but also for particle-particle interactions, say for the formation of doublets and

Liquid Films and Interactions between Particle and Surface

191

multiplets (flocs) from drops in emulsions [24]. For larger particles the gravitational force Fg, which represents the difference between the particle weight and the buoyancy (Archimedes) force, may give a contribution to the force balance in Eq. (5.13), [22,23]: 7D"c2 H -"

2a'rc 7: + Fg,

Fg - Ap g Vp

(5.14)

Here Ap is the difference between the mass densities of the fluid particle and the outer liquid phase, g is the acceleration due to gravity and Vp is the volume of the particle.

5.1.3.

THERMODYNAMICSOF THIN LIQUID FILMS

First, we consider symmetric thin liquid films, like that depicted in Fig. 5.4. Since such films have two fluid surfaces, the respective thermodynamic equations sometimes differs from their analogues for wetting films (Section 5.1.2) by a multiplier 2; these differences will be noted in the text below. Symmetric films appear between two attached similar drops or bubbles, as well as in foams. As in Fig. 5.2, Pin is the pressure in the fluid particles and Pt is the pressure in the outer liquid phase (in the case of f o a m - that is the liquid in the Plateau borders). The force balances per unit area of the film surface and per unit length of the contact line (see the lefthand side of Fig. 5.4) lead again to Eqs. (5.2)-(5.6). It should be noted that two different, but supplementary, approaches (models) are used in the macroscopic description of a thin liquid film. These are the "detailed approach", used until now, and the "membrane approach"; they are illustrated, respectively, on the left- and righthand side of Fig. 5.4. As described above, the "detailed approach" models the film as a liquid layer of thickness h and surface tension cr f . In contrast, the "membrane approach", treats the film as a membrane of zero thickness and total tension, T, acting tangentially to the membrane - see the right-hand side of Fig. 5.4. By making the balance of the forces acting on a plate of unit width along the y-axis (in Fig. 5.4 the profile of this plate coincides with the z-axis) one obtains the Rusanov [25] equation: 7' = 2or f + Pch

(Pc = P i n - Pt)

(5.15)

Chapter 5

192

Equation (5.15) expresses a condition for equivalence between the membrane and detailed models with respect to the lateral force. In the framework of the membrane approach the film can be treated as a single surface phase, whose Gibbs-Duhem equation reads [23,25,26]: k

dT--sf

d T - Z Fidlai

(5.16)

i=l

where )' is the film tension, T is temperature, s f is excess entropy per unit area of the film, Fi and lai are the adsorption and the chemical potential of the i-th component. The Gibbs-Duhem equations of the liquid phase (1) and the "inner" phase (in) read k

dPz = Svx d T + ~,niX dl-ti,

Z - l , in

(5.17)

i=l

where Svz and n/z are entropy and number of molecules per unit volume, and P72 is pressure in the respective phase. Since Pc - Pin - Pl, from Eq. (5.17) one can obtain an expression for dPc. Further, we multiply this expression by h and subtract the result from the Gibbs-Duhem equation of the film, Eq. (5.16). The result reads k

d T : -'~dT + h dPc - ~_, r-'id].li

(5.18)

i=l

where

s'~ = s f +(s ~ - s v' )h,

Fi - F + (n~ - n[ )h,

i=1 ..... k

(5.19)

An alternative derivation of the same equations is possible, after Toshev and Ivanov [27]. Imagine two equidistant planes separated at a distance h. The volume confined between the two planes is thought to be filled with the bulk liquid phase "/". Taking surface excesses with respect to the bulk phases, one can derive Eqs. (5.18) and (5.19) with ~" and F i being the excess surface entropy and adsorption ascribed to the surfaces of this liquid layer. A comparison between Eqs. (5.18) and (5.16) shows that there is one additional term in Eq. (5.18), viz. h dPc. It corresponds to one supplementary degree of freedom connected with the

Liquid Films and Interactions between Particle and Surface

193

choice of the parameter h. To specify the model one needs an additional equation to determine h. For example, let this equation be F~ - 0

(5.20)

Equation (5.20) requires h to be the thickness of a liquid layer from phase "/", containing the same amount of component 1 as the real film. This thickness is called the thermodynamic thickness of the film [28]. It can be of the order of the real film thickness if component 1 is chosen in an appropriate way, say, to be the solvent in the film phase. Combining Eqs. (5.3), (5.18) and (5.20) one obtains [27] k

(5.21)

d y - -'~dT + hdrI - ~ [-)U~LIi i=2

Note that the summation in the latter equation starts from i=2, and that the number of differentials in Eqs. (5.16) and (5.21) is the same. A corollary from Eq. (5.2t) is the Frumkin equation [29] /OT~

=h

(5.22)

For thin liquid films h is a relatively small quantity (h < 10-5 cm)" therefore Eq. (5.22) predicts a rather weak dependence of the film tension 7' on the disjoining pressure, l-I, in equilibrium thin films. By means of Eqs. (5.3) and (5.15) one can transform Eq. (5.21) to read [28] k

(5.23)

2dry f - -'~dT - FId h - ~_~ I-'id Jxi i=2

From Eq. (5.23) the following useful relations can be derived [27,28]

tehl.2

2( c?~Yf

=-I-I

(symmetric film)

(5.24)

cr i ( h ) - r +-~ f H(h)dh

(symmetric film)

(5.25)

. . . . . ]-t k

c,o

h

Note that the latter two equations differ from the respective relationships for a wetting film,

Chapter 5

194

Eqs (5.7) and (5.8), with multipliers 2 and 1/2; as already mentioned, this is due to the presence of two fluid surfaces in the case of a symmetric liquid film. Note also that the above thermodynamic equations are corollaries from the Gibbs-Duhem equation in the membrane approach, Eq. (5.16). The detailed approach, which treats the two film surfaces as separate surface phases with their own fundamental equations [25,27,30]; thus for a flat symmetric film one postulates k

dU f - T d S f + 2o- f d a + ~_~l.tidNf - H A d h ,

(5.26)

i=I

where A is area; U f , S f and N f are excesses of the internal energy, entropy and number of molecules ascribed to the film surfaces. Compared with the fundamental equation of a simple surface phase [31 ], Eq. (5.26) contains an additional term, -IIAdh, which takes into account the dependence of the film surface energy on the film thickness. Equation (5.26) provides an alternative thermodynamic definition of the disjoining pressure:

I The thin liquid films formed in foams or emulsions exist in a permanent contact with the bulk liquid in the Plateau borders, encircling the film. From a macroscopic viewpoint, the boundary film/Plateau border can be treated as a three-phase contact line: the line, at which the two surfaces of the Plateau border (the two concave menisci) intersect at the plane of the film, see the right-hand side of Fig. 5.4. The angle t~), subtended between the two meniscus surfaces, represents the thin film contact angle corresponding to the membrane approach. The force balance at each point of the contact line is given by the Neumann-Young equation, Eq. (2.73) with o-w = 7', and o-u = o-v = 6. The effect of the line tension, ~c, can be also taken into account, see Eq. (2.70). Thus for a symmetrical flat film with circular contact line (Fig. 5.4) one obtains [14] K"

7' + - - = 2o" c o s a 0 r0 where r0 is the radius of the respective contact line.

(5.28)

Liquid Films and Interactions between Particle and Surface

'

re2

d

~i ;

1 ; 0/

rcl

/

195

)

'

fluid 2

_/'_-IN

/

\

\'~

\

fluidl

~

Plateau

border

/

!

Fig. 5.5. Schematic presentation of the force balances in each point of the two contact lines at the boundary between a spherical film and the Plateau border, see Eq. (5.32); after Refs. [23,32]. There are two film surfaces and two contact lines in the detailed approach, see the left-hand side of Fig. 5.4. They can be treated thermodynamically as linear phases; further, an onedimensional analogue of Eq. (5.26) can be postulated [ 14]: (5.29)

d U L - T d S L + 2 ~ d L + E l . t i d N i c +'cdh i

Here Uc, SL and N~ are linear excesses, ~ is the line tension in the detailed approach and

(5.30) L ~ Oh

is a thermodynamical definition of the transversal tension, which is apparently an onedimensional analogue of the disjoining pressure II - cf. Eqs. (5.27) and (5.30). The vectorial force balance per unit length of the contact lines of a symmetric film, with account for the line tension effect, is [14] ~ +_o-r+ $ + c~:= O,

I~r{= ~ / r c

(5.31)

Chapter 5

196

A

OuterSet of Interference Fringes B

_ ~ Air._./.-.------__./ if

/

~ r = n C ~ ~ I ~

~

InnerSet of Interference Water Fringes

Asymmetric PlaneParallel Cell-Water-Air FoamFilm Film

~ TCR AmAb

? OtherProteins I Glycocallx

Fig. 5.6. Operation principle of the Film Trapping Technique. (A) A photograph of leukemic Jurkat cell trapped in a foam (air-water-air) film. The cell is observed in reflected monochromatic light; a pattern of alternating dark and bright interference fringes appears. (B) Sketch of the cell trapped in the film. The inner set of fringes corresponds to the region of contact of the cell with the protein adsorption layer (C). From the radii of the interference fringes one can restore the shapes of the liquid meniscus and the cell, and calculate the contact angle, o~, the cell membrane tension, CYc,and the tension of the cell-water-air film, T; from Ivanov et al. [ 12].

Liquid Films and Interactions between Particle and Surface

197

see Fig. 5.4; the vector cy~, expressing the line tension effect, is directed toward the center of curvature of the contact line, see Chapter 2 for details. In the case of a curved or non-symmetric film (film formed between two different fluid phases) Eq. (5.31) can be generalized as follows [23]: r

K

+ -~f + $i + _ci = 0,

i = 1,2

(5.32)

see Fig. 5.5 for the notation. Equation (5.32) represents a generalization of the NeumannYoung equation, Eq. (2.73), expressing the vectorial balance of forces at each point of the respective contact line. Equation (5.32) finds applications for determining contact angles of liquid films, which in their own turn bring information about the interaction energy per unit area of the film, see Eq. 5.10. Experimentally, information about the shape of fluid interfaces can be obtained by means of interferometric techniques and subsequent theoretical analysis of the interference pattern [33]. This approach can be applied also to biological cells. For example, as illustrated in Fig. 5.6, human T cells have been trapped in a liquid film, whose surfaces represent adsorption monolayers of monoclonal antibodies acting as specific ligands for the receptors expressed on the cell surface. From the measured contact angle the cell-monolayer adhesive energy was determined and information about the ligand-receptor interaction has been obtained [ 12].

5.1.4.

DERJAGUIN APPROXIMATION FOR FILMS OF UNEVEN THICKNESS

In the previous sections of this chapter we considered planar liquid films. Here we present a popular approximate approach, proposed by Derjaguin [34], which allows one to calculate the interaction between a particle and an interface across a film of nonuniform thickness, like that depicted in Fig. 5.1 b, assuming that the disjoining pressure of a plane-parallel film is known. Following the derivation by Derjaguin [2, 34], let us consider the zone of contact between a particle and an interface; in general, the latter is curved, see Fig. 5.7a. The "interface" could be the surface of another particle. The Derjaguin approximation is applicable to calculate the interaction between any couple of colloidal particles, either solid, liquid or gas bubbles. The only assumption is that the characteristic range of action of the surface forces is much smaller than any of the surface curvature radii in the zone of contact.

198

Chapter 5

,4,~,, .

(II)

.

.

.

.

.

')/ .

.

.

.

"]~' 'x l

y9

[h_

"

s " "~fD v

(I)

"

~x \

(a)

(b)

Fig. 5.7. (a) The zone of contact of two macroscopic bodies; h0 is the shortest surface-to-surface distance. (b) The directions of the principle curvatures of the two surfaces, in general, subtend some angle co. The length of the segment

OIO2

in Fig. 5.7a, which is the closest distance between the two

surfaces, is denoted by h0. The z-axis is oriented along the segment O102. In the zone of contact the shapes of the two surfaces can be approximated with paraboloids [2, 34]: z, = 71q x( + 89

z2 = 7c2x 2 1 2 +2c2y2,1 ..,. 2

2,

(5.33)

p

Here c~ and c~ are the principal curvatures of the first surface in the point O1" likewise,

C2

and

c~ are the principal curvatures of the second surface in the point O2; the coordinate plane xiYi passes through the point Oi, i = 1,2. The axes xi and Yi are oriented along the principal directions of the curved surface Si in the point Oi. In general, the directions of the principle curvatures of the two surfaces subtend some angle co (0 < co < 180~ see Fig. 5.7b: x 2 -- x 1 COS(D +

y] sinco,

Y2 = - x ]

sinco +

Yl COSCO

(5.34)

The local width of the gap between the two surfaces is (Fig. 5.7a) h = h0 + zl +

Z2

(5.35)

Combining Eqs. (5.33)-(5.35) one obtains [2, 34] h = ho + 89a x 2 +-~1 B y2 + C x l y]

(5.36)

Liquid Films and Interactions between Particle and Surface

199

where A, B and C are coefficients independent of x~ and yl: A = cl+

p

C2 COS2(D + C2 sin2o9

p

B = c~ + C = (c 2 -

(5.37)

p

C2 sin2o9 + C2 COS20) p c 2 ) cos(_/)

(5.38)

sinm

(5.39)

Equation (5.36) expresses h(xl,Yl) as a bilinear form; the latter, as known from the linear algebra, can be represented as a quadratic form by means of a special coordinate transformation

(Xl, yl) --> (x, y): h -- h 0 + 1 c x 2 - k - l c t y 2

(5.40)

This is equivalent to bringing of the symmetric matrix (tensor) of the bilinear form into diagonal form:

/

~/

1

'

(5.41)

Since the determinant of a tensor is invariant with respect to coordinate transformations, one can write (5.42)

c c" = A B - C 2

Further, we assume that the interaction free energy (due to the surface forces) per unit area of a plane-parallel film of thickness h is known: this is the function f(h) defined by Eq. (5.9). The "core" of the Derjaguin approximation is the assumption that the energy of interaction, U, between the two bodies (I and II in Fig. 5.7a) across the film is given by the expression U

(5.43)

; ; f (h(x, y)) dxdy

where h = ho + 89 x 2 +lc'y2. Further, let us introduce polar coordinates in the plane xy:

x

-~cccosq9,

y-

cosq9

Since h depends only on p, Eq. (4.43) acquires the form

(5.44)

200

Chapter 5

21r

oo

U = ~ f f ( h ( p ) ) pdpdtp

(h = ho + 1/92)

(5.45)

o o Integrating with respect to tp and using the relationship dh = p dp one finally obtains [2, 34] oo

2Jr ~f(h)dh, U(h~ = - ~ h o

(interaction energy)

(5.46)

E - cc'=c,c; +c2c 2 + (c,c 2 +c;c2)sin z co +(c,c 2 +c;c2)cos 2 co

(5.47)

The last expression is obtained by substitution of Eqs (5.37)-(5.39) into Eq. (5.42). We recall that oJ is the angle subtended between the directions of the principle curvatures of the two approaching surfaces. It has been established, both experimentally [3] and theoretically [35], that Eq. (5.46) provides a good approximation for the interaction energy in the range of its validity. The interaction force between two bodies, separated at a surface-to-surface distance h0, can be obtained by differentiation of Eq. (5.46):

F(ho)=

~U = ~-~ 2/r f (ho) Oho

(interaction force)

(5.48)

Next, we consider various cases of special geometry:

Sphere-Wall: This is the configuration depicted in Fig. 5.1b - particle of radius R p

situated at a surface-to-surface distance h0 from a planar solid surface. In such a case Cl = c 1 = p

1/R, whereas c2 - c 2 = 0. Then from Eqs. (5.46)-(5.47) one deduces oo

U(h0)= 2zcg ~f(h)dh, ho

(sphere-wall)

(5.49)

Truncated Sphere - Wall: For this configuration, see Fig. 5.1 a, the interaction across the plane-parallel film of radius rc should be also taken into account [36-39]: oo

U(h o )= 2JrR f f(h)dh + ~'r~f (ho)

(truncated s p h e r e - wall)

(5.50)

h0

Two Spheres: For two spherical particles of radii RI and t surface distance h0 one has cl = c 1 = 1/RI and c2 = c 2 p

=

R2

separated at a surface-to-

1/R2. Then Eqs. (5.46)-(5.47) yield

Liquid Films and Interactions between Particle and Surface

U (ho ) - 2JrR1R2 i R, + R 2 f (h ) dh

201

(two spheres)

(5.51)

In the limit R1---~R and R 2 ~ , Eq. (5.51) reduces to Eq. (5.49), as it should be expected.

Two Crossed Cylinders: For two infinitely long cylinders (rods) of radii rl and r2, which are separated at a transversal surface-to-surface distance h0, and whose axes subtend an angle p

p

co, one has Cl = 1/r~, c I = O, c2 = 1/r2 and c 2 = 0. Then Eqs. (5.46)-(5.47) lead to [2]

U(ho)- 2zt'-~lre i f ( h ) d h sin co

(two cylinders)

(5.52)

h0

The latter equation is often used to interpret data obtained by means of the surface force apparatus, which operates with crossed cylinders [3]. For parallel cylinders, that is for co-->0, Eq. (5.52) gives U ~ , , ; this divergence is not surprising because the contact zone between two parallel cylinders is infinitely long, whereas the interaction energy per unit length is finite. In the surface force apparatus usually co = 90 ~ and then sin co = 1. The interaction force can be calculated by a mere differentiation of Eqs. (5.49)-(5.52) in accordance with Eq. (5.48). The Derjaguin approximation is applicable to any type of force law (attractive, repulsive, oscillatory) if only the range of the forces is much smaller than the particle radii. Moreover, it is applicable to any kind of surface force, irrespective of its physical origin: van der Waals, electrostatic, steric, oscillatory-structural, etc. forces, which are described in the next section.

5.2.

I N T E R A C T I O N S IN THIN LIQUID F I L M S

5.2.1.

OVERVIEW OF THE TYPES OF SURFACE FORCES

As already mentioned, if a liquid film is sufficiently thin (thinner than c.a. 100 rim) the interaction of the two neighboring phases across the film is not negligible. The resulting disjoining pressure, H(h), may contain contributions from various kinds of molecular interactions. The first successful theoretical model of the interactions in liquid films and the stability of

202

Chapter 5

colloidal dispersions was created by Derjaguin & Landau [ 16], and Verwey & Overbeek [17]; it is often termed "DLVO theory" after the names of the authors. This model assumes that the disjoining pressure is a superposition of electrostatic repulsion and van der Waals attraction, see Eq. (5.12), Fig. 5.3 and Sections 5.2.2 and 5.2.4 below. In many cases this is the correct physical picture and the DLVO theory provides a quantitative description of the respective effects and phenomena. Subsequent studies, both experimental and theoretical, revealed the existence of other surface forces, different from the conventional van der Waals and electrostatic (double layer) interactions. Such forces appear as deviations from the DLVO theory and are sometimes called "non-DLVO surface forces" [3]. An example is the hydrophobic attraction which brings about instability of aqueous films spread on a hydrophobic surface, see Section 5.2.3. Another example is the hydration repulsion, which appears as a considerable deviation from the DLVO theory in very thin (h < 10 rim) films from electrolyte solutions, see Section 5.2.5. Oscillations of the surface force with the surface-to-surface distance were first detected in films from electrolyte solutions sandwiched between solid surfaces [3,40]. This oscillatory

structural force appears also in thin liquid films containing small colloidal particles like surfactant micelles, polymer coils, protein macromolecules, latex or silica particles [41 ]. For larger particle volume fractions the oscillatory force is found to stabilize thin films and dispersions, whereas at low particle concentrations it degenerates into the depletion attraction, which has the opposite effect, see Section 5.2.7. When the surfaces of the liquid film are covered with adsorption layers form nonionic surfactants, like those having polyoxiethylene moieties, the overlap of the formed polymer brushes give rise to a steric interaction [3, 42], which is reviewed in Section 5.2.8. The surfactant adsorption monolayers on liquid interfaces and the lipid lamellar membranes are involved in a thermally exited motion, which manifests itself as fluctuation capillary waves. When such two interfaces approach each other, the overlap of the interfacial corrugations causes a kind of steric interaction (though a short range one), termed the fluctuation force [3], see Section 5.2.9. The approach of a fluid particle (emulsion drop or gas bubble) to a phase boundary might be

Liquid Films and Interactions between Particle and Surface

203

accompanied with interfacial deformations: dilatation and bending. The latter also do contribute to the overall particle-surface interaction, see Section 5.2.10. In a final reckoning, the total energy of interaction between a particle and a surface, U, can be expressed as a sum of contributions of different origin: from the interfacial dilatation and bending, from the van der Waals, electrostatic, hydration, oscillatory-structural, steric, etc. surface forces as follows [43]: U = Udil + Ubend + Uvw + Uel + Uhydr + Uosc + Ust + ""

(5.53)

Below we present theoretical expressions for calculating the various terms in the right-hand side of Eq. (5.53). In addition, in the next Chapter 6 we consider also the surface forces of

hydrodynamic origin, which are due to the viscous dissipation of energy in the narrow gap between two approaching surfaces in liquid (Section 6.2). In summary, below in this chapter we present a brief description of the various kinds of surface forces. The reader could find more details in the specialized literature on surface forces and thin liquid films [2, 3, 42-45]

5.2.2.

VAN DER WAALS SURFACE FORCE

The van der Waals forces represent an averaged dipole-dipole interaction, which is a superposition of three contributions: (i) orientation interaction between two permanent dipoles: effect of Keesom [46]; (ii) induction interaction between one permanent dipole and one induced dipole: effect of Debye [47]; (iii) dispersion interaction between two induced dipoles: effect of London [48]. The energy of van der Waals interaction between molecules i and j obeys the law [49] uij (r) --

a iJ r6

(5.54)

where uij is the potential energy of interaction, r is the distance between the two molecules and o~ij is a constant characterizing the interaction. In the case of two molecules in a gas phase one has [3, 49]

Chapter 5

204

p2

2

_ ~ i Pj -1" (p20~Oj -[- p j2 O~Oi )jr. 37~O~oiO~ojhpViV j a ij -3kT v i nt-V j

where Pi and O(.oiare molecular dipole moment and electronic polarizability, hp = 6.63•

(5.55)

TM J.s

is the Planck constant and vi can be interpreted as the orbiting frequency of the electron in the Bohr atom; see Refs. [3, 50] for details. The van der Waals interaction between two macroscopic bodies can be found by integration of Eq. (5.54) over all couples of interacting molecules followed by subtraction of the interaction energy at infinite separation between the bodies. The result of integration depends on the geometry of the system. For a plane-parallel film located between two semiinfinite phases the van der Waals interaction energy per unit area and the respective disjoining pressure, stemming from Eq. (5.54), are [51 ]: An fvw =-12:rt.h-------T,

Hvw = -

~ fvw An ol-----~=-6tch------T

(5.56)

where, as usual, h is the thickness of the film and AH is the Hamaker constant [44, 51 ]; about the calculation of A H - see Eqs. (5.65)-(5.74) below. By integration over all couples of interacting molecules Hamaker [51 ] has derived the following expression for the energy of van der Waals interaction between two spheres of radii R1 and R2:

Uvw(ho)=

At/

Y

X2 + x y + x

_~ x 2

y

+xy+x+y

X 2 -k-xy-]-X ) +21nx2 + x y + x + y

(5.57)

where x = h 0 / 2R 1,

y = R 2 ] R~ < 1

(5.58)

as before, h0 is the shortest surface-to-surface distance. For x ~

I 12 E i -- E j

kT Ei -[-~-'J

+

(5.69)

16-v/2-(n2 +nj

where ei and ej are the dielectric constants of phases i and j; ni and nj are the respective refractive indices for visible light; as usual, hp is the Planck constant; Ve is the main electronic absorption frequency which is ~ 3.0xl015Hz for water and the most organic liquids [3]. The first term in the right-hand side of Eq. (5.69), A/}v=~ , the so called zero frequency term, expresses the contribution of the orientation and induction interactions. Indeed, these two contributions to the van der Waals force represent electrostatic effects. Equation (5.69) shows that this zero-frequency term can never exceed 3 kT -- 3

x 10 -21

J. The last term in Eq. (5.69),

A~v>~ , accounts for the dispersion interaction. If the two phases, i and j, have comparable densities (as it is for emulsion systems, say oil-water-oil), then a/}v>~ and a/}v=~ are comparable by magnitude. If one of the phases, i or j, has low density (gas, vacuum), as a rule A~v>~ >> A~y=~ in this respect the macroscopic and microscopic theories often give different predictions for the value of AH. For the more general configuration of phases i and k, interacting across a film from phase j, the macroscopic (Lifshitz) theory provides the following expression [3]

Liquid Films and Interactions between Particle and Surface

AH = Aijk = "~ij~

+ ~lijk

--

kT

209

J

(5.70)

+

Upon substitution k = i Eq. (5.70) reduces to Eq. (5.69). Equation (5.70) can be simplified if the following approximate relationship is satisfied: (5.71) that is the arithmetic and geometric mean of the respective quantities are approximately equal. Substitution of Eq. (5.7 l) into (5.70) yields a more compact expression: 3hpVe(?l 2 --?12)(?l 2 - n ~ ) AH = Aij ~ =

kT '~i + •j

e k + ej

+ 164~-(n2 + n~

)3/4 (n 2 + nj2)3/4

(5.72)

Comparing Eqs. (5.69) and (5.72) one obtains the following c o m b i n i n g relations: (V=o, _ A ijk L iji

[A(V=O)A(V=O, ]1,2 kjk

(5.73)

(v>0) (v>0) A (V>~ = [Aiji Akjk ] 1/

(5.74)

The latter two equations show that according to the macroscopic theory the Hamaker a(V-~ (orientation + approximation, Eq. (5.67), holds separately for the zero-frequency term, "ijk induction interactions) and for the dispersion interaction term, a(v>0) "ijk 9 Effect o f e l e c t r o m a g n e t i c

retardation.

The asymptotic behavior of the dispersion

interaction at large intermolecular separations does not obey Eq. (5.54); instead u 0 o~ 1/r 7 due to the electromagnetic retardation effect established by Casimir and Polder [59]. Experimentally this effect has been first detected by Derjaguin and Abrikossova [60] in measurements of the interaction between two quartz glass surfaces in the distance range 100-400 rim. Various expressions have been proposed to account for this effect in the Hamaker constant; one convenient formula for the case of symmetric films has been derived by Prieve and Russel, see

210

Chapter 5

Ref. [42]:

i/l+

2

where, as usual, h is the film thickness; the dimensionless thickness h is defined by the expression "~

2 ]1/2 2 7 W e

h - nj (n2 + n j ,

h

,

(5.76)

c

where c = 3.0 • 10 l~ cm/s is the speed of light; the integral in Eq. (5.75) is to be solved numerically; for estimates one can use the approximate interpolating formula [42]:

~O+2hz)exp(-2hZ)dz= 0

(1+2z2) 2

rc ~

rch 1+ - ~

(5.77)

For small thickness A/~v>~ , as given by Eqs. (5.75), is constant, whereas for large thickness h one obtains A/~ >~

h -1. For additional information about the electromagnetic retardation

e f f e c t - see Refs. [3,42,52]. It is interesting to note that this relativistic effect essentially influences the critical thickness of rupture of foam and emulsion films, see Section 6.2 below.

Screening of the orientation and induction interactions in electrolyte solutions. As already mentioned, the orientation and induction interactions (unlike the dispersion interaction) are electrostatic effects; so, they are not subjected to electromagnetic retardation. Instead, they are influenced by the Debye screening due to the presence of ions in the aqueous phase. Thus for the interaction across an electrolyte solution the screened Hamaker constant is given by the expression [50]

A H =A (v=~ (2tch)e -2rh +A (v>~

(5.78)

where A (v-'--~ denotes the contribution of orientation and induction interaction into the Hamaker constant in the absence of any electrolyte" A (v>~ is the contribution of the dispersion interaction; tr is the Debye screening parameter defined by Eqs. (1.56) and (1.64). Additional information about this effect can be found in Refs. [3, 42, 50].

Liquid Films and Interactions between Particle and Surface

5.2.3.

211

LONG-RANGE HYDROPHOBIC SURFACE FORCE

The experiment sometimes gives values of the Hamaker constant, which are markedly larger than the values predicted by the theory. This fact could be attributed to the action of a strong attractive hydrophobic force, which is found to appear across thin aqueous films sandwiched between two hydrophobic surfaces [61-63].

The experiments showed that the nature of the

hydrophobic force is different from the van der Waals interaction [61-69]. It turns out that the hydrophobic interaction decays exponentially with the increase of the film thickness, h. The hydrophobic free energy per unit area of the film can be described by means of the equation [3]

fhydrophobic= - 2 y e -h / 2~)

( 5.79 )

where typically y = 10-50 mJ/m 2, and 20 = 1-2 nm in the range 0 < h < 10 nm. Larger decay length, 2o - 12-16 nm, was reported by Christenson et al. [69] for the range 20 n m < h < 90 nm. This long-ranged attraction entirely dominates over the van der Waals forces. The fact that the hydrophobic attraction can exist at high electrolyte concentrations, of the order of 1 M, means that this force cannot have electrostatic origin [69-74]. In practice, this attractive interaction leads to a rapid coagulation of hydrophobic particles in water [75, 76] and to rupturing of water films spread on hydrophobic surfaces [77]. It can play a role in the adhesion and fusion of lipid bilayers and biomembranes [78]. The hydrophobic interaction can be completely suppressed if the adsorption of surfactant, dissolved in the aqueous phase, converts the surfaces from hydrophobic into hydrophilic. There is no generally accepted explanation of the hydrophobic force [79]. One of the possible mechanisms is that an orientational ordering, propagated by hydrogen bounds in water and other associated liquids, could be the main underlying factor [3, 80]. Another hypothesis for the physical origin of the hydrophobic force considers a possible role of formation of gaseous capillary bridges between the two hydrophobic surfaces [65, 3, 72], see Fig. 2.6a. In this case the hydrophobic force would be a kind of capillary-bridge force; see Chapter 11 below. Such bridges could appear spontaneously, by nucleation (spontaneous dewetting), when the distance between the two surfaces becomes smaller than a certain threshold value, of the order of several hundred nanometers, see Table 11.2 below. Gaseous bridges could appear even if there is no dissolved gas in the water phase; the pressure inside a bridge can be as low as the equilibrium

Chapter 5

212

vapor pressure of water (23.8 mm Hg at 25~

owing to the high interfacial curvature of

nodoid-shaped bridges, see Chapter 11. A number of recent studies [81-88] provide evidence in support of the capillary-bridge origin of the long-range hydrophobic surface force. In particular, the observation of "steps" in the experimental data was interpreted as an indication for separate acts of bridge nucleation [87].

5.2.4.

ELECTROSTATIC SURFACE FORCE

The electrostatic (double layer) interactions across an aqueous film are due to the overlap of the double electric layers formed at two charged interfaces. The surface charge can be due to dissociation of surface ionizable groups or to the adsorption of ionic surfactants (Fig. 1.4) and polyelectrolytes [2,3]. Note however, that sometimes electrostatic repulsion is observed even between interfaces covered by adsorption monolayers of nonionic surfactants [89-92]. First, let us consider the electrostatic (double layer) interaction between two identical charged plane parallel surfaces across a solution of an electrolyte (Fig. 5.9). If the separation between the two planes is very large, the number concentration of both counterions and coions would be equal to its bulk value, no, in the middle of the film. However, at finite separation, h, between the surfaces the two electric double layers overlap and the counterion and coion concentrations in the middle of the film, t/lm and t/Zm, are not equal. As pointed out by Langmuir [93], the electrostatic disjoining pressure, Fief, can be identified with the excess osmotic pressure in the

middle of the film: I-Iel -- kT(nlm + n2m -- 2n 0 )

(5.80)

One can deduce Eq. (5.80) starting from a more general definition of disjoining pressure [2,23]: I-I = P N -- Pbulk

(5.81 )

where PN is the normal (with respect to the film surface) component of the pressure tensor P and Pbulk is the pressure in the bulk of the electrolyte solution. The condition for mechanical equilibrium, V.P = 0, yields OPN/OZ = 0, that is PN = const, across the film; the z-axis is directed

Liquid Films and Interactions between Particle and Surface

213

Z

vl

--ram

~)

~

e

e

film

"--~ - " ~

@ 0

bulk x solution

|

/

l',X,,.

/,'

vs

@

(a)

(b)

Fig. 5.9. (a) Schematic presentation of a liquid film from electrolyte solution between two identical charged surfaces; the film is equilibrated with the bulk solution. (b) Distribution ~(z) of the electric potential across the liquid film (the continuous line): ~,,, is the minimum value of ~t(z) in the middle of the film; the dashed lines show the electric potential distribution created by the respective charged surfaces in contact with a semiinfinite electrolyte solution. perpendicular to the film surfaces, Fig. 5.9a. Hence H, defined by Eq. (5.81), has a constant value for a given liquid film at a given thickness. For a liquid film from electrolyte solution one can use Eq. (1.17) to express PN :

PN = ezz = Po ( Z ) - 87[7~ dz

(5.82)

where, as usual, ~ z ) is the potential of the electric field, e is the dielectric permittivity of the solution, Po(z) is the pressure in a uniform phase, which is in chemical equilibrium with the bulk electrolyte solution and has the same composition as the film at level z. Considering the electrolyte solution as an ideal solution, and using the known expression for the osmotic pressure, we obtain Po(z) - Pbulk = k T [ n l ( z ) + n2(z) - 2n0]

(5.83)

where n~(z) and n2(z) are local concentrations of the counterions and coions inside the film. The combination of Eqs. (5.81)-(5.83) yields

214

Chapter 5

kT[nl(z)

1-Iel-"

+ r/2(z) -

2n0] - ~

(5.84)

Equation (5.84) represents a general definition for the electrostatic component of disjoining pressure, which is valid for symmetric and non-symmetric electrolytes, as well as for identical and nonidentical film surfaces. The same equation was derived by Derjaguin [44] in a different, thermodynamic manner. Note that I-Ie~, defined by Eq. (5.84), must be constant, i.e. independent of the coordinate z. To check that one can use the equations of Boltzmann and Poisson:

ni(z) = no exp[-Zie~z)/kT]

(5.85)

d2~ _ dz 2

(5.86)

~

m

D

~

47r Z Z i e n i ( z ) e i

Let us multiply Eq. (5.86) with d~t/dz, substitute n~(z) from Eq. (5.85) and integrate with respect to z; the result can be presented in the form

8~ ~,-~z ) _ kT ~"ni(z)i

= const.

(5.87)

The latter equation, along with Eq. (5.84), proves the constancy of 1-Iel across the film. If the film has identical surfaces, the electric potential has an extremum in the midplane of the film, (du//dz)z=o = 0, see Fig. 5.9b. Then from Eq. (5.87) one obtains e 8~

dN / (~,-~z ) - ~r[n~(z) + n2(z)] = - kr(nlm + n2m)

(5.88)

where nim - ni(O), i = 1,2. One can check that the substitution of Eq. (5.88) into Eq. (5.84) yields the Langmuir expression for 1-Id, that is Eq. (5.80). To obtain the dependence of 1-Id on the film thickness h, one has to first determine the dependence of nlm and n2m on h by solving the Poisson-Boltzmann equation, and then to substitute the result in the definition (5.80). This was done rigorously by Derjaguin and Landau [16], who obtained an equation in terms of elliptic integrals, see also Refs. [2, 44]. However,

Liquid Films and Interactions between Particle and Surface

215

for applications it is much more convenient to use the asymptotic form of this expression: l-Iel(h) =

C exp(-~'h)

for exp(-~h) L0~f3

(5.122)

where L0 is defined by Eq. (5.115). The first term in the right-hand side of Eq. (5.121) comes from the osmotic repulsion between the brushes; the second term is negative and accounts effectively for the decrease of the elastic energy of the initially extended chains with the decrease of the film thickness, h. The boundary between the power-law regime (/st ~ 1/h 2) and the exponential decay regime is at h - L0,4r3 -- 1.7 L0, the latter being slightly less than 2L0 which is the intuitively expected beginning of the steric overlap. In the case of a good solvent the disjoining pressure

list

=

-dfst/dh can be calculated by means

of an expression stemming from the theory by Alexander and de Gennes [ 186-188]:

ns,(h)=/~rr3/2

E//94//341 _

h

for h < 2Lg"

Lg-N(F15~/3

(5.123)

where Lg is the thickness of a brush in a good solvent [186]. The positive and the negative terms in the right-hand side of Eq. (5.123) correspond to osmotic repulsion and elastic attraction. The validity of Alexander-de Gennes theory was experimentally confirmed by Taunton et al. [ 189] who measured the forces between two brush layers grafted on the surfaces of two crossed mica cylinders, see also Ref. [3]. Theoretical expressions, which are applicable to the case when intersegment attraction is present (the solvent is poor, see Fig. 5.17) are reviewed by Russel et al. [42].

5.2.9.

UNDULATIONAND PROTRUSION FORCES

Adsorption monolayers at fluid interfaces and bilayers of amphiphilic molecules in solution (phospholipid membranes, surfactant lamellas) are involved in a fluctuation wave motion. The configurational confinement of such thermally exited modes within the narrow space between two approaching interfaces gives rise to short-range repulsive surface forces, called fluctuation

forces, which are briefly presented below.

Chapter 5

236

-h

(a)

h1% c 6Q6~176 oQ c-x c?Qc x- ?c QoQc %

(b)

Fig. 5.18. Fluctuation wave forces due to configurational confinement of thermally excited modes into a thin liquid film. (a) The undulation force is related to the bending mode of membrane fluctuations. (b) The protrusion force is caused by the spatial overlap of protrusions of adsorbed amphiphilic molecules.

Undulation force. The undulation force arises from the configurational confinement related to the bending mode of deformation of two fluid bilayers, like surfactant lamellas or lipid membranes.

This mode consists in undulation of the bilayer at constant area and

thickness, Fig. 5.18a. Helfrich et al. [ 191,192] established that two such undulated "tensionfree" bilayers, separated at a mean surface-to-surface distance h, experience a repulsive disjoining pressure: Hund(h ) = 3zc2(kT)2

(5.124)

64kth 3 Here kt is the total bending elastic modulus of the bilayer as a whole; the experiment shows that

kt is of the order of 10-19 J for lipid bilayers [193]. The undulation force was measured and the dependence FIund o~ h -3 was confirmed experimentally [ 194-196]. In lamellar phases present in concentrated solutions of nonionic amphiphiles the undulation repulsion opposes the van der Waals attraction thus producing a stabilizing effect [ 197-199].

Protrusion force. The protrusion of an amphiphilic molecule from an adsorption monolayer (or micelle) may fluctuate about the equilibrium position of the molecule owing to the thermal motion, Fig. 5.18b.

In other words, the adsorbed molecules are involved in a

discrete wave motion, which differs from the continuous mode of deformation related to the

Liquid Films and Interactions between Particle and Surface

237

undulation force. The molecular protrusions from lipid membranes and adsorption monolayers have been detected by means of NMR, neutron diffraction and X-ray synchrotron diffraction [200,201]. In relation to the micelle kinetics, Aniansson et al. [202,203] found that the energy of protrusion of an amphiphilic molecule can be modeled as a linear function: u(z) = ~ z, where z is the distance out of the surface (z > 0); they determined o~= 3 • 10-11 J/m for single-chained surfactants. By using a mean-field approach Israelachvili and Wennerstr6m [99] derived an expression for the protrusion disjoining pressure which appears when two protrusion zones overlap (Fig. 5.18b): Flprotr(h)_ r a : r

(h/,a,)exp(-h/&) l (l + h /,a,)exp(- h /

.

x - kTo~

has the meaning of characteristic protrusion decay length; 2, = 0.14 nm at 25~

(5.125)

F denotes the

number of protrusion sites per unit area. I-Iprotr is positive and corresponds to repulsion; it decays exponentially for h >> 24 in the other limit, h ~) one obtains R = 2R. In general, the total force acting on the particle, F, can be expressed as a sum of some external driving force, Fe, and the surface force Fs:

F = F e - Fs,

dU(h) Fs - - ~ dh

(6.21)

where U is defined by Eq. (5.53). The opposite signs of FE and Fs stem from the convention, that the "external" force FE pushes the particle toward the interface (the other particle), whereas a repulsive "surface" force, Fs > 0, opposes the thinning of the gap. (Attractive surface force,

Fs < 0, is also possible.) The external force FE can be the gravitational, buoyancy or Brownian force. The time of mutual approach of two particles (the drainage time of a liquid film) is [ 18]

h! dh r,~-

V(h)

where V denotes velocity and

(6.22)

hA is some initial value of the surface-to-surface distance. For

260

Chapter 6

constant F, the substitution of

h for V(h) in Eq. (6.22) yields ~'a ~ 0% i.e. infinitely long

gTa or

time is needed for the two surfaces to come into direct contact. On the other hand, if the force at short distances is dominated by the van der Waals interaction, then in view of Eqs. (5.61) and (6.21) F ~ Fs or 1/h 2,

VTa oc

1/h, and Eq. (6.22) gives a finite value for the time of

approach ~'a.

6.2.2.

INVERSION THICKNESS FOR FLUID PARTICLES

Two fluid particles (drops, bubbles) approaching each other are initially spherical. With the decrease of the distance between them, the interfacial shape in the gap changes from convex to concave. The thickness corresponding to this inversion of the sign of the interfacial curvature is called the inversion thickness, hinv. From a physical viewpoint this is the beginning of the deformation of the droplets (bubbles) in the contact zone, with subsequent formation of a thin film between them (see Fig. 5.19). One can estimate the inversion thickness from the following expression [ 18, 25, 26] F hinv -- ~ , 2/r~-

(6.23)

where ~- is related to the interfacial tensions of the two fluid particles, 05 and 0-2, by means of Eq. (6.12). If one of the particles is solid (0-1 ~ 0% 0-2 = 0-), then ~ - - 20-. Equation (6.23) is valid for relatively large surface-to-surface distances between the two drops, for which the surface forces can be neglected (F -- Fe). A generalization of Eq. (6.23), taking into account the effects of the surface forces and the particle size, was reported in Ref. [27]: F R hinv = ~ + ~hi,vFI(hinv)" 2~r~- 2~-

(6.24)

as usual, 1-I(h) is disjoining pressure. In general, Eq. (6.24) holds for two dissimilar droplets of radii R~ and R2, and surface tensions 0-j and 0-2" see Eqs. (6.12) and (6.17). One can determine hinv by numerical solution of Eq. (6.24) if the dependencies 1-I(h) and F(h) are given, see e.g. Ref. [51.

Particles at Interfaces: Deformations and Hydrodynamic Interactions

6.2.3.

261

REYNOLDSREGIME OF PARTICLEAPPROACH

For h < hinv a liquid film is formed in the zone of contact of the two surfaces (Fig. 5.19). The viscous dissipation of energy in this film is strong enough to dominate the net hydrodynamic force. In such case the rate of approach of two fluid particles obeys the Reynolds formula, which describes the rate of thinning of a planar film between two solid discs [28, 23]: 2h3F

VRe - 3rot/r4

(6.25)

h is the distance between the discs (the film thickness), rc is the radius of the disc (film) radius. In the case of fluid particles rc can be estimated from Eqs. (6.14)-(6.17). Since VRe = dh/dt, by integration of Eq. (6.25) one can deduce an expression for the time needed to bring two parallel discs (the two film surfaces) from an initial separation h~ to a final separation h2 under the action of a constant force F:

t-

3:r/:r/r4 ( 1 4F h 22

13 h(

(6.26)

The latter equation was derived by Stefan [29] in 1874. One can combine Eqs. (6.25) and (6.17) to obtain [5]: 8~-2h 3 VRe = 3r/~_ZF

(6.27)

It is interesting to note, that in Reynolds regime (in which there is flattening and Eq. 6.27 holds) the velocity VRe decreases with the rise of the driving force F. This tendency is exactly the opposite to that for the particle motion in Stokes or Taylor regimes, cf. Eqs. (6.18) and (6.20). The latter fact leads to a non-monotonic dependence of the droplet life-time, 7:,, on the drop radius R; see Fig. 6.6 below.

6.2.4.

TRANSITIONFROM TAYLOR TO REYNOLDS REGIME

It is possible to describe smoothly the transition from Taylor to Reynolds regime, i.e. the transition from spherical to deformed fluid particles. The following generalized expression was derived in Ref. [30]:

Chapter 6

262

1 + hR + h 2~2

F = -23TcrlV--s

(6.28)

where R is defined by Eq. (6.17). For small film radii, rc---)0, Eq. (6.28) reduces to the Taylor's Eq. (6.20), whereas for large films, rc2/(h-R) >> 1, Eq. (6.28) yields the Reynolds' Eq. (6.25). Expressing the velocity from Eq. (6.28) one obtains [5] 1

1

1

1

- - = ~ + - ~--g gTa 4VTaVRe gRe

(6.29)

To calculate the life time of a doublet from two emulsion drops moving towards each other under the action of a constant force F one can use the expression [5]

Ta --

JV(h) her

= ~

,,

2F

ln~+

her

_

herR

/ / r4 / cr/l 1--

-~A

+

2--'~2hc,:R

1--

(6.30)

hA

which is derived by integration of Eq. (6.28)" her denotes the critical thickness of rupture of the liquid film; as before, hA is an initial thickness of the film. In the case of coalescence of an oil drop with its homophase (oil drop below a fiat oil-water interface, see Fig. 5.19a) one has R = 2R, where R is the radius of the drop, which experiences 3g a p , with g and Ap being the gravity acceleration and the density a buoyancy force F b -- _.47fR 3 difference. Setting F -- Fb, and combining Eqs. (6.16) and (6.30), one can calculate the dependence ~:a - Ta(R) if an estimate for the critical thickness, her, is available" see Eq. (6.36) below. The calculations show that the curves of Ta vs. R should exhibit a minimum in the region R = 1 0 - 200 gin. To check the predictions of the theory experiments with soybean oil droplets in aqueous solution of the protein bovine serum albumin (BSA) have been carried out by Basheva et al. [31]. The oil drops of various size have been released by means of a syringe in the aqueous solution; then the drops move upwards under the action of the buoyancy force and approach a horizontal oil-water interface. The life-time ~'a of the drops beneath the interface was measured as a function of the drop radius, R. The data are presented in Fig. 6.6. The theoretical curve is calculated by means of Eqs. (6.16) and (6.30). For all drops ha = 15 lktm was used.

Particles at Interfaces: Deformations and Hydrodynamic Interactions

160

_

263

{

140 120 . ( -l=,l

~

100

_

I ~

80

_

60

_

-1='4

~

~'~ 0 ~

40 2o

. ', _

o

1

_

I

0

~

I

100

~

I

200

~

I

300

,

I

400

,

I

500

,

I

600

Droplet radius

~

I

,

700

I

800

,

I

900

,

I

1000

,

I

1100

(t~m)

Fig. 6.6. Life time % plotted versus the radius, R, of oil-in-water drops approaching from below the water-oil interface. The circles are experimental points for aqueous solutions of bovine serum albumin (BSA) with 0.15 M NaC1; the oil phase is soybean oil [31]. The theoretical curve is drawn by means of Eqs. (6.16) and (6.30). The arbitrariness of this choice does not affect substantially the results for %. The critical thickness, her, was calculated by means of Eq. (24) in Ref. [5] assuming predominant van der Waals forces in the film. One sees in Fig. 6.6 that the theory agrees well with the experiment. The left branch of the curve corresponds to the Taylor regime (non-deformed droplets), whereas the right branch corresponds to the Reynolds regime (planar film between the droplets)" for details see Refs. [5, 31 ].

6.2.5.

FLUID PARTICLES OF COMPLETELY MOBILE SURFACES (NO SURFACTANTS)

If the surface of an emulsion droplet is mobile, it can transmit the motion of the outer fluid to the fluid within the droplet. This leads to a circulation pattern of the inner fluid and affects the dissipation of energy in the system. The problem about the approach of two nondeformed

(spherical) drops or bubbles in the absence of surfactants has been investigated by many authors [32-41] and a number of solutions, generalizing the Taylor equation (6.20), have been obtained. For example, the velocity of central approach of two spherical drops in pure liquid, Vp, is related to the total force, F, by means of a Pad6-type expression derived by Davis et al.

[40]

264

Chapter 6

1+1.711~ +0.461~ 2 - VTa

1 + 0.402 ~

rlout [R'

~ -

Flin ~/ 2h

(6.31)

where, as usual, h is the closest surface-to-surface distance between the two drops; Fli, and Flou, are the viscosities of the liquids inside and outside the droplets. In the limiting case of solid particles one has Flin-----)c,oand Eq. (6.31) reduces to the Taylor equation, Eq. (6.20). Note that in the case of close approach of two drops (h---~0 and ~ >> 1) the velocity Vp is proportional to ~/-h. Consequently, the integral in Eq. (6.22) is convergent and the two drops can come into contact (h = 0) in a finite period of time (~'~< oo) under the action of constant force F. In contrast, in the case of immobile interface (Flin-->ooand ~ ~ofor F = const. In the limiting case of two spherical gas bubbles (Flin---->0) in pure liquid, Eq. (6.31) cannot be used; instead, Vp can be calculated from the expression due to Beshkov et al. [37] F Vp = 2~Flout~_ln(~_/h)

(6.32)

Note that in this case Vp ~ (lnh) -~ and the integral in Eq. (6.22) is convergent, that is the hydrodynamic theory predicts a finite lifetime of a doublet of two colliding spherical bubbles in pure liquid. Of course, the real lifetime of a doublet of bubbles or drops is affected by the surface forces for h < 100 rim, which should be accounted for in F, see Eq. (6.21); this may lead to the formation of a thin film in the zone of contact, as discussed above.

6.2.6.

FLUID PARTICLES WITH PARTIALLY MOBILE SURFACES (SURFACTANT IN CONTINUOUS PHASE)

The presence of surfactant in the continuous phase and at the surface of fluid particles decreases their surface mobility. This is due mostly to the effect of Gibbs elasticity, Ec;, which leads to the appearance of surface tension gradients (Marangoni effect). The latter oppose the viscous stresses due to the hydrodynamic flow and suppress the two-dimensional flow throughout the phase boundary. In the limit Ec,---~0 the interface becomes tangentially immobile. When the effect of the driving force F is small compared to that of the capillary pressure of the droplets/bubbles, the deformation of the two spherical fluid particles upon collision is only a small perturbation in the zone of contact. Then the film thickness and the

Particles at Interfaces: Deformations and Hydrodynamic Interactions

265

pressure within the gap can be presented as a sum of a non-perturbed part and a small perturbation. Solving the resulting linearized hydrodynamic problem for negligible interfacial viscosity, an analytical formula for the velocity of approach was derived by Ivanov et al. [ 16]: V _ h s d___llln(d + 1)-1 gTa 2h

]1

(6.33)

where, as usual, VTa is the Taylor velocity given by Eq. (6.20); the dimensionless parameter d and the characteristic surface diffusion thickness h s are defined as follows

d - h(1 + b) '

6outs

h,. - ~ , E G

b -

out (O }e 3 out / }e EG

~

q

=

r

~

q

(6.34)

and D denotes the bulk diffusivity of the surfactant (dissolved in the continuous phase); D, is its surface diffusivity; as before, cr and EG are the surface tension and surface (Gibbs) elasticity, c and F are surfactant concentration and adsorption; the subscript "eq" denotes equilibrium values. In the limiting case of very large EG (tangentially immobile interface) the parameter d tends to zero and one can verify that Eq. (6.33) predicts V ~ VTa, as it should be expected. Equation (6.33) is applicable when the surfactant is dissolved in the continuous phase. In contrast, if the surfactant is dissolved in the emulsion-drop phase, it can efficiently saturate the drop surface and to suppress the effect of surface elasticity [42, 43]. In such case, the drop surface behaves as almost completely mobile and one could apply Eq. (6.31) to estimate the velocity of approach [5]. The relative solubility of the surfactant in the water and oil phases is characterized by the hydrophile-lipophile balance (HLB) - see the book by Krugljakov [44].

6.2.7.

CRITICAL THICKNESS OF A LIQUID FILM

The surface of a fluid particle is corrugated by capillary waves due to thermal fluctuations or other perturbations. The interfacial shape can be expressed mathematically as a superposition of Fourier components with different wave numbers and amplitudes. If attractive disjoining pressure is present, it enhances the amplitude of corrugations in the zone of contact of two droplets (Fig. 5.19) [45-48]. For e v e r y Fourier component there is a film thickness, called transitional thickness, htr, at which the r e s p e c t i v e surface fluctuation becomes unstable and this surface corrugation begins to grow spontaneously [18, 26]. For htr > h > her the film continues

Chapter6

266

to thin, while the instabilities grow, until the film ruptures at a certain critical thickness h = hcr. The transitional thickness of the film between two deformed drops (Fig. 5.19b) can be computed solving the following transcendental equation [5, 27]" 2+d htr t'? [I-/'(htr )] 2 1 +---ff= 8~-[2~/R-- 1--[(htr)]' As before,

rc denotes

H,

0n - ~9h

(6.35)

the radius of the film formed between the two fluid particles. The effect

of surface mobility is characterized by the parameter d, see Eq. (6.34); note that d depends on htr, viz. d -

(hs/htr)/(1 + b);

for tangentially immobile interfaces h,--~0 and hence d-->0. In

addition, Eq. (6.35) shows that the disjoining pressure significantly influences the transitional thickness htr; this equation is valid for FI < 2 ~ - / R , i.e. for a film which thins and ruptures before reaching its equilibrium thickness, corresponding to H = P c - 2 ~ - / R " see. Eq. (5.1). The calculation of the transitional thickness htr is a prerequisite for computing the critical thickness hcr. For the case of two

identical

attached fluid particles of surface tension o" and

radius R (Fig. 5.19b) the critical thickness can be obtained as a solution of the equation [48, 49]

__ 2kT exp II(hcr,htr ) ] hZr - l(hcr,htr ) -4~ where

(6.36)

I(htr,hcr) stands for the following function I(hcr,htr ) -

Here ~v is a

dh hc ~(h)[2cr/R-l-I(h)] htr

I-I t

lrI'(htr )r 2 f

mobility factor

(6.37)

accounting for the tangential mobility of the surface of the fluid

particle; expressions for ~v can be found in Ref. [22]. In the special case of tangentially immobile interfaces and large film (negligible effect of the transition zone) one has ~v(h) - 1" then the integration in Eq. (6.37) can be carried out analytically [48, 49]:

I(hcr' h t r ) -

1--['(htr)r 2 lnI2~y/R-H(hcr) 1 2/ R-~ 1-Iy (htr)

(6.38)

Equations (6.35)-(6.38) hold for an emulsion film formed between two attached liquid drops, and for a foam film intervening between two gas bubbles. In Fig. 6.7 we compare the prediction of Eqs. (6.35)-(6.38) with experimental data for her vs. r,., obtained by Manev et al. [50] for free foam films formed from aqueous solution of 0.43 mM SDS + O. 1 M NaCI. It turns

Particles at Interfaces." Deformations and Hydrodynamic Interactions

267

48 46 44 =

42 40

9 . ,,.,.~

..c::

.,-.~ -9 L)

38 36 34 32 30 28

26 I

~

I

~

5

6

7

8

i

I

,

,

,

1 0 -1

,

i

1.5

,

i

=,

i

,

i

2

~

, i , , , , i

2.5

,

3

i

,

I

T

4

5

Film radius, rc (mm)

Fig. 6.7. Critical thickness, hcr, vs. radius, rc, of a foam film formed from aqueous solution of 0.43 mM SDS + 0.1 M NaCI: comparison between experimental points, measured by Manev et al. [50], with the theoretical model based on Eqs. (6.35)-(6.38) - the solid line; no adjustable parameters. The dot-dashed line shows the best fit obtained using the simplifying assumptions that hcr = htr and that the electromagnetic retardation effect is negligible. out that for this system the solution-air surface behaves as tangentially immobile, and then O v - 1, see Ref. [22]. The disjoining pressure was attributed to the van der Waals attraction" II =-AH/(6rth3), where AH was calculated with the help of Eq. (5.75) to take into account the electromagnetic retardation effect. The solid line in Fig. 6.7 was calculated by means of Eqs. (6.35)-(6.38) without using any adjustable parameters; one sees that there is an excellent agreement between this theoretical model and the experiment [22]. The dot-dashed line in Fig. 6.7 shows the best fit obtained if the retardation effect is neglected (AH = const.) and if the critical thickness is approximately identified with the transitional thickness (hcr ~ htr), cf. Ref. [51]. The difference between the two fits shows that the latter two effects are essential and should not be neglected. In particular, the retardation effect turns out to be important in the experimental range of critical thicknesses, which is 25 n m < hcr < 50 nm in this specific case.

268

6.3.

Chapter 6

DETACHMENT OF OIL DROPS FROM A SOLID SURFACE

The subject of this section is the detachment of oil drops from a solid substrate by mechanical and physicochemical factors, such as shear flow in the adjacent aqueous phase and modification of the interfaces due to adsorption of surfactants. These processes have practical importance for enhanced oil recovery [52,53], detergency [54] and membrane emulsification [55-57]. Analogous experiments on deformation and detachment in shear flow have been carried our to explore the mechanical properties of biological cells and their adhesion to substrates [58, 59]. Despite its importance, the drop detachment has been investigated only in few studies. Our purpose here is to briefly review the available works, to systematize and discuss the accumulated information and to indicate some non-resolved research problems.

6.3.1.

DETACHMENT OF DROPS EXPOSED TO SHEAR FLOW

The detachment of solid colloidal particles from a flat surface (substrate) is studied better than the analogous problem for liquid drops. Hydrodynamic flows normal and parallel to the substrate were considered. The incipient motion of a detaching particle can be described as a superposition of three modes: sliding, rolling and lifting. Expressions for the hydrodynamic force and torque acting on an attached spherical particle were derived. The comparison of the computed and experimentally measured critical hydrodynamic force for particle release show a good agreement, indicating that the essential physics of the problem has been captured in the model; for details see the studies by Hubbe [60], Sharma et al. [61], and the literature cited therein. What concerns the more complicated problem about the detachment of liquid drops from substrates, specific theoretical difficulties arise from the deformability of the drops and from the boundary conditions at the three-phase contact line. Technologically motivated studies [62, 63] established linkages between the value of the interfacial tension and the removal of oil drops. Thompson [54] examined experimentally the effects of the oil-water interfacial tension and the three-phase contact angle on the efficiency of washing of fabrics; in that study the mechanism of oil detachment was not directly observed.

Particles at Interfaces: Deformations and Hydrodynamic Interactions

269

Mahd et al. [64-66] investigated experimentally the detachment of alkane drops from a glass substrate by shear flow in the aqueous phase. According to them, a liquid drop detaches when the exerted hydrodynamic drag equals the maximum retentive capillary force (the integral of the oil-water surface tension along the contact line) [64]. The hydrodynamic drag force, Fn, was estimated by means of a formula due to Goldman et al. [67]: Fn ~: 7"/~R=

(6.39)

where 7/is the viscosity of the continuous (water) phase; R is the radius of the oil droplet; ~-Ovx/Oz characterizes the rate of the applied shear flow (the x and z axes are oriented, respectively, tangential and normal to the substrate). On the other hand, the adhesion force FA has been evaluated by means of a formula derived by Dussan and Chow [68]: FA = o'L(coSOA - cOSOR)

(6.40)

where L is the width of the drop, 0a and OR are the advancing and receding contact angles (see Fig. 6.9 below)" as usual, o" is the interfacial tension. According to Mahd et al., the critical shear rate, ~)c, corresponds to FH = FA

(integral criterion for drop detachment)

(6.41)

Equating (6.39) and (6.40) and setting L ~ rc one obtains [64] ~)c R2 ~ o- re (COS0A-- COS0R) 77

(6.42)

As usual, rc is the radius of the contact line, see Fig. 5.19a. Experimental plots of 7c R2 vs. rc showed a good linear dependence [64, 66], as predicted by Eq. (6.42). This theoretical modeling seems adequate; note however, that it has not yet been proven whether or not the slopes of the experimental straight lines are proportional to O'(COS0A-- COS0R)/r/. For the time being, the "integral" criterion for drop detachment, Eq. (6.41), is a hypothesis, whose validity needs additional experimental proofs. There is neither detailed theoretical model, nor systematic experimental data about the detachment of oil drops in tangential shear flow (note that the studies by Mah6 et al. are focused mostly on attachment, rather than on detachment, of drops). Moreover, there could be an alternative "local" criterion for detachment

Chapter 6

270 EMULSIFICATION MECHANISM (Destabilization of the Oil-Water Interface)

Water ~ 0 ~

Water

-,/./-..//(a)

/

"///../r,/ (b)

(c)

Fig. 6.8. Scheme of the emulsification mechanism of oil-drop detachment by a shear flow. (a) An oil drop attached to the boundary water-solid. (b) If shear flow is present in the water phase, the hydrodynamic drag force deforms the drop, which could acquire unstable shape and (c)could be split on two parts: residual and emulsion drop, the latter being drawn by the flow away. of the drop (related to a local violation of the Young equation), which is discussed below. Basu et al. [69] described theoretically the sliding of an oil drop along a solid surface in shear flow. This is a special pattern of motion of an already detached drop; however the mechanism and criteria of detachment have not been investigated in Ref. [69]. It should be noted that from a theoretical viewpoint the drop detachment from a solid substrate resembles the hydrodynamic problem for sliding of a liquid drop down an inclined plate [68, 70-73]. Another, related problem is the detachment of emulsion drops from the orifices of pores; this is a central issue in the method of emulsification by means of microporous glass and ceramic membranes, which has found various practical applications [55-57].

Hydrodynamic mechanisms of drop detachment. Based on the preceding studies one may conclude that two major hydrodynamic mechanisms for detachment of a liquid drop from a solid substrate by a shear flow can be distinguished [54]" (a) Emulsification mechanism due to destabilization of the oil-water interface; (b) Rolling-up mechanism related to destabilization of the three-phase contact line.

(a) The emulsification mechanism (Fig. 6.8) involves a deformation of the attached oil drop by the shear flow until a unstable configuration is reached. Then the oil drop splits into an

Particles at Interfaces: Deformationsand Hydrodynamic Interactions

271

emulsion drop convected by the shear flow, and a residual drop, which remains attached to the substrate. Lower oil-water interfacial tension and greater contact angle (measured across the oil phase) are found to facilitate the drop detachment by emulsification. At our best knowledge, the emulsification mechanism, termed also the "necking and drawing" mechanism, was first explicitly formulated by Dillan et al. [62].

(b) The rolling-up mechanism, as a disbalance of the interfacial tensions acting at the three-phase contact line, was proposed by Adam [74] long ago. This mechanism is related to the notion of advancing and receding contact angle. Let 0 be the contact angle measured across the oil. If oil is added to a quiescent oil drop, its volume and contact angle increase until a threshold value, the static advancing angle 0 = 0A, is reached (Fig. 6.9a). Then the contact line begins to expand and the oil spreads over the solid; usually the dynamic advancing angle, 0(Ad) , is smaller than the threshold static advancing angle, 0a. In this aspect, there is an analogy with

static friction (body dragged over a surface). Moreover, some theoretical studies attribute the hysteresis of contact angle to static friction [71,72]. Likewise, if oil is sucked out from a quiescent oil drop, its volume and contact angle decrease until a threshold value 0 = OR, the static receding angle, is reached (Fig. 6.9b). Then the contact line begins to shrink" usually the dynamic receding angle, 0(Rd~ , is larger than the threshold static receding angle, OR; again there is an analogy with static friction. The hysteresis of the contact angle consists in the fact that for quiescent drops OR < 0 < 0a.

Receding drop

Advancing drop

a)

Water~

b)

Water

Fig. 6.9. (a) The static advancing angle 0A is the threshold value of the contact angle just before the advance of the contact line. (b) The static receding angle OR is the threshold value of the contact angle just before the receding of the contact line.

Chapter 6

272

Static drop on inclined plane

01~0~02 OR < 01 < 02 < 0 A

Hysteresis of contact angle (equivalent to static friction)

Fig. 6.10 An immobile liquid drop over an inclined plate. A liquid drop is able to rest over an inclined plate owing to the fact that the contact angle can vary along the contact line [70]; in general, 01 < 0 < 02, see Fig. 6.10. The necessary condition the contact line to be immobile is OR < 01 < 02 < 0A. Similarly, if a liquid drop is exposed to a shear flow (Fig. 6.11a), the contact line will be immobile if OR < 01 0A at the leeward side of the drop, Fig. 6.1 lb, the contact line will advance in this zone and the oil-wet area will increase, i.e. the shear will produce a spreading of the oil drop (rather than detachment). If 0A ~ 180 ~ then the contact line at the leeward zone remains immobile, but the deformed oil drop could form a water film in this zone, Fig. 6.1 lc. Such events have been observed by Mah6 et al. [64]. When the magnitude of the shear increases, the contact angle 01 at the stream-ward edge of the drop decreases. At the instant when 0~ = OR the contact line in this zone begins to recede and the oil-wet area decreases (Fig. 6.1 l d). Further, two scenarios are possible: (A) Progressive shrinkage of the oil-wet area until full detachment of the oil drop; this has been observed by Mah6 et al. [64]. (B) During the shrinkage of the oil-wet area the contact line 01 could become again greater than OR, and the shrinking of the oil-wet area ceases. Further, oil-drop detachment is possible at higher shear rate by means of the emulsification mechanism, i.e. with the appearance of a

Particles at Interfaces." Deformations and Hydrodynamic Interactions

273

ROLLING-UP MECHANISM (Destabilization of a Three-Phase C o n t a c t Line)

a) The oil-water interface is stable ~

Water strcamward ~

/

leeward

,

,

I I

J I

/;")

b) 0 2 > 0 A =:~ Spreading without detachment Water

,

/7

/;, ..1//" / ,

~, I

1, / . . / . , / , , , I

"/I

I I

C) 0 A ~

/7"

" '''y

"

'/

I I

180 ~ ~ Formation o f water film without detachment

~

/

Water

.I I

I I

d) For 01 < OR =} D e t a c h m e| n t of the contact line and rolling-up o f the drop

I v/

//. 2

1

~o~n~-~,

/,/, .." /,. ,2

KEY: 01 < OR is a sufficient condition for rolling-up

Fig. 6. l 1. (a) For O1 > ORand 02 < OAthe flow cannot cause motion of the contact line. (b) For 02 > OA the contact line advances at the leeward side and the oil-wet area increases. (c) For Oa~ 180~ the deformation of the drop leads to the formation of a water film at the leeward side. (d) For Oi < ORthe contact line at the leeward side recedes and the oil-wet area decreases.

Chapter 6

274

residual drop, see the photographs in Fig. 6.12. In other words, this is a mixed mechanism of drop detachment.

Discussion. Coming back to the mechanisms for destabilization of an attached oil drop, we can summarize their features in the following way: (i) Emulsification mechanism: Unstable shape (necking) of the oil drop in the shear flow, see Fig. 6.8 and 6.12. (ii) Rolling up mechanism with an "integral" criterion for the onset of drop detachment, Eq. (6.41): The total hydrodynamic drag force exerted on the oil drop becomes greater than the retentive capillary force [64]. In other words, this is a violation of the integral balance of forces acting on the drop. (iii) Rolling up mechanism with a local criterion for the onset of drop detachment: The contact angle at the stream-ward side becomes smaller than the threshold receding angle, 01 < OR

(local criterion for drop detachment)

(6.43)

Thus the contact line begins to recede, the oil-wet area decreases, and eventually the drop detaches (Fig. 6.11 d). In other words, this is a violation of the local balance of forces acting per unit length on the contact line at the stream-ward side. Intuitively, one may expect that in some cases the criterion (iii) could be satisfied for lower shear rates, as compared to criterion (ii). It is necessary to verify, both theoretically and experimentally, which is the real mechanism of drop detachment, (i), (ii), (iii) or a combination of them. It may happen that for different systems different mechanisms are operative. As an illustration, in Fig. 6.12 we present consecutive video-frames of the detachment of an oil drop in shear flow; photos taken by Marinov [75]. The water phase is a 0.5 mM solution of sodium dodecyl sulfate (SDS) + 50 mM NaC1. The oil drop is from triolein, a triglyceride which is completely insoluble in the surfactant solution. The oil-water interfacial tension is o ' 20 mN/m. The substrate is a glass plate, representing the bottom of the experimental channel. The latter has height Hc = 5 mm and width Wc = 6 mm; the height of the oil drop is mm. For this geometry the Reynolds number can be estimated as follows

Hd =

1.7

Particles

at Interfaces:

Deformations

and Hydrodynamic

275

Interactions

i~ ~!~ !i!~!!i!i!i!!!i~i~i~i~i~i!i !! i I i 84~i 84 ~!i ~~iii~ Q = 0 cm3/s

Q = 0.88 cm3/s

h ,

,,

,

Q - 1.64 cm3/s

Q = 1.61 cm3/s ,,i!g:,::I,.~.i,,., i . ~ , ~ : . , . ~

",

)~,i'~".,~" , ' ' i ~ ~~~i~",,~2:!";;

,

,

I

Qc~ = 1.76 cm3/s (detachment- frame # 1)

(detachment- frame # 2) ,,

.

.

.

.

.

.

.

.

.

.

.

,,,,

,,

,,

.

(detachment- frame # 3)

(detachment- frame # 4)

Fig. 6.12. Consecutive stages of detachment of a triolein drop exposed to shear flow. The water phase is a solution of 0.5 mM SDS with 50 mM NaC1 at 25~ o'-- 20 mN/m. Each photo corresponds to a given rate of water delivery Q. The first four frames show steady state configurations, whereas the last four frames, taken at the same Q = Qcr, show stages of the drop detachment (Recr = 112) [75].

276

Chapter 6

R e = pwQHd r/wWcHc

(6.44)

where Pw and r/w are the mass density and the dynamic viscosity of water; Q (cm3/s) is the rate of water delivery in the channel. In the absence of hydrodynamic flow (Q = 0) the oil-water interface is spherical. The videoframes in Fig. 6.12 show the variation of the drop shape with the increase of Q. The photos taken at Q = 1.61 and 1.64 cm3/s show that the contact line on the stream-ward side has moved and the area wet by oil has shrunk; however, the drop configuration is still stationary (no detachment occurs). The detachment happens at a critical value Qcr = 1.76 cm3/s; at this rate of water delivery the oil-water interface becomes unstable, necking is observed and eventually a residual drop remains on the substrate; see the last four photos in Fig. 6.12, all of them taken at Q = Qcr. Hence, in this experimental system the final stage of drop detachment follows the

emulsification mechanism. The critical value of the Reynolds number, estimated by means of Eq. (6.44) for 71w/Pw= 0.89 x 10- 2 cmZ/s at temperature 25~ 6.3.2.

is Rec~ -- 112.

DETACHMENT OF OIL DROPS PROTRUDING FROM PORES

If an oil drop is located at the orifice of a pore, there is a strong hysteresis of the contact angle. The experimental video-frames shown in Figs. 6.13 and 6.14 show two mechanisms of detachment of oil drops exposed to shear flow. Note that during these experiments the volume of the oil drops has been fixed (no supply of additional oil through the orifice).

Hydrophobic orifice of the pore. To mimic such pore we used a glass capillary with hydrophobic inner wall and inner diameter 0.6 mm, Fig. 6.13. The aqueous and oil phases, and the temperature are the same as in Fig. 6.12. When carrying out the experiments special measures have been taken to prevent an entry of the surfactant solution in the capillary, which would cause hydrophilization of its inner wall. The first three photos in Fig. 6.13 show stationary configurations of the drop corresponding to increasing values of the rate of water supply Q. The last three frames, taken at the same Q = Qcr, represent consecutive stages of the drop detachment, which again follows the emulsification mechanism. The height and width of the channel are Hc = 3 mm and Wc = 5 ram; the height of the oil drop is Ha-- 1.3 ram. From Eq. (6.44) with Q~r- 1.39 cm3/s we estimate Re~-~ 135.

Particles at Interfaces." Deformations and Hydrodynamic Interactions

277

flow

Q = 0 cm3/s

Q = 0.59 cm3/s . . . . . .

~

. . . . .

Q = 1.17 cm3/s

Qcr= 1.39 cm3/s (detachment- frame # 1)

(detachment- frame # 2)

(detachment- frame # 3)

..

Fig. 6.13. Oil drop at the tip of a glass capillary with hydrophobized orifice of inner diameter 0.6 ram: consecutive stages of drop detachment due to applied shear flow. The drop has a fixed volume. The aqueous and oil phases are as in Fig. 6.12. The first three frames show stationary configurations at three fixed rates of water delivery, Q. The last three frames, taken at the same Q = Qcr, show stages of the drop detachment (Reef = 135) [75].

278

Chapter

6

flow

!

,

,

,

Q = 0 cm3/s

'

,

,

Q = 0.88 cmB/s

.... ...................

Qcr = 1.05 cm3/s (detachment- frame # 1)

...............:!!

(detachment- frame # 2)

, '

"'i

,

;'"'

:""i~

,

'

'

'

'

,

.~.,..,,

'

4 ;i;"

""~,~

. . . .

,;,, ., .

,,,,:

:,:, ~.,. ,, . . ~ , :.,., : ,:~, ~ :, s: ,,~ . i"27,:';o

%

,, :v, ~ff,,

i77~~i,' .' ~ t; ,5;gL,,

( d e t a c h m e n t - frame # 3)

: :'.,

(detachment - I)ame # 41)

Fig. 6.14. Oil drop at the tip of a glass capillary with h y d r o p h i l i T e d orifice of inner diameter 0.6 mm: consecutive stages of drop detachment due to applied .,,hear flov~. The oily and aqueous phases are the same as in Figs. 6.12 and 6.13, with the only difference that the concentration of SDS is 20 times higher; the interfacial tension is o-= 5 mN/m, The first two frames show stationary configurations at two fixed rates of water delivery, Q. The last four fi'ames, taken at the same Q = Qc~, show stages of the drop detachment (Re,, = 42) [7 5_i.

Particles at Interfaces: Deformations and Hydrodynamic Interactions

279

Hydrophilic orifice of the pore. Figure 6.14 shows consecutive video-frames of the detachment of an oil drop protruding from a capillary with hydrophilized orifice. To achieve hydrophobization, first aqueous surfactant solution was let to fill the upper part of the capillary, where its inner wall was hydrophilized owing to the adsorption of surfactant. Next, some amount of oil was supplied to form a protruding oil drop; simultaneously, a water film, sandwiched between oil and glass, was formed in the hydrophilized zone. This water film essentially facilitates the detachment of the oil drop by the shear flow, see Figs. 6.14 and 6.15. The protruding drop is not attached to the solid edge. At higher shear rates, the drop, deformed by the flow, is cut at the edge of the capillary; we could call this the "edge-cut" mechanism. In Fig. 6.14 the height and width of the channel are Hc = 2 mm and Wc = 12.5 mm; the height of the oil drop is Hd -- 0.9 mm. From Eq. (6.44) with Qcr = 1.05 cm3/s we estimate Recr = 42 (compare the latter value with Recr-- 135 for the hydrophobic capillary). We may conclude that the hydrophilization essentially facilitates the detachment of an oil drop protruding from an orifice.

EDGE-CUT MECHANISM

a) Water ~ k ""

water ~/film

b) F

Water

d)

Water

.,

C)

~/

Water

~

drop ,...

,

,/,,

///

F

unstable filmx / ~ . . . . . _ ~ J

Fig. 6.15 Scheme of the edge-cut mechanism. (a) In the zone, where the inner wall of the pore is hydrophilized by the surfactant solution, a thin aqueous film separates the oil and solid. (b)In shear flow the oil drop deforms easier because it is not attached to the solid edge. (c) The latter cuts the drop on two parts at a higher shear rate. (d) Even a rounded solid edge could cause splitting of the drop in shear flow because of the instability of the formed oily film.

280

Chapter 6

The situation becomes more complicated when oil is continuously supplied through the capillary (pore) and oil drops are blown out one after another. The experiments show that the radius of the formed drops is from 3.0 to 3.5 times larger than the radius of the capillary, if there is no coalescence of the drops after their formation [55-57]. The latter fact has not yet been explained theoretically. Moreover, it has been observed [76] that if a shear flow is applied, the size of the drops essentially decreases with the rise of the shear rate for Re > 100.

6.3.3.

PHYSICOCHEMICAL FACTORS INFLUENCING THE DETACHMENT OF OIL DROPS

Up to here we considered mostly the role of mechanical factors: drag force due to shear flow and retention force related to surface tension and stress balance at the contact line. These factors presume an input of mechanical energy in the system. However, even for a great energy input some residual oil drops could remain on the substrate, see Figs. 6.8c and 6.12, i.e. complete removal of the oil may not be achieved. An alternative way to accomplish detachment of oil drops is to utilize the action of purely

physicochemical factors. One of them is related to the mechanism of the disjoining film, which is described briefly below. Historically, such a mechanism has been first observed for polycrystalline solids immersed in liquid, see Fig. 6.16a. If the tension of the solid-liquid interface, o",/, is small enough to satisfy the relationship 2o'sl< o'g, where o-g is the surface tension at the boundary between two crystalline grains, then a liquid film penetrates between the grains and splits the polycrystal to small monocrystals. This phenomenon is observed with Zn in liquid Ga, C'u in liquid Bi, NaCI in water [77]. An analogous phenomenon (penetration of disioining water film) has been observed by Powney [78], Stevenson [79, 80] and Kao eta!. [81] I\)r a drop of oil attached to a solid substrate. It is termed also the "diffusional" mechanism. The c~ndition for penetration of di~.joining water film between oil and solid is O'ow + O'sw < Oso

(6.45)

see Fig. 6.16b for the notation. Equation (6.45) means that a Neumann-Young triangle does not

Particles at Interfaces: Deformations and Hydrodynamic Interactions

DISJOINING-FILM

a) In Polycrystallites liquid

.>c;,s'tiJli ~'ir~stai': ~

281

MECHANISM

2CYsl< Cyg1 liquid

.,"Z;",~ :,,: )~, I l~disjoining;"

b) Attached Drops flow + O'sw < ~ s o 1

Water

W a t e r ~ disjoining,film ~~,:,~ (easy detachment ~ ~..~ # / in shear flow) ,,'/~////// . //// "/////~/~//.. //~ /////~ ~.//////.

KEY: Micellar solutions are found to promote the formation of disjoining film

Fig. 6.16. Scheme of the disjoining film mechanism with (a) polycrystallites and (b) oil drop attached to a substrate. exist, see Chapter 2. For that reason the solid-oil interface is exchanged with a water film, whose surfaces have tensions Oow and Osw. Equation (6.45) shows that the formation of such film is energetically favorable. This can happen if a "strong" surfactant, dissolved in the aqueous phase, sufficiently lowers the oil-water and solid-water surface tensions. In the experiments of Kao et al. [81] drops of crude oil have been detached from glass in solutions of 1 wt% Cl6-alpha-olefin-sulfonate + 1 wt% NaC1. These authors have observed directly the dynamics of water-film penetration. Once the disjoining film has been formed, even a weak shear flow is enough to detach the oil drop from the substrate. The study in Ref. [81] was related to the enhanced oil recovery; however, similar mechanism can be very important

282

Chapter 6

also for oil-drop detachment in other applications of detergency. It is worthwhile noting that not every surfactant could cause penetration of disjoining water film. For each specific system one should clarify which surfactants and surfactant blends give rise to penetration of disjoining films between oil and solid, and how sensitive is their action to the type of oil and substrate. The major advantage of the disjoining-film mechanism is that it strongly reduces the input of mechanical energy in washing, and effectuates complete washing, i.e. no residual oil drops remain on the substrate. A drawback of this mechanism is that the "strong" surfactant could produce undesirable changes in the properties of the substrate (change of the color of fabrics, irritation action on skin, etc.). 6.4.

SUMMARY

In this chapter we consider some aspects of the interaction of colloidal particles with an interface, which involve deformations of a fluid phase boundary and/or hydrodynamic flows. First, from a thermodynamic viewpoint, we discuss the energy changes accompanying the deformation of a fluid particle (emulsion drop of gas bubble) upon its collision with an interface or another particle. Formally, the interaction energy depends on two parameters: the surface-to-surface distance h and the radius rc of the film formed in the collision zone: U = U(h,rc), see Eq. (6.1). If the interaction is governed by the surface dilatation and the DLVO

forces (van der Waals attraction and electrostatic repulsion), the energy may exhibit a minimum, which corresponds to the formation of a floc of two attached fluid particles with a liquid film between them, see Fig. 6.1a. The depth of this minimum increases if the electrostatic repulsion is suppressed by addition of electrolyte, or if the size of the fluid particle is greater, Fig. 6.lb. When oscillatory-structural forces are operative, then the surface U(h,rc) exhibits a series of minima separated by energy barriers, Fig. 6.2. When the height of such barrier is greater than kT, it can prevent the Brownian flocculation of the fluid particles and may decelerate the creaming in emulsions, Fig. 6.3. The radius of the liquid film formed between a fluid particle and an interface can be determined by means of force balance considerations. The theory predicts that for small contact angles the film radius must be proportional to the squared radius of the particle, Eq. (6.15). The latter equation agrees excellently with experimental data (Fig. 6.5).

Particles at Interfaces: Deformations and Hydrodynamic bzteractions

283

Next we consider the hydrodynamic interactions of a colloidal particle with an interface (or another particle), which are due to hydrodynamic flows in the viscous liquid medium. Each particle is subjected to the action of a driving force F, which is a sum of an external force (gravitational, Brownian, etc.) and the surface force operative in the zone of contact (the thin liquid film), see Eq. (6.21). The theory relates the driving force with the velocity of mutual approach of the two surfaces. The respective relationships depend on the shape of the particle, its deformability and surface mobility. For example, if the particle is spherical and its surface is tangentially immobile, then the velocity is given by the Taylor formula, Eq. (6.20). If the particle is a drop or bubble, it deforms in the collision zone when the width of the gap becomes equal to a certain distance hinv called the "inversion thickness", see Eq. (6.23). After a liquid film of uniform thickness is formed, then the velocity of particle approach is determined by the Reynolds formula, Eq. (6.25). The transition from Taylor to Reynolds regime is also considered, see Eq. (6.28) and Fig. 6.6. If the surface of an emulsion drop is tangentially mobile (no adsorbed surfactant), then the streamlining by the outer liquid gives rise to a circulation of the inner liquid, which makes the relation between velocity and force dependent on the viscosities of the two liquid phases, see Eq. (6.31). The most complicated is the case when the mobility of the particle surface is affected by the presence of adsorbed soluble surfactant. In this case the connection between velocity and force is given by Eq. (6.33), which takes into account the effects of the Gibbs elasticity, and of the surface and bulk diffusivity of the surfactant molecules. The gradual mutual approach of two fluid particles may terminate when the thickness of the gap between them reaches a certain critical value, at which fluctuation capillary waves spontaneously grow and cause rupturing of the liquid film and coalescence of the fluid particles, see Section 6.2.7. Finally, we consider the factors and mechanisms for detachment of an oil drop from a solid surface - this is a crucial step in the process of washing. In the presence of shear flow in the adjacent aqueous phase, the oil drop deforms, the oil-water interface acquires a unstable configuration and eventually the drop splits on two parts; this is known as the emulsification mechanism of drop removal, see Figs. 6.8 and 6.12. Alternatively, the deformation might be

accompanied with destabilization of the contact line (violation of the Young equation), which would lead to detachment of the drop from the substrate: rolling-up mechanism, see Fig. 6.11.

Chapter 6

284

Special attention is paid to the detachment of oil drops from the orifice of a pore, which essentially depends on whether the inner surface of the pore is hydrophobic or hydrophilic, see Figs. 6.13 - 6.15. The adsorption of some surfactants is able to modi~ the interfacial tensions in such a way, that an aqueous (disjoining) film can penetrate between the oil drop and the solid surface thus causing drop detachment without any input of mechanical energy: disjoining-

film mechanism. The latter purely physicochemical mechanism is illustrated in Fig. 6.16. 6.5.

REFERENCES

1.

N.D. Denkov, D.N. Petsev, K.D. Danov, J. Colloid Interface Sci. 176 (1995) 189.

2.

N.D. Denkov, D.N. Petsev, K.D. Danov, Phys. Rev. Lett. 71 (1993) 3226.

3.

K.D. Danov, D.N. Petsev, N.D. Denkov, R. Borwankar, J. Chem. Phys. 99 (1993) 7179.

4.

P.A. Kralchevsky, N.D. Denkov, K.D. Danov, D.N. Petsev, "Effect of Droplet Deformability and Surface Forces on Flocculation", In: Proceedings of the 2nd World Congress on Emulsion (Paper 2-2-150), Bordeaux, 1997.

5. 6.

I.B. Ivanov, K.D. Danov, P.A. Kralchevsky, Colloids Surf. A, 152 (1999) 161. K.G. Marinova, T.D. Gurkov, G.B. Bantchev, P.A. Kralchevsky, "Role of the Oscillatory Structural Forces for the Stability of Emulsions", in: Proceedings of the 2nd World Congress on Emulsions (Paper 2-3-151), Bordeaux, 1997.

7.

K.G. Marinova, T.D. Gurkov, T.D. Dimitrova, R.G. Alargova, D. Smith, Langmuir 14 (1998) 2011. 8. K.D. Danov, I.B. Ivanov, T.D. Gurkov, R.P. Borwankar, J. Colloid Interface Sci. 167 (1994) 8. I.B. Ivanov, D.S. Dimitrov, Thin Film Drainage, in: "Thin Liquid Films", I.B. Ivanov (Ed.), Marcel Dekker, New York, 1988; p. 379. 10. B.V. Derjaguin, M.M. Kussakov, Acta Physicochim. USSR, 10 (1939) 153. 11. R.S. Allan, G.E. Charles, S.G. Mason, J. Colloid Sci. 16 (1961) 150. .

12. S. Hartland, R.W. Hartley, "Axisymmetric Fluid-Liquid Interfaces", Elsevier, Amsterdam, 1976. 13. P.A. Kralchevsky, I.B. Ivanov, A.D. Nikolov, J. Colloid Interface Sci. 112 (1986) 108. 14. E.S. Basheva, Faculty of Chemistry, University of Sofia, private communication. 15. S.A.K. Jeelani, S. Hartland, J. Colloid Interface Sci. 164 (1994) 296. 16. I.B. Ivanov, D.S. Dimitrov, P. Somasundaran, R.K. Jain, Chem. Eng. Sci. 40 (1985) 137. 17. V.G. Levich, "Physicochemical Hydrodynamics", Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 18. I.B. Ivanov, Pure Appl. Chem. 52 (1980) 1241. 19. D.A. Edwards, H. Brenner, D.T. Wasan, "Interfacial Transport Processes and Rheology", Butterworth-Heinemann, Boston, 1991.

Particles at Interfaces: l)eformations and Hydrodynamic Interactions

285

20. I.B. Ivanov, P.A. Kralchevsky, Colloids Surf. A, 128 (1997) 155. 21. D. M6bius, R. Miller (Eds.) "Drops and Bubbles in Interfacial Research", Elsevier, Amsterdam, 1998. 22. K.D. Danov, P.A. Kralchevsky, I.B. Ivanov. in: Encyclopedic Handbook of Emulsion Technology, J. Sj6blom (Ed.), Marcel Dekker, New York, 2001. 23. L.D. Landau, E.M. Lifshitz, "Fluid Mechanics", Pergamon Press, Oxford, 1984. 24. P. Taylor, Proc. Roy. Soc. (London) A108 (1924) 11. 25. 1.B. Ivanov, B.P. Radoev, T. Traykov, D. Dimitrov, E. Manev, Chr. Vassilieff, in: "Proceedings of the International Conference on Colloid and Surface Science", E. Wolfram (Ed.), Vol.1, p.583, Akademia Kiado, Budapest, 1975. 26. P.A. Kralchevsky, K.D. Danov, I.B. Ivanov, Thin Liquid Film Physics, in: "Foams: Theory, Measurements and Applications", R.K. Prud'homme (Ed.), M. Dekker, New York, 1995, p.86. 27. K.D. Danov, I.B. Ivanov, Critical Film Thickness and Coalescence in Emulsions, in: Proceedings of the 2rid World Congress on Emulsion (Paper No. 2-3-154), Bordeaux, 1997. 28. O. Reynolds, Phil. Trans. Roy. Soc. (London) A177 (1886) 157. 29. M.J. Stefan, Sitzungsberichte der Mathematish-naturwissenschaften Klasse der Kaiserlichen Akademie der Wissenschaften, II. Abteilung (Wien), Vol. 69 (1874) 713. 30. K.D. Danov, N.D. Denkov, D.N. Petsev, R. Borwankar, Langmuir 9 (1993) 1731. 31. E.S. Basheva, T.D. Gurkov, I.B. Ivanov, G.B. Bantchev, B. Campbell, R.P. Borwankar, Langmuir 15 (1999) 6764. 32. E. Rushton, G.A. Davies, Appl. Sci. Res. 28 (1973) 37. 33. S. Haber, G. Hetsroni, A. Solan, Int. J. Multiphase Flow 1 (1973) 57. 34. L.D. Reed, F.A. Morrison, Int. J. Multiphase Flow 1 (1973) 573. 35. G. Hetsroni, S. Haber, Int. J. Multiphase Flow 4 (1978) 1. 36. F.A. Morrison, L.D. Reed, Int. J. Multiphase Flow 4 (1978) 433. 37. V.N. Beshkov, B.P. Radoev, I.B. Ivanov, Int. J. Multiphase Flow 4 (1978) 563. 38. D.J. Jeffrey, Y. Onishi, J. Fluid Mech. 139 (1984) 261. 39. Y.O. Fuentes, S. Kim, D.J. Jeffrey, Phys. Fluids 31 (1988) 2445. 40. R.H. Davis, J.A. Schonberg, J.M. Rallison, Phys. Fluids A1 (1989) 77. 41. X. Zhang, R.H. Davis, J. Fluid Mech. 230 (1991 ) 479. 42. T.T. Traykov, I.B. Ivanov, International J. Multiphase Flow, 3 (1977) 471. 43. T.T. Traykov, E.D. Manev, I.B. Ivanov, International J. Multiphase Flow, 3 (1977) 485. 44. P.M. Krugljakov, Hydrophile-Lipophile Balance, in: "Studies in Interface Science", Vol. 9, D. M6bius and R. Miller (Eds.), Elsevier, Amsterdam, 2000. 45. A.J. Vries, Rec. Trav. Chim. Pays-Bas 77 (1958) 44. 46. A. Scheludko, Proc. K. Akad. Wetensch. B, 65 (1962) 87. 47. A. Vrij, Disc. Faraday Soc. 42 (1966) 23. 48. I.B. Ivanov, B. Radoev, E. Manev, A. Scheludko, Trans. Faraday Soc. 66 (1970) 1262.

286

Chapter 6

49. I.B. Ivanov, D. S. Dimitrov, Colloid Polymer Sci. 252 (1974) 982. 50. E.D. Manev, S.V. Sazdanova, D.T. Wasan, J. Colloid Interface Sci. 97 (1984) 591. 51. A.K. Malhotra, D.T. Wasan, Chem. Eng. Commun. 48 (1986) 35. 52. N. Munyan, World Oil 8 (1981) 42. 53. M.V. Ostrovsky, E. Nestaas, Colloids Surf. 26 (1987) 351. 54. L. Thompson, J. Colloid Interface Sci. 163 (1994) 61. 55. K. Kandori, Application of Microporous Glass Membranes: Membrane Emulsification, in: "Food Processing: Recent Developments", A. Gaonkar (Ed.), Elsevier, Amsterdam, 1995. 56. V. Schr6der, H. Schubert, "Production of Emulsions with Ceramic Membranes", Proc. 2nd World Congress on Emulsion, Vol. 1, Paper No. 1-2-290, Bordeaux, 1997. 57. V. Schr6der, O. Behrend, H. Schubert, J. Colloid Interface Sci. 202 (1998) 334. 58. E.A. Evans, Biophys. J. 13 (1973) 941. 59. E.A. Evans, R.M. Hochmuth, J. Membr. Biol. 30 (1977) 351. 60. M.A. Hubbe, Colloids Surf. 12 (1984) 151. 61. M.M. Sharma, H. Chamoun, D.S.H. Sita Rama Sarma, R.S. Schechter, J. Colloid Interface Sci. 149 (1992) 121. 62. K.W. Dillan, E.D. Goddard, D.A. McKenzie, J. Am. Oil. Chem. Soc. 56 (1979) 59. 63. M.C. Gum, E.D. Goddard, J. Am. Oil. Chem. Soc. 59 (1982) 142. 64. M. Mah6, M. Vignes-Adler, A. Rosseau, C.G. Jacquin, P.M. Adler, J. Colloid Interface Sci. 126 (1988) 314. 65. M. Mah6, M. Vignes-Adler, P. M. Adler, J. Colloid Interface Sci. 126 (1988) 329. 66. M. Mah6, M. Vignes-Adler, P. M. Adler, J. Colloid Interface Sci. 126 (1988) 337. 67. A.J. Goldmann, R.G. Cox, H. Brenner, Chem. Eng. Sci. 22 (1967) 653. 68. E.B. Dussan, R.T.-P. Chow, J. Fluid Mech. 137 (1983) 1. 69. S. Basu, K. Nandakumar, J.H. Masliyah, J. Colloid Interface Sci. 190 (1997) 253. 70. R. Finn, "Equilibrium Capillary Surfaces", Springer Verlag, Berlin, 1986. 71. R. Finn, M. Shinbrot, J. Math. Anal. Appl. 123 (1987) 1. 72. S.D. Iliev, J. Colloid Interface Sci. 194 (1997) 287. 73. S.D. Iliev, J. Colloid Interface Sci. 213 (1999) 1. 74. N.K. Adam, J. Soc. Dyers Colour. 53 (1937) 121. 75. G.S. Marinov, Faculty of Chemistry, Univ. of Sofia, private communication. 76. C.A. Paraskevas, Chem. Engineering Department., Univ. Patras, private communication. 77. E.D. Shchukin, A.V. Pertsov, E.A. Amelina, "Colloid Chemistry", Moscow University Press, Moscow, 1982. 78. J. Powney, J. Text. Inst. 40 (1949) 519. 79. D.C. Stevenson, J. Text. Inst. 42 (1951) 194. 80. D.G. Stevenson, J. Text. Inst. 44 (1953) 548. 81. R.L. Kao, D.T. Wasan, A.D. Nikolov, D.A. Edwards, Colloids Surf. 34 (1988) 389.

287

CHAPTER 7

LATERAL CAPILLARY FORCES BETWEEN PARTIALLY IMMERSED BODIES

This chapter describes results from theoretical and experimental studies on lateral capillary forces. Such forces emerge when the contact of particles, or other bodies, with a fluid phase boundary causes perturbations in the interracial shape. The latter can appear around floating particles, semi-immersed vertical cylinders, particles confined in a liquid film, inclusions in the membranes of lipid vesicles or living cells, etc. Except the case of floating particles (see Chapter 8), whose weight produces the meniscus deformations, in all other cases the deformations are due to the surface wetting properties of partially immersed bodies or particles. The "immersion" capillary forces, resulting from the overlap of such interfacial perturbations, can be large enough to cause the two-dimensional aggregation and ordering of small colloidal particles observed in many experiments. The lateral capillary force between similar bodies is attractive, whereas between dissimilar bodies it is repulsive. Energy and force approaches, which are alternative but equivalent, can be used for the theoretical description of the lateral capillary interactions. Both approaches require the Laplace equation of capillarity to be solved and the meniscus profile around the particles to be determined. The energy approach accounts for contributions due to the increase of the meniscus area, gravitational energy and/or energy of wetting. The second approach is based on calculating the net force exerted on the particle, which can originate from the hydrostatic pressure and interfacial tension. For small perturbations, the superposition approximation can be used to derive an asymptotic formula for the capillary forces, which has been found to agree well with the experiment. In all considered configurations of particles and interfaces the lateral capillary interaction originates from the overlap of interfacial deformations and is subject to a unified theoretical treatment, despite the fact that the characteristic particle size can vary from 1 cm down to 1 nm. (Protein molecules of nanometer size can be treated as "particles" insofar as they are considerably larger than the solvent (water) molecules.)

288

Chapter 7

7.1.

PHYSICAL ORIGIN OF THE LATERAL CAPILLARY FORCES

7. ]. ].

TYPES OF CAPILLARY FORCES AND RELATED STUDIES

The experience from experiment and practice shows that particles floating on a fluid interface attract each other and form clusters. Such effects are observed and utilized in some extraction and separation flotation processes [1,2]. Nicolson [3] developed an approximate theory of these lateral capillary forces taking into consideration the deformation of the interface due to the particle weight and buoyancy force. The shape of the surface perturbations created by floating particles has been studied by Hinsch [4] by means of a holographic method. Allain and Jouher [5], and in other experiment Allain and Cloitre [6], have studied the aggregation of spherical particles floating at the surface of water. Derjaguin and Starov [7] calculated theoretically the capillary force between two parallel vertical plates, or between two inclined plates, which are partially immersed in a liquid. Additional interest in the capillary forces has been provoked by the fact that small colloidal particles and protein macromolecules confined in liquid films also exhibit attraction and do form clusters and larger ordered domains (2-dimensional arrays) [8-13]. The weight of such tiny particles is too small to create any substantial surface deformation. In spite of that, they also produce interfacial deformations because of the confinement in the liquid film combined with the effect of wettability of the particle surfaces. The wettability is related to the thermodynamic requirement that the interface must meet the particle surface at a given angle the contact angle. The overlap of such wetting-driven deformations also gives rise to a lateral capillary force [ 14]. As already mentioned, the origin of the lateral capillary forces is the deformation of the liquid surface, which is supposed to be flat in the absence of particles. The larger the interfacial deformation created by the particles, the stronger the capillary interaction between them. Two similar particles floating on a liquid interface attract each other [3,15-17] - see Fig. 7.1a. This attraction appears because the liquid meniscus deforms in such a way that the gravitational potential energy of the two particles decreases when they approach each other. One sees that the origin of this force is the particle weight (including the Archimedes buoyancy force).

289

Lateral Capillary Forces between Partially Immersed Bodies

IMMERSION FORCES (effect driven by wetting)

FLOTATION FORCES

(effect driven by gravity) (a)

~

~

(b)

~

] s{nvlsi: aV~ > 01 (C)

~

(d)

~

~

~ ~ ~ Q ~

:: 100 nm, in which the disjoining pressure 1-I (the interaction between the two adjacent phases across the liquid film) becomes negligible. In fact, the gravity keeps the interface planar (horizontal) far from the particle when the film is

thick. On the contrary, when the film is thin,

the existence of a positive disjoining pressure (repulsion between the two film surfaces) keeps the film plane-parallel far from the particle, supposedly the substrate is planar. The condition for stable mechanical equilibrium of this film is

296

Chapter 7

=

=

1

(7.59)

where the superscript "out" means that Eq. (7.59) is valid in the outer asymptotic region of nottoo-small interparticle separations, (qa) 2 >_ 1, in which the superposition approximation can be applied. The indices "l" and "r" denote the left- and right-hand side particles, in particular

rl2 = (x + S) 2 -k- y2,

rr2= ( X - S) 2 q- y2.

(7.60)

In the outer region, for relatively long distances, (rc/S) 2 o,,for m 4: 0, i.e. the liquid climbs up in the narrow gap between the two infinitely long vertical cylinders; this limiting result could be qualitatively correct for the considered idealized situation, but it could hardly be quantitatively correct insofar as the presumption for small meniscus slope, IVlI~] 2 2

(r2[$2 0;

(7.148)

here, as usual, Q2 = r2 sin~2 and r = (x 2 + y2)]/2. In particular, Eq. (7.148) allows one to determine the shape of the contact line on the wall [18], ~'l(y) - q-lcotal + Q2 [2K0(Iqyl) + ln(1 - a2/y2)],

(7.149)

as well as the increase of the wet area on the wall due to the presence of the spherical particle:

~4 - ~ [~] (y)--~, (oo)]dy = 2m/-~Q2(1 - qa)

(7.150)

-oo

The form of Eq. (7.148) shows that if the contact angle at the wall is a~ = 0, then the shape of the meniscus is the same as that of the meniscus around two identical particles separated at a distance s, each of them being the mirror image of the other one with respect to the wall. For that reason in such a case the capillary interaction between particle and wall is equivalent to the interaction of the particle with its mirror image. In this aspect there is an analogy with the image forces in electrostatics; the same analogy is present also in the case of floating particle, see Chapter 8 below. In the considered case of small meniscus slope the projection of the particle contact line on the plane xy can be approximately considered as a circumference of radius re. When the distance s between the particle and the wall varies, then both r2 and ~2 alter. The values of r2, ~2 and of the meniscus elevation at the particle contact line,

h2, c a n

be determined for each given s by

solving numerically the equation [ 18] h2 =

q-tcotal exp(-qs)

- r2

sin ~t2 ln[(yeq) 2 (s + a)r2]4],

(7.151)

in which r2 and ~t2 are expressed as functions of h2 as follows: rz(h2) = [(/0 + hz)(2R2 - l 0 - h2)] 1/2,

~t2(h2) = arcsin(r2(h2)/R2)

-

0~2,

Lateral Capillary Forces between Partially Immersed Bodies

345

5.0 ~2"--" lO

4.0 ~,

25___~ ~ \ ",\ ~X~

3.0

"7 r-,-I

x

2.0

10= 0.5/zm R2 = 1 / , m a = 40 m N / m

"",~

7", = 0-01~

1.0 0.0

i 1

t

,

I

50

,

~

D

,

I

t

,

,

100

,

I

150

,

i

i

i

200

s [ktm] Fig. 7.24. Calculated in Ref. [18] capillary force, ~k)= ~k,)+ ~kp), exerted on a particle of radius R2 = 1 gm which is situated at a distance s from a vertical wall, see Fig. 7.23. The two curves correspond to particle contact angles a2 = 1~ and 25~ the other parameter values are: lo = 0.5 gm, cr = 40 mN/m and gt~ = 0.01 ~ cf. Eqs. (7.92) and (7.99). The values of the geometrical parameters thus determined can be further used to calculate the force of particle-wall interaction. The contributions of the meniscus surface tension and the hydrostatic p r e s s u r e , Fx~ka) a n d F(xkp), can be calculated substituting Eq. (7.148) for 1: - z2 = ln[(s + a)/r2] into Eqs. (7.132)-(7.133) and (7.138) and carrying out numerically the integration with respect to co. As an illustration Fig. 7.24 presents the calculated force F(xk) = F~ ~~ + F~ kp) plotted against the particle-to-wall distance s" the two curves correspond to two values of the particle contact angle" a2 = 1~ and 25 ~ The other parameter values are R2 = 1 lam, gtl = 0.01 o and 10 = 0.5 gm. The calculated capillary force (Fig. 7.24) corresponds to attraction between the spherical particle and the wall.

7.5.

SUMMARY

Lateral capillary forces appear when the contact of particles (or other bodies) with a fluid phase boundary brings about perturbations in the interfacial shape. The capillary interaction is due to the overlap of such perturbations. The latter can appear around floating particles (Fig. 7.1a,c), particles confined in a liquid film (Figs. 7.1b,d,f), particles attached to holders (Fig. 7.7),

346

Chapter 7

vertical cylinders (Fig. 7.10), inclusions in lipid membranes (Fig. 7.18c), etc., and can be both attractive (between similar particles) and repulsive (between dissimilar particles). The asymptotic law of the capillary interaction, Eq. (7.13) or Eq. (7.145), in its approximate form given by Eq. (7.14), resembles the Coulomb's law in electrostatics. Following the latter analogy one can introduce "capillary charges" of the attached particles (see Eq. 7.9), which can be both positive and negative. Except the case of floating particles (see Chapter 8), whose weight causes the meniscus deformations, in all other cases the deformations are governed by the surface wetting properties of partially immersed bodies or particles. The resulting "immersion" capillary forces can be large enough (Fig. 7.15) to cause two-dimensional aggregation and ordering of small colloidal particles, which has been observed in many experiments. There are two equivalent theoretical approaches to the lateral capillary interactions: energy and force approaches. Both of them require the Laplace equation of capillarity to be solved and the meniscus profile around the particles to be determined, see Section 7.2.1. The energy approach accounts for contributions due to the alteration of the meniscus area, gravitational energy and/or energy of wetting, see Eq. (7.16). The second approach is based on calculating the net force exerted on the particle which can originate from the hydrostatic pressure and interfacial tension, see Eqs. (7.21)-(7.23) and (7.132)-(7.138). In the case of small overlap of the interfacial perturbations created by two interacting bodies, the superposition approximation can be combined with the energy or force approach to derive an asymptotic formula for the lateral capillary force, see Sections 7.1.3 and 7.4.2. This formula has been found to agree well with the experiment (Figs. 7.5 and 7.8). Using the method of the matched asymptotic expansions one can derive analytical expressions for the capillary elevation of the contact line, hk, and the shape of the contact line, (c,k(c0), see Sections 7.2.2 and 7.2.3. The energy of capillary immersion interaction between two vertical

cylinders turns out to be equal to a half of the energy of wetting and can be expressed in terms of h~, cf. Eqs. (7.83) and (7.84). The expression for the energy of interaction between two

spherical particles, Eq. (7.98), is similar, but it should be taken into account that the radius of the contact lines on the particles alters when the interparticle distance is varied, see

Lateral Capillary Forces between Partially Immersed Bodies

347

Eqs. (7.99)-(7.102). In a similar way one can calculate the energy of capillary interaction between cylinder and sphere, see Eq. (7.104) and Fig. 7.17. The energy approach has been also applied to the case, when the position of the contact line (rather than the magnitude of the contact angle) is fixed at the particle surface (Section 7.3.4). This can happen when the contact line is attached to some edge, or to the boundary between hydrophilic and hydrophobic zones on the particle surface, see Fig. 7.18. The derived analytical expressions, Eqs. (7.106) and (7.120), predict that the capillary interaction at fixed meniscus elevation is weaker than that at fixed meniscus slope; however in both cases it corresponds to attraction between similar bodies and its energy can be much larger than the thermal energy kT, see Fig. 7.19. For small particles, that is for (qrk) 2 1.2 nm can be considered as being much larger. The pronounced difference in the strength of the two types of capillary interactions, see Fig. 8.3, is due to the different magnitude of the interfacial deformation. The small floating particles are too light to create a substantial deformation of the liquid surface and then the lateral capillary force becomes negligible. In the case of immersion forces the particles are restricted in the vertical direction by the solid substrate (see Fig. 7. lb) or by the two surfaces of the liquid film (Fig. 7.1f). Therefore, as the film becomes thinner, the liquid surface deformation increases, thus giving rise to a strong interparticle attraction. For that reason, as already mentioned, the immersion forces may be one of the main factors causing the observed self assembly of lam-sized and sub-~tm colloidal particles and protein macromolecules confined in thin liquid films or lipid bilayers. In conclusion, the different physical origin of the flotation and immersion lateral capillary forces results in different magnitudes of the "capillary charges" Qk, which depend in a different way on the interfacial tension cr and the particle radius R, see Eqs. (8.4) and (8.20). In this respect these two kinds of capillary force resemble the electrostatic and gravitational forces, which obey the same power law, but differ in the physical meaning and magnitude of the force constants (charges, masses) [7,8].

8.1.4.

MORE ACCURATE CALCULATION OF THE CAPILLARY INTERACTION ENERGY

To derive Eq. (8.2) we have used the superposition approximation, i.e. we approximately presented the meniscus around two floating particles as a superposition of the axisymmetric menisci around each particle in isolation. Although this approximation works well for long interparticle separations, at short distances it is not quantitatively correct; in particular, it leads to violation of the boundary condition for constancy of the contact angle at the particle surface (the Young equation). We can overcome this problem using the more rigorous expressions derived in Section 7.2 by means of bipolar coordinates. Second problem with the superposition

359

Lateral Capillary Forces Between Floating Particles

approximation is that the interaction energy is assumed to be identical to the gravitational energy of one of the two floating particles, thus apriori neglecting the contributions of the meniscus surface energy and the energy of wetting. The latter assumption needs to be verified and validated. The above two issues are elucidated below following Ref. [4].

Gravitational,

wetting

and meniscus

contributions.

Our starting point are Eqs.

(7.16)-(7.18) which express the energy of the system (the grand thermodynamic potential) f2 as a sum of gravitational, wetting and meniscus contributions, f~ = Wg + Ww + Win. The meniscus surface energy is given again by Eq. (7.93), which can be represented in the form A Wm ~- Wm - Wmoo -- TC(y Z

(Q *h k - r~ ) - Apglv - Wmoo ,

(8.21)

k=l,2

where Wm~ is the limiting value of Wm for infinite interparticle separation (L--->oo); Iv is given by Eq. (7.72). Substituting the geometrical expressions for the portions of the particle area wet by fluids I and II in Eq. (7.18) one obtains [4]:

Ww = 2rt ~_~[crk~iRkb k + O'k,iiRk (2R k -b~ )]

(8.22)

k=l,2

Using the Young equation, ok,i~ - crk,i = cr coso~, and the fact that the particle radius R~ is independent of L from Eq. (8.22) one obtains AWw = -2rtcr Z R k b k cosa~ - Wwoo

(8.23)

k=l,2

where Wwoois the limiting value of the first term in the right-hand side of Eq. (8.23) for L---~oo. AWw expresses the contribution of wetting to the capillary interaction energy. Finally, using Eq. (7.17) and geometrical considerations for the system depicted in Fig. 8.2 one can derive the gravitational contribution to the capillary interaction energy [4]: AWg = Apg{Iv + ~ [ (DkVs (k~- Vl(k))Zk (c~ + rt(r~2 + 2hkZ)r~2/4]} - Wgoo

(8.24)

k=l.2

where Wgoois the limiting value of Wg for L~oo, and Z~(c~ = h k - bk + Rk

(k = 1,2)

(8.25)

360

Chapter 8

is the z-coordinate of the (mass) center of the k-th sphere. Note that when summing up Eqs. (8.21) and (8.24) the terms with Iv cancel each other. In view of Eq. (7.16) the total energy of capillary interaction between two floating particles can be expressed in the form [4] (8.26)

zxn = AWw + AWm + AW~

where AWw is given by Eq. (8.23); AWm and AWg are defined without the canceling/v-term as follows kl~ m = ~o" E (Q*hk - rff ) - I~'~

(8.27)

k=l,2 1

AWg = -7c0" ~_~{2Qkh k

q2[ 88

__

d +(sDkR~ - R k b k + g b k )(R k - b k

(8.28)

k=l,2

To obtain Eq. (8.28) from Eq. (8.24) we have used Eqs. (8.6), (8.9) and (8.25); the constants Win= and Wg= are defined in such a way that for L--+oo both AWm and AWg tend to zero. Asymptotic expression f o r the capillary flotation force.

It has been proven by Chan et

al. [3] that when the particles are small, the meniscus slope is also small, that is (qR~) 2 > rk] substituting Q~ = Q~") and Fk "- Fk(n).

(iii) Then from Eq. (8.41) one obtains the next approximation r~(n+~) which is to be substituted in Eq. (7.40) to get r~n+l) . Next, step (ii) is repeated again to give h~"2~ , etc. This iteration procedure is quickly convergent when (qrk) 2 in Eq. (8.41) is a small parameter. With the values of Q~, rk and hk thus obtained one next calculates ~ and b~:

Lateral Capillary Forces Between Floating Particles

gtk = arcsin(Qk/rk),

363

bk = R~[1 + cos(a~ + gtk)].

(8.43)

Finally, from Eqs. (8.23) and (8.26)-(8.28) one determines the energy of capillary interaction Aft(L); the capillary flotation force can be obtained by differentiation: F = - d(Af~)/dL. For R~ < 100 gm one can calculate Af~ and F from Eqs. (8.35) and (8.36). Note that the latter two equations are more general than Eqs. (8.37) and (8.38) which are subject to the additional restriction L >> r~. Calculated

energy

and force

of

capillary

interaction.

The

capillary

force,

F =-d(Af~)/dL, calculated by numerical differentiation of Eq. (8.26) [along with Eqs. (7.48), (8.23), (8.27) and (8.28)], can be compared with the capillary force calculated from the asymptotic formula, Eq. (8.37), along with Eq. (8.16). For that purpose the ratio, q~, of the values of F calculated by means of the more rigorous and less rigorous equations, = F(Eq.(8.26)) , F(Eq.(8.37))

(8.44)

has been calculated as a function of the interparticle distance L. Figure 8.4 shows plots of 9 vs. L/2R obtained in Ref. [4] for two identical floating particles using the following parameter values" R~

= R2 =

R = 10 lam, Pl

= P2

=

3 g/cm3; water-air interface 9p~ = 1 g/cm 3, PH = 0,

o = 70 mN/m. The three curves in Fig. 8.4 correspond to three different values of the particle contact angle: O~l = a2 = o~= 30 ~ 60 ~ and 90 ~ For L/2R > 3 one has 9 -~ 1, that is the two equations predict practically identical values of the force F, see Fig. 8.4. This result could be anticipated, because Eq. (8.37) is an asymptotic expression for the capillary force derived for the case of large interparticle separations. On the other hand, Fig. 8.4 shows that for shorter distances (L/2R < 2) and for a > 30 ~ the asymptotic formula, Eq. (8.37), considerably underestimates the capillary force; hence in such cases the usage of the more rigorous Eq. (8.26) has to be recommended for calculation of F. Figure 8.5 shows plot of AU~/kT vs. L calculated by means of Eq. (8.26) for two identical particles. The temperature is 25~

the values of the other parameters are shown in the figure.

Chapter 8

364

4 . 0

E

i/

(1) = F (Eq.(8.26)) F (Eq.(8.27))

3.0

o.0

i

Pk = 3 . 0 g . e m -3 R - 10/_tin

,

cx = 30~

,

,,,,

,

.

.

.

.

.

.

9

....

.

.

-2.0

pk= 1.5 g . c m -3

~

-4.0

"

i-"7

=

~

"

~

~

-6.0

'

. . . .

2.0 . . . .

S Pk = 2.0 g. c m -3

-t3.o O'q.O

.

.

.

.

.

'

J

-

~=70 mN.m" R=10.0 ~m cz=60 o p~=1.0g.cm 3

....I f

-

.

Pn = 0

2.o [[ \-\ \~ . _ _ . _....... ~-6~176 o _ a=90 o 1.0

.

g . c m -3

A~/kT

p, = 1.0 g . e m -3 f\

9

Pk = i . 0 5

3'.0 '

'

'

' 4 0

,

lo 1

i ,

i

L

i

L

I

p.=O i

/

/

I

,

i ,

L

i

i

i

lOZ

i

i

i

i

103

L [/~m]

L/(eR) Fig. 8.4. Plot of (I) vs. distance L scaled by the particle diameter 2R obtained in Ref. [4] for two identical particles; (I) is the ratio of the values of the capillary force calculated by means of the more rigorous Eq. (8.26) and the approximate Eq. (8.37).

Fig. 8.5. Dependence of the capillary interaction energy A~/kT on the distance L between two identical floating particles calculated in Ref. [4] for three different values of the particle mass density Pk.

The curves in Fig. 8.5 are close to straight lines, which can be explained with the fact that Eqs. (7.54) and (8.38) predict A ~ o~ ln(qL) for qL 0, i.e. to repulsion. Note also that the magnitude of the interaction energy in Fig. 8.9 is much greater that that in Fig. 8.5. The reason is that the particle radii in Fig. 8.9 are larger than those in Fig. 8.5, and that the capillary flotation force increases very strongly with the rise of the particle size:

A~2 ~

Q1Q2 ~

R3R 3, see Eqs. (8.16) and (8.38). Thus for two identical particles, Rl

one obtains Af2

~

R6

and F

,,~

= R2 =

R,

R 6, cf. Eqs. (8.20) and (8.37). This strong dependence of the

flotation force on the particle size makes it to vanish for Brownian particles of radius R < 1 pm, see Fig. 8.3. In other words, the macroscopic effect. On the other hand, the

flotation force turns out to be an essentially

immersion force is operative between both sub-

micrometer (Brownian) and larger particles.

8.2.

PARTICLE-WALL

8.2.1.

ATTRACTIVE AND REPULSIVE CAPILLARY IMAGE FORCES

INTERACTION:

CAPILLARY IMAGE FORCES

In this section we consider a particle of radius in the range between 5 pm and 1 mm, which is floating on a liquid surface in the vicinity of a vertical wall. The overlap of the meniscus around the floating particle with the meniscus on a vertical wall gives rise to a particle-wall interaction, which can be both repulsive and attractive, as explained below. We will use subscripts "1" and "2" to denote parameters characterizing the wall and particle, respectively. First, following Refs. [9,10], we consider the simplest case, when the contact angle at the wall is C~l = 90 ~ In such a case the meniscus would be flat if the floating particle (Fig. 8.10a) were removed. Let us denote by O~l = 90 ~ the function

~o(x,y) the meniscus shape in the presence of particle. Since

~o(x,y) must satisfy the boundary condition (~'0/~x)x_=0 = 0 at the wall

surface. Using considerations for symmetry one realizes that the meniscus shape

~o(x,y) in Fig.

8.10a would be the same if (instead of a wall at a distance s) one has a second particle (mirror image) floating at a distance 2s from the original particle. The "image" must be identical to the original particle with respect to its size, weight and contact angle; in other words, the particle

Chapter 8

368

(a)

----..-.-.-

1 e- . . . . . . . . .

. . . . . .

_

.:.........

9

i

c~2 ...-~L..... ...... -Lv~-:"."-l--~ .............. ..,~-az""-__"_ :__t.__-. . . . . s

~z I|O

,

,

,a..___ ~__

II

l:,ar/icle

[z ,,

1(3.

.

.

~.........-

.

.

.

'*"-. ~

"

'"

'", image

""

s

I~ ~

-I!~-'

(b)

,

I 2r , F zq

!1 ~-

.

, 2r , ~ 21 a..___~,

,

i

!

.

.

.

l

(II) x

..~_ ~ . . ~ . ~ . , _

O)

I .

1 . - . ..a ~ .. ..,

.

..~

~ (II)

. . . . . . ..

x

..~

0)

s pat4icle

Fig. 8.10. Sketch of the meniscus profile, ~'0(r), around a particle floating in the vicinity of a vertical wall; a: and re are the particle contact angle and contact line radius; ~2 is the meniscus slope angle at the particle contact line; (a) fixed contact angle at the wall (a~ = 90 ~ corresponding to attractive capillary image force; (b) fixed contact line at the wall (~'0 = 0 for x = 0) which leads to repulsive capillary image force [9]. and its image have identical "capillary charges", equal to Qe. Note, that the capillary charge of the floating particle can be estimated by means of Eq. (8.16) above. As mentioned earlier, the lateral capillary force between two identical particles is always attractive. Hence, the particle and its mirror image depicted in Fig. 8.10a will attract each other, which in fact means that the wall will attract the floating particle; the resulting force will (asymptotically) obey Eq. (8.37) with Q1 = Q:. The boundary condition ~'0(x=0) = 0 represents a requirement for a zero elevation of the contact line at the wall. As noticed in Ref. [9] this can be experimentally realized if the contact line is attached to the edge of a vertical plate, as shown in Fig. 8.10b, or to the boundary between a hydrophobic and a hydrophilic domain on the wall. As before, we assume that the meniscus would be flat if the floating particle (Fig. 8.10b) were removed. Using again considerations for symmetry, one realizes that the meniscus shape ~o(x,y) in Fig. 8.10b would be the same if

Lateral Capillary Forces Between Floating Particles

369

(instead of a wall at a distance s) one has a second particle (image) of the opposite capillary charge (Ql = - Q 2 ) at a distance 2s from the original particle. In such a case the capillary force is repulsive, i.e. in reality the wall will repel the floating particle [9]. The above considerations imply that one can use Eq. (8.37) with L = 2s to describe the asymptotic behavior of the particle-wall interaction for the two configurations depicted in Figs. 8.10a and 8.10b: F = (-1)4 2rccr Q~ qKl(2qs)[1 + O(q2Rk2)],

r~ e2 or el < e2. Coming back to the capillary image forces, we notice that the configurations depicted in Fig. 8.10 represent a very special case, insofar as we have assumed that the interface is horizontal in the absence of floating article (o~1 = 90~ The usual experimental situation is that an inclined meniscus (al ~ 90 ~ is formed in a vicinity of the vertical wall. Below, following Ref. [9], we consider that more general case.

8.2.2.

THE CASE OF INCLINED MENISCUS AT THE WALL Qualitative dependence of energy vs. distance. Figure 8.11a shows the capillary

meniscus formed in the neighborhood of a vertical planar wall. Now the contact angle O~l at the wall is different from 90 ~. In such case, the gravitational force will tend to slip the particle

Chapter 8

370

--~

x

(a) 19 j.

s

_!

x

r,

Af~

(b)

0

s

Fig. 8.11. (a) Sketch of a heavy particle floating at a distance s from a vertical wall of fixed three-phase contact angle ~ 1 ,

(8.69)

(qa) 2 > r2) (8.86)

The range of validity of Eq. (8.86) is verified in Fig. 8.14 below.

8.2.5. APPLICATION OF THE FORCE APPROACH TO QUANTIFY THE PARTICLE-WALL INTERACTION

General equations.

Our purpose is to directly calculate the x-component of the force

exerted on the floating particle in Figs. 8.11 and 8.12. In agreement with Eqs. (7.21)-(7.23) one obtains

(8.87)

F~ = Fx (~ + F~ ~")

F(O~ = ex. ~ dl G_,

F~(p) = ex. ~ ds ( - n P ) ,

L2

(8.88)

$2

where _~ is the vector of surface tension, P is hydrostatic pressure, L 2 denotes the contact line on the particle surface $2, the latter having a running unit normal n,

dl and ds

are linear and surface

elements. The gravitational force is directed along the z-axis, and consequently, it does not (directly) contribute to Fx (although it contributes indirectly to Fx through

Fx(m, see

below). To

calculate Fx ~') one can use Eq. (7.137), that is

F(xP) - Apgr2 i

~"2 (qg) cos qgdq9

(8.89)

0

Note that in view of Eq. (8.49) ~" = ~'0 + ~'l. Usually ~'0 is expressed in terms of the bipolar coordinate co: ~'0 = ~'0(co). Then to carry out the integration in Eq. (8.89) one can use the following relationships between the azimuthal angle q9 and co [17]:

cosco -

s cos q9 + r 2

,

s + r 2 cos q9 where 0 < o9 < rt and 0 < q9 < ft.

do) a ~ = dq9 s + r 2 cos q9

(8.90)

Chapter 8

380

H

X'Fig. 8.13. Sketch of an auxiliary cylinder of radius r2, whose generatrix is orthogonal to the surface ~'~(x) of the non-disturbed meniscus at the wall and passes through the contact line on the particle surface. The angle between the running unit normal n to the surface of this cylinder and the vector of surface tension ~ is equal to ~2 in each point of the contact line; t is unit vector tangential to the contact line and b - t x n . Next, we continue with the calculation of the force Fx{m which is due to the vector of surface tension _c integrated along the contact line. First, let us consider an auxiliary cylinder of radius r2, whose generatrix is orthogonal to the surface ~'~(x) and passes through the contact line on the particle surface, see Fig. 8.13o The angle between the running unit normal to the surface of this cylinder, n, and the surface tension vector cr is equal to g2 in each point of the contact line. Let us introduce a coordinate system ( x ' , y', z'), whose z'-axis coincides with the axis of the cylinder in Fig. 8.13. The unit basis vectors of the new coordinate system are e" =exCOS~+ezsin~,

p

e~ = e y ,

e~ = e z c o s ~ - e x s i n ~ t .

(8.91)

where ~ = ~ s ) is the local slope of the meniscus on the wall, see Eq. (8.85) and Fig. 8.13. The linear element dl along the contact line and its running unit tangent t are expressed as follows

dl= r2,%'dq~,

t-

Z -

1+ ~ ~ ~r 2 dq9

- e ~ sin (p +ey cosq~ +e~ - - ~ r 2 d~oj

(8.92)

(8.93)

Lateral Capillary Forces Between Floating Particles

381

The running unit normal to the surface of the cylinder n and the running binormal b are defined as follows (Fig. 8.13): n = e x' c o s ~

b - tx n

+ e,,' s i n ~ 0 ,

(8.94)

The vector of surface tension o belongs to the plane formed by the vectors n and b: = 0.(b singt2 + n cosgt2)

(8.95)

Combining Eqs. (8.91)-(8.95) one obtains:

0.x - ex.O = 0.

sin 11/2 sin gt + -

rzZ drp

sin I//2 sin q~+ cosgt 2 cosq~ cosgt

(8.96)

Finally, we combine Eqs. (8.88) with Eqs. (8.92) and (8.96) to derive [9]

F) ~ - ~ 0.,dl - 2/r0.r2 sin I//2 sin I//- 20 sin I//2 f d~'0 sin (pd(p + AF,~~

~o d~p

L2

(8.97)

where AF~(a) = 20"r2 cosgt2 cos I/tl X cos (pd(p ~ -~-2! o ~, dq~

cos (pd(p

(8.98)

In view of Eqs. (8.5), (8.85), (8.87), (8.89), (8.97) and (8.98) the net capillary force exerted on the floating particle in the vicinity of the wall is [9,18]

F,~ Fx = -27r,0.Q2q~l(S) + (0./r2) j [ 2Q2~'0((p) + (d~o/dq~)2 + q2r22~2]cosq)dq)

(8.99)

0 Note that ~"= ~'o + ~'~" in the case of fixed contact angle ~ and ~'0 are given by Eqs. (8.51) and (8.53); in the case of fixed contact line ~l and ~'0 are given by Eqs. (8.52) and (8.62); to derive Eq. (8.99) we have used integration by parts in Eq. (8.97). The integral in Eq. (8.99) is to be taken numerically. In Ref. [18] Eq. (8.99) was applied to interpret experimental data for the equilibrium distance between floating particle and vertical wall, see Section 8.2.7 for details.

Asymptotic expression for long distances. For long distances (s >> r2) the last two terms in Eq. (8.97) yield Eq. (7.145), where L = 2s and QIQ2 = (-1))VQ22, see Eq. (8.47) and Fig. 8.10; then in view of Eq. (8.85) we obtain the respective asymptotic form of Eq. (8.97)"

Chapter 8

382

Fx {m = -21tcy[Q2q~l(s) - (-1)XqQ22Kl(2qs)]

(S >> r2)

(8.100)

For not extremely small angle ~ and not-too-large capillary charge Q2 one can estimate Fx ~'~ using the following approximation for the shape of the contact line:

g(~~176176

(8.101)

The substitution of Eq. (8.101) into Eq. (8.89), in view of Eq. (8.85), yields Fx (p) ~ /~o'(qr2)2[ ~ (d~/dx) ]x=s = - l ~ q [ q r 2 ~l (s) ] 2,

(s >> r2, sin21/t 0.05 the lipid membrane usually breaks. Equation (10.17a) describes the transition from "logarithmic" to "linear" regime of dilatation. 10.2.3. BENDING MODE OF DEFORMATION AND CURVATURE ELASTIC MODULI

We consider flexural deformations of a lipid bilayer (membrane) under the condition for small deviations from planarity. In such a case the work of flexural deformation per unit area, Awb, can be expressed in terms of the Helfrich [59] phenomenological expression Awb = 2kt H 2 + kt K

(10.22) m

Here k, is the bending elastic modulus of the bilayer as a whole; k, is torsion or Gaussian curvature elastic modulus, H and K are the mean and the Gaussian curvatures of the bilayer midsurface, see Section 3.1.2 for details. Below, following Ref. [45], we derive an equation of the type of Eq. (10.22) using the "sandwich" model of the lipid membrane, and then comparing the coefficients multiplying

H2

and K we obtain expressions for the curvature elastic moduli k, and k t . In the framework of this model Awb can be presented in the form Awb = Aws + AWin,

(10.23)

where Aws and Awin are contributions due to the bilayer surfaces and bilayer interior (chain region), respectively. The latter two contributions are considered separately below. The f l e x u r a l d e f o r m a t i o n o f the bilayer interior can be characterized by the equation of

the shape of the bilayer midplane: z - ~(x,y),

(10.24)

see Fig. 10.5. The initial state is assumed to be a planar bilayer, like those depicted in Fig. 10.4. The bending of the hydrocarbon chain region will transform the "rectangles" in Fig. 10.4 into the "trapezia" in Fig. 10.5. The bilayer subjected to such deformation cannot exhibit its two dimensional fluidity (viscous slip between chains of neighboring lipids). For that reason the

Mechanics of Lipid Membranes and bzteraction between Inclusions

439

Z

/

Tr y)

Fig. 10.5. Bending deformation of an initially planar lipid bilayer of thickness h; z = ~(x,y) is the equation describing the shape of the bilayer midsurface after the deformation. chain region can be treated as an incompressible elastic

medium (elastic plate) when

considering a purely flexural deformation [45]. Then one can use directly the expressions for the components of the strain tensor (in linear approximation) derived in Ref. [55], see Eq. (1.4) therein: ~2~

.

Uxx = --Zo 03X2

'

Uyy

ux~=uy:-O,

~

.

--g & 2

'

~ blxy

.

-z 0x0y

uz~=z~ 0x 2 + - ~

(10.25)

(10.26)

The relative dilatation of the lower and upper bilayer surfaces, al and c~2, and the change in its thickness, Ah, are related to the components of the strain tensor by the expressions [45]: hi2 0(, 1

=

( U x x "Jr" IAg),),) I z=-h/2

"

o~,- - (uxx + u,,y) " I :=-~,~ "

Ah-

I

UZZ. dz

(10.27)

-hi2

Substituting uxx, Uyy and u:~ from Eqs. (10.25)-(10.26) into Eq. (10.27) one obtains o~2= - o h ,

Ah = 0 ,

(10.28)

which means that the lower surface is extended, the upper surface is compressed and the membrane thickness does not change (in linear approximation) during the considered flexural deformation. The stress tensor for an incompressible isotropic elastic medium is [55]

rij = 2~, uij

(i, j = x, y, z)

(10.29)

where, as usual, ~ is the coefficient of shear elasticity. The free energy per unit area of the bilayer is given by a standard expression from the theory of elasticity [55]:

Chapter" 10

440 h/2

(~o.3o)

AWin = 1 I Z Tijuijdg -h/2 i,j

Next we substitute Eqs. (10.25), (10.26) and (10.29) into Eq. (10.30) and after some transformations we obtain [60,55]:

(~o.31)

Awin = 3 I~h3H2 - -6l l~h3 K

where we have used the fact that in linear approximation the mean and Gaussian curvatures can be expressed as follows:

2H- 0-~+

o.q v 2'

K-------

~X 2 & - 4-

(10.32) ~,OqX&

Equation (10.3 l) gives the sought-for contribution of the bilayer interior to the work of flexural deformation.

The flexural deformation of the bilayer surfaces is accompanied by a change in the energy of the system, which can be derived from the thermodynamic expression for the work of interfacial deformation per unit area [cf. Eq. (3.1)]"

dws = ~_~[~dCZk + ~kdflk + BkdHk + OkdDk]

(10.33)

k=l,2

Here k = 1 for the lower bilayer surface and k = 2 for the upper bilayer surface (Fig. 10.5); O~kand flk are the relative dilatation and shear of the k-th surface; these deformations are related to the trace and deviator of the two-dimensional strain tensor (Uuv = durst, see Section 4.2.2):

ak=a~Vu~v,

flk=q~Vu~v,

forz=(--1)kh/2

(l.t, V=x, y)

(10.34)

(k = 1, 2); a uv is the metric tensor in the respective curved surface, and qUV is its curvature deviatoric tensor [cf. Eq. (4.14)]:

q~V = (b~V _ Hk a~V)/Dk

(10.35)

where b ~v are components of the curvature tensor, Hk = 89 1) + c~2)) and Dk - - 1~ (C~I) _ C~2) ) (2)

are the mean and deviatoric curvatures of the respective bilayer surfaces with c~~) and c k

being the two principal curvatures; Bk and Ok in Eq. (10.33) are the respective surface bending and torsion moments; ~ and ~k are the thermodynamic surface tension and shearing tension,

Mechanics of Lipid Membranes and Interaction between Inclusions

441

which are related to the respective mechanical surface and shearing tensions, o'~, and r k , as follows [see Eq.(4.81)]: I B~H~ + l |

~

_

rl~ + 51 BkD~ + 7l|

'

(10.36)

In linear approximation Eq. (10.34)-(10.36) considerably simplify. First we note that Ah = 0, cf. Eq. (10.28), and then [45] -HI = H2 = H ,

-D1 = D2 - D ,

(10.37)

where H and D refer to the bilayer midsurface; the curvatures of the two bilayer surfaces have the opposite sign because the z-projections of the respective outer surface normals, directed from the chains toward the head-groups, have the opposite signs. In general, D 2 = H 2 - K, see Eq. (3.4); then using Eq. (10.32) in linear approximation one obtains

4 ax 2 o>

tax#

/

(lO.38)

Further, in linear approximation the components of the curvature tensor are

a2C" b"V= buy = Ox, OXv

(Xl ----X; X2 ----y)

(10.39)

As known (Section 3.1.2), 2H and K, are equal to the trace and determinant of the curvature tensor b or, cf. Eqs. (10.32) and (10.39). Substituting Eqs. (10.25) and (10.35) into Eq. (10.34) and using Eqs. (10.32) and (10.37)-(10.39) one can derive [45] ak = - h H k ;

flk = -hDk ;

k = 1, 2.

(10.40)

Not only the bilayer as a whole, but also its surfaces can be considered as Helfrich surfaces, for which the energy of flexural deformation (per unit area) can be expressed in the form w f - 2kc(H- H0) 2 4. k2 K

(10.41)

m

Here k~ and k C are the bending and torsion curvature elastic moduli for the film surfaces" H0 is their "spontaneous curvature". Differentiating Eq. (10.41) and using the identity K - H 2 - D: one obtains

Chapter10

442

Bk .

.

I/ f/1D H=H~.

B0 .+ 2(2kc + k .C)Hk,

|.

-

-~

H D=Dk

2kcD~

(10.42)

where B0 =-4kcHo is the bending moment of a planar bilayer surface, see Eq. (3.10). The expressions for Bk and Oh in Eq. (10.42) can be considered as truncated power expansions for low curvature. Of course, B0 cannot depend on Hk, but it can depend on the surface dilatation ak [45]: Bo(o0- Boo + ~ ~

o~k +O(o~ 2)

(10.43)

a=0

Next, combining Eqs. (10.40) and (10.43) with Eq. (10.42) we obtain the linear approximation for the bending moment B~ : Bk = Boo + (4kc + 2k C - Boh)H~" "

B o' - ~(3B~ -~

-

(10.44)

The mechanical surface tension o'~, also depends on both dilatation and curvature; in linear approximation one obtains ok = cy0 + Eoo~k + l BooHk, where the last term is often called the "Tolman term", see Eq. (4.87). Then in linear approximation Eq. (10.36) acquires the form

= Cro+ Ec,o~k+ BooHk,

i BooD~, ~ = --i

(10.45)

where we have used Eqs. (10.42) and (10.44) and have substituted 77 = 0 for a fluid interface (isotropic two-dimensional stress tensor). Further, using Eqs. (10.37), (10.40) and (10.45) we obtain the contributions from the dilatation and shearing into the work of surface deformation: O~k

Z I rkd~l~-(EGh2-BOOh)H2" k=l,2 0

/~/,.

E I~kdfll'----1Bo0hD2

(10.46)

k=l,2 0

Likewise, using Eqs. (10.37), (10.42) and (10.44) we obtain the contributions from the bending and torsion into the work of surface deformation: Hk

E k=l,2

I Bk dHk - (4kc + 2k~.- B'oh)H 2" 0

E

IQk dDk --2k~ .D2

(10.47)

k=l,2 0

Next, we integrate Eq. (10.33) and substitute Eqs. (10.46)-(10.47); using again the identity D 2 = H z - K we present the result into the form [45]:

Mechanics of Lipid Membranes and Interaction between Inclusions

443

Aws = [4kc- (v3 Boo + B o)h + EGh2]H 2 + (2 k c + 7I Booh)K

(~0.48)

Finally, we substitute Eqs. (10.31) and (10.48) into Eq. (10.23) and compare the result with Eq. (10.22); thus we obtain the sought-for expressions for the curvature elastic moduli of the bilayer as a whole [45]" kt - 2k~. - ( ~3Boo + 71 Bo, )h + -s EGh2 + 71~h 3

_ _ i Boohkt - 2kc + -55_

-~2h 3

(bending elastic modulus)

(10.49)

(torsion elastic modulus)

(~o.5o)

The first terms in the right-hand sides of Eqs. (10.49) and (10.50), 2kc and 2 k C , obviously stem from the bending and torsion elasticities of the two bilayer surfaces" the terms with Booh and B0 h are contributions from the bending moment (spontaneous curvature) of the these surfaces; 1 Ec, h 2 in Eq. (10.49) was first obtained the contribution of the surface (monolayer) elasticity -~

by Evans and Skalak [15], who derived kt = 7 Ech 2 by means of model considerations; the term proportional to ,~h3 accounts for the elastic effect of the bilayer interior (the hydrocarbon-chain region). In Ref. [45] typical parameter values have been used to estimate the magnitude of the contributions of the various terms in Eqs. (10.49) and (10.50): h = 3.6 rim, EG = 40 mN/m, X= 3 • 106 Pa, Boo __ 7 • 10-~ N, kc - 4 x 10-21 J; to estimate k C the relationship k C = - 71

kc

from Refs. [61,62] can be used; then k--,. = - 5 x 10-22 J- finally, B 0 can be assessed by means of the connection between B0 and the AV (Volta) potential [62] assuming that the value of B 0 is determined mostly by electrostatic interactions [45]" " c~B~ -~---~--AV oAV~= - 3 . 2 x 1 0 - ' 1 N B~ = o~---~- 4zr cga

(10.51)

At the last step experimental data for the dependence of AV vs. a for dense lipid monolayers have been used: from Fig. 3 in Ref. [63] one obtains AV-- 350 mV, OAV/Oo~ ~ - - 3 2 3

mV;

dielectric constant ~ = 32 has been adopted for the headgroup region. [When using Eq. (10.51) AV must be substituted in CGSE units, i.e. the value of AV in volts must be divided by 300.]

444

Chapter" 10

With the above parameters values one can estimate the magnitude of the various terms in Eqs. (10.49) and (10.50); below we list their values (x 10-19 J)" 1.83" 2kc-~ 0.08"

kt ~

k- t

--0.99;

1 B 0')h---1.31" -(~-3B0o+ -~

2k- C ---0.04;

5I Booh ~ 1.26;

~-EGh 2 ~ 2 . 5 9 ;

5l,~h3--0.47

--~/~h 3 -0.23

(10.52) (10.53)

One sees that the value of kt is determined mostly by the competition between the positive surface stretching elasticity term, 1 E G h 2 , and the negative surface bending moment term l B o')h. On the other hand, the value of k t is dominated by the positive surface -(-~ Boo + ~bending moment term, 71Booh. The chain elasticity contribution or fl,h 3 is about 25% of the magnitude of kt and k t , which is consonant with the discussion in Ref. [56]. In summary, the "sandwich" model provides expressions for calculating the stretching, bending and torsion elastic constants, Ks, kt and k t , in terms of the chain elasticity constant ,~ and of the properties of the respective lipid m o n o l a y e r s

at oil-water interface (EG, Boo, B o , k,., k C, etc.),

see Eqs. (10.21), (10.49) and (10.50). The estimates show that the quantitative predictions of the model are reasonable, although additional experiments are necessary to determine more precisely the values of the parameters.

10.3.

DESCRIPTION OF MEMBRANE DEFORMATIONS CAUSED BY INCLUSIONS

1 0 . 3 . 1 . SQUEEZING (PERISTALTIC) MODE OF DEFORMATION." RHEOLOGICAL MODEL

The deformation of a lipid bilayer around a cylindrical inclusion (say a transmembrane protein), having a hydrophobic belt of width 10, represents a variation of the bilayer thickness at planar midplane (Fig. 10.6). Such a mode of deformation corresponds to the s q u e e z i n g (peristaltic) mode observed with thin liquid films [64]. This type of deformation appears if there is a "mismatch", h~. - ( l o - h ) / 2 r O, between the hydrophobic zones of the inclusion and bilayer; here, as usual, h is the thickness of the non-disturbed bilayer far from the inclusion. The extension of the lipid hydrocarbon chains along the z-axis is greater for molecules situated closer to the inclusion (Fig. 10.6). The chain region of a separate lipid molecule (one of the

Mechanics of Lipid Membranes and Interaction between Inclusions

445

many small rectangles depicted in Fig. 10.6) exhibits an elastic response to extensioncompression; therefore it can be modeled as a stretchable elastic body of fixed volume. On the other hand, lateral slip between molecules (neighboring rectangles in Fig. 10.6) is not accompanied with any elastic effects because of the two-dimensional fluidity of the membrane. Both these properties are accounted for in the following mechanical constitutive relation for the stress tensor rij [45]:

cgz '

"cij - -P(~ij,

(i, j ) r ( z , z )

(10.54)

i, j - x, y , z

Here ~j is the Kroneker symbol, p has the meaning of pressure characterizing the bilayer as a two-dimensional fluid; u: is the z-component of the displacement vector u; the coordinate system

is depicted in Fig. 10.6. The above relationship between ~,z and Ouz/Oz is a typical

constitutive relation for an elastic body, cf. Eqs. (10.11) and (10.29). On the other hand, the tangential stresses ~;j (i, j = x, y) in Eq. (10.54) are isotropic as it should be for a twodimensional fluid. The condition for hydrostatic equilibrium and the continuity equation yield [551: ~

Oxi

=0,

j - 1,2,3;

V.u-O

(Xl = x, x2= y, x3= z),

(10.55)

t; rc

m

--"-

Fig. 10.6. Sketch of the deformation around a cylindrical inclusion (membrane protein) of radius r~ and width of the hydrophobic belt /o; h is the thickness of the non-disturbed bi|ayer; ~" is the perturbation in the bilayer thickness caused by the inclusion; h~. is the mismatch between the hydrophobic regions of the inclusion and the bilayer; n and fi are unit vectors normal to the membrane surface and inclusion surface, respectively; m is unit vector in direction of the bilayer surface tension.

446

Chapter 10

where rij is to be substituted from Eq. (10.54); as usual, summation over the repeated indices is assumed. In this way, the mechanical problem is formulated: Eqs. (10.55) represent a set of 4 equations for determining the 4 unknown functions ux, u>., u~ and p. Below, following Ref. [45] we present the solution of this mechanical problem.

] 0.3.2. DEFORMATIONS IN THE HYDROCARBON-CHAIN REGION Considerations of symmetry imply that uz must be an odd function of z which has to satisfy the boundary condition

u~ = ~(x, y) for z = h/2,

(10.56)

where z = ~(x,y) describes the shape of the upper bilayer surface, see Fig. 10.6. Equation (10.55) forj = z, along with Eq. (10.54), yields

c?2uz = 0

(10.57)

&2

Combining Eqs. (10.56) and (10.57) one obtains 2z

(~0.ss)

< = i, ((x,y) The continuity (incompressibility) equation V-u = 0 can be expressed in the form VlI'UII -"

~Hz &

(10.59)

where UlI is the projection of the displacement vector u in the plane xy and, as usual, Vn is the gradient operator in the same plane: UII-- exHx

+ eyU,. "

VII = e x - x - + e > .

Ox

_,

Oy

(10.60)

One can seek Un in the form [45]

uu = -VII g(x, y,z)

( 10.61 )

where g is an unknown scalar function. The substitution of Eqs. (10.58) and (10.61) into Eq. (10.59) yields an equation for determining g :

g~g -h~

(10.62)

Mechanics of Lipid Membranes and Interaction between Inclusions

447

In addition, substituting Eq. (10.54) into (10.55) for i, j = x, y gives Vilp = 0, and consequently p is independent of x and y. Further, since zij expresses a perturbation and the bilayer far from the inclusion(s) is not perturbed, one can conclude that p is identically zero [45]" p-0

(10.63)

To determine g from Eq. (10.62) one can use the boundary condition of impermeable inclusion surface [45]" fi.un =

c?g =0 3~

(at the inclusion surface)

(10.64)

Here fi is an outer unit normal to the inclusion surface (Fig. 10.6) and Og/O~ is a directional derivative. Additional boundary conditions, which have to be imposed at the surfaces of the lipid bilayer, are considered below.

10.3.3. DEFORMATION OF THE BILAYER SURFACES

Since the bilayer surfaces are symmetric with respect to a planar midsurface (Fig. 10.6), it is sufficient to determine the shape z = ~(x,y) of the upper bilayer surface. We do not impose any restrictions on the number and mutual positions of the cylindrical inclusions. The mechanical description can be based on the theory of liquid films of uneven thickness developed in Refs. [51,65]. In particular, we will employ the equation for the balance of all forces applied to the upper surface of such a film (the interfacial balance of the linear momentum) [51,65]" V ~ . ~ - n - ( T i - TII)I z=h/2 + I-I(e:-n)ez = 0

(10.65)

where _~ is the surface stress tensor (the usual surface tension is equal to a half of the trace of _~), V~ is the two-dimensional gradient operator of the film surface, which is to be distinguished from the gradient operator Vii in the plane xy (the midsurface)" n is the running outer unit normal of the bilayer surface, which can be expressed in the form [65] n = ( e : - Vn~)(1 + IVII~'I2)-1/2"

(10.66)

T~ and Tn are respectively the stress tensors inside and outside the bilayer, which can be expressed in the form [45]"

T ~ - - PN e= e : - Pr U~ + _~"

TII =-PoU 9

(10.67)

448

Chapter 10

U is the spatial unit tensor, UH = ex ex + e,, ey is the unit tensor in the plane xy, Po is the pressure in the aqueous phase, PN and PT characterize the stresses in a plane-parallel bilayer (see Fig. 10.3), the components vii of the tensor I: are defined by Eq. (10.54)" I: accounts for the additional stresses due to the deformation in the bilayer. The disjoining pressure H is due to the conventional surface forces (like the van der Waals ones), whereas _!: accounts for the elastic stresses. The general form of the surface stress tensor is [66,67] (10.68)

cy = a~avCr uv + a ~ n o 'u~n)

see Eq~ (4.26); here and hereafter the Greek indices take values 1 and 2, summation over repeated indices is assumed, a~ and a2 are vectors of the surface local basis, which at each point are tangential to the bilayer surface; see Refs. [68,69] about the formalism of differential geometry; o"v and o"~n~ are the respective components of the surface stress tensor; o"~n~ are known as surface transversal shear stress resultants [15]. Next, using Eqs. (10.66)-(10.68) one can obtain the normal and tangential projections of the vectorial balance, Eq. (10.65), with respect to the bilayer surface z = ~(x,y) [45]" b~vcr TM+ o'V(n),v - [(PN - PT)IVII~ "12 - 17I]( 1 + ]VII~'12) -1 4- No + n-_x.n

O,,V._bVuo,U(n)=(pT_pN_H)(1 + IVII~-I2)-I/2 ~-.v + n._,!:.a v

(10.69) ( v = 1,2)

(10.70)

As before, buy are components of the curvature tensor, the comma denotes covariant differentiation [68,69], and H0 = P o - PN is the disjoining pressure of the non-deformed planeparallel bilayer. The normal projection of the stress balance, Eq. (10.69), presents a generalization of the Laplace equation of capillarity; it will be used below to determine the shape of the bilayer surface. The tangential projection, Eq. (10.70), allows one to determine the variation of the surface tension along the deformed surface; it will be utilized below to calculate the interaction between two inclusions. First of all, we transform and simplify the normal projection of the surface force balance, Eq. (10.69). Because of the two-dimensional fluidity of the lipid bilayer, the surface stress tensor ~ must be tangentially isotropic [45]" o " v - o a uv

(aUV _ aU.aV)

( 10.71 )

Mechanics of Lipid Membranes and Interaction between Inclusions

449

where o is the usual scalar surface tension and a/~v are components of the surface metric tensor [68,69]. Then one obtains (10.72)

b~2vO"uv- a~'Vb~,va - 2 H a = a V ~I (

At the last step we have used the fact that in linear approximation (for small g") the mean curvature is determined by the expression 2H = g~i ~', cf. Eq. (10.32). In addition, just as in Chapters 7 and 9 for small deformations we will expand the disjoining pressure in series keeping the linear terms: FI = FI0 + 2FI'~'+ ..."

FI' = (dH/dh)l r

(10.73)

To express the transverse shear stress resultants will employ an equation, which stems from the surface balance of the a n g u l a r momentum, see Eq. (4.49) and Refs. [ 15,66,67,70]" O p(n) =

-MUV'v

(10.74)

Here M ~v are components of the tensor of the surface moments (see Section 4.2.3), which can be expressed as a sum of isotropic and deviatoric parts [70,71]" M~V= 71 (B a uv + 0 q~V)

(10.75)

see also Eq. (10.35). For a Helfrich interface the bending and torsion moments, B and O, are given by Eq. (10.42)" then Eq. (10.75) acquires the form MUV= [(2k~. + k c )H + Bo/2]a ~ v - k c D q uv ,

(10.76)

In view of Eq. (10.43) one obtains p

t

Bo = Boo + B o a = Boo - (2 B o/h) ~ ,

a=-2~/h

"

(10.77)

the last expression for the dilatation o~ of the bilayer surface follows from Eqs. (10.11), (10.16) and (10.58). Next, we differentiate Eq. (10.76) with the help of Eqs. (10.35), (10.77) and the identity b ~v v = 2 H "~ [70] as a result we obtain [45] ov(n~ - - m U V ' v

- -2kc H'~ + ( Bo/h)~'u

(10.78)

Further, we substitute Eqs. (10.54), (10.58), (10.72), (10.73) and (10.78) into the normal stress balance, Eq.(10.69), to obtain its linearized form [45]

450

Chapter 10

t~ 0 V 2u ~'- kcV.4 ~'= 2(2X/h - 1-I' )~" 9

( ~o =- C~o+ Bo/h)

(10.79)

where or0 is the value of o for the non-disturbed plane-parallel bilayer. Equation (10.79) plays the role of a generalized Laplace equation for the bilayer surfaces.

10.3.4.

THE GENERALIZED LAPLACE EQUATION FOR THE BILA YER SURFACES

Equation (10.79) is a fourth order differential equation, which can be represented in the form [45]" (VII - q22 ) ( V n2 - q()~'= 0

(10.80)

Here q2 and q22 are roots of the biquadratic equation

kcq 4 - ~0q 2 + 2(2/~/h - H ' ) = 0

(10.81)

which gives --.02 _ 8kc(2X/h - H' )] 1/2 } /(2kc) q122 = { 6 0 _+ [ O"

(10.82)

Depending of the sign of the discriminant in Eq. (10.82), Eq. (10.81) may have four real or four complex roots for q. Complex q leads to decaying oscillatory profiles for ~(x,y), resembling those obtained in Ref [38] for model inclusions of translational symmetry. For not-too-flaccid membranes the discriminant in Eq. (10.82) is positive, .--2 O" 0 > 8kc(2X/h- 1-1' )

(real roots)

(10.83)

In such case two positive roots for q2 are obtained and the bilayer profile around an inclusion,

~(x,y) in Fig. 10.6, will decay without oscillations. Using Eq. (10.4) one can estimate that H' is typically about 2 x 1013 N/m 3 which is negligible compared with 2X/h. Indeed, with h = 3 nm and ~ = 2 x 106 N/m 2 one obtains 22/h = 1.33 x 1015 N/m 3. Further, assuming kc -- 4 x 10-21 J

(see Ref. [72]) from Eq. (10.83) one obtains c~0 > 6.5 mN/m. The latter inequality can be fulfilled for not-too-flaccid bilayers. Note that the bilayer surface tension o0, as introduced in Section 10.2.1 above, is usually much larger than the total tension of the bilayer, ~,, see the discussion after Eq. (10.7); the importance of the surface tension o'0 is discussed also at the end of Appendix 10A.

Mechanics of Lipid Membranes and Interaction between Inclusions

451

Following Ref. [45], below we will restrict our considerations to the case of

real q2,

in which

Eq. (10.83) is satisfied; the case of complex q2 is also possible and physically meaningful. As in Chapter 7, here q-l has the meaning of a characteristic capillary length determining the range of the deformation around an inclusion, and in turn, the range of the lateral capillary forces between inclusions (see below). With the above values of h, ~ and k~., and with c70 = 20 mN/m from Eq. (10.82) two possible decay lengths can be calculated [45]" q l l = 2 . 7 n m

and

q2-1 - 0 . 4 5 rim. The second decay length, q-~2, is smaller than the size of the headgroup of a phospholipid molecule (typically 0.8 nm); for that reason this decay length has been disregarded in Ref. [45]. Below we will work with the other decay length, that is with q2= q2 = { c7~ _ [c72_ 8kc(22/h -

FI')]~/2il(2kc)=

42/(hc70)

(10.84)

At the last step we expanded the square root in series for small kc. Disregarding the solution of Eq. (10.80) for q = q2 means that we have to seek ~"as a solution of the equation [45] V i2 ~"= q2 ~-

(10.85)

where q is determined by Eq. (10.84). All solutions of Eq. (10.85) satisfy also Eq. (10.80). The boundary conditions for Eq. (10.85) are ~"= hc at the lipid-protein boundary and ~'-->0 for r--->o,,, see Fig. 10.6. Note that Eq. (10.85) is almost identical to Eq. (7.6) with the only difference in the definitions of q. The account for the compressing stresses at the bilayer surfaces and the elastic stresses in the bilayer interior (see Section 10.2.1) leads to the appearance of the bilayer surface tension Go and the chain shear elasticity /~ in the expression for the decay length: q

-1

~ (h c~0/4/~)

1/2

Comparing Eqs. (10.62) and (10.85)one can determine g [45]" g - 2~'/(hq 2) + f(x, y, z),

(10.86)

V~,/= 0

(10.87)

Here f is unknown function (to be determined from the boundary conditions) which satisfies Eq. (10.87). Below we will determine ~" and f for the cases of one and two inclusions incorporated in a lipid membrane.

Chapter 10

452

10.3.5. SOLUTION OF THE EQUATIONS DESCRIBING THE DEFORMATION Single cylindrical inclusion. In this case ~"depends on the radial coordinate r (Fig. 10.6) and Eq. (10.85) acquires the form of a modified Bessel equation: r

r dr

= q2~,

(10.88)

---dTr

The boundary condition for fixed position of the contact line implies ~"= hc = const.

(at the contact line)

(10.89)

The solution of Eq. (10.88), which satisfies Eq. (10.89) and decays at infinity is [45]:

=

h~.

K0(qr),

r > r~.

(10.90)

K0(qrc) where rc is the radius of the cylindrical inclusion (Fig. 10.6) and Ko is the modified Bessel (Macdonald) function of zeroth order [73-75]. To completely quantify the deformation we have to determine also the components

Ur

and uz of

the displacement vector u. A substitution of Eq. (10.90) into Eq. (10.58) directly gives uz. To find Ur we first substitute Eq. (10.86) into the boundary condition (10.64): ( 2 d~"

c?f/

- 0

(10.91)

~q2-~r + c?r r=rc Then from Eq. (10.87) we obtain [45] f = A lnr

(10.92)

The integration constant A can be determined from the boundary condition

Kl(qrc) (-~r )r=rc - - t a n w c - - q h ~ . - -K0(qrc)

(10.93)

where gtc is the surface slope at the contact line (Fig. 10.6). Next we substitute Eqs. (10.92) and (10.93) into Eq. ( 10.91 ) and determine A - (2re tan ~)/(hq 2)

(10.94)

Mechanics of Lipid Membranes and Interaction between hzclusions

453

A substitution of ~"from Eq. (10.90) and f from Eq. (10.92) into Eq. (10.86) gives the function g, which is further substituted in (10.61) to obtain [45] Ur =

----K l ( q r ) - rc K l (qrc ) , qhKo(qr~. ) r

(10.95)

uo-O

Finally, the components of the strain tensor (in cylindrical coordinates) can be obtained using standard formulas from Ref. [55]:

Urr =

aU r

;

UO0 =

Ur

c?r

Couple

;

Uzz =

r

of identical

c?u. az

cylindrical

2~"

~ = -h

inclusions.

"

Urz =

z a~"

;

hOr

UrO = UOz --

0.

(10.96)

In this case it is convenient to introduce

bipolar (bicylindrical) coordinates as explained in Section 7.2.1, see Eq. (7.25). Then Eq. (10.85) acquires the form 2 +--~(_0 ~ 2 ~2' ) --(qa) 2 ~'(T, o9) " (cosh "c - cos co) 2 / ~--7--~ dT

a

= (L2/4- rc2) 1/2

(10.97)

where L is the distance between the axes of the two cylindrical inclusions, see Fig. 10.1. In contrast with Section 7.2.1, here in general (qa) 2 is not a small parameter, and therefore we cannot use asymptotic expansions to find analytical solution of Eq. (10.97). The latter can be solved by numerical integration. The domain of integration is a rectangle in the co't-plane bounded by the lines co = +n; and "c = +zc, where z,. = ln[(a + L / 2 ) / r c ] , cf. Eq. (7.57). Owing to the symmetry one can carry out the numerical integration only in a quarter of the integration domain: 0 < "t-< rc and 0 < co < re. The boundary conditions are:

~ I~-=~.==

2~dl fi

(1 + IVII~'I2)-I/20 "

(10.108)

Bo/h) ~ dl fi

(10.109)

c

F (B) - 2(kcq 2 -

IVn~'l 2 (1 + 17ii(12) -1

c

Note that the surface tension o" in Eq. (10.108) can vary along the contact line; the dependence of cy on deformation can be derived from the tangential projection of the stress balance, Eq. (10.70). To do that we first note that the vectors of a covariant local basis in the upper bilayer surface can be expressed in the form [65]

456

Chapter 10

a v = ev+ ez ~'v

( v = 1,2)

(10.110)

where el and e2 are the vectors of the local basis in the plane xv. Recalling that p = 0, with the help of Eqs. (10.54), (10.58), (10.66) and (10.110) one can derive [45] n. ~. av = (4X/h)~" ~',v (1 + IVII~'I2)-1/2

(10.111 )

Next, we substitute Eqs. (10.71), (10.73), (10.78) and (10.111) into the tangential balance, Eq. (10.70); as a result we obtain

VHcr+ (kcq 2 - Bo/h)(b.Vii ~) = (PT-- Po) VII(+ (2/]Jh - I-[')Vi~ "2

(10.112)

where higher order terms have been neglected. One can employ Eq. (10.39) to derive: b-Viii" --

(VllVli~).Vli ( = 1 VII IVil~'l2. On substituting the last result in Eq. (10.112) and integrating one obtains the sought-for expression for the variation of the bilayer surface tension o caused by the deformation [45]:

(7- (70- 7~ (kcq"-* Bo"/h)IVH~I 2 + (PT-- Po)~+ ( 2 M h - H')~ "2

(10.1 13)

Now we are ready to bring the expressions for F ((r) and F (B) in a form convenient for calculations. As we work with small deformations, we will keep only linear and quadratic terms with respect to ~"and its derivatives. We choose the x-axis to connect the two inclusions, as it is in Fig. 7.18; then F ('r) and F (8) have non-zero projection only along the x-axis, whose unit vector is denoted by e~. In addition, e~. fi = cos(/), where 4) is the azimuthal angle providing a parametrization of the contact line (see Fig. 7.20). Then from Eqs. (10.108), (10.109) and (10.113) we obtain [45] F~ (r) -ex-F(rr) = -(o0 + kcq 2 - Bo/h) 2~dl IVii~'lZcosO

(10.114)

c F~!B> - e, . F Cg) --(2B'o/h - 2k,.q 2) ~dl IV~'I2 cos0

(10.1 15)

c

It is convenient to introduce bipolar coordinates (09, r) and to use the relationships

d l - z,.dco,

c o s 0 - (coshr,.cosco- 1)Z,./a,

I

1 (~~ /z-=7:,

VIIi" ~'=z',--ez- Z

(10.116)

Mechanics of Lipid Membranes and Interaction between Inclusions

457

where Zc = a/(coshrc - cost0 ); cf. Eqs. (7.27) and (7.131). Finally, combining Eqs. (10.114) (10.116) we obtain the non-zero x-component of the lateral capillary force between the two inclusions [45]: Jr

.

.

.

(fro - kcq ) dco(cosh r c cosco- 1).

. a

where fro = q, + Bo/h

2

0

(10.117) --T c

has been introduced by Eq. (10.79). Note that the force Fx can be

attractive or repulsive depending on whether c~0 > kcq 2 or c~0 < kcq 2. The respective interaction energy can be obtained by integration: oo

An(L)- IFx(L)dL

(10.118)

L

As usual, L denotes the distance between the axes of the two cylindrical inclusions, see Fig. 10.1 and Eq. (10.97). To obtain numerical results one can first calculate the function ~'(c0,r) as explained in Section 10.3.5, and then to substitute (O~lOr) .... in Eq. (10.117) to calculate F~. (and further Aft) by means of numerical integration. Another (equivalent) approach, which yields directly expressions for calculating the interaction energy AE~, is described in the next section.

10.4.2.

THE ENERGY APPROACH

Mechanics and thermodynamics provide general expressions for the variation of the grand thermodynamic potential, ~if~, rather than for f~ itself. One can find f~ by integrating 8f~, however such an integration is straightforward only for uniform fluid phases [78,79] or isotropic elastic bodies [55]. In the case of curved interfaces 8f~ depends on three independent variations: ~', ux and u,., see e.g. Ref. [80], Eqs. (5.7)-(5.8) therein. In our case of a lipid bilayer, the solution of the mechanical problem for the hydrocarbon-chain interior, along with the boundary conditions at the bilayer surfaces, leads to connections between ux, Uy and ~'. These connections enable one to obtain a posteriori an expression for f~ in terms of ~" only. As demonstrated in Ref. [45] as a

Chapter 10

458

starting equation one can use the expression for the grand thermodynamic potential of a thin liquid film of surface tension cy and reference pressure of the film interior

~-2Ids~s

Pr,

cf. Ref. [81 ]"

I d V P r - IdVPo V,n Vout

(10.119)

Here S stands for the bilayer surface, and Vin and

Vout denote,

respectively, the volume of the

bilayer interior and of the outer aqueous phase. Equation (10.119) expresses the grand thermodynamic potential for a

lipid bilayer

if cy is substituted from Eq. (10.113). The latter

equation is nothing else than the integrated tangential stress balance at the bilayer surface, Eq. (10.70), in which the chain elasticity is involved through the elastic stress tensor 3; see Eqs. (10.54) and (10.58). In other words, Eq. (10.113) contains in a "condensed" form the information about the bilayer mechanics for the "squeezing" mode of deformation. To demonstrate that we first transform the volume integrals:

~dVP:r + IdVPo- 2~ds Vm

Vout

dzPr + ~dePo

S0

0

(10.120)

h/2+g

where So denotes the projection of the bilayer surface S on the bilayer midplane; in other words, So is the whole xy-plane except the area excluded by the incorporated proteins; the exact position of the plane z = z~ is not important because it does not affect the final result. In addition, using Eq. (10.113) we obtain

ds cy = I ds (1 + IV,,~'l2) 1,2[o.0 S

_

, -~, ( k c q2 - B0/h)lV.r

[2

+(Pr-Po)~+(2,Vh-n

)gel (10.121)

So

The substitution of Eqs. (10.120) and (10.121) into Eq. (10.119), after some transformations, yields a relatively compact expression for the bilayer grand thermodynamic potential [45]:

= ~ds [(c~0 - kcq 2) IVIt~l2 + 2(2fl/h

--

1-It)~-2] + const.

(10.122)

So

The validity of Eq. (10.122) can be confirmed by checking the correctness of its implications. First of all, imposing the requirement ~ to be minimum for any variations ~" with

fixed

boundaries, from Eq. (10.122) one derives [45]"

8 o V~ ~- k,.qZV~ ~'= 2(2Mh

- II')~"

(10.123)

Mechanics of Lipid Membranes and Interaction between Inclusions

459

The last equation is equivalent to Eq. (10.79) in view of Eq. (10.85). Moreover, in Ref. [45] it is proven that using the formula Fx = -Sf~/~L and variations at movable boundaries, from Eq. (10.122) one can deduce the expression for the lateral force, Eq. (10.117). 2 With the help of the identity IV,~'I2 - Vn.(~'V.~') - ~"VIE ~"and the Green integral theorem [69]

Eq. (10.122) can be further transformed:

f~= 2(~o-kcq2)~dI (-fi).(~V.~)- Ids[~o viz ~-kcq2V,Z ~-2(2X/h - FI')~']~" C

(10.124)

S0

The multiplier 2 before the curvilinear integral comes from the two identical contours corresponding to the two inclusions. The integrand of the surface integral in Eq. (10.124) is zero owing to Eq. (10.123). Therefore, Eq. (10.124) can be presented in the simple form [45] ff2(L) = 4rt( ~o - kc q2)rc he.tanqJc(L)

(10.125)

where tanWc(L)- 2rcrc ~dl (-ft. V,I~')- ~ 1 i dco(O~"/r ~,c?'r C

-Jr

(10.126) =v c

expresses the average slope of bilayer surface at the contact line, cf. Fig. 10.6. Then the energy of interaction between the two inclusions can be written in the form Aft(L) - ~(L) - f~(oo) = 4rc(c70 -kcq2)rchc[tanUgc(L)-tanUgc(oo)]

(10.127)

p

If the interfacial curvature effects are negligible (B 0 = 0, kc = 0), then Eq. (10.127) reduces to Eq. (7.106) with an additional multiplier 2 accounting for the two contact lines per inclusion. The slope angle at infinite separation, W,.(oo), can be identified with the angle qtc in Eq. (10.93). To determine We(L) one has to first calculate ~'(c0,r) by numerical integration of Eq. (10.97), and then to carry out numerically the integration in Eq. (10.126). Alternatively, one can use the asymptotic formula

i qrcK ~(qL)

tanq'c(L) = qhc K~(qrc)-5-

(10.128)

Ko(qr~.)+Ko(qL) which has been derived in Ref. [45] utilizing the method of reflections [82]. Substituting Eqs. (10.93) and (10.128) into Eq. (10.127) one obtains an asymptotic formula for Af2(L) [45]:

Chapter 10

460

Aft(L) - 4Jr (c~o -kcq-)qrch~ 2

K l ( q r c ) - s q1r c K o (qL) Ko (qrc) + Ko (qL) - K0 KI (qr':) (qrc) 1

(10.129)

The numerical test of Eq. (10.129) shows that it gives Aft(L) with a good accuracy, see the next section.

10.5.

N U M E R I C A L RESULTS FOR MEMBRANE PROTEINS

To illustrate the theoretical predictions in this section we present results from the numerical calculations [45] of the energy of interaction between two membrane proteins incorporated into a flat lipid bilayer. For this purpose parameters of the bacteriorhodopsin molecule, determined by means of electron microscopy [2,83], have been used: rc = 1.5 nm and l0 = 3.0 nm; see Fig. 10.6 for the notation. It is assumed that the hydrophobic o~-helix regions of the bacteriorhodopsin molecule are imbedded inside the lipid bilayer. The following values of the bilayer mechanical parameters have been used: Z = 2

x 10 6

p

N/m 2, (7o = 35 mN/m and B 0 = -3.2

p

p

x 10-ll N" with h = 3 nm one calculates B o/h = -11 raN/m, ~o = (7o + Bo/h = 24 mN/m and q

-1

= 3 nm; in this case the term kcq 2 = 0.4 mN/m is negligible compared to ~Y0.

The mismatch between the height of the cylindrical inclusion, 10, and the thickness of the nondisturbed layer, h, can be characterized by the quantity hc = (lo - h)/2, see Fig. 10.6. In the experiments of Lewis and Engelman [30] l0 was fixed, whereas h was varied by using lipids of various chain lengths. The respective experimental values of h have been used in our calculations: they are denoted on the respective curves in Figs.

10.7a,b, all of them

corresponding to the same value of 10 (to the same protein). The calculated curves of A ~ / k T vs. L/(2rc) for hc > 0 are shown in Fig. 10.7a, whereas those for hc < 0 are shown in Fig. 10.7b. In general, one sees that the strength of the lateral capillary attraction increases with the increase of the magnitude of the mismatch, Ihcl. Af2 can be larger than the thermal energy kT both for hc > 0 and he. < 0, except the cases with too small mismatch (h = 2.6 and 3.4 nm). For the curves with the largest mismatch, those with h = 1.55 nm in Fig. 10.7a and h = 3.75 nm in Fig. 10.7b, the calculated A ~ (5-8 kT at close contact, L = 2rc) is high enough to cause aggregation of the membrane protein molecules. Indeed, only in the latter two bilayers (h = 1.55 and 3.75 nm) did Lewis and Engelman [30] observe protein aggregation.

Mechanics of Lipid Membranes and Interaction between Inclusions

461

0.0

-2.0 I

a f2 kT

(a) -4.0

-

"

2 -

9

[

ao = 35 mN.m-

/

h = 1.55 nm

-6.0

i

0.0

N

~ 3 . 2 x 1 0 -ll

[ I

i

i

I

t

2.0

1.0

J

3.0

L .0

L/2r c 0.0

,

I

'

I

'

I

-1.0

'

.

.

.

.

.

.

(b)

kT -2.0

h=3.75 nm -3.0 0.0

1.0

2.0

3.0

4.0

L/2r c

Fig. 10.7. Calculated in Ref. [45] interaction energy between two inclusions, AlL scaled by kT, vs. the separation L, scaled by rc; the geometrical parameters of bacteriorhodopsin molecule taken from Refs. [2, 83] are rc = 1.5 nm, l0 = 3.0 nm; the values of h correspond to the experiments in Ref. [30]; (a) thinner bilayer, h - l 0 < 0; (b) thicker bilayer, h - l 0 > 0. Comparing the curves with the same magnitude, but opposite signs of hc (he = 0.2 for the curve with h - 2.6 nm in Fig. 10.7a, while he. = - 0 . 2 for the curve with h = 3.4 nm in Fig. 10.7b), one can conclude that Af~ has larger magnitude and longer range in the case of he < 0, that is for a bilayer which is thicker than the inclusion, all other physical parameters being the same. This result is also consonant with the experimental observations of Lewis and Engelman [30]. Illustration of the same effect is given in Fig. 10.8, where it appears as a slight asymmetry of the Af~ vs. h/l,, curves with respect to the vertical line h/l,, = 1. The curves in Fig. 10.8 are calculated for fixed distance,

L = 2r,

corresponding

to

close

contact

between the two

462

Chapter" 10 0.0[

Z

~

"

I

-1.0 ...............~ . - . ; ; : ~ .....................~ k J

-3.0:

-'~

........................................

./.,(;" o'0=32 m N m-y.,,)

'(i'.~ ",:'~

- 5 0 - 00=36 m N m -1.' i -6.0 -

o0=40 m N m-l/

\)~

/"

-7.0 -8.0

0.2

,

,

,

0.4

0.6

0.8

1.0

,

i

1.2

1.4

1.6

h/l o Fig. 10.8. Dimensionless interaction energy, A~21kT, vs. the dimensionless bilayer thickness, hllo, calculated in Ref. [45] for two bacteriorhodopsin molecules at close contact, L = 2rc. The three curves correspond to three different values of the bilayer surface tension Go; the other parameters are as in Fig. 10.7.

0.0 _

.

p~_ ~ ~

~

" ~~

"="

"="

m m

"

--

. . . . . . k9T ..... 99-; ~ / , ~ " ~ - 99- - 9- - - - - - - . . . . . . . . . . . . . . . . . . . . -2.0 t~'--

exact result

/"

,.~ -4.o

h = 1.55 n m rc= 1 . 5 n m

0,

S =-- ( J A W - (JOW -- (JOA

(S - spreading coefficient)

(14.10)

A criterion for instability of an oil bridge was formulated by Garrett [45]: B > 0,

2 2 W-- (JOA 2 B - (JAW + O'O

(B - bridging coefficient)

( 14.11 )

see Figs. 14.10 - 14.12. Below we discuss the physical meaning of the coefficients E, S and B, as well as their relation to the foam-breaking action of oil droplets. As illustrated in Fig. 14.10, the particle entry is related to the disappearance of two interfaces of surface tensions (JAW and (jow, and by the appearance of a new interface of tension (JOA; this is reflected in the form of the definition of the entering coefficient E, Eq. (14.9). If E > 0, then the

Effect of Oil Drops and Particulates on the Stabili O, of Foams

607

E ~(~AW "[- (J'OW --O'oA

__.

I"ENTERING"

Air

--------------------.---------------'----

Fig. 14.10. The entering of an oil drop at the air-water interface leads to the disappearance of two interfaces of surface tensions CrAW and Crow, and to the appearance of a new interface of tension CrOA; this is reflected in the definition of the entering coefficient E. For E > 0 the entering could happen spontaneously if there is no high disjoining-pressure barrier to drop entry.

....

S ~ O'AW -- O'ow -- O ' O A

Air

~ O W

-

r

(YAW

..........

2

2

2

B - o" AW -[- O'ow -- O'OA

Air (5"OA ~ APoA= Po - PA AP~w~PA-Pw= 0 ~

::-Wat er ??-77-_---:~?_-???-----:_--:-:?-?:?-?:

0 < re/2 ~ B

>0

Fig.

14.11. A lens can rest in equilibrium on the air-water interface if only the Neumann triangle, formed by the vectors of the three interfacial tensions, CRAW, crow and CrOA, exists. For S > 0 such a triangle does not exist; then one observes a spontaneous spreading of oil over the air-water interface, instead of lens formation.

Fig. 14.12. Sketch of an oil bridge formed inside a plane-parallel foam film. The bridge can rest in equilibrium if APow = APoA. The latter requirement cannot be satisfied if 0 < 7c/2 as depicted in the figure; this corresponds to positive bridging coefficient, B > 0, see Eq. (14.12); in such a case the bridge has a nonequilibrium configuration and causes film rupture.

608

Chapter 14

entry of the particle is energetically favorable. However, E > 0 does not guarantee drop entry. Indeed, a necessary condition for effectuation of drop entry is rupturing of the asymmetric oilwater-air film, separating the drop from the air-water interface. This asymmetric film could be stabilized by the action of electrostatic (double layer), steric or oscillatory structural forces [6], see Section 14.3.1 below. They create a barrier to drop entry, which can be manifested as existence of a maximum (or multiple maxima) in the disjoining pressure isotherm, see Chapter 5 for details. If this barrier is high enough, drop entry will not happen, despite the fact that it is energetically favorable (E > 0). This situation is analogous to an exothermic chemical reaction, which does not eventuate because of the existence of a high activation-energy barrier. As an illustration, values of the entering coefficient E for a shampoo-type system are shown in Table 14.1; E has a minimum for a given surfactant composition (at Betaine molar fraction about 0.6), which is due to a synergistic effect for the used couple of surfactants [17]. For all compositions of this surfactant blend the entering coefficient E is positive (Table 14.1), which means that the oil-drop entry is energetically favorable; moreover, the other two coefficients are also positive: S > 0 and B > 0. However, in this system the oil drops exhibit only a weak and slow antifoaming action [ 17], which indicates the existence of a barrier to drop entry, as discussed above. An oil drop located at the air-water interface acquires a lens-shape, Fig. 14.11. Such a lens can rest in equilibrium if only the Neumann triangle, formed by the three interfacial tensions, O'AW, O'ow and O'OA, does exist (see Chapter 2 for details). As known, such a triangle cannot exist if one of its sides is longer than the sum of the other two sides, say O'AW> O'OW+ C~OA,that is S > 0, see Eq. (14.10). If the spreading coefficient is positive (S > 0), one observes a spontaneous spreading of the oil over the air-water interface; in contrast, negative spreading coefficient (S < 0) corresponds to the formation of equilibrium oil lenses [51 ]. Often the sign of S depends on whether the interface is preequilibrated with the oil phase (see e.g. Table 14.1): S could be positive for a non-preequilibrated interface, whereas S could become negative after the equilibration. This is due to the decrease of O'AW caused by the molecular spreading of oil. The values of C~AW"without oil" and "equilibrated with oil" in

Effect of Oil Drops and Particulates on the Stability of Foams

609

Table 14.1. Measured interfacial tensions and calculated entering, spreading and bridging coefficients, E, S and B; Seq is the spreading coefficient after the equilibration with oil. The data are obtained in Ref. [ 17] for mixed surfactant solutions of Betaine (dodecyl-amide-propyl betaine) and SDP3S (sodium dodecyl-trioxyetylene-sulfate) at total concentration 0.1 M; the hydrophobic phase is silicon oil. Molar

O'OA

O'OW

O'AW

(YAW mN/m

E

S

B

Seq

Betaine

mN/m

mN/m

(no oil)

(equilibrated with oil)

mN/m

mN/m

(raN/m) 2

mN/m

0.0

19.8

8.45

32.7

25.5

21.4

4.45

749

2.75

0.2

19.8

7.10

30.4

23.9

17.7

3.50

582

3.00

0.4

19.8

6.40

29.0

23.0

15.6

2.80

490

3.20

0.5

19.8

5.70

28.9

23.1

14.8

3.40

476

-2.40

0.6

19.8

5.50

28.8

23.1

14.5

3.50

468

-2.20

0.8

19.8

5.70

29.0

23.5

14.9

3.50

482

-2.00

1.0

19.8

6.65

31.6

26.3

18.5

5.15

651

-0.15

part of

mN/m

Table 14.1 are measured, respectively, before and after dropping locally a small amount of oil on the surface of the investigated surfactant solution. Note that S > 0 automatically implies E > 0, cf. Eqs. (14.9) and (14.10). On the other hand, a high barrier to drop entry can prevent both the drop entering and the subsequent spontaneous spreading of oil. To introduce the bridging coefficient B, Garrett [45] considered the balance of the pressures in the case of an oil capillary bridge formed in a foam film, Fig. 14.12. For the sake of simplicity it was assumed, that the film (air-water) surfaces are plane-parallel. Then the pressure change across the air-water interface is (approximately) equal to zero, that is APAw - P A - P w - 0. The latter fact implies, that the pressure differences across the oil-water (APow - P o - P w ) and oilair (APoA - P o - P A ) interfaces must be approximately equal, i.e. APow ~ APoA, for an equilibrium bridge.

Chapter 14

610

The latter requirement certainly cannot be satisfied if the oil-air interface is

convex

(Z~I19OA> 0),

whereas the oil-water interface is concave (APow < 0), see Fig. 14.12. Hence, such a bridge cannot be in mechanical equilibrium, and its destruction will cause rupturing of the foam film. As seen in Fig. 14.12 this non-equilibrium configuration corresponds to 0 < rt/2, that is to cos0 > 0. This is the same angle 0, which appears in the Neumann triangle in Fig. 14.11. Using the cosine theorem for this triangle one obtains [45]: 2

2

2

B - CrAW+Crow-- CrOA = 2CRAWCrowCOS0

(14.12)

Then it is obvious that the condition for non-equilibrium configuration, cos0 > 0, is equivalent to B > 0, cf. Eq. (14.11). On the other hand, an equilibrium configuration is possible when both the oil-air and oil-water interfaces are convex, and consequently APow = APoA > 0. One may check that this configuration corresponds to cos0 < 0 and B < 0. For the sake of simplicity let us denote x = CRAW,Y -- CrOWand z = CrOA. Then, in view of Eq. (14.12) the relationship B > 0 can be presented in the following equivalent forms:

B=xZ + yZ-z2>O

r

(x + y ) Z - z Z > 2xy

(x + y + Z)(X + y - Z) > 2xy

r

(x + y + z)E > 2xy

r

(14.13) (14.14)

where at the last step we used the definition of the entering coefficient E, Eq. (14.9). Since x, y and z are positive, Eq. (14.14) implies that E must be also positive. In other words, from B > 0 it follows E > 0, [52]. On the other hand, from E > 0 it does not necessarily follow B > 0. The experiment shows, that sometimes bridges with B > 0 can be (meta)stable (like the "fish eyes" in Fig. 14.8) in contrast with the prediction of the criterion Eq. (14.11). This can be due to the fact that in reality the foam film is not plane-parallel in a vicinity of the oil bridge [35], as it is assumed when deriving Eq. (14.11). The data in Table 14.1 shows that for a shampoo-type system all three coefficients are positive (E > 0, S > 0 and B > 0), and one could expect that the drop entry and oil spreading occur spontaneously, and the formed oil bridges are unstable. In contrast, the experiment shows that the oil drops in this system exhibit a rather weak antifoaming action. As already discussed, this apparent discrepancy can be attributed to the existence of a high disjoining pressure barrier to

Effect of Oil Drops and Particulates on the Stabili O, of Foams

611

drop entry. Note that the drop (particle) entry is a necessary step in each of the antifoaming mechanisms shown in Fig 14.9. Hence, the information about E, S and B should be combined with data about the stability of the asymmetric oil-water-air films in order to predict the antifoaming activity for a given system [6, 17].

14.2.3. SPREADING MECHANISM

As mentioned earlier, after entering the air-water interface an oil drop forms a lens. At the same time, spreading of a molecularly thin oil film can happen. If the spreading coefficient is positive (S > 0), then spontaneous spreading of thick oil film could also happen, which would strongly destabilize the foam films. The foam-destabilizing action of oil spreading was pointed out in the studies by Ross and McBain [39] and Ross [40], in which the spreading mechanism was formulated. It was noted there that the spreading may lead to bridging. As a possible scenario it has been suggested that the foam-destructive role of oil consists in spreading of an oil duplex film on both sides of the foam film, thereby driving out the aqueous phase and leaving an oil film, which is unstable and easily breaks [39]. The importance of oil spreading for the antifoaming action has been emphasized in subsequent works {41-43,49,53-59]. Kulkarni et al. [5] have noted, that the major advantage of the silicone antifoams over their organic counterparts arises by virtue of the low surface tension and spontaneous spreading of the silicone oil over most aqueous foaming systems. The organic oils, in general, cannot spread effectively on aqueous surfactant solutions, on which, on the other hand, the silicone oils have positive spreading coefficient (S > 0) [5]. The mechanism of foam destruction by silicone-oil droplets in a shampoo-type system has been directly observed by Basheva et al. [17] in experiments with vertical films formed in the threeleg frame, see Fig. 14.3. Silicone-oil droplets of average size 11 lam (volume fraction 0.001 in the emulsion) have been dispersed in 0.1 M solution of sodium dodecyl-trioxyethylene sulfate (SDP3S). After the simultaneous creation of three vertical films in the frame, one first observes their regular thinning (Fig. 14.13a). The oil droplets are expelled from the foam films and accumulated in the Plateau border (Fig. 14.13b). The Plateau border also thins due to the drainage of water. At a certain moment one observes entry of an oil drop at the surface of the

612

Chapter 14

Plateau border, which is accompanied by a fast oil spreading over the neighboring foam films (Fig. 14.13c). The spreading of oil causes hydrodynamic instabilities, which quickly propagate over the whole film area (Fig. 14.13d). The film ruptures several seconds after the drop entry.

r .

(a)

(b)

(c)

(d)

,

Fig. 14.13. Vertical films formed in a three-leg frame (see Fig. 14.3): consecutive video-frames taken by Basheva et al. [17]. The films are produced from 0.1 M solution of SDP3S containing silicone-oil droplets of average size 11 ~tm. (a) Initially, the foam films are regularly thinning. (b) The oil droplets are expelled from the films and accumulated in the Plateau border, which also thins due to the outflow of water. (c) At a certain moment, an oil drop is observed to enter the surface of the Plateau border and spreading of oil over the neighboring foam films takes place. (d) This causes hydrodynamic instabilities followed by film rupture.

Effect of Oil Drops and Particulates on the Stabili O, of Foams

613

Consequently, in this system the antifoaming mechanism follows the route A---~F~G---~C---~D in Fig. 14.9. Although the importance of oil spreading has been widely recognized, many authors notice that there is no simple correlation between spreading and antifoaming action [4-6]. Many materials spread without showing antifoaming action, whereas others do not spread but nevertheless exhibit a foam-destructive effect. This situation is understandable having in mind the sequence of stages in the antifoaming mechanisms (Fig. 14.9). Indeed, since entering is a prerequisite for spreading, an oily material with high positive spreading coefficient cannot exhibit its antifoaming activity if there is a high barrier to oil-drop entry. On the other hand, nonspreading materials can have foam-breaking performance, insofar as there are other antifoaming mechanisms, alternative to spreading, like the bridging-dewetting and bridgingstretching mechanisms.

14.2.4. BRIDGING-DEWE777NG MECHANISM

As already mentioned, the possibility for bridging of foam films by antifoam particles has been discussed long ago by Ross and McBain [39]. As a separate mechanism, especially for hydrophobic solid particles alone, the bridging-dewetting mechanism (the transition E---~K in Fig. 14.9) was formulated by Garrett [44, 45], and was accepted in many subsequent studies for the cases of solid and liquid particles [4, 6, 15, 23, 46, 47].

Illumination

/ -3

Observation

Fig. 14.14. Experimental cell used by Dippenaar [46] to study the rupture of liquid films by solid particles. A liquid film is formed in the interior of a short glass capillary (1) initially filled with aqueous solution. The thickness of the formed film is controlled by ejection or injection of liquid through the side orifice (2) and syringe-needle (3). The formed film is observed in transmitted light through the optical glass plate (4) to avoid the aberration due to the cylindrical wall. The cell is closed in container (not shown) to prevent evaporation of water and convection of air.

614

Chapter 14

Dippenaar [46] directly recorded bridging-dewetting events with hydrophobic particles in water films (without surfactant) with the help of high-speed cinematography. In his experiments he used a version of the Scheludko cell, made of glass, whose cylindrical wall is optically connected to a vertical plane-parallel glass plate (Fig. 14.14). The observation of the foam films across the latter plate allows one to avoid optical distortions due to the cylindrical wall of the cell. In the case of liquid antifoaming particles it was suggested [6, 15, 23, 33, 47, 60] that the lens, formed after the oil-drop entry at the air-water interface (in the film or Plateau border), enters also the opposite air-water surface, which leads to the formation of an oil bridge. Alternatively, such a bridge can be created by breaking of the oil-water-oil film formed between two lenses, attached to two air-water interfaces, as it is in the experiments of Wang et al. [48].

Air

i i Waterii ii

ii ii ii ii ii ii il

Fig. 14.15. An oil lens, initially attached to the upper film surface, enters the lower film surface. The Laplace pressure in the contact zone drives the liquid away from the lens thus dewetting its lower surface. As a rule the foam systems contain surfactants, which adsorb at any hydrophobic surfaces rendering them hydrophilic. For that reason one can expect that the surface of any antifoam particle is hydrophilized by the surfactant. In other words, the surfactant promotes wetting (rather than dewetting) of antifoam-particles. In spite of that, the bridging by a hydrophilized oily drop can have a foam-destructive effect. The curvature of the film surfaces in the neighborhood of a bridging oil lens gives rise to a capillary pressure, which drives the water away from the lens (Fig. 14.15), until finally the two three-phase contact lines coincide. This is equivalent to dewetting of the oil lens, which is immediately followed by hole formation and film rupture [4, 17, 47]. Alternatively, the oil bridge itself can be mechanically unstable and can break in its central part after stretching (without dewetting), see Section 14.2.5.

Effect of Oil Drops and Particulates on the Stability of Foams

615

14.2.5. BRIDGING-STRETCHING MECHANISM

Ross [40] mentioned the bridging-stretching mechanism (the transition E--~L in Fig. 14.9) as one of the possible scenarios of foam destruction by oily drops. The existence of this mechanism was directly proven and experimentally investigated by Denkov et al. [18, 35] with the help of a high-speed video camera (1000 frames per second). Foam films with oily bridges were formed in the experimental cell of Dippenaar (Fig. 14.14) in the following way [18, 35]:

Air

Air

(a)

(b)

water

Air

Water

(c) Fig. 14.16. Sketch of the system configuration (on the left) and consecutive video-frames (on the right) of an oil capillary bridge formed in a foam film; experimental results of Denkov et al. [18]. (a) A capillary bridge with "neck" is formed after an oil lens, situated at the upper surface of the aqueous layer, touches its lower surface. (b)The capillary bridge stretches with time. (c) Unstable oil film appears in its central zone, which ruptures breaking the whole foam film.

616

Chapter 14

First the cylindrical experimental cell has been loaded with the investigated aqueous surfactant solution, which acquires the shape of a biconcave liquid layer. Then an oil drop (of diameter about 100 Jam) is placed on the upper concave meniscus; the oil forms a floating lens situated in the central zone of the meniscus. Next, some amount of the aqueous solution is gradually sucked out from the biconcave liquid layer, which leads to a decrease of its thickness. An oil capillary bridge forms when the oil lens situated at the upper surface of the aqueous layer touches its lower surface (Fig. 14.16a). The observations show that this capillary bridge stretches with time (Fig. 14.16b) and an oily film appears in its central zone (Fig. 14.16c). The oily film is unstable: it ruptures and breaks the whole foam film. The total period of existence of these unstable oil bridges in foam films is only several milliseconds [ 18, 35]. It is worthwhile noting that the oil capillary bridges of relatively small size turn out to be mechanically stable. On the other hand, the larger bridges are unstable. This behavior is consonant with the theoretical predictions [35]. Initially small stable bridges could be latter transformed into unstable ones due to the action of the following two factors. (i) The characteristic length, determining whether a capillary bridge is small or large, is scaled by the thickness of the foam film; when the thickness (the length scale) decreases due to the drainage of water an oily bridge of fixed volume may undergo a transition from small stable into large unstable. (ii) It has been established [35] that oil can be transferred from a pre-spread oil layer (over the air-water interface) toward the oil bridge; thus the size of the bridge actually increases and it can undergo a transition from stable state to unstable state. In the experiments by Denkov et al. [18, 35] the lifetime of the small stable bridges has been up to several seconds; this is the time elapsed between the moments of bridge formation and destabilization. As already mentioned, the lifetime of the larger unstable bridges is only few milliseconds and it can be recorded with the help of a high-speed video technique. The latter enables one to establish whether the oil bridge ruptures the film following the stretching or dewetting mechanisms.

Effect of Oil Drops and Particulates on the Stabilio, of Foams

14.3.

617

S T A B I L I T Y OF A S Y M M E T R I C F I L M S : THE KEY FOR C O N T R O L OF FOAMINESS

14.3.1. THERMODYNAMIC AND KINETIC STABILIZING FACTORS The formation of a stable or unstable foam depends on the stability of the separate air-water-

air films. In addition, a colloidal particle (say, an oil droplet) can exhibit antifoaming action if only the asymmetric particle-water-air film is unstable. The rupture of the latter film is equivalent to particle entry, which is a necessary step of the spreading and bridging mechanisms (Fig. 14.9). Consequently, the stability of the respective liquid films has a primary importance for both foamability and antifoaming action. The factors, which govern the stability, are similar for symmetric and asymmetric liquid films; these factors, and the related mechanisms of film rupture, are considered below in this section. The major thermodynamic stabilizing factor is the action of a repulsive disjoining pressure, l-I, within the liquid film. A stable equilibrium state of a liquid film can exist if only the following two conditions are satisfied [61 ]: and

=

\ o n )n =PA

1-Imax; most probably this could happen in the center of the film (around r = 0), where the viscous contribution to PA is maximal. With typical parameter values, je 6 x 1017 cm-2s -1 [107], Vw= 30/~ 2, R = 0 . 1 cm, h = 100 nm and r / - 0.01 poises one obtains 3rlVwjeR2/h3= 5.4 x 105 Pa

(14.22)

which is really a considerable effect. The same effect may strongly facilitate the entry of oil drops (captured in the foam) at the water-air interface. For the respective oil-water-air films both R and h are expected to be smaller than for the foam films. This would lead again to a large viscous contribution to PA insofar as R 2 and h 3 enter, respectively, the numerator and denominator in Eq. (14.22) and the decreases in the values of these two parameters tend to compensate each other. In conclusion, the evaporation of water from the foam leads to a strong increase in the applied capillary pressure P A due to viscous effects, which may cause overcoming of the disjoining pressure barrier(s) and possible film rupture.

626

Chapter 14

The physical picture can be quite different if the surfactant solution contains micelles of low surface electric charge. In this case the evaporation-driven influx of water brings surfactant micelles in the foam film, just as it is depicted in Fig. 13.33, and, moreover, the electrostatic repulsion is not strong enough to expel the newcomers from the film. The water evaporates, but the micelles remain in the film; this leads to an increase of the micelle local concentration, and could even cause formation of surfactant liquid crystal structures within the film. This has been observed with mixed solutions of anionic surfactant with amphoteric one (betaine) [108]. The accumulation of surfactant within foam films has a stabilizing effect and can protect the films from rupturing.

14.4.

SUMMARY AND CONCLUSIONS

Foams are produced in many processes in industry and every-day life. In some cases the foaminess is desirable, while in other cases it is not wanted. The fact that oil droplets, solid particles and their combination exhibit antifoaming action can be used as a tool for control of foam stability. In this aspect the knowledge about the mechanism of antifoaming action could be very helpful. The antifoaming action can be investigated in experiments with single films in the cells of Scheludko (Fig. 14.2) and Dippenaar (Fig. 14.14), as well as with vertical films formed in a frame (Figs. 14.3 and 14.13). Direct observations show that when the foam decay is slow (from minutes to hours, see Fig. 14.6), the antifoam particles are expelled from the foam films into the Plateau borders. The breakage of foam cells happens when the surfaces of the thinning Plateau borders press the captured particles. The low rate of thinning of the Plateau borders is the reason for the low rate of foam decay in this case. In contrast, when the particles exhibit a fast antifoaming action, they are observed to break directly the foam films, which thin much faster than the Plateau borders; see Figs. 14.8 and 14.16. This leads to a greater rate of foam decay. Three different mechanisms of antifoaming action have been established: spreading mechanism, bridging-dewetting and bridging-stretching mechanism, see Fig. 14.9. All ot: them involve as a necessary step the entering of an antifoam particle at the air-water interface, which is equivalent to rupture of the asymmetric particle-water-air film. Criteria for the entering, spreading and bridging to happen spontaneously have been proposed in terms of the respective

Effect of Oil Drops and Particulates on the Stability of Foams

627

entering, spreading and bridging coefficients, see Eqs. (14.9)-(14.11). The experiment shows that the key determinant for antifoaming action is the stability of the asymmetric particle-waterair film, see the discussion concerning Table 14.1. Repulsive interactions in this film may create a high barrier to drop entry. The major thermodynamic factors, which stabilize the asymmetric film, are related to the presence of (i) barrier due to the electrostatic (double layer) repulsion, (ii) multiple barriers due to the oscillatory structural forces in micellar surfactant solution, (iii) barrier created by the steric polymer-chain repulsion in the presence of adsorbed nonionic surfactants. In addition, there are kinetic stabilizing factors, which damp the instabilities in the liquid films; such are (i) the surface (Gibbs) elasticity, (ii) the surface

viscosity of the adsorption monolayers, (iii) the adsorption relaxation time related to the diffusion supply of surfactant from the bulk of solution. On the other hand, a foam-destabilizing factor can be any attractive force operative in the liquid films, as well as any factor suppressing the effect of the aforementioned stabilizing factors. For example, the addition of salt reduces the height of the electrostatic and oscillatory-structural barriers in the case of ionic surfactant. Oscillatory-structural barriers due to nonionic-surfactant micel|es are suppressed by the rise of temperature [69]. Solid particulates of irregular shape, adsorbed at the oil-water interface, have a "piercing effect" on asymmetric oil-water-air films and on symmetric oil-water-oil films as well. It is worthwhile noting that some factors may have stabilizing or destabilizing effect depending on the specific conditions. For example, at low concentration the surfactant micelles have destabilizing effect because they give rise to the depletion attraction; on the other hand, at high concentration they exhibit stabilizing effect owing to the barriers of the oscillatory structural force. Likewise, oil droplets located in the Plateau borders of a foam have a foam-breaking effect when they are large enough (larger than 1 0 - 20 gin); on the other hand, smaller oil drops may block the outflow of water along the Plateau channels thus producing a foam-stabilizing effect. A third example is the effect of water evaporation from a foam: in the absence of surfactant micelles the evaporation-driven flux of water within the foam film creates strong viscous pressure, which helps to overcome the disjoining-pressure barrier(s), see Eq. (14.21); on the other hand, if micelles are present in the solution, the evaporation may increase their

628

Chapter 14

concentration within the foam film and can create a stabilizing surfactant-structural barrier to film rupture. The variety of factors and mechanisms may leave the discouraging impression that: it is virtually impossible to predict and control the stability of foams and the antifoaming action of colloid particles. Accepting an optimistic viewpoint, we believe that it is still possible to give definite prescriptions and predictions based on the accumulated knowledge about the mechanisms of foam destruction. In this aspect, the role of an expert in foam stability resembles that of a medical doctor, who establishes the diagnosis and formulates prescriptions after a careful examination of each specific case.

14.5.

REFERENCES

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P.R. Garrett (Ed.), Defoaming: Theory and Industrial Applications, Marcel Dekker, New York, 1993.

5. R.D. Kulkarni, E.D. Goddard, P. Chandar, in: R.K. Prud'homme & S.A. Khan (Eds.) Foams: Theory, Measurements and Applications, Marcel Dekker, New York, 1995, p. 555. 6. D.T. Wasan, S.P. Christiano, in: K.S. Birdi (Ed.) Handbook of Surface and Colloid Chemistry, CRC Press, New York, 1997, pp. 179-215. 7.

T.G. Rubel, Antifoaming and Defoaming Agents, Noyes Data Corp., Park Ridge, NJ, 1972.

8. H.T. Kerner, Foam Control Agents, Noyes Data Corp., Park Ridge, NJ, 1976. 9.

J.C. Colbert, Foam and Emulsion Control Agents and Processes, Recent Developments, Noyes Data Corp., Park Ridge, NJ, 198 I.

10. R. Aveyard, B.P. Binks, P.D.I. Fletcher, C.E. Rutherford, J. Dispersion Sci. Technol. 15 (1994) 251. 11. R.E. Patterson, Colloids Surf. A, 74 (1993) 115. 12. R.D. Kulkarni, E.D. Goddard, B. Kanner, Ind. Eng. Chem. Fund. 16 (1977) 472. 13. M.A. Ott, Modern Paints Coatings 67 (1977) 31.

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629

14. I. Lichtman, T. Gammon, in: M. Grayson (Ed.), Kirk-Othmer Encyclopedia of Science and Technology, Vol. 7, Wiley-Interscience, New York, 1980, pp. 430-448. 15. R. Aveyard, P. Cooper, P.D.I. Fletcher, C.E. Rutherford, Langmuir 9 (1993) 604. 16. R. Aveyard, B.P. Binks, P.D.I. Fletcher, T.G. Peck, C.E. Rutherford, Adv. Colloid Interface Sci. 48 (1994) 93. 17. E.S. Basheva, D. Ganchev, N.D. Denkov, K. Kasuga, N. Satoh, K. Tsujii, Langmuir, 16 (2000 ) 1000. 18. N.D. Denkov, P. Cooper, J.-Y. Martin, Langmuir, 15 (1999) 8514. 19. R. Koczo, B. Ludanyi, Gy. Racz, Period. Polytech. Chem. Eng. 31 (1987) 83. 20. R.K. Prud'homme & S.A. Khan (Eds.) Foams: Theory, Measurements and Applications, Marcel Dekker, New York, 1995. 21. D. Exerowa, P.M. Kruglyakov, Foam and Foam Films, Elsevier, Amsterdam, 1998. 22. I.B. Ivanov, B.P. Radoev, T.T. Traykov, D.S. Dimitrov, E.D. Manev, C.S. Vassilieff, in E. Wolfram (Ed.) Proceedings of the International Conference on Colloid and Surface Science, Akademia Kiado, Budapest, 1975; p. 473. 23. D.T. Wasan, K. Koczo, A.D. Nikolov, Foams: Fundamentals and Application in the Petroleum Industry, in: L.L. Schram (Ed.) ACS Symp. Set. No. 242, American Chemical Society, Washington D.C., 1994. 24. A. Scheludko, D. Exerowa, Kolloid-Z. 165 (1959) 148. 25. A. Scheludko, Adv. Colloid Interface Sci. 1 (1967) 391. 26. J. Mysels, J. Phys. Chem. 68 (1964) 3441. 27. M. Born, E. Wolf, Principles of Optics, 4th Ed., Pergamon Press, Oxford, 1970. 28. A. Vasicek, Optics of Thin Films, North Holland Publishing Co., Amsterdam, 1960. 29. A. Scheludko, Kolloid-Z. 155 (1957) 39. 30. E.S. Basheva, K.D. Danov, P.A. Kralchevsky, Langmuir 13 (1997) 4342. 31. K. Koczo, G. Racz, Colloids Surfaces 22 (1987) 97. 32. M.P. Aronson, Langmuir 2 (1986) 653. 33. K. Koczo, J.K. Koczone, D.T. Wasan, J. Colloid Interface Sci. 166 (1994) 225. 34. I.B. Ivanov, D.S. Dimitrov, Thin Film Drainage, in I.B. Ivanov (Ed.) Thin Liquid Films, Marcel Dekker, New York, 1988. 35. N.D. Denkov, Langmuir 15 (1999) 8530. 36. I.B. Ivanov, Pure Appl. Chem. 52 (1980) 1241.

630

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63. J.N. Israelachvili, Intermolecular & Surface Forces, Academic Press, London, 1992. 64. J.A. de Feijter, A. Vrij, J. Colloid Interface Sci. 70 (1979) 456. 65. A.D. Nikolov, D.T. Wasan, P.A. Kralchevsky, I.B. Ivanov, in: N. Ise and I. Sogami (Eds.), Ordering and Organisation in Ionic Solutions, World Scientific, Singapore, 1988. 66. A.D. Nikolov, D.T. Wasan, J. Colloid Interface Sci. 133 (1989) 1. 67. A. D. Nikolov, P. A. Kralchevsky, I. B. Ivanov, D. T. Wasan, J. Colloid Interface Sci. 133 (1989) 13. 68. P.A. Kralchevsky, A.D. Nikolov, D.T. Wasan, I. B. Ivanov, Langmuir 6 (1990) 1180. 69. A.D. Nikolov, D.T. Wasan, N.D. Denkov, P.A. Kralchevsky, I.B. Ivanov, Prog. Colloid Polym. Sci. 82 (1990) 87. 70. D.T. Wasan, A.D. Nikolov, P.A. Kralchevsky, I.B. Ivanov, Colloids Surf. 67 (1992) 139. 71. V. Bergeron, C.J. Radke, Langmuir 8 (1992) 3020. 72. M.L. Pollard, C.J. Radke, J. Chem. Phys. 101 (1994) 6979. 73. X.L. Chu, A.D. Nikolov, D.T. Wasan, Langmuir 10 (1994) 4403. 74. X.L. Chu, A.D. Nikolov, D.T. Wasan, J. Chem. Phys. 103 (1995) 6653. 75. P.A. Kralchevsky, N.D. Denkov, Chem. Phys. Lett. 240 (1995) 385. 76. K.G. Marinova, T.D. Gurkov, T.D. Dimitrova, R.G. Alargova, D. Smith, Langmuir 14 (1998) 2011. 77. I.B. Ivanov, P.A. Kralchevsky, Colloids Surfaces A, 128 (1997) 155. 78. K.D. Danov, P.A. Kralchevsky, I.B. Ivanov, Equilibrium and Dynamics of Surfactant Adsorption Monolayers and Thin Liquid Films, in: U. Zoller and G. Broze (Eds.) Handbook of Detergents, Vol. 1: Properties, Chapter 9; M. Dekker, New York, 1999. 79. A.J. Vries, Rec. Trav. Chim. Pays-Bas 77 (1958) 44. 80. A. Scheludko, Proc. K. Akad. Wetensch. B, 65 (1962) 87. 81. A. Vrij, Disc. Faraday Soc. 42 (1966) 23. 82. I.B. Ivanov, B. Radoev, E. Manev, A. Scheludko, Trans. Faraday Soc. 66 (1970) 1262. 83. I.B. Ivanov, D.S. Dimitrov, Colloid Polymer Sci. 252 (1974) 982. 84. P.A. Kralchevsky, K.D. Danov, I.B. Ivanov, Thin Liquid Film Physics, in: R.K. Prud'homme and S.A. Khan (Eds.) Foams, M. Dekker, New York, 1995, p. 1. 85. I.B. Ivanov, K.D. Danov, P.A. Kralchevsky, Colloids and Surfaces A, 152 (1999) 161. 86. B.V. Derjaguin, Y.V. Gutop, Kolloidn. Zh. 24 (1962) 431.

632

Chapter 14

87. B.V. Derjaguin, A.V. Prokhorov, J. Colloid Interface Sci. 81 (1981) 108. 88. A.V. Prokhorov, B.V. Derjaguin, J. Colloid Interface Sci. 125 (1988) 111. 89. D. Kashchiev, D. Exerowa, J. Colloid Interface Sci. 77 (1980) 501. 90. D. Kashchiev, D. Exerowa, Biochim. Biophys. Acta 732 (1983) 133. 91. D. Kashchiev, Colloid Polymer Sci. 265 (1987) 436. 92. Y.A. Chizmadzhev, V.F. Pastushenko, Electrical Breakdown of Bilayer Lipid Membranes, in: I.B. Ivanov (Ed.) Thin Liquid Films, M. Dekker, New York, 1988; p. 1059. 93. L.V. Chernomordik, M.M. Kozlov, G.B. Melikyan, I.G. Abidor, V.S. Markin, Y.A. Chizmadzhev, Biochim. Biophys. Acta 812 (1985) 643. 94. L.V. Chernomordik, G.B. Melikyan, Y.A. Chizmadzhev, Biochim. Biophys. Acta 906 (1987) 309. 95. A. Kabalnov, H. Wennerstr6m, Langmuir 12 (1996) 276. 96. I.B. Ivanov, S.K. Chakarova, B. I. Dimitrova, Colloids Surf. 22 (1987) 311. 97. B.I. Dimitrova, I.B. Ivanov, E. Nakache, J. Dispers. Sci. Technol. 9 (1988) 321. 98. K.D. Danov, I.B. Ivanov, Z. Zapryanov, E. Nakache, S. Raharimalala, in: M.G. Velarde (Ed.) Proceedings of the Conference of Synergetics, Order and Chaos, World Scientific, Singapore, 1988, p. 178. 99. C.V. Sterling, L.E. Scriven, AIChE J. 5 (1959) 514. 100. S.P. Lin, H.J. Brenner, J. Colloid Interface Sci. 85 (1982) 59. 101. J.L. Castillo, M.G. Velarde, J. Colloid Interface Sci. 108 (1985) 264. 102. D.S. Valkovska, P.A. Kralchevsky, K.D. Danov, G. Broze, A. Mehreteab, Langmuir 16 (2000) - in press. 103. J.K. Angarska, K.D. Tachev, P.A. Kralchevsky, A. Mehreteab, B. Broze, J. Colloid Interface Sci. 200 (1998) 31. 104. P.R. Garrett, J. Davis, H.M. Rendall, Colloids Surf. A, 85 (1994) 159. 105. A. Pouchelon, A. Araud, J. Dispersion Sci. Technol. 14 (1993) 447. 106. N.D. Denkov, K.G. Marinova, C. Christova, A. Hadjiiski, P. Cooper, Langmuir 16 (2000) 2515. 107. C.D. Dushkin, H. Yoshimura, K. Nagayama, Chem. Phys. Lett. 204 (1993) 455. 108. D. Ganchev, E. Ahmed, N. Denkov, Fac. Chem., Univ. Sofia, private communication.

Appendices

633

APPENDIX 1A"

EQUIVALENCE OF THE TWO FORMS OF THE GIBBS ADSORPTION EQUATION

Following Ref. [1] here we derive Eq. (1.70) from Eq. (1.68). We consider a solution of various species (i = 1,2 ..... N), both amphiphilic and non-amphiphilic. As before we will use index "1" to denote the surfactant ions, whose adsorption determines the sign of the surface electric charge and potential. A substitution of a/~ from Eq. (1.71) into Eq. (1.68) yields

do- = ~ F/d In ais + kT i=1

zi['i

~s

(1A.1)

i=1

Since the solution as a whole is electroneutral, one can write [2-4] N

(1A.2)

-o i=1

From Eqs. (1.69) and (1A.2) one obtains the following expression for the surface electric charge density p.," N

N

Ps = Ps _ E Z,~. - - E ziA' Zle i=1 i=l

(1A.3)

Further, in view of Eqs. (1.69), (1.71) and (1A.2) one can transform Eq. (1A.1) to read kT = ~-~ Fid In ais + i~1A i d In aioo i=1

"=

ziA i

rlps

(1A.4)

i=1

With the help of Eqs. (1.49), (1.69) and (1A.3) one can bring Eq. (1A.4) in the form N

N

do- = E F i d l n a i s kT i=l

+ ZXidai~ i=1

+ Psd~s

(1A.5)

where oo

A, = Ai = I[exp(-zi*)-1]dz ai~ o

(1A.6)

On the other hand, integrating Eq. (1.57), along with Eq. (1A.6), one can deduce

F ---

k,

- EaiooXi,__,

Differentiating Eq. (IA.7) one obtains

(1A.7)

Appendices

634 N

N

(1A.8)

~ F :~ E X i (~7li~176 "+ E a i~ ~ X i i=1 i=1

where "~" denotes a variation of the respective thermodynamic parameter corresponding to a small variation in the composition of the solution. Further, with the help of Eqs. (1A.6) and (1.55) one obtains N

~176

Z aioo~7~i- - r E ai~oZiexp(-ziOP)~Pdz i=1

0 i=1

2 ~d20 0

=

(1A.9)

Kc 0 a z

7-2 ~ K'c

z=O

t~I)s-----~J |--~Z Kc 0 k

dx

Then combining Eqs. (1A.7), (1A.9) and (1.59) one obtains N E a ioo(~A i -- p s ( ~ s - t~F i=l

( 1 A . lO)

A substitution ofEq. (1A.10) into Eq. (1A.8) yields N

25F - E A , ~/i~, q- PsC~s

(1A.11)

i=1

Next, the substitution of Eq. (1A. 11) into the Gibbs adsorption equation (1A.5) leads to + 2F - ~ Fi d In ai,

(T = const.)

(1A. 12)

i=1

Comparing the definition of F, Eq. (1A.7), with Eq. (1.61) one finds that 2F = -c~dkT. The substitution of the latter result into Eq. (1A. 12), along with Eq. (1.19), finally gives the sought for Eq. (1.70). REFERENCES: APPENDIX

IA

1. P.A. Kralchevsky, K.D. Danov, G. Broze, A. Mehreteab, Langmuir 15 (1999) 2351. 2.

S. Hachisu, J. Colloid Interface Sci. 33 (1970) 445.

3. D.G. Hall, in: D.M. Bloor, E. Wyn-Jones (Eds.) The Structure, Dynamics and Equilibrium Properties of Colloidal Systems, Kluwer, Dordrecht, 1990; p. 857. 4.

D.G. Hall, Colloids Surf. A, 90 (1994) 285.

Appendices

635

APPENDIX 8A: DERIVATIONOF EQUATION (8.31)

Following Ref. [1] we consider the configuration of two floating particles depicted in Fig. 8.2, where the meaning of the notation is explained. For small particles, (qRe) 2